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\begin{document}
\title{Formal Deformation Theory}
\maketitle
\phantomsection
\label{section-phantom}
\tableofcontents
\section{Introduction}
\label{section-introduction}
\noindent
This chapter develops formal deformation theory in a form applicable
later in the Stacks project, closely following Rim \cite[Exposee VI]{SGA7-I}
and Schlessinger \cite{Sch}. We strongly encourage the reader new to
this topic to read the paper by Schlessinger first, as it is sufficiently
general for most applications, and Schlessinger's results are indeed
used in most papers that use this kind of formal deformation theory.
\medskip\noindent
Let $\Lambda$ be a complete Noetherian local ring with residue field $k$,
and let $\mathcal{C}_\Lambda$ denote the category of Artinian local
$\Lambda$-algebras with residue field $k$. Given a functor
$F : \mathcal{C}_\Lambda \to \textit{Sets}$ such that $F(k)$
is a one element set, Schlessinger's paper introduced conditions
(H1)-(H4) such that:
\begin{enumerate}
\item $F$ has a ``hull'' if and only if (H1)-(H3) hold.
\item $F$ is prorepresentable if and only (H1)-(H4) hold.
\end{enumerate}
The purpose of this chapter is to generalize these results in two ways
exactly as is done in Rim's paper:
\begin{enumerate}
\item[(A)] The functor $F$ is replaced by a category $\mathcal{F}$ cofibered
in groupoids over $\mathcal{C}_\Lambda$, see
Section \ref{section-CLambda}.
\item[(B)] We let $\Lambda$ be a Noetherian ring and $\Lambda \to k$
a finite ring map to a field. The category $\mathcal{C}_\Lambda$ is
the category of Artinian local $\Lambda$-algebras $A$ endowed with a
given identification $A/\mathfrak m_A = k$.
\end{enumerate}
The analogue of the condition that $F(k)$ is a one element set is that
$\mathcal{F}(k)$ is the trivial groupoid. If $\mathcal{F}$ satisfies this
condition then we say it is a {\it predeformation category}, but in general
we do not make this assumption. Rim's paper \cite[Exposee VI]{SGA7-I} is the
original source for the results in this document. We also mention the useful
paper \cite{Vistoli}, which discusses deformation theory with groupoids but
in less generality than we do here.
\medskip\noindent
An important role is played by the ``completion''
$\widehat{\mathcal{C}}_\Lambda$ of the category $\mathcal{C}_\Lambda$.
An object of $\widehat{\mathcal{C}}_\Lambda$ is a Noetherian complete
local $\Lambda$-algebra $R$ whose residue field is identified with $k$, see
Section \ref{section-category-completion-CLambda}.
On the one hand $\mathcal{C}_\Lambda \subset \widehat{\mathcal{C}}_\Lambda$
is a strictly full subcategory and on the other hand
$\widehat{\mathcal{C}}_\Lambda$ is a full subcategory of the category
of pro-objects of $\mathcal{C}_\Lambda$. A functor
$\mathcal{C}_\Lambda \to \textit{Sets}$ is {\it prorepresentable}
if it is isomorphic to the restriction of a representable functor
$\underline{R} = \Mor_{\widehat{\mathcal{C}}_\Lambda}(R, -)$
to $\mathcal{C}_\Lambda$ where
$R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$.
\medskip\noindent
{\it Categories cofibred in groupoids} are dual to categories fibred in
groupoids; we introduce them in Section \ref{section-preliminary}.
A {\it smooth} morphism of categories cofibred in groupoids over
$\mathcal{C}_\Lambda$ is one that satisfies the infinitesimal lifting
criterion for objects, see
Section \ref{section-smooth-morphisms}.
This is analogous to the definition of a formally smooth ring map, see
Algebra, Definition \ref{algebra-definition-formally-smooth}
and is exactly dual to the notion in
Criteria for Representability, Section \ref{criteria-section-formally-smooth}.
This is an important notion as we eventually want to prove that certain
kinds of categories cofibred in groupoids have a smooth prorepresentable
presentation, much like the characterization of algebraic stacks in
Algebraic Stacks, Sections \ref{algebraic-section-stack-to-presentation} and
\ref{algebraic-section-smooth-groupoid-gives-algebraic-stack}.
A {\it versal formal object} of a category $\mathcal{F}$ cofibred
in groupoids over $\mathcal{C}_\Lambda$ is an object
$\xi \in \widehat{\mathcal{F}}(R)$ of the completion such that the
associated morphism
$\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$
is smooth.
\medskip\noindent
In
Section \ref{section-schlessinger-conditions},
we define conditions (S1) and (S2) on $\mathcal{F}$ generalizing
Schlessinger's (H1) and (H2). The analogue of Schlessinger's
(H3)---the condition that $\mathcal{F}$ has finite dimensional
tangent space---is not given a name.
A key step in the development of the theory is the existence of
versal formal objects for predeformation categories satisfying
(S1), (S2) and (H3), see
Lemma \ref{lemma-versal-object-existence}.
Schlessinger's notion of a {\it hull} for a functor
$F : \mathcal{C}_\Lambda \to \textit{Sets}$
is, in our terminology, a versal formal object $\xi \in \widehat{F}(R)$
such that the induced map of tangent spaces
$d\underline{\xi} : T\underline{R}|_{\mathcal{C}_\Lambda} \to TF$
is an isomorphism.
In the literature a hull is often called a ``miniversal'' object.
We do not do so, and here is why. It can happen that a functor has a
versal formal object without having a hull. Moreover, we show in
Section \ref{section-minimal-versal}
that if a predeformation category has a versal formal object, then
it always has a {\it minimal} one (as defined in
Definition \ref{definition-minimal-versal})
which is unique up to isomorphism, see
Lemma \ref{lemma-minimal-versal}.
But it can happen that the minimal versal formal object does not
induce an isomorphism on tangent spaces! (See
Examples \ref{example-do-not-get-S2} and
\ref{example-smooth-continued}.)
\medskip\noindent
Keeping in mind the differences pointed out above,
Theorem \ref{theorem-miniversal-object-existence}
is the direct generalization of (1) above: it recovers Schlessinger's
result in the case that $\mathcal{F}$ is a functor and it characterizes
minimal versal formal objects, in the presence of conditions
(S1) and (S2), in terms of the map
$d\underline{\xi} : T\underline{R}|_{\mathcal{C}_\Lambda} \to TF$
on tangent spaces.
\medskip\noindent
In Section \ref{section-RS-condition},
we define Rim's condition (RS) on $\mathcal{F}$ generalizing
Schlessinger's (H4). A {\it deformation category} is defined as a
predeformation category satisfying (RS).
The analogue to prorepresentable functors are the categories
cofibred in groupoids over $\mathcal{C}_\Lambda$ which have
a {\it presentation by a smooth prorepresentable groupoid in functors}
on $\mathcal{C}_\Lambda$, see
Definitions \ref{definition-groupoid-in-functors},
\ref{definition-prorepresentable-groupoid-in-functors}, and
\ref{definition-smooth-groupoid-in-functors}.
This notion of a presentation takes into account the groupoid structure
of the fibers of $\mathcal{F}$. In
Theorem \ref{theorem-presentation-deformation-groupoid}
we prove that $\mathcal{F}$ has a presentation by a smooth prorepresentable
groupoid in functors if and only if $\mathcal{F}$ has a finite dimensional
tangent space and finite dimensional infinitesimal automorphism space.
This is the generalization of (2) above: it reduces to Schlessinger's result
in the case that $\mathcal{F}$ is a functor.
There is a final
Section \ref{section-minimality}
where we discuss how to use minimal versal formal objects
to produce a (unique up to isomorphism) minimal presentation
by a smooth prorepresentable groupoid in functors.
\medskip\noindent
We also find the following conceptual explanation for Schlessinger's
conditions. If a predeformation category $\mathcal{F}$ satisfies (RS),
then the associated functor of isomorphism classes
$\overline{\mathcal{F}}: \mathcal{C}_\Lambda \to \textit{Sets}$
satisfies (H1) and (H2)
(Lemmas \ref{lemma-RS-implies-S1-S2} and
\ref{lemma-S1-S2-associated-functor}).
Conversely, if a functor
$F : \mathcal{C}_\Lambda \to \textit{Sets}$
arises naturally as the functor of isomorphism classes of
a category $\mathcal{F}$ cofibered in groupoids, then it seems to happen in
practice that an argument showing $F$ satisfies (H1) and (H2) will also show
$\mathcal{F}$ satisfies (RS) (see
Artin's Axioms, Section \ref{artin-section-examples}
for examples). Moreover, if $\mathcal{F}$ satisfies (RS), then condition
(H4) for $\overline{\mathcal{F}}$ has a simple interpretation in terms of
extending automorphisms of objects of $\mathcal{F}$
(Lemma \ref{lemma-RS-associated-functor}).
These observations suggest that (RS) should be regarded as the
fundamental deformation theoretic glueing condition.
\section{Notation and Conventions}
\label{section-notations-conventions}
\noindent
A ring is commutative with $1$. The maximal ideal of a local ring $A$
is denoted by $\mathfrak{m}_A$. The set of positive integers is denoted
by $\mathbf{N} = \{1, 2, 3, \ldots\}$. If $U$ is an object of a
category $\mathcal{C}$, we denote by $\underline{U}$
the functor
$\Mor_\mathcal{C}(U, -): \mathcal{C} \to \textit{Sets}$, see
Remarks \ref{remarks-cofibered-groupoids} (\ref{item-definition-yoneda}).
Warning: this may conflict with the notation in other chapters where we
sometimes use $\underline{U}$ to denote $h_U(-) = \Mor_\mathcal{C}(-, U)$.
\medskip\noindent
Throughout this chapter $\Lambda$ is a Noetherian ring and
$\Lambda \to k$ is a finite ring map from $\Lambda$ to a field.
The kernel of this map is denoted $\mathfrak m_\Lambda$ and the
image $k' \subset k$. It turns out that $\mathfrak m_\Lambda$ is
a maximal ideal, $k' = \Lambda/\mathfrak m_\Lambda$ is a field, and
the extension $k' \subset k$ is finite. See discussion surrounding
(\ref{equation-k-prime}).
\section{The base category}
\label{section-CLambda}
\noindent
Motivation. An important application of formal deformation theory is
to criteria for representability by algebraic spaces. Suppose given a
locally Noetherian base $S$ and a functor
$F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$.
Let $k$ be a finite type field over $S$, i.e., we are given a
finite type morphism $\Spec(k) \to S$.
One of Artin's criteria is that for any element $x \in F(\Spec(k))$
the predeformation functor associated to
the triple $(S, k, x)$ should be prorepresentable. By
Morphisms, Lemma \ref{morphisms-lemma-point-finite-type}
the condition that $k$ is of finite type over $S$ means that there exists
an affine open $\Spec(\Lambda) \subset S$ such that $k$
is a finite $\Lambda$-algebra. This motivates why we work throughout
this chapter with a base category as follows.
\begin{definition}
\label{definition-CLambda}
Let $\Lambda$ be a Noetherian ring and let $\Lambda \to k$ be a finite
ring map where $k$ is a field. We define {\it $\mathcal{C}_\Lambda$} to be
the category with
\begin{enumerate}
\item objects are pairs $(A, \varphi)$ where $A$ is an Artinian local
$\Lambda$-algebra and where $\varphi : A/\mathfrak m_A \to k$ is a
$\Lambda$-algebra isomorphism, and
\item morphisms $f : (B, \psi) \to (A, \varphi)$ are local $\Lambda$-algebra
homomorphisms such that $\varphi \circ (f \bmod \mathfrak m) = \psi$.
\end{enumerate}
We say we are in the {\it classical case} if $\Lambda$ is a Noetherian
complete local ring and $k$ is its residue field.
\end{definition}
\noindent
Note that if $\Lambda \to k$ is surjective and if $A$ is an Artinian local
$\Lambda$-algebra, then the identification $\varphi$, if it exists,
is unique. Moreover, in this case any $\Lambda$-algebra map $A \to B$ is
going to be compatible with the identifications. Hence in this case
$\mathcal{C}_\Lambda$ is just the category of local Artinian $\Lambda$-algebras
whose residue field ``is'' $k$. By abuse of notation we also denote objects of
$\mathcal{C}_\Lambda$ simply $A$ in the general case. Moreover, we will
often write $A/\mathfrak m = k$, i.e., we will pretend all rings in
$\mathcal{C}_\Lambda$ have residue field $k$ (since all ring maps in
$\mathcal{C}_\Lambda$ are compatible with the given identifications this
should never cause any problems).
Throughout the rest of this chapter the base ring $\Lambda$ and the
field $k$ are fixed. The category $\mathcal{C}_\Lambda$ will be the base
category for the cofibered categories considered below.
\begin{definition}
\label{definition-small-extension}
Let $f: B \to A$ be a ring map in $\mathcal{C}_\Lambda$. We say $f$
is a {\it small extension} if it is surjective and $\Ker(f)$ is a nonzero
principal ideal which is annihilated by $\mathfrak{m}_B$.
\end{definition}
\noindent
By the following lemma we can often reduce arguments involving surjective ring
maps in $\mathcal{C}_\Lambda$ to the case of small extensions.
\begin{lemma}
\label{lemma-factor-small-extension}
Let $f: B \to A$ be a surjective ring map in $\mathcal{C}_\Lambda$.
Then $f$ can be factored as a composition of small extensions.
\end{lemma}
\begin{proof}
Let $I$ be the kernel of $f$. The maximal ideal $\mathfrak{m}_B$ is
nilpotent since $B$ is Artinian, say $\mathfrak{m}_B^n = 0$. Hence we get a
factorization
$$
B = B/I\mathfrak{m}_B^{n-1} \to B/I\mathfrak{m}_B^{n-2} \to
\ldots \to B/I \cong A
$$
of $f$ into a composition of surjective maps whose kernels are annihilated by
the maximal ideal. Thus it suffices to prove the lemma when $f$ itself is such
a map, i.e.\ when $I$ is annihilated by $\mathfrak{m}_B$. In this case
$I$ is a $k$-vector space, which has finite dimension, see
Algebra, Lemma \ref{algebra-lemma-artinian-finite-length}.
Take a basis $x_1, \ldots, x_n$ of $I$ as a $k$-vector space to get a
factorization
$$
B \to B/(x_1) \to \ldots \to B/(x_1, \ldots, x_n) \cong A
$$
of $f$ into a composition of small extensions.
\end{proof}
\noindent
The next lemma says that we can compute the length of a module over a local
$\Lambda$-algebra with residue field $k$ in terms of the length over
$\Lambda$. To explain the notation in the statement, let $k' \subset k$
be the image of our fixed finite ring map $\Lambda \to k$. Note
that $k/k'$ is a finite extension of rings. Hence $k'$ is a field
and $k'/k$ is a finite extension, see
Algebra, Lemma \ref{algebra-lemma-integral-under-field}.
Moreover, as $\Lambda \to k'$ is surjective we see that its kernel
is a maximal ideal $\mathfrak m_\Lambda$. Thus
\begin{equation}
\label{equation-k-prime}
[k : k'] = [k : \Lambda/\mathfrak m_\Lambda] < \infty
\end{equation}
and in the classical case we have $k = k'$. The notation
$k' = \Lambda/\mathfrak m_\Lambda$ will be fixed throughout this chapter.
\begin{lemma}
\label{lemma-length}
Let $A$ be a local $\Lambda$-algebra with residue field $k$.
Let $M$ be an $A$-module. Then
$[k : k'] \text{length}_A(M) = \text{length}_\Lambda(M)$.
In the classical case we have
$\text{length}_A(M) = \text{length}_\Lambda(M)$.
\end{lemma}
\begin{proof}
If $M$ is a simple $A$-module then $M \cong k$ as an $A$-module, see
Algebra, Lemma \ref{algebra-lemma-characterize-length-1}.
In this case $\text{length}_A(M) = 1$ and
$\text{length}_\Lambda(M) = [k' : k]$, see
Algebra, Lemma \ref{algebra-lemma-dimension-is-length}.
If $\text{length}_A(M)$ is finite, then the result follows on
choosing a filtration of $M$ by $A$-submodules with simple quotients
using additivity, see
Algebra, Lemma \ref{algebra-lemma-length-additive}.
If $\text{length}_A(M)$ is infinite, the result follows from the obvious
inequality $\text{length}_A(M) \leq \text{length}_\Lambda(M)$.
\end{proof}
\begin{lemma}
\label{lemma-surjective}
Let $A \to B$ be a ring map in $\mathcal{C}_\Lambda$.
The following are equivalent
\begin{enumerate}
\item $f$ is surjective,
\item $\mathfrak m_A/\mathfrak m_A^2 \to \mathfrak m_B/\mathfrak m_B^2$
is surjective, and
\item $\mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)
\to \mathfrak m_B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)$ is surjective.
\end{enumerate}
\end{lemma}
\begin{proof}
For any ring map $f : A \to B$ in $\mathcal{C}_\Lambda$ we have
$f(\mathfrak m_A) \subset \mathfrak m_B$ for example because
$\mathfrak m_A$, $\mathfrak m_B$ is the set of nilpotent elements of
$A$, $B$. Suppose $f$ is surjective. Let $y \in \mathfrak m_B$.
Choose $x \in A$ with $f(x) = y$. Since $f$ induces an isomorphism
$A/\mathfrak m_A \to B/\mathfrak m_B$ we see that $x \in \mathfrak m_A$.
Hence the induced map
$\mathfrak m_A/\mathfrak m_A^2 \to \mathfrak m_B/\mathfrak m_B^2$
is surjective. In this way we see that (1) implies (2).
\medskip\noindent
It is clear that (2) implies (3). The map $A \to B$ gives rise
to a canonical commutative diagram
$$
\xymatrix{
\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \otimes_{k'} k \ar[r] \ar[d] &
\mathfrak m_A/\mathfrak m_A^2 \ar[r] \ar[d] &
\mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2) \ar[r] \ar[d] & 0 \\
\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \otimes_{k'} k \ar[r] &
\mathfrak m_B/\mathfrak m_B^2 \ar[r] &
\mathfrak m_B/(\mathfrak m_\Lambda B + \mathfrak m_B^2) \ar[r] & 0
}
$$
with exact rows. Hence if (3) holds, then so does (2).
\medskip\noindent
Assume (2). To show that $A \to B$ is surjective it suffices by
Nakayama's lemma (Algebra, Lemma \ref{algebra-lemma-NAK})
to show that $A/\mathfrak m_A \to B/\mathfrak m_AB$ is surjective.
(Note that $\mathfrak m_A$ is a nilpotent ideal.)
As $k = A/\mathfrak m_A = B/\mathfrak m_B$ it suffices to show that
$\mathfrak m_AB \to \mathfrak m_B$ is surjective. Applying
Nakayama's lemma once more we see that it suffices to see that
$\mathfrak m_AB/\mathfrak m_A\mathfrak m_B \to \mathfrak m_B/\mathfrak m_B^2$
is surjective which is what we assumed.
\end{proof}
\noindent
If $A \to B$ is a ring map in $\mathcal{C}_\Lambda$, then the map
$\mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)
\to \mathfrak m_B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)$
is the map on relative cotangent spaces. Here is a formal definition.
\begin{definition}
\label{definition-tangent-space-ring}
Let $R \to S$ be a local homomorphism of local rings. The
{\it relative cotangent space}\footnote{Caution: We will see later
that in our general setting the tangent
space of an object $A \in \mathcal{C}_\Lambda$ over $\Lambda$ should
not be defined simply as the $k$-linear dual of the relative
cotangent space. In fact, the correct definition of the relative
cotangent space is
$\Omega_{S/R} \otimes_S S/\mathfrak m_S$.} of $R$ over $S$ is the
$S/\mathfrak m_S$-vector space
$\mathfrak m_S/(\mathfrak m_R S + \mathfrak m_S^2)$.
\end{definition}
\noindent
If $f_1: A_1 \to A$ and $f_2: A_2 \to A$ are two ring maps, then the fiber
product $A_1 \times_A A_2$ is the subring of $A_1 \times A_2$ consisting of
elements whose two projections to $A$ are equal. Throughout this chapter we
will be considering conditions involving such a fiber product when $f_1$
and $f_2$ are in $\mathcal{C}_\Lambda$. It isn't always the case that the
fibre product is an object of $\mathcal{C}_\Lambda$.
\begin{example}
\label{example-fibre-product}
Let $p$ be a prime number and let $n \in \mathbf{N}$.
Let $\Lambda = \mathbf{F}_p(t_1, t_2, \ldots, t_n)$ and let
$k = \mathbf{F}_p(x_1, \ldots, x_n)$ with map $\Lambda \to k$ given
by $t_i \mapsto x_i^p$. Let $A = k[\epsilon] = k[x]/(x^2)$.
Then $A$ is an object of $\mathcal{C}_\Lambda$. Suppose that
$D : k \to k$ is a derivation of $k$ over $\Lambda$, for example
$D = \partial/\partial x_i$. Then the map
$$
f_D : k \longrightarrow k[\epsilon], \quad
a \mapsto a + D(a)\epsilon
$$
is a morphism of $\mathcal{C}_\Lambda$. Set $A_1 = A_2 = k$ and set
$f_1 = f_{\partial/\partial x_1}$ and $f_2(a) = a$. Then
$A_1 \times_A A_2 = \{a \in k \mid \partial/\partial x_1(a) = 0\}$
which does not surject onto $k$. Hence the fibre product isn't
an object of $\mathcal{C}_\Lambda$.
\end{example}
\noindent
It turns out that this problem can only occur if the residue field
extension $k' \subset k$ (\ref{equation-k-prime}) is inseparable
and neither $f_1$ nor $f_2$ is surjective.
\begin{lemma}
\label{lemma-fiber-product-CLambda}
Let $f_1 : A_1 \to A$ and $f_2 : A_2 \to A$ be ring maps in
$\mathcal{C}_\Lambda$. Then:
\begin{enumerate}
\item If $f_1$ or $f_2$ is surjective, then
$A_1 \times_A A_2$ is in $\mathcal{C}_\Lambda$.
\item If $f_2$ is a small extension, then so is
$A_1 \times_A A_2 \to A_1$.
\item If the field extension $k' \subset k$ is separable, then
$A_1 \times_A A_2$ is in $\mathcal{C}_\Lambda$.
\end{enumerate}
\end{lemma}
\begin{proof}
The ring $A_1 \times_A A_2$ is a $\Lambda$-algebra via the map
$\Lambda \to A_1 \times_A A_2$ induced by the maps
$\Lambda \to A_1$ and $\Lambda \to A_2$. It is a local ring with unique
maximal ideal
$$
\mathfrak m_{A_1} \times_{\mathfrak m_A} \mathfrak m_{A_2} =
\Ker(A_1 \times_A A_2 \longrightarrow k)
$$
A ring is Artinian if and only if it has finite length as a module
over itself, see
Algebra, Lemma \ref{algebra-lemma-artinian-finite-length}.
Since $A_1$ and $A_2$ are Artinian, Lemma \ref{lemma-length} implies
$\text{length}_\Lambda(A_1)$ and $\text{length}_\Lambda(A_2)$,
and hence $\text{length}_\Lambda(A_1 \times A_2)$, are all finite. As
$A_1 \times_A A_2 \subset A_1 \times A_2$ is a $\Lambda$-submodule, this
implies
$\text{length}_{A_1 \times_A A_2}(A_1 \times_A A_2) \leq
\text{length}_\Lambda(A_1 \times_A A_2)$ is finite. So $A_1
\times_A A_2$ is Artinian. Thus the only thing that is keeping
$A_1 \times_A A_2$ from being an object of $\mathcal{C}_\Lambda$ is
the possibility that its residue field maps to a proper subfield of $k$
via the map $A_1 \times_A A_2 \to A \to A/\mathfrak m_A = k$ above.
\medskip\noindent
Proof of (1). If $f_2$ is surjective, then the projection
$A_1 \times_A A_2 \to A_1$ is surjective. Hence the composition
$A_1 \times_A A_2 \to A_1 \to A_1/\mathfrak m_{A_1} = k$ is surjective
and we conclude that $A_1 \times_A A_2$ is an object of $\mathcal{C}_\Lambda$.
\medskip\noindent
Proof of (2). If $f_2$ is a small extension then $A_2 \to A$ and
$A_1 \times_A A_2 \to A_1$ are both surjective with the same kernel.
Hence the kernel of $A_1 \times_A A_2 \to A_1$ is a $1$-dimensional
$k$-vector space and we see that $A_1 \times_A A_2 \to A_1$ is a small
extension.
\medskip\noindent
Proof of (3). Choose $\overline{x} \in k$ such that
$k = k'(\overline{x})$ (see
Fields, Lemma \ref{fields-lemma-primitive-element}).
Let $P'(T) \in k'[T]$ be the minimal polynomial of $\overline{x}$ over $k'$.
Since $k/k'$ is separable we see that
$\text{d}P/\text{d}T(\overline{x}) \not = 0$.
Choose a monic $P \in \Lambda[T]$ which maps to $P'$ under the surjective map
$\Lambda[T] \to k'[T]$. Because $A, A_1, A_2$ are henselian, see
Algebra, Lemma \ref{algebra-lemma-local-dimension-zero-henselian},
we can find $x, x_1, x_2 \in A, A_1, A_2$ with $P(x) = 0, P(x_1) = 0,
P(x_2) = 0$ and such that the image of $x, x_1, x_2$ in $k$ is $\overline{x}$.
Then $(x_1, x_2) \in A_1 \times_A A_2$ because $x_1, x_2$
map to $x \in A$ by uniqueness, see
Algebra, Lemma \ref{algebra-lemma-uniqueness}.
Hence the residue field of
$A_1 \times_A A_2$ contains a generator of $k$ over $k'$ and we win.
\end{proof}
\noindent
Next we define essential surjections in $\mathcal{C}_\Lambda$. A necessary
and sufficient condition for a surjection in $\mathcal{C}_\Lambda$ to be
essential is given in Lemma \ref{lemma-essential-surjection}.
\begin{definition}
\label{definition-essential-surjection}
Let $f: B \to A$ be a ring map in $\mathcal{C}_\Lambda$. We say $f$
is an {\it essential surjection} if it has the following properties:
\begin{enumerate}
\item $f$ is surjective.
\item If $g: C \to B$ is a ring map in $\mathcal{C}_\Lambda$ such that
$f \circ g$ is surjective, then $g$ is surjective.
\end{enumerate}
\end{definition}
\noindent
Using Lemma \ref{lemma-surjective}, we can characterize
essential surjections in $\mathcal{C}_\Lambda$ as follows.
\begin{lemma}
\label{lemma-essential-surjection-mod-squares}
Let $f: B \to A$ be a ring map in $\mathcal{C}_\Lambda$.
The following are equivalent
\begin{enumerate}
\item $f$ is an essential surjection,
\item the map $B/\mathfrak m_B^2 \to A/\mathfrak m_A^2$ is an essential
surjection, and
\item the map
$B/(\mathfrak m_\Lambda B + \mathfrak m_B^2) \to
A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)$ is an essential surjection.
\end{enumerate}
\end{lemma}
\begin{proof}
Assume (3). Let $C \to B$ be a ring map in $\mathcal{C}_\Lambda$ such
that $C \to A$ is surjective. Then
$C \to A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)$ is surjective
too. We conclude that $C \to B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)$
is surjective by our assumption. Hence $C \to B$ is surjective by applying
Lemma \ref{lemma-surjective} (2 times).
\medskip\noindent
Assume (1). Let $C \to B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)$
be a morphism of $\mathcal{C}_\Lambda$ such that
$C \to A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)$ is surjective. Set
$C' = C \times_{B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)} B$
which is an object of $\mathcal{C}_\Lambda$ by
Lemma \ref{lemma-fiber-product-CLambda}.
Note that $C' \to A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)$
is still surjective, hence $C' \to A$ is surjective by
Lemma \ref{lemma-surjective}.
Thus $C' \to B$ is surjective by our assumption. This implies
that $C' \to B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)$ is
surjective, which implies by the construction of $C'$ that
$C \to B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)$ is surjective.
\medskip\noindent
In the first paragraph we proved (3) $\Rightarrow$ (1) and in the second
paragraph we proved (1) $\Rightarrow$ (3). The equivalence of
(2) and (3) is a special case of the equivalence of (1) and (3), hence
we are done.
\end{proof}
\noindent
To analyze essential surjections in $\mathcal{C}_\Lambda$ a bit more
we introduce some notation. Suppose that $A$ is an object
of $\mathcal{C}_\Lambda$. There is a canonical exact sequence
\begin{equation}
\label{equation-sequence}
\mathfrak m_A/\mathfrak m_A^2 \xrightarrow{\text{d}_A}
\Omega_{A/\Lambda} \otimes_A k \to
\Omega_{k/\Lambda} \to 0
\end{equation}
see
Algebra, Lemma \ref{algebra-lemma-differential-seq}.
Note that $\Omega_{k/\Lambda} = \Omega_{k/k'}$ with $k'$ as
in (\ref{equation-k-prime}). Let $H_1(L_{k/\Lambda})$
be the first homology module of the naive cotangent complex of $k$
over $\Lambda$, see
Algebra, Definition \ref{algebra-definition-naive-cotangent-complex}.
Then we can extend (\ref{equation-sequence})
to the exact sequence
\begin{equation}
\label{equation-sequence-extended}
H_1(L_{k/\Lambda}) \to
\mathfrak m_A/\mathfrak m_A^2 \xrightarrow{\text{d}_A}
\Omega_{A/\Lambda} \otimes_A k \to
\Omega_{k/\Lambda} \to 0,
\end{equation}
see
Algebra, Lemma \ref{algebra-lemma-exact-sequence-NL}.
If $B \to A$ is a ring map in $\mathcal{C}_\Lambda$ then we obtain a
commutative diagram
\begin{equation}
\label{equation-sequence-functorial}
\vcenter{
\xymatrix{
H_1(L_{k/\Lambda}) \ar[r] \ar@{=}[d] &
\mathfrak m_B/\mathfrak m_B^2 \ar[r]_{\text{d}_B} \ar[d] &
\Omega_{B/\Lambda} \otimes_B k \ar[r] \ar[d] &
\Omega_{k/\Lambda} \ar[r] \ar@{=}[d] & 0 \\
H_1(L_{k/\Lambda}) \ar[r] &
\mathfrak m_A/\mathfrak m_A^2 \ar[r]^{\text{d}_A} &
\Omega_{A/\Lambda} \otimes_A k \ar[r] &
\Omega_{k/\Lambda} \ar[r] & 0
}
}
\end{equation}
with exact rows.
\begin{lemma}
\label{lemma-H1-separable-case}
There is a canonical map
$$
\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \longrightarrow H_1(L_{k/\Lambda}).
$$
If $k' \subset k$ is separable (for example if the characteristic
of $k$ is zero), then this map induces an isomorphism
$\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \otimes_{k'} k = H_1(L_{k/\Lambda})$.
If $k = k'$ (for example in the classical case), then
$\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 = H_1(L_{k/\Lambda})$.
The composition
$$
\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \longrightarrow
H_1(L_{k/\Lambda}) \longrightarrow \mathfrak m_A/\mathfrak m_A^2
$$
comes from the canonical map $\mathfrak m_\Lambda \to \mathfrak m_A$.
\end{lemma}
\begin{proof}
Note that $H_1(L_{k'/\Lambda}) = \mathfrak m_\Lambda/\mathfrak m_\Lambda^2$
as $\Lambda \to k'$ is surjective with kernel $\mathfrak m_\Lambda$.
The map arises from functoriality of the naive cotangent complex.
If $k' \subset k$ is separable, then $k' \to k$ is an \'etale ring map, see
Algebra, Lemma \ref{algebra-lemma-etale-over-field}.
Thus its naive cotangent complex has trivial homology groups, see
Algebra, Definition \ref{algebra-definition-etale}.
Then
Algebra, Lemma \ref{algebra-lemma-exact-sequence-NL}
applied to the ring maps $\Lambda \to k' \to k$ implies that
$\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \otimes_{k'} k = H_1(L_{k/\Lambda})$.
We omit the proof of the final statement.
\end{proof}
\begin{lemma}
\label{lemma-essential-surjection}
Let $f: B \to A$ be a ring map in $\mathcal{C}_\Lambda$.
Notation as in (\ref{equation-sequence-functorial}).
\begin{enumerate}
\item The equivalent conditions of
Lemma \ref{lemma-essential-surjection-mod-squares}
characterizing when $f$ is surjective are also equivalent to
\begin{enumerate}
\item $\Im(\text{d}_B) \to \Im(\text{d}_A)$ is surjective, and
\item the map $\Omega_{B/\Lambda} \otimes_B k \to
\Omega_{A/\Lambda} \otimes_A k$ is surjective.
\end{enumerate}
\item The following are equivalent
\begin{enumerate}
\item $f$ is an essential surjection,
\item the map $\Im(\text{d}_B) \to \Im(\text{d}_A)$ is an
isomorphism, and
\item the map $\Omega_{B/\Lambda} \otimes_B k \to
\Omega_{A/\Lambda} \otimes_A k$ is an isomorphism.
\end{enumerate}
\item If $k/k'$ is separable, then $f$ is an essential surjection if
and only if the map
$\mathfrak m_B/(\mathfrak m_\Lambda B + \mathfrak m_B^2) \to
\mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)$
is an isomorphism.
\item If $f$ is a small extension, then $f$ is not essential if and only if
$f$ has a section $s: A \to B$ in $\mathcal{C}_\Lambda$
with $f \circ s = \text{id}_A$.
\end{enumerate}
\end{lemma}
\begin{proof}
Proof of (1). It follows from (\ref{equation-sequence-functorial})
that (1)(a) and (1)(b) are equivalent. Also, if
$A \to B$ is surjective, then (1)(a) and (1)(b) hold. Assume (1)(a).
Since the kernel of $\text{d}_A$ is the image of
$H_1(L_{k/\Lambda})$ which also maps to
$\mathfrak m_B/\mathfrak m_B^2$ we conclude that
$\mathfrak m_B/\mathfrak m_B^2 \to \mathfrak m_A/\mathfrak m_A^2$
is surjective. Hence $B \to A$ is surjective by
Lemma \ref{lemma-surjective}. This finishes the proof of (1).
\medskip\noindent
Proof of (2). The equivalence of (2)(b) and (2)(c) is immediate from
(\ref{equation-sequence-functorial}).
\medskip\noindent
Assume (2)(b). Let $g : C \to B$ be a ring map in $\mathcal{C}_\Lambda$
such that $f \circ g$ is surjective. We conclude that
$\mathfrak m_C/\mathfrak m_C^2 \to \mathfrak m_A/\mathfrak m_A^2$
is surjective by
Lemma \ref{lemma-surjective}.
Hence
$\Im(\text{d}_C) \to \Im(\text{d}_A)$ is surjective
and by the assumption we see that
$\Im(\text{d}_C) \to \Im(\text{d}_B)$ is surjective.
It follows that $C \to B$ is surjective by (1).
\medskip\noindent
Assume (2)(a). Then $f$ is surjective and we see that
$\Omega_{B/\Lambda} \otimes_B k \to \Omega_{A/\Lambda} \otimes_A k$
is surjective. Let $K$ be the kernel. Note that
$K = \text{d}_B(\Ker(\mathfrak m_B/\mathfrak m_B^2 \to
\mathfrak m_A/\mathfrak m_A^2))$ by (\ref{equation-sequence-functorial}).
Choose a splitting
$$
\Omega_{B/\Lambda} \otimes_B k =
\Omega_{A/\Lambda} \otimes_A k \oplus K
$$
of $k$-vector space. The map $\text{d} : B \to \Omega_{B/\Lambda}$
induces via the projection onto $K$ a map $D : B \to K$. Set
$C = \{b \in B \mid D(b) = 0\}$. The Leibniz rule shows that this is
a $\Lambda$-subalgebra of $B$. Let $\overline{x} \in k$. Choose $x \in B$
mapping to $\overline{x}$. If $D(x) \not = 0$, then we can find an element
$y \in \mathfrak m_B$ such that $D(y) = D(x)$. Hence $x - y \in C$ is
an element which maps to $\overline{x}$. Thus $C \to k$ is surjective
and $C$ is an object of $\mathcal{C}_\Lambda$. Similarly, pick
$\omega \in \Im(\text{d}_A)$. We can find $x \in \mathfrak m_B$
such that $\text{d}_B(x)$ maps to $\omega$ by (1). If $D(x) \not = 0$, then
we can find an element $y \in \mathfrak m_B$ which maps to zero
in $\mathfrak m_A/\mathfrak m_A^2$ such that $D(y) = D(x)$.
Hence $z = x - y$ is an element of $\mathfrak m_C$ whose
image $\text{d}_C(z) \in \Omega_{C/k} \otimes_C k$ maps to $\omega$.
Hence $\Im(\text{d}_C) \to \Im(\text{d}_A)$ is surjective.
We conclude that $C \to A$ is surjective by (1). Hence $C \to B$ is
surjective by assumption. Hence $D = 0$, i.e., $K = 0$, i.e., (2)(c) holds.
This finishes the proof of (2).
\medskip\noindent
Proof of (3). If $k'/k$ is separable, then
$H_1(L_{k/\Lambda}) =
\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \otimes_{k'} k$, see
Lemma \ref{lemma-H1-separable-case}.
Hence $\Im(\text{d}_A) =
\mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)$
and similarly for $B$. Thus (3) follows from (2).
\medskip\noindent
Proof of (4). A section $s$ of $f$ is not surjective (by definition a
small extension has nontrivial kernel), hence $f$ is not essentially
surjective. Conversely, assume $f$ is a small surjection but not an
essential surjection. Choose a ring map $C \to B$ in $\mathcal{C}_\Lambda$
which is not surjective, such that $C \to A$ is surjective. Let
$C' \subset B$ be the image of $C \to B$. Then $C' \not = B$ but
$C'$ surjects onto $A$. Since $f : B \to A$ is a small extension,
$\text{length}_C(B) = \text{length}_C(A) + 1$. Thus
$\text{length}_C(C') \leq \text{length}_C(A)$ since
$C'$ is a proper subring of $B$. But $C' \to A$ is surjective, so in
fact we must have $\text{length}_C(C') = \text{length}_C(A)$ and
$C' \to A$ is an isomorphism which gives us our section.
\end{proof}
\begin{example}
\label{example-essential-surjection}
Let $\Lambda = k[[x]]$ be the power series ring in $1$ variable over $k$.
Set $A = k$ and $B = \Lambda/(x^2)$. Then $B \to A$ is an essential
surjection by
Lemma \ref{lemma-essential-surjection}
because it is a small extension and the map $B \to A$ does not have a
right inverse (in the category $\mathcal{C}_\Lambda$). But the map
$$
k \cong \mathfrak m_B/\mathfrak m_B^2
\longrightarrow
\mathfrak m_A/\mathfrak m_A^2 = 0
$$
is not an isomorphism. Thus in
Lemma \ref{lemma-essential-surjection} (3)
it is necessary to consider the map of relative cotangent spaces
$\mathfrak m_B/(\mathfrak m_\Lambda B + \mathfrak m_B^2) \to
\mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)$.
\end{example}
\section{The completed base category}
\label{section-category-completion-CLambda}
\noindent
The following ``completion'' of the category $\mathcal{C}_\Lambda$ will serve
as the base category of the completion of a category cofibered in groupoids
over $\mathcal{C}_\Lambda$
(Section \ref{section-formal-objects}).
\begin{definition}
\label{definition-completion-CLambda}
Let $\Lambda$ be a Noetherian ring and let $\Lambda \to k$ be a finite
ring map where $k$ is a field. We define {\it $\widehat{\mathcal{C}}_\Lambda$}
to be the category with
\begin{enumerate}
\item objects are pairs $(R, \varphi)$ where $R$ is a Noetherian complete
local $\Lambda$-algebra and where $\varphi : R/\mathfrak m_R \to k$ is a
$\Lambda$-algebra isomorphism, and
\item morphisms $f : (S, \psi) \to (R, \varphi)$ are local $\Lambda$-algebra
homomorphisms such that $\varphi \circ (f \bmod \mathfrak m) = \psi$.
\end{enumerate}
\end{definition}
\noindent
As in the discussion following
Definition \ref{definition-CLambda}
we will usually denote an object of $\widehat{\mathcal{C}}_\Lambda$
simply $R$, with the identification $R/\mathfrak m_R = k$ understood.
In this section we discuss some basic properties of objects and morphisms
of the category $\widehat{\mathcal{C}}_\Lambda$ paralleling our discussion of
the category $\mathcal{C}_\Lambda$ in the previous section.
\medskip\noindent
Our first observation is that any object $A \in \mathcal{C}_\Lambda$
is an object of $\widehat{\mathcal{C}}_\Lambda$ as an Artinian local
ring is always Noetherian and complete with respect to its maximal ideal
(which is after all a nilpotent ideal). Moreover, it is clear from the
definitions that
$\mathcal{C}_\Lambda \subset \widehat{\mathcal{C}}_\Lambda$
is the strictly full subcategory consisting of all Artinian rings.
As it turns out, conversely every object of
$\widehat{\mathcal{C}}_\Lambda$ is a limit of objects of
$\mathcal{C}_\Lambda$.
\medskip\noindent
Suppose that $R$ is an object of $\widehat{\mathcal{C}}_\Lambda$.
Consider the rings $R_n = R/\mathfrak m_R^n$ for $n \in \mathbf{N}$.
These are Noetherian local rings with a unique nilpotent prime ideal, hence
Artinian, see
Algebra, Proposition \ref{algebra-proposition-dimension-zero-ring}.
The ring maps
$$
\ldots \to R_{n + 1} \to R_n \to \ldots \to R_2 \to R_1 = k
$$
are all surjective. Completeness of $R$ by definition means
that $R = \lim R_n$. If $f : R \to S$ is a ring map in
$\widehat{\mathcal{C}}_\Lambda$ then we obtain a system of ring maps
$f_n : R_n \to S_n$ whose limit is the given map.
\begin{lemma}
\label{lemma-surjective-cotangent-space}
Let $f: R \to S$ be a ring map in $\widehat{\mathcal{C}}_\Lambda$.
The following are equivalent
\begin{enumerate}
\item $f$ is surjective,
\item the map
$\mathfrak m_R/\mathfrak m_R^2 \to \mathfrak m_S/\mathfrak m_S^2$
is surjective, and
\item the map
$\mathfrak m_R/(\mathfrak m_\Lambda R + \mathfrak m_R^2) \to
\mathfrak m_S/(\mathfrak m_\Lambda S + \mathfrak m_S^2)$
is surjective.
\end{enumerate}
\end{lemma}
\begin{proof}
Note that for $n \geq 2$ we have the equality of relative cotangent spaces
$$
\mathfrak m_R/(\mathfrak m_\Lambda R + \mathfrak m_R^2)
=
\mathfrak m_{R_n}/(\mathfrak m_\Lambda R_n + \mathfrak m_{R_n}^2)
$$
and similarly for $S$. Hence by
Lemma \ref{lemma-surjective}
we see that $R_n \to S_n$ is surjective for all $n$.
Now let $K_n$ be the kernel of $R_n \to S_n$. Then the sequences
$$
0 \to K_n \to R_n \to S_n \to 0
$$
form an exact sequence of directed inverse systems. The system $(K_n)$ is
Mittag-Leffler since each $K_n$ is Artinian. Hence by
Algebra, Lemma \ref{algebra-lemma-ML-exact-sequence}
taking limits preserves exactness. So
$\lim R_n \to \lim S_n$ is surjective, i.e., $f$ is surjective.
\end{proof}
\begin{lemma}
\label{lemma-CLambdahat-pushouts}
The category $\widehat{\mathcal{C}}_\Lambda$ admits pushouts.
\end{lemma}
\begin{proof}
Let $R \to S_1$ and $R \to S_2$ be morphisms of
$\widehat{\mathcal{C}}_\Lambda$. Consider the ring
$C = S_1 \otimes_R S_2$.
This ring has a finitely generated maximal ideal
$\mathfrak m = \mathfrak m_{S_1} \otimes S_2 +
S_1 \otimes \mathfrak m_{S_2}$ with residue field $k$.
Set $C^\wedge$ equal to the completion of $C$ with respect to $\mathfrak m$.
Then $C^\wedge$ is a Noetherian ring complete with respect to
the maximal ideal $\mathfrak m^\wedge = \mathfrak mC^\wedge$
whose residue field is identified with $k$, see
Algebra, Lemma \ref{algebra-lemma-completion-Noetherian}.
Hence $C^\wedge$ is an object of $\widehat{\mathcal{C}}_\Lambda$.
Then $S_1 \to C^\wedge$ and $S_2 \to C^\wedge$ turn $C^\wedge$
into a pushout over $R$ in $\widehat{\mathcal{C}}_\Lambda$ (details omitted).
\end{proof}
\noindent
We will not need the following lemma.
\begin{lemma}
\label{lemma-CLambdahat-coproducts}
The category $\widehat{\mathcal{C}}_\Lambda$ admits coproducts
of pairs of objects.
\end{lemma}
\begin{proof}
Let $R$ and $S$ be objects of $\widehat{\mathcal{C}}_\Lambda$.
Consider the ring $C = R \otimes_\Lambda S$. There is a canonical
surjective map $C \to R \otimes_\Lambda S \to k \otimes_\Lambda k \to k$
where the last map is the multiplication map. The kernel of
$C \to k$ is a maximal ideal $\mathfrak m$. Note that $\mathfrak m$
is generated by $\mathfrak m_R C$, $\mathfrak m_S C$ and finitely many
elements of $C$ which map to generators of the kernel of
$k \otimes_\Lambda k \to k$. Hence $\mathfrak m$ is a finitely
generated ideal. Set
$C^\wedge$ equal to the completion of $C$ with respect to $\mathfrak m$.
Then $C^\wedge$ is a Noetherian ring complete with respect to
the maximal ideal $\mathfrak m^\wedge = \mathfrak mC^\wedge$
with residue field $k$, see
Algebra, Lemma \ref{algebra-lemma-completion-Noetherian}.
Hence $C^\wedge$ is an object of $\widehat{\mathcal{C}}_\Lambda$.
Then $R \to C^\wedge$ and $S \to C^\wedge$ turn $C^\wedge$
into a coproduct in $\widehat{\mathcal{C}}_\Lambda$ (details omitted).
\end{proof}
\noindent
An empty coproduct in a category is an initial object of the category.
In the classical case $\widehat{\mathcal{C}}_\Lambda$ has an initial
object, namely $\Lambda$ itself. More generally, if $k' = k$, then
the completion $\Lambda^\wedge$ of $\Lambda$ with respect to
$\mathfrak m_\Lambda$ is an initial object. More generally still, if
$k' \subset k$ is separable, then $\widehat{\mathcal{C}}_\Lambda$ has an
initial object too. Namely, choose a monic polynomial $P \in \Lambda[T]$
such that $k \cong k'[T]/(P')$ where $p' \in k'[T]$ is the image
of $P$. Then $R = \Lambda^\wedge[T]/(P)$ is an initial object, see proof of
Lemma \ref{lemma-fiber-product-CLambda}.
\medskip\noindent
If $R$ is an initial object as above, then we have
$\mathcal{C}_\Lambda = \mathcal{C}_R$ and
$\widehat{\mathcal{C}}_\Lambda = \widehat{\mathcal{C}}_R$ which effectively
brings the whole discussion in this chapter back to the classical case.
But, if $k' \subset k$ is inseparable, then an initial object does not
exist.
\begin{lemma}
\label{lemma-derivations-finite}
Let $S$ be an object of $\widehat{\mathcal{C}}_\Lambda$.
Then $\dim_k \text{Der}_\Lambda(S, k) < \infty$.
\end{lemma}
\begin{proof}
Let $x_1, \ldots, x_n \in \mathfrak m_S$ map to a $k$-basis
for the relative cotangent space
$\mathfrak m_S/(\mathfrak m_\Lambda S + \mathfrak m_S^2)$.
Choose $y_1, \ldots, y_m \in S$ whose images in $k$ generate $k$
over $k'$. We claim that $\dim_k \text{Der}_\Lambda(S, k) \leq n + m$.
To see this it suffices to prove that if $D(x_i) = 0$ and
$D(y_j) = 0$, then $D = 0$. Let $a \in S$. We can find a
polynomial $P = \sum \lambda_J y^J$ with $\lambda_J \in \Lambda$
whose image in $k$ is the same as the image of $a$ in $k$.
Then we see that $D(a - P) = D(a) - D(P) = D(a)$ by our assumption
that $D(y_j) = 0$ for all $j$. Thus we may assume $a \in \mathfrak m_S$.
Write $a = \sum a_i x_i$ with $a_i \in S$. By the Leibniz rule
$$
D(a) = \sum x_iD(a_i) + \sum a_iD(x_i) = \sum x_iD(a_i)
$$
as we assumed $D(x_i) = 0$. We have $\sum x_iD(a_i) = 0$
as multiplication by $x_i$ is zero on $k$.
\end{proof}
\begin{lemma}
\label{lemma-derivations-surjective}
Let $f : R \to S$ be a morphism of $\widehat{\mathcal{C}}_\Lambda$.
If $\text{Der}_\Lambda(S, k) \to \text{Der}_\Lambda(R, k)$ is injective,
then $f$ is surjective.
\end{lemma}
\begin{proof}
If $f$ is not surjective, then
$\mathfrak m_S/(\mathfrak m_R S + \mathfrak m_S^2)$ is nonzero by
Lemma \ref{lemma-surjective-cotangent-space}.
Then also $Q = S/(f(R) + \mathfrak m_R S + \mathfrak m_S^2)$ is nonzero.
Note that $Q$ is a $k = R/\mathfrak m_R$-vector space via $f$. We turn
$Q$ into an $S$-module via $S \to k$. The quotient
map $D : S \to Q$ is an $R$-derivation: if $a_1, a_2 \in S$, we can write
$a_1 = f(b_1) + a_1'$ and $a_2 = f(b_2) + a_2'$ for some
$b_1, b_2 \in R$ and $a_1', a_2' \in \mathfrak m_S$. Then
$b_i$ and $a_i$ have the same image in $k$ for $i = 1, 2$ and
\begin{align*}
a_1a_2 & = (f(b_1) + a_1')(f(b_2) + a_2') \\
& = f(b_1)a_2' + f(b_2)a_1' \\
& = f(b_1)(f(b_2) + a_2') + f(b_2)(f(b_1) + a_1') \\
& = f(b_1)a_2 + f(b_2)a_1
\end{align*}
in $Q$ which proves the Leibniz rule. Hence $D : S \to Q$ is a
$\Lambda$-derivation which is zero on composing with $R \to S$.
Since $Q \not = 0$ there also exist derivations $D : S \to k$ which
are zero on composing with $R \to S$, i.e.,
$\text{Der}_\Lambda(S, k) \to \text{Der}_\Lambda(R, k)$ is not injective.
\end{proof}
\begin{lemma}
\label{lemma-m-adic-topology}
Let $R$ be an object of $\widehat{\mathcal{C}}_\Lambda$. Let $(J_n)$ be a
decreasing sequence of ideals such that $\mathfrak m_R^n \subset J_n$.
Set $J = \bigcap J_n$. Then the sequence $(J_n/J)$ defines the
$\mathfrak m_{R/J}$-adic topology on $R/J$.
\end{lemma}
\begin{proof}
It is clear that $\mathfrak m_{R/J}^n \subset J_n/J$. Thus it suffices
to show that for every $n$ there exists an $N$ such that
$J_N/J \subset \mathfrak m_{R/J}^n$. This is equivalent to
$J_N \subset \mathfrak m_R^n + J$. For each $n$ the ring $R/\mathfrak m_R^n$
is Artinian, hence there exists a $N_n$ such that
$$
J_{N_n} + \mathfrak m_R^n = J_{N_n + 1} + \mathfrak m_R^n = \ldots
$$
Set $E_n = (J_{N_n} + \mathfrak m_R^n)/\mathfrak m_R^n$.
Set $E = \lim E_n \subset \lim R/\mathfrak m_R^n = R$.
Note that $E \subset J$ as for any $f \in E$ and any $m$
we have $f \in J_m + \mathfrak m_R^n$ for all $n \gg 0$, so
$f \in J_m$ by Artin-Rees, see
Algebra, Lemma \ref{algebra-lemma-intersect-powers-ideal-module-zero}.
Since the transition maps $E_n \to E_{n - 1}$ are all surjective,
we see that $J$ surjects onto $E_n$. Hence for $N = N_n$ works.
\end{proof}
\begin{lemma}
\label{lemma-limit-artinian}
Let $\ldots \to A_3 \to A_2 \to A_1$ be a sequence of surjective
ring maps in $\mathcal{C}_\Lambda$. If
$\dim_k (\mathfrak m_{A_n}/\mathfrak m_{A_n}^2)$ is bounded, then
$S = \lim A_n$ is an object in $\widehat{\mathcal{C}}_\Lambda$
and the ideals $I_n = \Ker(S \to A_n)$ define the
$\mathfrak m_S$-adic topology on $S$.
\end{lemma}
\begin{proof}
We will use freely that the maps $S \to A_n$ are surjective for all $n$.
Note that the maps
$\mathfrak m_{A_{n + 1}}/\mathfrak m_{A_{n + 1}}^2 \to
\mathfrak m_{A_n}/\mathfrak m_{A_n}^2$ are surjective, see
Lemma \ref{lemma-surjective-cotangent-space}.
Hence for $n$ sufficiently large the dimension
$\dim_k (\mathfrak m_{A_n}/\mathfrak m_{A_n}^2)$ stabilizes to an
integer, say $r$.
Thus we can find $x_1, \ldots, x_r \in \mathfrak m_S$ whose images in
$A_n$ generate $\mathfrak m_{A_n}$. Moreover, pick $y_1, \ldots, y_t \in S$
whose images in $k$ generate $k$ over $\Lambda$. Then we get a ring map
$P = \Lambda[z_1, \ldots, z_{r + t}] \to S$, $z_i \mapsto x_i$ and
$z_{r + j} \mapsto y_j$ such that the composition
$P \to S \to A_n$ is surjective for all $n$. Let $\mathfrak m \subset P$
be the kernel of $P \to k$. Let $R = P^\wedge$ be the $\mathfrak m$-adic
completion of $P$; this is an object of $\widehat{\mathcal{C}}_\Lambda$.
Since we still have the compatible system of (surjective) maps $R \to A_n$
we get a map $R \to S$. Set $J_n = \Ker(R \to A_n)$.
Set $J = \bigcap J_n$. By
Lemma \ref{lemma-m-adic-topology}
we see that $R/J = \lim R/J_n = \lim A_n = S$
and that the ideals $J_n/J = I_n$ define the $\mathfrak m$-adic topology.
(Note that for each $n$ we have $\mathfrak m_R^{N_n} \subset J_n$ for
some $N_n$ and not necessarily $N_n = n$, so a renumbering of the ideals
$J_n$ may be necessary before applying the lemma.)
\end{proof}
\begin{lemma}
\label{lemma-power-series}
Let $R', R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$. Suppose that
$R = R' \oplus I$ for some ideal $I$ of $R$. Let $x_1, \ldots, x_r \in I$
map to a basis of $I/\mathfrak m_R I$. Set $S = R'[[X_1, \ldots, X_r]]$
and consider the $R'$-algebra map $S \to R$ mapping $X_i$ to $x_i$.
Assume that for every $n \gg 0$ the map
$S/\mathfrak m_S^n \to R/\mathfrak m_R^n$ has a left inverse in
$\mathcal{C}_\Lambda$. Then $S \to R$ is an isomorphism.
\end{lemma}
\begin{proof}
As $R = R' \oplus I$ we have
$$
\mathfrak m_R/\mathfrak m_R^2 =
\mathfrak m_{R'}/\mathfrak m_{R'}^2 \oplus I/\mathfrak m_RI
$$
and similarly
$$
\mathfrak m_R/\mathfrak m_R^2 =
\mathfrak m_{R'}/\mathfrak m_{R'}^2 \oplus \bigoplus kX_i
$$
Hence for $n > 1$ the map $S/\mathfrak m_S^n \to R/\mathfrak m_R^n$
induces an isomorphism on cotangent spaces. Thus a left inverse
$h_n : R/\mathfrak m_R^n \to S/\mathfrak m_S^n$ is surjective by
Lemma \ref{lemma-surjective-cotangent-space}.
Since $h_n$ is injective as a left inverse it is an isomorphism.
Thus the canonical surjections $S/\mathfrak m_S^n \to R/\mathfrak m_R^n$
are all isomorphisms and we win.
\end{proof}
\section{Categories cofibered in groupoids}
\label{section-preliminary}
\noindent
In developing the theory we work with categories {\it cofibered} in groupoids.
We assume as known the definition and basic properties of categories
{\it fibered} in groupoids, see
Categories, Section \ref{categories-section-fibred-groupoids}.
\begin{definition}
\label{definition-category-cofibred-groupoids}
Let $\mathcal{C}$ be a category. A {\it category cofibered in groupoids over
$\mathcal{C}$} is a category $\mathcal{F}$ equipped with a functor
$p: \mathcal{F} \to \mathcal{C}$ such that $\mathcal{F}^{opp}$ is a category
fibered in groupoids over $\mathcal{C}^{opp}$ via
$p^{opp}: \mathcal{F}^{opp} \to \mathcal{C}^{opp}$.
\end{definition}
\noindent
Explicitly, $p: \mathcal{F} \to \mathcal{C}$ is cofibered in groupoids if
the following two conditions hold:
\begin{enumerate}
\item For every morphism $f: U \to V$ in $\mathcal{C}$ and every object
$x$ lying over $U$, there is a morphism $x \to y$ of $\mathcal{F}$ lying
over $f$.
\item For every pair of morphisms $a: x \to y$ and $b: x \to z$
of $\mathcal{F}$ and any morphism $f: p(y) \to p(z)$ such that $p(b) = f
\circ p(a)$, there exists a unique morphism $c: y \to z$ of $\mathcal
F$ lying over $f$ such that $b = c \circ a$.
\end{enumerate}
\begin{remarks}
\label{remarks-cofibered-groupoids}
Everything about categories fibered in groupoids translates directly to the
cofibered setting. The following remarks are meant to fix notation.
Let $\mathcal{C}$ be a category.
\begin{enumerate}
\item We often omit the functor $p: \mathcal{F} \to \mathcal{C}$ from the
notation.
\item The fiber category over an object $U$ in $\mathcal{C}$ is denoted by
$\mathcal{F}(U)$. Its objects are those of $\mathcal{F}$ lying over $U$ and its
morphisms are those of $\mathcal{F}$ lying over $\text{id}_U$.
If $x, y$ are objects of $\mathcal{F}(U)$, we sometimes write
$\Mor_U(x, y)$ for $\Mor_{\mathcal{F}(U)}(x, y)$.
\item The fibre categories $\mathcal{F}(U)$ are groupoids, see
Categories, Lemma \ref{categories-lemma-fibred-groupoids}.
Hence the morphisms in $\mathcal{F}(U)$ are all isomorphisms.
We sometimes write $\text{Aut}_U(x)$ for $\Mor_{\mathcal{F}(U)}(x, x)$.
\item
\label{item-pushforward}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}$, let $f: U \to V$ be a morphism in $\mathcal{C}$, and
let $x \in \Ob(\mathcal{F}(U))$.
A {\it pushforward} of $x$ along $f$ is a morphism
$x \to y$ of $\mathcal{F}$ lying over $f$. A pushforward
is unique up to unique isomorphism (see the discussion following
Categories, Definition \ref{categories-definition-cartesian-over-C}).
We sometimes write $x \to f_*x$ for ``the'' pushforward of $x$
along $f$.
\item A {\it choice of pushforwards for $\mathcal{F}$} is the choice of
a pushforward of $x$ along $f$ for every pair $(x, f)$ as above. We can make
such a choice of pushforwards for $\mathcal{F}$ by the axiom of choice.
\item Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}$. Given a choice of pushforwards for $\mathcal{F}$, there
is an associated pseudo-functor $\mathcal{C} \to \textit{Groupoids}$.
We will never use this construction so we give no details.
\item
\label{item-cofibered-morphism}
A morphism of categories cofibered in groupoids over $\mathcal{C}$ is a
functor commuting with the projections to $\mathcal{C}$. If $\mathcal{F}$
and $\mathcal{F}'$ are categories cofibered in groupoids over
$\mathcal{C}$, we denote the morphisms from $\mathcal{F}$ to $\mathcal{F}'$
by $\Mor_\mathcal{C}(\mathcal{F}, \mathcal{F}')$.
\item
\label{item-definition-cofibered-groupoids-2-category}
Categories cofibered in groupoids form a $(2, 1)$-category
$\text{Cof}(\mathcal{C})$. Its 1-morphisms are the morphisms described in
(\ref{item-cofibered-morphism}). If $p : \mathcal{F} \to C$ and
$p': \mathcal{F}' \to \mathcal{C}$ are categories cofibered in groupoids
and $\varphi, \psi : \mathcal{F} \to \mathcal{F}'$ are $1$-morphisms, then
a 2-morphism $t : \varphi \to \psi$ is a morphism of functors such that
$p'(t_x) = \text{id}_{p(x)}$ for all $x \in \Ob(\mathcal{F})$.
\item
\label{item-construction-associated-cofibered-groupoid}
Let $F : \mathcal{C} \to \textit{Groupoids}$ be a functor. There
is a category cofibered in groupoids $\mathcal{F} \to \mathcal{C}$
associated to $F$ as follows. An object of $\mathcal{F}$ is a pair $(U, x)$
where $U \in \Ob(\mathcal{C})$ and $x \in \Ob(F(U))$. A
morphism $(U, x) \to (V, y)$ is a pair $(f, a)$ where
$f \in \Mor_\mathcal{C}(U, V)$ and
$a \in \Mor_{F(V)}(F(f)(x), y)$.
The functor $\mathcal{F} \to \mathcal{C}$ sends $(U, x)$ to $U$. See
Categories, Section \ref{categories-section-presheaves-groupoids}.
\item
\label{item-associated-functor-isomorphism-classes}
Let $\mathcal{F}$ be cofibered in groupoids over $\mathcal{C}$.
For $U \in \Ob(\mathcal{C})$ set $\overline{\mathcal{F}}(U)$ equal to
the set of isomorphisms classes of the category $\mathcal{F}(U)$.
If $f : U \to V$ is a morphism of $\mathcal{C}$, then we obtain a
map of sets $\overline{\mathcal{F}}(U) \to \overline{\mathcal{F}}(V)$ by
mapping the isomorphism class of $x$ to the isomorphism class of a pushforward
$f_*x$ of $x$ see (\ref{item-pushforward}). Then
$\overline{\mathcal{F}} : \mathcal{C} \to \textit{Sets}$ is a
functor. Similarly, if $\varphi : \mathcal{F} \to \mathcal{G}$ is a
morphism of cofibered categories, we denote by
$\overline{\varphi}: \overline{\mathcal{F}} \to \overline{\mathcal{G}}$
the associated morphism of functors.
\item
\label{item-convention-cofibered-sets}
Let $F: \mathcal{C} \to \textit{Sets}$ be a functor. We can think of a
set as a discrete category, i.e., as a groupoid with only identity morphisms.
Then the construction (\ref{item-construction-associated-cofibered-groupoid})
associates to $F$ a category cofibered in sets. This defines a fully
faithful embedding of the category of functors $\mathcal{C} \to \textit{Sets}$
to the category of categories cofibered in groupoids over $\mathcal{C}$.
We identify the category of functors with its image under this embedding.
Hence if $F : \mathcal{C} \to \textit{Sets}$ is a functor, we denote the
associated category cofibered in sets also by $F$; and if
$\varphi : F \to G$ is a morphism of functors, we denote still by $\varphi$
the corresponding morphism of categories cofibered in sets, and vice-versa.
See Categories, Section \ref{categories-section-fibred-in-sets}.
\item
\label{item-definition-yoneda}
Let $U$ be an object of $\mathcal{C}$. We write $\underline{U}$ for the
functor
$\Mor_\mathcal{C}(U, -): \mathcal{C} \to
\textit{Sets}$. This defines a fully faithful embedding of $\mathcal
C^{opp}$ into the category of functors $\mathcal{C} \to
\textit{Sets}$. Hence, if $f : U \to V$ is a morphism, we are
justified in denoting still by $f$ the induced morphism $\underline{V}
\to \underline{U}$, and vice-versa.
\item
\label{item-fibre-product}
Fiber products of categories cofibered in groupoids: If $\mathcal{F}
\to \mathcal{H}$ and $\mathcal{G} \to \mathcal{H}$ are morphisms
of categories cofibered in groupoids over $\mathcal{C}_\Lambda$, then a
construction of their 2-fiber product is given by the construction for their
2-fiber product as categories over $\mathcal{C}_\Lambda$, as described in
Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}.
\item
\label{item-definition-restricting-base-category}
Restricting the base category: Let $p : \mathcal{F} \to \mathcal{C}$ be a
category cofibered in groupoids, and let $\mathcal{C}'$ be a full
subcategory of $\mathcal{C}$. The restriction $\mathcal{F}|_{\mathcal{C}'}$
is the full subcategory of $\mathcal{F}$ whose objects lie over
objects of $\mathcal{C}'$. It is a category cofibered in groupoids via
the functor
$p|_{\mathcal{C}'}: \mathcal{F}|_{\mathcal{C}'} \to \mathcal{C}'$.
\end{enumerate}
\end{remarks}
\section{Prorepresentable functors and predeformation categories}
\label{section-cofibered-groupoids}
\noindent
Our basic goal is to understand categories cofibered in groupoids over
$\mathcal{C}_\Lambda$ and $\widehat{\mathcal{C}}_\Lambda$. Since
$\mathcal{C}_\Lambda$ is a full subcategory of
$\widehat{\mathcal{C}}_\Lambda$ we can restrict categories cofibred in
groupoids over $\widehat{\mathcal{C}}_\Lambda$ to $\mathcal{C}_\Lambda$, see
Remarks \ref{remarks-cofibered-groupoids}
(\ref{item-definition-restricting-base-category}).
In particular we can do this with functors, in particular with
representable functors. The functors on $\mathcal{C}_\Lambda$
one obtains in this way are called
prorepresentable functors.
\begin{definition}
\label{definition-prorepresentable}
Let $F : \mathcal{C}_\Lambda \to \textit{Sets}$ be a functor.
We say $F$ is {\it prorepresentable} if there exists an isomorphism
$F \cong \underline{R}|_{\mathcal{C}_\Lambda}$ of functors for some
$R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$.
\end{definition}
\noindent
Note that if $F : \mathcal{C}_\Lambda \to \textit{Sets}$ is prorepresentable
by $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$, then
$$
F(k) = \Mor_{\widehat{\mathcal{C}}_\Lambda}(R, k) = \{*\}
$$
is a singleton. The categories cofibered in groupoids over
$\mathcal{C}_\Lambda$ that are arise in deformation theory will often satisfy
an analogous condition.
\begin{definition}
\label{definition-predeformation-category}
A {\it predeformation category} $\mathcal{F}$ is a category cofibered
in groupoids over $\mathcal{C}_\Lambda$ such that $\mathcal{F}(k)$ is
equivalent to a category with a single object and a single morphism,
i.e., $\mathcal{F}(k)$ contains at least one object and there is a
unique morphism between any two objects. A {\it morphism of predeformation
categories} is a morphism of categories cofibered in groupoids over
$\mathcal{C}_\Lambda$.
\end{definition}
\noindent
A feature of a predeformation category is the following.
Let $x_0 \in \Ob(\mathcal{F}(k))$. Then every object of
$\mathcal{F}$ comes equipped with a unique morphism to $x_0$.
Namely, if $x$ is an object of $\mathcal{F}$ over $A$, then we
can choose a pushforward $x \to q_*x$ where $q : A \to k$ is the quotient
map. There is a unique isomorphism $q_*x \to x_0$ and the composition
$x \to q_*x \to x_0$ is the desired morphism.
\begin{remark}
\label{remark-predeformation-functor}
We say that a functor $F: \mathcal{C}_\Lambda \to \textit{Sets}$
is a {\it predeformation functor} if the associated cofibered set is a
predeformation category, i.e.\ if $F(k)$ is a one element set. Thus if
$\mathcal{F}$ is a predeformation category, then $\overline{\mathcal{F}}$ is a
predeformation functor.
\end{remark}
\begin{remark}
\label{remark-localize-cofibered-groupoid}
Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in
groupoids, and let $x \in \Ob(\mathcal{F}(k))$. We denote by
$\mathcal{F}_x$ the category of objects over $x$.
An object of $\mathcal{F}_x$ is an arrow $y \to x$.
A morphism $(y \to x) \to (z \to x)$ in $\mathcal{F}_x$ is a commutative
diagram
$$
\xymatrix{
y \ar[rr] \ar[dr] & & z \ar[dl] \\
& x &
}
$$
There is a forgetful functor $\mathcal{F}_x \to \mathcal{F}$. We define
the functor $p_x : \mathcal{F}_x \to \mathcal{C}_\Lambda$ as the
composition
$\mathcal{F}_x \to \mathcal{F} \xrightarrow{p} \mathcal{C}_\Lambda$.
Then $p_x : \mathcal{F}_x \to \mathcal{C}_\Lambda$ is a
predeformation category (proof omitted). In this way we can pass from an
arbitrary category cofibered in groupoids over $\mathcal{C}_\Lambda$
to a predeformation category at any $x \in \Ob(\mathcal{F}(k))$.
\end{remark}
\section{Formal objects and completion categories}
\label{section-formal-objects}
\noindent
In this section we discuss how to go between categories cofibred in
groupoids over $\mathcal{C}_\Lambda$ to categories cofibred in
groupoids over $\widehat{\mathcal{C}}_\Lambda$ and vice versa.
\begin{definition}
\label{definition-formal-objects}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. The {\it category $\widehat{\mathcal{F}}$ of formal
objects of $\mathcal{F}$} is the category with the following objects and
morphisms.
\begin{enumerate}
\item A {\it formal object $\xi = (R, \xi_n, f_n)$ of $\mathcal{F}$}
consists of an object $R$ of $\widehat{\mathcal{C}}_\Lambda$, and a collection
indexed by $n \in \mathbf{N}$ of objects $\xi_n$ of
$\mathcal{F}(R/\mathfrak m_R^n)$ and morphisms
$f_n : \xi_{n + 1} \to \xi_n$ lying over the projection
$R/\mathfrak m_R^{n + 1} \to R/\mathfrak m_R^n$.
\item Let $\xi = (R, \xi_n, f_n)$ and $\eta = (S, \eta_n, g_n)$ be
formal objects of $\mathcal{F}$. A {\it morphism $a : \xi \to \eta$ of
formal objects} consists of a map $a_0 : R \to S$ in
$\widehat{\mathcal{C}}_\Lambda$ and a collection $a_n : \xi_n \to \eta_n$
of morphisms of $\mathcal{F}$ lying over
$R/\mathfrak m_R^n \to S/\mathfrak m_S^n$,
such that for every $n$ the diagram
$$
\xymatrix{
\xi_{n + 1} \ar[r]_{f_n} \ar[d]_{a_{n + 1}} & \xi_n \ar[d]^{a_n} \\
\eta_{n + 1} \ar[r]^{g_n} & \eta_n
}
$$
commutes.
\end{enumerate}
\end{definition}
\noindent
The category of formal objects comes with a functor $\widehat{p}:
\widehat{\mathcal{F}} \to \widehat{\mathcal{C}}_\Lambda$ which sends an
object $(R, \xi_n, f_n)$ to $R$ and a morphism
$(R, \xi_n, f_n) \to (S, \eta_n, g_n)$ to the map $R \to S$.
\begin{lemma}
\label{lemma-completion-cofibred}
Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in
groupoids. Then
$\widehat{p} : \widehat{\mathcal{F}} \to \widehat{\mathcal{C}}_\Lambda$
is a category cofibered in groupoids.
\end{lemma}
\begin{proof}
Let $R \to S$ be a ring map in $\widehat{\mathcal{C}}_\Lambda$.
Let $(R, \xi_n, f_n)$ be an object of $\widehat{\mathcal{F}}$.
For each $n$ choose a pushforward $\xi_n \to \eta_n$ of $\xi_n$
along $R/\mathfrak m_R^n \to S/\mathfrak m_S^n$. For each $n$ there
exists a unique morphism $g_n : \eta_{n + 1} \to \eta_n$ in $\mathcal{F}$
lying over $S/\mathfrak m_S^{n + 1} \to S/\mathfrak m_S^n$ such that
$$
\xymatrix{
\xi_{n + 1} \ar[d] \ar[r]_{f_n} & \xi_n \ar[d] \\
\eta_{n + 1} \ar[r]^{g_n} & \eta_n
}
$$
commutes (by the first axiom of a category cofibred in groupoids).
Hence we obtain a morphism $(R, \xi_n, f_n) \to (S, \eta_n, g_n)$
lying over $R \to S$, i.e., the first axiom of a category cofibred in
groupoids holds for $\widehat{\mathcal{F}}$. To see the second axiom
suppose that we have morphisms
$a : (R, \xi_n, f_n) \to (S, \eta_n, g_n)$ and
$b : (R, \xi_n, f_n) \to (T, \theta_n, h_n)$ in $\widehat{\mathcal{F}}$
and a morphism $c_0 : S \to T$ in $\widehat{\mathcal{C}}_\Lambda$ such that
$c_0 \circ a_0 = b_0$. By the second axiom of a category cofibred in groupoids
for $\mathcal{F}$ we obtain unique maps $c_n : \eta_n \to \theta_n$
lying over $S/\mathfrak m_S^n \to T/\mathfrak m_T^n$ such that
$c_n \circ a_n = b_n$. Setting $c = (c_n)_{n \geq 0}$ gives the desired
morphism $c : (S, \eta_n, g_n) \to (T, \theta_n, h_n)$ in
$\widehat{\mathcal{F}}$ (we omit the verification that
$h_n \circ c_{n + 1} = c_n \circ g_n$).
\end{proof}
\begin{definition}
\label{definition-completion}
Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in
groupoids. The category cofibered in groupoids
$\widehat{p} : \widehat{\mathcal F} \to \widehat{\mathcal{C}}_\Lambda$
is called the {\it completion of $\mathcal{F}$}.
\end{definition}
\noindent
If $\mathcal{F}$ is a category cofibered in groupoids over $\mathcal
C_\Lambda$, we have defined $\widehat{\mathcal{F}}(R)$ for $R \in
\Ob(\widehat{\mathcal{C}}_\Lambda)$ in terms of the filtration of
$R$ by powers of its maximal ideal. But suppose $\mathcal{I} = (I_n)$ is a
filtration of $R$ by ideals inducing the $\mathfrak{m}_R$-adic topology. We
define $\widehat{\mathcal{F}}_\mathcal{I}(R)$ to be the category with the
following objects and morphisms:
\begin{enumerate}
\item An object is a collection $(\xi_n, f_n)_{n \in \mathbf{N}}$ of
objects $\xi_n$ of $\mathcal{F}(R/I_n)$ and morphisms
$f_n : \xi_{n + 1} \to \xi_n$ lying over the projections
$R/I_{n + 1} \to R/I_n$.
\item A morphism $a : (\xi_n, f_n) \to (\eta_n, g_n)$ consists of a
collection $a_n : \xi_n \to \eta_n$ of morphisms in
$\mathcal{F}(R/I_n)$, such that for every $n$ the diagram
$$
\xymatrix{
\xi_{n + 1} \ar[r]^{f_n} \ar[d]_{a_{n + 1}} & \xi_n \ar[d]^{a_n} \\
\eta_{n + 1} \ar[r]^{g_n} & \eta_n
}
$$
commutes.
\end{enumerate}
\begin{lemma}
\label{lemma-formal-objects-different-filtration}
In the situation above, $\widehat{\mathcal{F}}_\mathcal{I}(R)$ is equivalent
to the category $\widehat{\mathcal{F}}(R)$.
\end{lemma}
\begin{proof}
An equivalence
$\widehat{\mathcal{F}}_\mathcal{I}(R) \to \widehat{\mathcal{F}}(R)$
can be defined as follows. For each $n$, let $m(n)$ be the least $m$
that $I_m \subset \mathfrak m_R^n$. Given an object
$(\xi_n, f_n)$ of $\widehat{\mathcal{F}}_\mathcal{I}(R)$, let
$\eta_n$ be the pushforward of $\xi_{m(n)}$ along
$R/I_{m(n)} \to R/\mathfrak m_R^n$. Let $g_n : \eta_{n + 1} \to \eta_n$
be the unique morphism of $\mathcal{F}$ lying over
$R/\mathfrak m_R^{n + 1} \to R/\mathfrak m_R^n$ such that
$$
\xymatrix{
\xi_{m(n + 1)} \ar[rrr]_{f_{m(n)} \circ \ldots \circ f_{m(n + 1) - 1}} \ar[d]
& & & \xi_{m(n)} \ar[d] \\
\eta_{n + 1} \ar[rrr]^{g_n} & & & \eta_n
}
$$
commutes (existence and uniqueness is guaranteed by the axioms of a
cofibred category). The functor
$\widehat{\mathcal{F}}_\mathcal{I}(R) \to \widehat{\mathcal{F}}(R)$
sends $(\xi_n, f_n)$ to $(R, \eta_n, g_n)$. We omit the
verification that this is indeed an equivalence of categories.
\end{proof}
\begin{remark}
\label{remark-different-sequence-ideals}
Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in
groupoids. Suppose that for each
$R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$ we are given a filtration
$\mathcal{I}_R$ of $R$ by ideals. If
$\mathcal{I}_R$ induces the $\mathfrak m_R$-adic topology on $R$ for all $R$,
then one can define a category
$\widehat{\mathcal{F}}_\mathcal{I}$ by mimicking
the definition of $\widehat{\mathcal{F}}$. This category comes equipped with a
morphism
$\widehat{p}_\mathcal{I} : \widehat{\mathcal{F}}_\mathcal{I} \to
\widehat{\mathcal{C}}_\Lambda$ making it into a category cofibered in
groupoids such that $\widehat{\mathcal{F}}_\mathcal{I}(R)$ is isomorphic to
$\widehat{\mathcal{F}}_{\mathcal{I}_R}(R)$ as defined above. The categories
cofibered in groupoids $\widehat{\mathcal{F}}_\mathcal{I}$ and
$\widehat{\mathcal{F}}$ are equivalent, by using over an object
$R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$
the equivalence of
Lemma \ref{lemma-formal-objects-different-filtration}.
\end{remark}
\begin{remark}
\label{remark-completion-functor}
Let $F: \mathcal{C}_\Lambda \to \textit{Sets}$ be a functor.
Identifying functors with cofibered sets, the completion of $F$ is the functor
$\widehat{F} : \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$
given by $\widehat{F}(S) = \lim F(S/\mathfrak{m}_S^{n})$. This agrees
with the definition in Schlessinger's paper \cite{Sch}.
\end{remark}
\begin{remark}
\label{remark-restrict-completion}
Let $\mathcal{F}$ be a category cofibred in groupoids over
$\mathcal{C}_\Lambda$. We claim that there is a canonical
equivalence
$$
can :
\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda}
\longrightarrow
\mathcal{F}.
$$
Namely, let $A \in \Ob(\mathcal{C}_\Lambda)$ and let
$(A, \xi_n, f_n)$ be an object of
$\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda}(A)$.
Since $A$ is Artinian there is a minimal $m \in \mathbf{N}$
such that $\mathfrak m_A^m = 0$. Then $can$ sends $(A, \xi_n, f_n)$ to $\xi_m$.
This functor is an equivalence of categories cofibered in groupoids by
Categories, Lemma \ref{categories-lemma-equivalence-fibred-categories}
because it is an equivalence on all fibre categories by
Lemma \ref{lemma-formal-objects-different-filtration}
and the fact that the $\mathfrak m_A$-adic topology on a local
Artinian ring $A$ comes from the zero ideal. We will frequently identify
$\mathcal{F}$ with a full subcategory of $\widehat{\mathcal{F}}$ via a
quasi-inverse to the functor $can$.
\end{remark}
\begin{remark}
\label{remark-completion-morphism}
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories
cofibered in groupoids over $\mathcal{C}_\Lambda$. Then there is an induced
morphism
$\widehat{\varphi}: \widehat{\mathcal{F}} \to \widehat{\mathcal{G}}$
of categories cofibered in groupoids over $\widehat{\mathcal{C}}_\Lambda$.
It sends an object $\xi = (R, \xi_n, f_n)$ of
$\widehat{\mathcal{F}}$ to $(R, \varphi(\xi_n), \varphi(f_n))$, and it sends a
morphism $(a_0 : R \to S, a_n : \xi_n \to \eta_n)$ between
objects $\xi$ and $\eta$ of $\widehat{\mathcal{F}}$ to
$(a_0 : R \to S, \varphi(a_n) : \varphi(\xi_n) \to \varphi(\eta_n))$.
Finally, if $t : \varphi \to \varphi'$ is a $2$-morphism between
$1$-morphisms $\varphi, \varphi': \mathcal{F} \to \mathcal{G}$ of
categories cofibred in groupoids, then we obtain a $2$-morphism
$\widehat{t} : \widehat{\varphi} \to \widehat{\varphi}'$. Namely, for
$\xi = (R, \xi_n, f_n)$ as above we set
$\widehat{t}_\xi = (t_{\varphi(\xi_n)})$. Hence completion defines a
functor between $2$-categories
$$
\widehat{~} :
\text{Cof}(\mathcal{C}_\Lambda)
\longrightarrow
\text{Cof}(\widehat{\mathcal{C}}_\Lambda)
$$
from the $2$-category of categories cofibred in groupoids over
$\mathcal{C}_\Lambda$ to the $2$-category of categories cofibred
in groupoids over $\widehat{\mathcal{C}}_\Lambda$.
\end{remark}
\begin{remark}
\label{remark-completion-restriction-adjoint}
We claim the completion functor of
Remark \ref{remark-completion-morphism}
and the restriction functor
$|_{\mathcal{C}_\Lambda} : \text{Cof}(\widehat{\mathcal{C}}_\Lambda)
\to \text{Cof}(\mathcal{C}_\Lambda)$ of
Remarks \ref{remarks-cofibered-groupoids}
(\ref{item-definition-restricting-base-category})
are ``2-adjoint'' in the following precise sense. Let
$\mathcal{F} \in \Ob(\text{Cof}(\mathcal{C}_\Lambda))$
and let
$\mathcal{G} \in \Ob(\text{Cof}(\widehat{\mathcal{C}}_\Lambda))$.
Then there is an equivalence of categories
$$
\Phi :
\Mor_{\mathcal{C}_\Lambda}(
\mathcal{G}|_{\mathcal{C}_\Lambda}, \mathcal{F})
\longrightarrow
\Mor_{\widehat{\mathcal{C}}_\Lambda}(\mathcal{G}, \widehat{\mathcal{F}})
$$
To describe this equivalence, we
define canonical morphisms
$\mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda}}$ and
$\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ as follows
\begin{enumerate}
\item Let $R \in \Ob(\widehat{\mathcal{C}}_\Lambda))$ and let $\xi$
be an object of the fiber category $\mathcal{G}(R)$.
Choose a pushforward $\xi \to \xi_n$ of $\xi$ to
$R/\mathfrak m_R^n$ for each $n \in \mathbf{N}$, and let
$f_n : \xi_{n + 1} \to \xi_n$ be the induced morphism.
Then $\mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda}}$
sends $\xi$ to $(R, \xi_n, f_n)$.
\item This is the equivalence
$can : \widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$
of
Remark \ref{remark-restrict-completion}.
\end{enumerate}
Having said this, the equivalence
$\Phi : \Mor_{\mathcal{C}_\Lambda}(
\mathcal{G}|_{\mathcal{C}_\Lambda}, \mathcal{F}) \to
\Mor_{\widehat{\mathcal{C}}_\Lambda}(\mathcal{G},
\widehat{\mathcal{F}})$
sends a morphism
$\varphi : \mathcal{G}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$
to
$$
\mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda}}
\xrightarrow{\widehat{\varphi}} \widehat{\mathcal{F}}
$$
There is a quasi-inverse
$\Psi :
\Mor_{\widehat{\mathcal{C}}_\Lambda}(
\mathcal{G}, \widehat{\mathcal{F}}) \to
\Mor_{\mathcal{C}_\Lambda}(
\mathcal{G}|_{\mathcal{C}_\Lambda}, \mathcal{F})$
to $\Phi$ which sends $\psi : \mathcal{G} \to \widehat{\mathcal{F}}$ to
$$
\mathcal{G}|_{\mathcal{C}_\Lambda} \xrightarrow{\psi|_{\mathcal{C}_\Lambda}}
\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda} \to \mathcal{F}.
$$
We omit the verification that $\Phi$ and $\Psi$ are quasi-inverse.
We also do not address functoriality of $\Phi$ (because it would
lead into 3-category territory which we want to avoid at all cost).
\end{remark}
\begin{remark}
\label{remark-completion-restriction-cofset-adjoint}
For a category $\mathcal{C}$ we denote by $\text{CofSet}(\mathcal{C})$ the
category of cofibered sets over $\mathcal{C}$. It is a $1$-category
isomorphic the category of functors $\mathcal{C} \to \textit{Sets}$.
See Remarks \ref{remarks-cofibered-groupoids}
(\ref{item-convention-cofibered-sets}).
The completion and restriction functors restrict to functors
$\widehat{~} : \text{CofSet}(\mathcal{C}_\Lambda) \to
\text{CofSet}(\widehat{\mathcal{C}}_\Lambda)$ and
$|_{\mathcal{C}_\Lambda} : \text{CofSet}(\widehat{\mathcal{C}}_\Lambda) \to
\text{CofSet}(\mathcal{C}_\Lambda)$ which we denote by the same symbols.
As functors on the categories of cofibered sets, completion and restriction
are adjoints in the usual 1-categorical sense: the same construction as in
Remark \ref{remark-completion-restriction-adjoint} defines a functorial
bijection
$$
\Mor_{\mathcal{C}_\Lambda}(G|_{\mathcal{C}_\Lambda}, F)
\longrightarrow
\Mor_{\widehat{\mathcal{C}}_\Lambda}(G, \widehat{F})
$$
for $F \in \Ob(\text{CofSet}(\mathcal{C}_\Lambda))$ and
$G \in \Ob(\text{CofSet}(\widehat{\mathcal{C}}_\Lambda))$.
Again the map $\widehat{F}|_{\mathcal{C}_\Lambda} \to F$ is an
isomorphism.
\end{remark}
\begin{remark}
\label{remark-restrict-complete-continuous-functor}
Let $G : \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$
be a functor that commutes with limits.
Then the map $G \to \widehat{G|_{\mathcal{C}_\Lambda}}$ described in
Remark \ref{remark-completion-restriction-adjoint}
is an isomorphism. Indeed, if $S$ is an object of
$\widehat{\mathcal{C}}_\Lambda$, then we have canonical bijections
$$
\widehat{G|_{\mathcal{C}_\Lambda}}(S) =
\lim_n G(S/\mathfrak{m}_S^n) =
G(\lim_n S/\mathfrak{m}_S^n) = G(S).
$$
In particular, if $R$ is an object of $\widehat{\mathcal{C}}_\Lambda$ then
$\underline{R} = \widehat{\underline{R}|_{\mathcal{C}_\Lambda}}$ because
the representable functor $\underline{R}$ commutes with limits by definition
of limits.
\end{remark}
\begin{remark}
\label{remark-formal-objects-yoneda}
Let $R$ be an object of $\widehat{\mathcal{C}}_\Lambda$. It defines a functor
$\underline{R}: \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$
as described in
Remarks \ref{remarks-cofibered-groupoids} (\ref{item-definition-yoneda}).
As usual we identify this functor with the
associated cofibered set. If $\mathcal{F}$ is a cofibered category over
$\mathcal{C}_\Lambda$, then there is an equivalence of categories
\begin{equation}
\label{equation-formal-objects-maps}
\Mor_{\mathcal{C}_\Lambda}(
\underline{R}|_{\mathcal{C}_\Lambda}, \mathcal{F})
\longrightarrow
\widehat{\mathcal{F}}(R).
\end{equation}
It is given by the composition
$$
\Mor_{\mathcal{C}_\Lambda}(
\underline{R}|_{\mathcal{C}_\Lambda}, \mathcal{F})
\xrightarrow{\Phi}
\Mor_{\widehat{\mathcal{C}}_\Lambda}(
\underline{R}, \widehat{\mathcal{F}})
\xrightarrow{\sim}
\widehat{\mathcal{F}}(R)
$$
where $\Phi$ is as in
Remark \ref{remark-completion-restriction-adjoint}
and the second equivalence comes from the 2-Yoneda lemma
(the cofibered analogue of
Categories, Lemma \ref{categories-lemma-yoneda-2category}).
Explicitly, the equivalence sends a morphism
$\varphi : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$
to the formal object
$(R, \varphi(R \to R/\mathfrak{m}_R^n), \varphi(f_n))$ in
$\widehat{\mathcal{F}}(R)$, where
$f_n : R/\mathfrak m_R^{n + 1} \to R/\mathfrak m_R^n$ is the projection.
\medskip\noindent
Assume a choice of pushforwards for $\mathcal{F}$ has been made.
Given any $\xi \in \Ob(\widehat{\mathcal{F}}(R))$ we construct
an explicit
$\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$
which maps to $\xi$ under (\ref{equation-formal-objects-maps}).
Namely, say $\xi = (R, \xi_n, f_n)$. An object $\alpha$ in
$\underline{R}|_{\mathcal{C}_\Lambda}$ is the same thing as a morphism
$\alpha : R \to A$ of $\widehat{\mathcal{C}}_\Lambda$ with $A$
Artinian. Let $m \in \mathbf{N}$ be minimal such that $\mathfrak m_A^m = 0$.
Then $\alpha$ factors through a unique $\alpha_m : R/\mathfrak m_R^m \to A$
and we can set $\underline{\xi}(\alpha) = \alpha_{m, *}\xi_m$.
We omit the description of $\underline{\xi}$ on morphisms and we
omit the proof that $\underline{\xi}$ maps to $\xi$
via (\ref{equation-formal-objects-maps}).
\medskip\noindent
Assume a choice of pushforwards for $\widehat{\mathcal{F}}$ has been made.
In this case the proof of
Categories, Lemma \ref{categories-lemma-yoneda-2category}
gives an explicit quasi-inverse
$$
\iota :
\widehat{\mathcal{F}}(R) \longrightarrow
\Mor_{\widehat{\mathcal{C}}_\Lambda}(
\underline{R}, \widehat{\mathcal{F}})
$$
to the 2-Yoneda equivalence which takes $\xi$ to the morphism
$\iota(\xi) : \underline{R} \to \widehat{\mathcal{F}}$ sending
$f \in \underline{R}(S) = \Mor_{\mathcal{C}_\Lambda}(R, S)$
to $f_*\xi$. A quasi-inverse to (\ref{equation-formal-objects-maps})
is then
$$
\widehat{\mathcal{F}}(R)
\xrightarrow{\iota}
\Mor_{\widehat{\mathcal{C}}_\Lambda}(
\underline{R}, \widehat{\mathcal{F}})
\xrightarrow{\Psi}
\Mor_{\mathcal{C}_\Lambda}(
\underline{R}|_{\mathcal{C}_\Lambda}, \mathcal{F})
$$
where $\Psi$ is as in
Remark \ref{remark-completion-restriction-adjoint}.
Given $\xi \in \Ob(\widehat{\mathcal{F}}(R))$ we have
$\Psi(\iota(\xi)) \cong \underline{\xi}$ where $\underline{\xi}$
is as in the previous paragraph, because both are mapped to $\xi$
under the equivalence of categories (\ref{equation-formal-objects-maps}).
Using $\underline{R} = \widehat{\underline{R}|_{\mathcal{C}_\Lambda}}$
(see Remark \ref{remark-restrict-complete-continuous-functor})
and unwinding the definitions of $\Phi$ and $\Psi$ we conclude that
$\iota(\xi)$ is isomorphic to the completion of $\underline{\xi}$.
\end{remark}
\begin{remark}
\label{remark-formal-objects-yoneda-map}
Let $\mathcal{F}$ be a category cofibred in groupoids over
$\mathcal{C}_\Lambda$. Let $\xi = (R, \xi_n, f_n)$ and
$\eta = (S, \eta_n, g_n)$ be formal objects of $\mathcal{F}$.
Let $a = (a_n) : \xi \to \eta$ be a morphism of formal objects, i.e.,
a morphism of $\widehat{\mathcal{F}}$. Let
$f = \widehat{p}(a) = a_0 : R \to S$ be the projection of $a$ in
$\widehat{\mathcal{C}}_\Lambda$. Then we obtain a $2$-commutative
diagram
$$
\xymatrix{
\underline{R}|_{\mathcal{C}_\Lambda} \ar[rd]_{\underline{\xi}} & &
\underline{S}|_{\mathcal{C}_\Lambda} \ar[ll]^f \ar[ld]^{\underline{\eta}} \\
& \mathcal{F}
}
$$
where $\underline{\xi}$ and $\underline{\eta}$ are the morphisms
constructed in
Remark \ref{remark-formal-objects-yoneda}.
To see this let $\alpha : S \to A$ be an object of
$\underline{S}|_{\mathcal{C}_\Lambda}$ (see loc.\ cit.).
Let $m \in \mathbf{N}$ be minimal such that $\mathfrak m_A^m = 0$.
We get a commutative diagram
$$
\xymatrix{
R \ar[d]^f \ar[r] & R/\mathfrak m_R^m \ar[d]_{f_m} \ar[rd]^{\beta_m} \\
S \ar[r] & S/\mathfrak m_S^m \ar[r]^{\alpha_m} & A
}
$$
such that the bottom arrows compose to give $\alpha$.
Then $\underline{\eta}(\alpha) = \alpha_{m, *}\eta_m$ and
$\underline{\xi}(\alpha \circ f) = \beta_{m, *}\xi_m$. The morphism
$a_m : \xi_m \to \eta_m$ lies over $f_m$ hence we obtain a canonical
morphism
$$
\underline{\xi}(\alpha \circ f) = \beta_{m, *}\xi_m
\longrightarrow
\underline{\eta}(\alpha) = \alpha_{m, *}\eta_m
$$
lying over $\text{id}_A$ such that
$$
\xymatrix{
\xi_m \ar[r] \ar[d]^{a_m} & \beta_{m, *}\xi_m \ar[d] \\
\eta_m \ar[r] & \alpha_{m, *}\eta_m
}
$$
commutes by the axioms of a category cofibred in groupoids. This defines
a transformation of functors $\underline{\xi} \circ f \to \underline{\eta}$
which witnesses the 2-commutativity of the first diagram of this remark.
\end{remark}
\begin{remark}
\label{remark-spell-out-formal-object}
According to Remark \ref{remark-formal-objects-yoneda}, giving a formal object
$\xi$ of $\mathcal{F}$ is equivalent to giving a prorepresentable functor
$U : \mathcal{C}_\Lambda \to \textit{Sets}$ and a morphism
$U \to \mathcal{F}$.
\end{remark}
\section{Smooth morphisms}
\label{section-smooth-morphisms}
\noindent
In this section we discuss smooth morphisms of categories
cofibered in groupoids over $\mathcal{C}_\Lambda$.
\begin{definition}
\label{definition-smooth-morphism}
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories
cofibered in groupoids over $\mathcal{C}_\Lambda$. We say $\varphi$ is
{\it smooth} if it satisfies the following condition: Let $B \to A$ be
a surjective ring map in $\mathcal{C}_\Lambda$. Let $y \in
\Ob(\mathcal{G}(B)), x \in \Ob(\mathcal{F}(A))$, and $y
\to \varphi(x)$ be a morphism lying over $B \to A$. Then there
exists $x' \in \Ob(\mathcal{F}(B))$, a morphism $x' \to x$
lying over $B \to A$, and a morphism $\varphi(x') \to y$ lying
over $\text{id}: B \to B$, such that the diagram
$$
\xymatrix{
\varphi(x') \ar[r] \ar[dr] & y \ar[d] \\
& \varphi(x)
}
$$
commutes.
\end{definition}
\begin{lemma}
\label{lemma-smoothness-small-extensions}
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories
cofibered in groupoids over $\mathcal{C}_\Lambda$. Then $\varphi$ is smooth
if the condition in Definition \ref{definition-smooth-morphism} is assumed to
hold only for small extensions $B \to A$.
\end{lemma}
\begin{proof}
Let $B \to A$ be a surjective ring map in $\mathcal{C}_\Lambda$.
Let $y \in \Ob(\mathcal{G}(B))$, $x \in \Ob(\mathcal{F}(A))$,
and $y \to \varphi(x)$ be a morphism lying over $B \to A$. By
Lemma \ref{lemma-factor-small-extension} we can factor $B \to A$ into
small extensions $B = B_n \to B_{n-1} \to \ldots \to B_0 = A$.
We argue by induction on $n$. If $n = 1$ the result is true by assumption.
If $n > 1$, then denote $f : B = B_n \to B_{n - 1}$ and denote
$g : B_{n - 1} \to B_0 = A$. Choose a pushforward
$y \to f_* y$ of $y$ along $f$, so that the morphism $y \to \varphi(x)$
factors as $y \to f_* y \to \varphi(x)$. By the induction hypothesis
we can find $x_{n - 1} \to x$ lying over $g : B_{n - 1} \to A$ and
$a : \varphi(x_{n - 1}) \to f_*y$ lying over
$\text{id} : B_{n - 1} \to B_{n - 1}$ such that
$$
\xymatrix{
\varphi(x_{n - 1}) \ar[r]_-a \ar[dr] & f_*y \ar[d] \\
& \varphi(x)
}
$$
commutes. We can apply the assumption to the composition
$y \to \varphi(x_{n - 1})$ of
$y \to f_*y$ with $a^{-1} : f_*y \to \varphi(x_{n - 1})$. We obtain
$x_n \to x_{n - 1}$ lying over $B_n \to B_{n - 1}$ and
$\varphi(x_n) \to y$ lying over $\text{id} : B_n \to B_n$ so that the diagram
$$
\xymatrix{
\varphi(x_n) \ar[r] \ar[d] & y \ar[d] \\
\varphi(x_{n - 1}) \ar[r]^-a \ar[dr] & f_*y \ar[d] \\
& \varphi(x)
}
$$
commutes. Then the composition $x_n \to x_{n - 1} \to x$ and
$\varphi(x_n) \to y$ are the morphisms required by the definition of
smoothness.
\end{proof}
\begin{remark}
\label{remark-smoothness-2-categorical}
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories
cofibered in groupoids over $\mathcal{C}_\Lambda$. Let $B \to A$ be a
ring map in $\mathcal{C}_\Lambda$. Choices of pushforwards along $B
\to A$ for objects in the fiber categories $\mathcal{F}(B)$ and
$\mathcal{G}(B)$ determine functors $\mathcal{F}(B) \to \mathcal{F}(A)$
and $\mathcal{G}(B) \to \mathcal{G}(A)$ fitting into a $2$-commutative
diagram
$$
\xymatrix{
\mathcal{F}(B) \ar[r]^{\varphi} \ar[d] & \mathcal{G}(B) \ar[d] \\
\mathcal{F}(A) \ar[r]^{\varphi} & \mathcal{G}(A) .
}
$$
Hence there is an induced functor $\mathcal{F}(B) \to \mathcal{F}(A)
\times_{\mathcal{G}(A)} \mathcal{G}(B)$. Unwinding the definitions shows that
$\varphi : \mathcal{F} \to \mathcal{G}$ is smooth if and only if this
induced functor is essentially surjective whenever $B \to A$ is
surjective (or equivalently, by
Lemma \ref{lemma-smoothness-small-extensions},
whenever $B \to A$ is a small extension).
\end{remark}
\begin{remark}
\label{remark-compare-smooth-schlessinger}
The characterization of smooth morphisms in
Remark \ref{remark-smoothness-2-categorical}
is analogous to Schlessinger's notion of
a smooth morphism of functors, cf.\ \cite[Definition 2.2.]{Sch}. In
fact, when $\mathcal{F}$ and $\mathcal{G}$ are cofibered in sets
then our notion is equivalent to Schlessinger's. Namely, in this case
let $F, G : \mathcal{C}_\Lambda \to \textit{Sets}$ be the corresponding
functors, see
Remarks \ref{remarks-cofibered-groupoids}
(\ref{item-convention-cofibered-sets}).
Then $F \to G$ is smooth if and only if for every surjection of rings
$B \to A$ in $\mathcal{C}_\Lambda$ the map $F(B) \to F(A) \times_{G(A)} G(B)$
is surjective.
\end{remark}
\begin{remark}
\label{remark-smooth-to-iso-classes}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. Then the morphism
$\mathcal{F} \to \overline{\mathcal{F}}$ is smooth.
Namely, suppose that $f : B \to A$ is a ring map in $\mathcal{C}_\Lambda$.
Let $x \in \Ob(\mathcal{F}(A))$ and let
$\overline{y} \in \overline{\mathcal{F}}(B)$
be the isomorphism class of $y \in \Ob(\mathcal{F}(B))$ such that
$\overline{f_*y} = \overline{x}$. Then we simply take $x' = y$, the
implied morphism $x' = y \to x$ over $B \to A$, and the equality
$\overline{x'} = \overline{y}$ as the solution to
the problem posed in Definition \ref{definition-smooth-morphism}.
\end{remark}
\noindent
If $R \to S$ is a ring map $\widehat{\mathcal{C}}_\Lambda$, then there
is an induced morphism $\underline{S} \to \underline{R}$ between the
functors $\underline{S}, \underline{R}: \widehat{\mathcal{C}}_\Lambda
\to \textit{Sets}$. In this situation, smoothness of the
restriction $\underline{S}|_{\mathcal{C}_\Lambda} \to
\underline{R}|_{\mathcal{C}_\Lambda}$ is a familiar notion:
\begin{lemma}
\label{lemma-smooth-morphism-power-series}
Let $R \to S$ be a ring map in $\widehat{\mathcal{C}}_\Lambda$. Then
the induced morphism
$\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$
is smooth if and only if $S$ is a power series ring over $R$.
\end{lemma}
\begin{proof}
Assume $S$ is a power series ring over $R$. Say $S = R[[x_1, \ldots, x_n]]$.
Smoothness of
$\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$
means the following (see Remark \ref{remark-compare-smooth-schlessinger}):
Given a surjective ring map $B \to A$ in
$\mathcal{C}_\Lambda$, a ring map $R \to B$, a ring map $S \to A$ such that
the solid diagram
$$
\xymatrix{
S \ar[r] \ar@{..>}[rd] & A \\
R \ar[u] \ar[r] & B \ar[u]
}
$$
is commutative then a dotted arrow exists making the diagram commute.
(Note the similarity with
Algebra, Definition \ref{algebra-definition-formally-smooth}.)
To construct the dotted arrow choose elements $b_i \in B$ whose images
in $A$ are equal to the images of $x_i$ in $A$. Note that
$b_i \in \mathfrak m_B$ as $x_i$ maps to an element of $\mathfrak m_A$.
Hence there is a unique $R$-algebra map $R[[x_1, \ldots, x_n]] \to B$
which maps $x_i$ to $b_i$ and which can serve as our dotted arrow.
\medskip\noindent
Conversely, assume
$\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$
is smooth. Let $x_1, \ldots, x_n \in S$ be elements whose images
form a basis in the relative cotangent space
$\mathfrak m_S/(\mathfrak m_R S + \mathfrak m_S^2)$ of $S$ over $R$.
Set $T = R[[X_1, \ldots, X_n]]$. Note that both
$$
S/(\mathfrak m_R S + \mathfrak m_S^2) \cong
R/\mathfrak m_R[x_1, \ldots, x_n]/(x_ix_j)
$$
and
$$
T/(\mathfrak m_R T + \mathfrak m_T^2) \cong
R/\mathfrak m_R[X_1, \ldots, X_n]/(X_iX_j).
$$
Let
$S/(\mathfrak m_R S + \mathfrak m_S^2) \to
T/(\mathfrak m_R T + \mathfrak m_T^2)$
be the local $R$-algebra isomorphism given by mapping
the class of $x_i$ to the class of $X_i$. Let
$f_1 : S \to T/(\mathfrak m_R T + \mathfrak m_T^2)$ be the
composition
$S \to S/(\mathfrak m_R S + \mathfrak m_S^2)
\to T/(\mathfrak m_R T + \mathfrak m_T^2)$.
The assumption that
$\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$
is smooth means we can lift $f_1$ to a map
$f_2 : S \to T/\mathfrak{m}_T^2$, then to a map
$f_3 : S \to T/\mathfrak{m}_T^3$, and so on, for all $n \geq 1$. Thus
we get an induced map $f : S \to T = \lim T/\mathfrak m_T^n$
of local $R$-algebras. By our choice of $f_1$, the map $f$ induces an
isomorphism
$\mathfrak m_S/(\mathfrak m_R S + \mathfrak m_S^2) \to
\mathfrak m_T/(\mathfrak m_R T + \mathfrak m_T^2)$
of relative cotangent spaces.
Hence $f$ is surjective by
Lemma \ref{lemma-surjective-cotangent-space}
(where we think of $f$ as a map in $\widehat{\mathcal{C}}_R$).
Choose preimages $y_i \in S$ of $X_i \in T$ under $f$. As $T$ is a
power series ring over $R$ there exists a local
$R$-algebra homomorphism $s : T \to S$ mapping $X_i$ to $y_i$.
By construction $f \circ s = \text{id}$. Then $s$ is injective.
But $s$ induces an isomorphism on relative cotangent spaces since
$f$ does, so it is also surjective by
Lemma \ref{lemma-surjective-cotangent-space}
again. Hence $s$ and $f$ are isomorphisms.
\end{proof}
\noindent
Smooth morphisms satisfy the following functorial properties.
\begin{lemma}
\label{lemma-smooth-properties}
Let $\varphi : \mathcal{F} \to \mathcal{G}$ and $\psi : \mathcal{G}
\to \mathcal{H}$ be morphisms of categories cofibered in groupoids over
$\mathcal{C}_\Lambda$.
\begin{enumerate}
\item If $\varphi$ and $\psi$ are smooth, then $\psi \circ \varphi$ is smooth.
\item If $\varphi$ is essentially surjective and $\psi \circ \varphi$ is
smooth, then $\psi$ is smooth.
\item If $\mathcal{G}' \to \mathcal{G}$ is a morphism of categories
cofibered in groupoids and $\varphi$ is smooth, then
$\mathcal{F} \times_\mathcal{G} \mathcal{G}' \to \mathcal{G}'$ is smooth.
\end{enumerate}
\end{lemma}
\begin{proof}
Statements (1) and (2) follow immediately from the definitions.
Proof of (3) omitted. Hints: use the formulation of smoothness given in
Remark \ref{remark-smoothness-2-categorical}
and use that $\mathcal{F} \times_\mathcal{G} \mathcal{G}'$
is the $2$-fibre product, see
Remarks \ref{remarks-cofibered-groupoids} (\ref{item-fibre-product}).
\end{proof}
\begin{lemma}
\label{lemma-smooth-morphism-essentially-surjective}
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a smooth morphism of
categories cofibered in groupoids over $\mathcal{C}_\Lambda$. Assume
$\varphi : \mathcal{F}(k) \to \mathcal{G}(k)$ is essentially surjective.
Then $\varphi : \mathcal{F} \to \mathcal{G}$ and
$\widehat{\varphi} : \widehat{\mathcal{F}} \to \widehat{\mathcal{G}}$
are essentially surjective.
\end{lemma}
\begin{proof}
Let $y$ be an object of $\mathcal{G}$ lying over
$A \in \Ob(\mathcal{C}_\Lambda)$. Let $y \to y_0$ be a pushforward
of $y$ along $A \to k$. By the assumption on essential surjectivity of
$\varphi : \mathcal{F}(k) \to \mathcal{G}(k)$ there exist an object
$x_0$ of $\mathcal{F}$ lying over $k$ and an isomorphism
$y_0 \to \varphi(x_0)$. Smoothness of $\varphi$ implies there exists
an object $x$ of $\mathcal{F}$ over $A$ whose image $\varphi(x)$
is isomorphic to $y$. Thus $\varphi : \mathcal{F} \to \mathcal{G}$
is essentially surjective.
\medskip\noindent
Let $\eta = (R, \eta_n, g_n)$ be an object of $\widehat{\mathcal{G}}$. We
construct an object $\xi$ of $\widehat{\mathcal{F}}$ with an isomorphism
$\eta \to \varphi(\xi)$. By the assumption on essential surjectivity of
$\varphi : \mathcal{F}(k) \to \mathcal{G}(k)$, there exists a morphism
$\eta_1 \to \varphi(\xi_1)$ in $\mathcal{G}(k)$ for some
$\xi_1 \in \Ob(\mathcal{F}(k))$. The morphism
$\eta_2 \xrightarrow{g_1} \eta_1 \to \varphi(\xi_1)$
lies over the surjective ring map $R/\mathfrak m_R^2 \to k$, hence
by smoothness of $\varphi$ there exists
$\xi_2 \in \Ob(\mathcal{F}(R/\mathfrak m_R^2))$, a
morphism $f_1: \xi_2 \to \xi_1$ lying over
$R/\mathfrak m_R^2 \to k$, and a morphism
$\eta_2 \to \varphi(\xi_2)$ such that
$$
\xymatrix{
\varphi(\xi_2) \ar[r]^{\varphi(f_1)} & \varphi(\xi_{1}) \\
\eta_2 \ar[u] \ar[r]^{g_1} & \eta_1 \ar[u] \\
}
$$
commutes. Continuing in this way we construct an object
$\xi = (R, \xi_n, f_n)$ of $\widehat{\mathcal{F}}$ and a morphism
$\eta \to \varphi(\xi) = (R, \varphi(\xi_n), \varphi(f_n))$
in $\widehat{\mathcal{G}}(R)$.
\end{proof}
\begin{remark}
\label{remark-cofibered-groupoid-projection-smooth}
Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in
groupoids. We can consider $\mathcal{C}_\Lambda$ as the trivial category
cofibered in groupoids over $\mathcal{C}_\Lambda$, and then $p$ is a morphism
of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. We say
$\mathcal{F}$ is {\it smooth} if its structure morphism $p$ is smooth.
This is the ``absolute'' notion of smoothness for a category cofibered in
groupoids over $\mathcal{C}_\Lambda$.
\end{remark}
\begin{example}
\label{example-smooth}
Let $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$.
When is $\underline{R}|_{\mathcal{C}_\Lambda}$ smooth?
In the classical case this means that $R$ is a power series
ring over $\Lambda$, see
Lemma \ref{lemma-smooth-morphism-power-series}.
(Strictly speaking this uses that
$\underline{\Lambda}|_{\mathcal{C}_\Lambda} = \mathcal{C}_\Lambda$
because $\Lambda$ is an initial object of $\widehat{\mathcal{C}}_\Lambda$
in the classical case.) In the general case we can construct examples
as follows. Pick an integer $n \geq 0$ and a maximal ideal
$\mathfrak m \subset \Lambda[x_1, \ldots, x_n]$ lying over
$\mathfrak m_\Lambda$ so that
$$
k' = \Lambda/\mathfrak m_\Lambda
\longrightarrow
\Lambda[x_1, \ldots, x_n]/\mathfrak m
$$
is isomorphic to $k' \to k$. Fix such an identification
$k = \Lambda[x_1, \ldots, x_n]/\mathfrak m$. Set
$R = \Lambda[x_1, \ldots, x_n]^\wedge$ equal to the $\mathfrak m$-adic
completion of $\Lambda[x_1, \ldots, x_n]$. Then $R$ is an object of
$\widehat{\mathcal{C}}_\Lambda$. Namely, it is a complete local
Noetherian ring (see
Algebra, Lemma \ref{algebra-lemma-completion-Noetherian-Noetherian})
and its residue field is identified with $k$.
We claim that $\underline{R}|_{\mathcal{C}_\Lambda}$ is smooth. To see
this we have to show: Given a surjection $B \to A$ in $\mathcal{C}_\Lambda$
and a map $R \to A$ there exists a lift of this map to $B$. This is clear
as we can first lift the composition
$\Lambda[x_1, \ldots, x_n] \to R \to A$ to a map
$\Lambda[x_1, \ldots, x_n] \to B$ and then observe that this latter map
factors through the completion $R$ as $B$ is complete (being Artinian).
In fact, it turns out that whenever $\underline{R}|_{\mathcal{C}_\Lambda}$
is smooth, then $R$ is isomorphic to a completion of a smooth
algebra over $\Lambda$, but we won't use this.
\end{example}
\begin{example}
\label{example-smooth-explicit}
Here is a more explicit example of an $R$ as in
Example \ref{example-smooth}.
Let $p$ be a prime number and let $n \in \mathbf{N}$.
Let $\Lambda = \mathbf{F}_p(t_1, t_2, \ldots, t_n)$ and let
$k = \mathbf{F}_p(x_1, \ldots, x_n)$ with map $\Lambda \to k$ given
by $t_i \mapsto x_i^p$. Then we can take
$$
R = \Lambda[x_1, \ldots, x_n]^\wedge_{(x_1^p - t_1, \ldots, x_n^p - t_n)}
$$
We cannot do ``better'' in this example, i.e., we cannot approximate
$\mathcal{C}_\Lambda$ by a smaller smooth object of
$\widehat{\mathcal{C}}_\Lambda$ (one can argue that the dimension of $R$
has to be at least $n$ since the map
$\Omega_{R/\Lambda} \otimes_R k \to \Omega_{k/\Lambda}$ is
surjective). We will discuss this phenomenon later in more detail.
\end{example}
\begin{remark}
\label{remark-smooth-on-top}
Suppose $\mathcal{F}$ is a predeformation category admitting a smooth morphism
$\varphi : \mathcal U \to \mathcal{F}$ from a predeformation category
$\mathcal{U}$. Then by
Lemma \ref{lemma-smooth-morphism-essentially-surjective}
$\varphi$ is essentially surjective, so by
Lemma \ref{lemma-smooth-properties}
$p: \mathcal{F} \to \mathcal{C}_\Lambda$ is smooth if and only if the
composition $\mathcal U \xrightarrow{\varphi} \mathcal{F} \xrightarrow{p}
\mathcal{C}_\Lambda$ is smooth, i.e.\ $\mathcal{F}$ is smooth if and only if
$\mathcal{U}$ is smooth.
\end{remark}
\noindent
Later we are interested in producing smooth morphisms from
prorepresentable functors to predeformation categories $\mathcal{F}$.
By the discussion in
Remark \ref{remark-formal-objects-yoneda}
these morphisms correspond to certain formal objects of $\mathcal{F}$
More precisely, these are the so-called versal formal objects of $\mathcal{F}$.
\begin{definition}
\label{definition-versal}
Let $\mathcal{F}$ be a category cofibered in groupoids. Let $\xi$ be a formal
object of $\mathcal{F}$ lying over $R \in \Ob(\widehat{\mathcal
C}_\Lambda)$. We say $\xi$ is {\it versal} if the corresponding morphism
$\underline{\xi}: \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$
of Remark \ref{remark-formal-objects-yoneda} is smooth.
\end{definition}
\begin{remark}
\label{remark-versal-object}
Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal
C_\Lambda$, and let $\xi$ be a formal object of $\mathcal{F}$. It follows
from the definition of smoothness that versality of $\xi$ is equivalent to the
following condition: If
$$
\xymatrix{
& y \ar[d] \\
\xi \ar[r] & x
}
$$
is a diagram in $\widehat{\mathcal{F}}$ such that $y \to x$ lies over a
surjective map $B \to A$ of Artinian rings (we may assume it is a small
extension), then there exists a morphism $\xi \to y$ such that
$$
\xymatrix{
& y \ar[d] \\
\xi \ar[r] \ar[ur] & x
}
$$
commutes. In particular, the condition that $\xi$ be versal does not depend on
the choices of pushforwards made in the construction of
$\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ in
Remark \ref{remark-formal-objects-yoneda}.
\end{remark}
\begin{lemma}
\label{lemma-versal-object-quasi-initial}
Let $\mathcal{F}$ be a predeformation category.
Let $\xi$ be a versal formal object of $\mathcal{F}$.
For any formal object $\eta$ of $\widehat{\mathcal{F}}$,
there exists a morphism $\xi \to \eta$.
\end{lemma}
\begin{proof}
By assumption the morphism
$\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$
is smooth. Then
$\iota(\xi) : \underline{R} \to \widehat{\mathcal{F}}$
is the completion of $\underline{\xi}$, see
Remark \ref{remark-formal-objects-yoneda}.
By
Lemma \ref{lemma-smooth-morphism-essentially-surjective}
there exists an object $f$ of $\underline{R}$ such that
$\iota(\xi)(f) = \eta$. Then $f$ is
a ring map $f : R \to S$ in $\widehat{\mathcal{C}}_\Lambda$. And
$\iota(\xi)(f) = \eta$ means that
$f_*\xi \cong \eta$ which means exactly that there is a morphism
$\xi \to \eta$ lying over $f$.
\end{proof}
\section{Schlessinger's conditions}
\label{section-schlessinger-conditions}
\noindent
In the following we often consider fibre products $A_1 \times_A A_2$
of rings in the category $\mathcal{C}_\Lambda$. We have seen in
Example \ref{example-fibre-product}
that such a fibre product may not always be an object of
$\mathcal{C}_\Lambda$. However, in virtually all cases below one of the
two maps $A_i \to A$ is surjective and $A_1 \times_A A_2$ will be
an object of $\mathcal{C}_\Lambda$ by
Lemma \ref{lemma-fiber-product-CLambda}.
We will use this result without further mention.
\medskip\noindent
We denote by $k[\epsilon]$ the ring of dual numbers over $k$. More
generally, for a $k$-vector space $V$, we denote by $k[V]$ the $k$-algebra
whose underlying vector space is $k \oplus V$ and whose multiplication is given
by $(a, v) \cdot (a', v') = (aa', av' + a'v)$. When $V = k$, $k[V]$ is the ring
of dual numbers over $k$. For any finite dimensional $k$-vector space $V$
the ring $k[V]$ is in $\mathcal{C}_\Lambda$.
\begin{definition}
\label{definition-S1-S2}
Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal
C_\Lambda$. We define {\it conditions (S1) and (S2)}
on $\mathcal{F}$ as follows:
\begin{enumerate}
\item[(S1)] Every diagram in $\mathcal{F}$
$$
\vcenter{
\xymatrix{
& x_2 \ar[d] \\
x_1 \ar[r] & x
}
}
\quad\text{lying over}\quad
\vcenter{
\xymatrix{
& A_2 \ar[d] \\
A_1 \ar[r] & A
}
}
$$
in $\mathcal{C}_\Lambda$ with $A_2 \to A$ surjective can be completed
to a commutative diagram
$$
\vcenter{
\xymatrix{
y \ar[r] \ar[d] & x_2 \ar[d] \\
x_1 \ar[r] & x
}
}
\quad\text{lying over}\quad
\vcenter{
\xymatrix{
A_1 \times_A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\
A_1 \ar[r] & A.
}
}
$$
\item[(S2)]
The condition of (S1) holds for diagrams in $\mathcal{F}$ lying over
a diagram in $\mathcal{C}_\Lambda$ of the form
$$
\xymatrix{
& k[\epsilon] \ar[d] \\
A \ar[r] & k.
}
$$
Moreover, if we have two commutative diagrams in $\mathcal{F}$
$$
\vcenter{
\xymatrix{
y \ar[r]_c \ar[d]_a & x_\epsilon \ar[d]^e \\
x \ar[r]^d & x_0
}
}
\quad\text{and}\quad
\vcenter{
\xymatrix{
y' \ar[r]_{c'} \ar[d]_{a'} & x_\epsilon \ar[d]^e \\
x \ar[r]^d & x_0
}
}
\quad\text{lying over}\quad
\vcenter{
\xymatrix{
A \times_k k[\epsilon] \ar[r] \ar[d] & k[\epsilon] \ar[d] \\
A \ar[r] & k
}
}
$$
then there exists a morphism $b : y \to y'$ in
$\mathcal{F}(A \times_k k[\epsilon])$ such that $a = a' \circ b$.
\end{enumerate}
\end{definition}
\noindent
We can partly explain the meaning of conditions (S1) and (S2) in terms of
fibre categories. Suppose that $f_1 : A_1 \to A$ and $f_2 : A_2 \to A$ are
ring maps in $\mathcal{C}_\Lambda$ with $f_2$ surjective.
Denote $p_i : A_1 \times_A A_2 \to A_i$ the projection maps.
Assume a choice of pushforwards for $\mathcal{F}$ has been made.
Then the commutative diagram of rings translates into a $2$-commutative diagram
$$
\xymatrix{
\mathcal{F}(A_1 \times_A A_2) \ar[r]_-{p_{2, *}} \ar[d]_{p_{1, *}} &
\mathcal{F}(A_2) \ar[d]^{f_{2, *}} \\
\mathcal{F}(A_1) \ar[r]^{f_{1, *}} & \mathcal{F}(A)
}
$$
of fibre categories whence a functor
\begin{equation}
\label{equation-compare}
\mathcal{F}(A_1 \times_A A_2) \to
\mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)
\end{equation}
into the $2$-fibre product of categories.
Condition (S1) requires that this functor be essentially surjective.
The first part of condition (S2) requires that this functor be a
essentially surjective if $f_2$ equals the map $k[\epsilon] \to k$.
Moreover in this case, the second part of (S2) implies that two objects
which become isomorphic in the target are isomorphic in the source
(but it is {\it not} equivalent to this statement).
The advantage of stating the conditions as in the definition
is that no choices have to be made.
\begin{lemma}
\label{lemma-S1-small-extensions}
Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal
C_\Lambda$. Then $\mathcal{F}$ satisfies (S1) if the condition of (S1)
is assumed to hold only when $A_2 \to A$ is a small extension.
\end{lemma}
\begin{proof}
Proof omitted. Hints: apply Lemma \ref{lemma-factor-small-extension}
and use induction similar to the proof of
Lemma \ref{lemma-smoothness-small-extensions}.
\end{proof}
\begin{remark}
\label{remark-compare-S1-S2-schlessinger}
When $\mathcal{F}$ is cofibered in sets, conditions (S1) and (S2) are exactly
conditions (H1) and (H2) from Schlessinger's paper \cite{Sch}.
Namely, for a functor $F: \mathcal{C}_\Lambda \to
\textit{Sets}$, conditions (S1) and (S2) state:
\begin{enumerate}
\item [(S1)] If $A_1 \to A$ and $A_2 \to A$ are maps in
$\mathcal{C}_\Lambda$ with $A_2 \to A$ surjective, then the induced
map $F(A_1 \times_A A_2) \to F(A_1) \times_{F(A)} F(A_2)$ is
surjective.
\item [(S2)] If $A \to k$ is a map in $\mathcal{C}_\Lambda$, then the
induced map
$F(A \times_k k[\epsilon]) \to F(A) \times_{F(k)} F(k[\epsilon])$
is bijective.
\end{enumerate}
The injectivity of the map
$F(A \times_k k[\epsilon]) \to F(A) \times_{F(k)} F(k[\epsilon])$
comes from the second part of condition (S2) and the fact that morphisms
are identities.
\end{remark}
\begin{lemma}
\label{lemma-S2-extensions}
Let $\mathcal{F}$ be a category cofibred in groupoids over
$\mathcal{C}_\Lambda$. If $\mathcal{F}$ satisfies (S2), then the
condition of (S2) also holds when $k[\epsilon]$ is replaced by $k[V]$
for any finite dimensional $k$-vector space $V$.
\end{lemma}
\begin{proof}
In the case that $\mathcal{F}$ is cofibred in sets, i.e., corresponds
to a functor $F : \mathcal{C}_\Lambda \to \textit{Sets}$ this follows
from the description of (S2) for $F$ in
Remark \ref{remark-compare-S1-S2-schlessinger}
and the fact that
$k[V] \cong k[\epsilon] \times_k \ldots \times_k k[\epsilon]$
with $\dim_k V$ factors. The case of functors is what we will use in
the rest of this chapter.
\medskip\noindent
We prove the general case by induction on $\dim(V)$. If $\dim(V) = 1$, then
$k[V] \cong k[\epsilon]$ and the result holds by assumption.
If $\dim(V) > 1$ we write $V = V' \oplus k\epsilon$. Pick a diagram
$$
\vcenter{
\xymatrix{
& x_V \ar[d] \\
x \ar[r] & x_0
}
}
\quad\text{lying over}\quad
\vcenter{
\xymatrix{
& k[V] \ar[d] \\
A \ar[r] & k
}
}
$$
Choose a morphism $x_V \to x_{V'}$ lying over $k[V] \to k[V']$
and a morphism $x_V \to x_\epsilon$ lying over $k[V] \to k[\epsilon]$.
Note that the morphism $x_V \to x_0$ factors as
$x_V \to x_{V'} \to x_0$ and as $x_V \to x_\epsilon \to x_0$.
By induction hypothesis we can find a diagram
$$
\vcenter{
\xymatrix{
y' \ar[d] \ar[r] & x_{V'} \ar[d] \\
x \ar[r] & x_0
}
}
\quad\text{lying over}\quad
\vcenter{
\xymatrix{
A \times_k k[V'] \ar[d] \ar[r] & k[V'] \ar[d] \\
A \ar[r] & k
}
}
$$
This gives us a commutative diagram
$$
\vcenter{
\xymatrix{
& x_\epsilon \ar[d] \\
y' \ar[r] & x_0
}
}
\quad\text{lying over}\quad
\vcenter{
\xymatrix{
& k[\epsilon] \ar[d] \\
A \times_k k[V'] \ar[r] & k
}
}
$$
Hence by (S2) we get a commutative diagram
$$
\vcenter{
\xymatrix{
y \ar[d] \ar[r] & x_\epsilon \ar[d] \\
y' \ar[r] & x_0
}
}
\quad\text{lying over}\quad
\vcenter{
\xymatrix{
(A \times_k k[V']) \times_k k[\epsilon] \ar[d] \ar[r] & k[\epsilon] \ar[d] \\
A \times_k k[V'] \ar[r] & k
}
}
$$
Note that
$(A \times_k k[V']) \times_k k[\epsilon] = A \times_k k[V' \oplus k\epsilon]
= A \times_k k[V]$. We claim that $y$ fits into the correct commutative
diagram. To see this we let $y \to y_V$ be a morphism lying over
$A \times_k k[V] \to k[V]$. We can factor the morphisms
$y \to y' \to x_{V'}$ and $y \to x_\epsilon$ through the morphism
$y \to y_V$ (by the axioms of categories cofibred in groupoids).
Hence we see that both $y_V$ and $x_V$ fit into commutative diagrams
$$
\vcenter{
\xymatrix{
y_V \ar[r] \ar[d] & x_\epsilon \ar[d] \\
x_{V'} \ar[r] & x_0
}
}
\quad\text{and}\quad
\vcenter{
\xymatrix{
x_V \ar[r] \ar[d] & x_\epsilon \ar[d] \\
x_{V'} \ar[r] & x_0
}
}
$$
and hence by the second part of (S2) there exists an isomorphism
$y_V \to x_V$ compatible with $y_V \to x_{V'}$ and $x_V \to x_{V'}$
and in particular compatible with the maps to $x_0$.
The composition $y \to y_V \to x_V$ then fits into the required commutative
diagram
$$
\vcenter{
\xymatrix{
y \ar[r] \ar[d] & x_V \ar[d] \\
x \ar[r] & x_0
}
}
\quad\text{lying over}\quad
\vcenter{
\xymatrix{
A \times_k k[V] \ar[d] \ar[r] & k[V] \ar[d] \\
A \ar[r] & k
}
}
$$
In this way we see that the first part of $(S2)$ holds with $k[\epsilon]$
replaced by $k[V]$.
\medskip\noindent
To prove the second part suppose given two commutative
diagrams
$$
\vcenter{
\xymatrix{
y \ar[r] \ar[d] & x_V \ar[d] \\
x \ar[r] & x_0
}
}
\quad\text{and}\quad
\vcenter{
\xymatrix{
y' \ar[r] \ar[d] & x_V \ar[d] \\
x \ar[r] & x_0
}
}
\quad\text{lying over}\quad
\vcenter{
\xymatrix{
A \times_k k[V] \ar[d] \ar[r] & k[V] \ar[d] \\
A \ar[r] & k
}
}
$$
We will use the morphisms $x_V \to x_{V'} \to x_0$ and
$x_V \to x_\epsilon \to x_0$ introduced in the first paragraph of the proof.
Choose morphisms $y \to y_{V'}$ and $y' \to y'_{V'}$
lying over $A \times_k k[V] \to A \times_k k[V']$. The axioms of a
cofibred category imply we can find commutative diagrams
$$
\vcenter{
\xymatrix{
y_{V'} \ar[r] \ar[d] & x_{V'} \ar[d] \\
x \ar[r] & x_0
}
}
\quad\text{and}\quad
\vcenter{
\xymatrix{
y'_{V'} \ar[r] \ar[d] & x_{V'} \ar[d] \\
x \ar[r] & x_0
}
}
\quad\text{lying over}\quad
\vcenter{
\xymatrix{
A \times_k k[V'] \ar[d] \ar[r] & k[V'] \ar[d] \\
A \ar[r] & k
}
}
$$
By induction hypothesis we obtain an isomorphism
$b : y_{V'} \to y'_{V'}$
compatible with the morphisms $y_{V'} \to x$ and $y'_{V'} \to x$,
in particular compatible with the morphisms to $x_0$.
Then we have commutative diagrams
$$
\vcenter{
\xymatrix{
y \ar[r] \ar[d] & x_\epsilon \ar[d] \\
y'_{V'} \ar[r] & x_0
}
}
\quad\text{and}\quad
\vcenter{
\xymatrix{
y' \ar[r] \ar[d] & x_\epsilon \ar[d] \\
y'_{V'} \ar[r] & x_0
}
}
\quad\text{lying over}\quad
\vcenter{
\xymatrix{
A \times_k k[\epsilon] \ar[d] \ar[r] & k[\epsilon] \ar[d] \\
A \ar[r] & k
}
}
$$
where the morphism $y \to y'_{V'}$ is the composition
$y \to y_{V'} \xrightarrow{b} y'_{V'}$ and where the morphisms
$y \to x_\epsilon$ and $y' \to x_\epsilon$ are the compositions of
the maps $y \to x_V$ and $y' \to x_V$ with the morphism $x_V \to x_\epsilon$.
Then the second part of (S2) guarantees the existence of an isomorphism
$y \to y'$ compatible with the maps to $y'_{V'}$, in particular compatible
with the maps to $x$ (because $b$ was compatible with the maps to $x$).
\end{proof}
\begin{lemma}
\label{lemma-S1-S2-associated-functor}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$.
\begin{enumerate}
\item If $\mathcal{F}$ satisfies (S1), then so does
$\overline{\mathcal{F}}$.
\item If $\mathcal{F}$ satisfies (S2), then so does
$\overline{\mathcal{F}}$ provided at least one of the following conditions is
satisfied
\begin{enumerate}
\item $\mathcal{F}$ is a predeformation category,
\item the category $\mathcal{F}(k)$ is a set or a setoid, or
\item for any morphism $x_\epsilon \to x_0$ of $\mathcal{F}$
lying over $k[\epsilon] \to k$ the pushforward map
$\text{Aut}_{k[\epsilon]}(x_\epsilon) \to \text{Aut}_k(x_0)$
is surjective.
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{proof}
Assume $\mathcal{F}$ has (S1).
Suppose we have ring maps $f_i : A_i \to A$ in $\mathcal{C}_\Lambda$
with $f_2$ surjective. Let $x_i \in \mathcal{F}(A_i)$ such that
the pushforwards $f_{1, *}(x_1)$ and $f_{2, *}(x_2)$ are isomorphic.
Then we can denote $x$ an object of $\mathcal{F}$ over $A$ isomorphic
to both of these and we obtain a diagram as in (S1). Hence we find
an object $y$ of $\mathcal{F}$ over $A_1 \times_A A_2$ whose pushforward
to $A_1$, resp.\ $A_2$ is isomorphic to $x_1$, resp.\ $x_2$. In this way
we see that (S1) holds for $\overline{\mathcal{F}}$.
\medskip\noindent
Assume $\mathcal{F}$ has (S2).
The first part of (S2) for $\overline{\mathcal{F}}$ follows as in
the argument above. The second part of (S2) for
$\overline{\mathcal{F}}$ signifies that the map
$$
\overline{\mathcal{F}}(A \times_k k[\epsilon]) \to
\overline{\mathcal{F}}(A)
\times_{\overline{\mathcal{F}}(k)} \overline{\mathcal{F}}(k[\epsilon])
$$
is injective for any ring $A$ in $\mathcal{C}_\Lambda$. Suppose that
$y, y' \in \mathcal{F}(A \times_k k[\epsilon])$. Using the axioms
of cofibred categories we can choose commutative diagrams
$$
\vcenter{
\xymatrix{
y \ar[r]_c \ar[d]_a & x_\epsilon \ar[d]^e \\
x \ar[r]^d & x_0
}
}
\quad\text{and}\quad
\vcenter{
\xymatrix{
y' \ar[r]_{c'} \ar[d]_{a'} & x'_\epsilon \ar[d]^{e'} \\
x' \ar[r]^{d'} & x'_0
}
}
\quad\text{lying over}\quad
\vcenter{
\xymatrix{
A \times_k k[\epsilon] \ar[d] \ar[r] & k[\epsilon] \ar[d] \\
A \ar[r] & k
}
}
$$
Assume that there exist isomorphisms
$\alpha : x \to x'$ in $\mathcal{F}(A)$ and
$\beta : x_\epsilon \to x'_\epsilon$ in $\mathcal{F}(k[\epsilon])$.
This also means there exists an isomorphism $\gamma : x_0 \to x'_0$
compatible with $\alpha$. To prove (S2) for $\overline{\mathcal{F}}$
we have to show that there exists an isomorphism $y \to y'$ in
$\mathcal{F}(A \times_k k[\epsilon])$.
By (S2) for $\mathcal{F}$ such a morphism will exist if we can
choose the isomorphisms $\alpha$ and $\beta$ and $\gamma$ such that
$$
\xymatrix{
x \ar[d]^\alpha \ar[r] & x_0 \ar[d]^\gamma &
x_\epsilon \ar[d]^\beta \ar[l]^e \\
x' \ar[r] & x'_0 & x'_\epsilon \ar[l]_{e'}
}
$$
is commutative (because then we can replace $x$ by $x'$ and $x_\epsilon$
by $x'_\epsilon$ in the previous displayed diagram). The left hand square
commutes by our choice of $\gamma$. We can factor $e' \circ \beta$ as
$\gamma' \circ e$ for some second map
$\gamma' : x_0 \to x'_0$. Now the question is whether we can arrange it so
that $\gamma = \gamma'$? This is clear if $\mathcal{F}(k)$ is a set, or a
setoid. Moreover, if
$\text{Aut}_{k[\epsilon]}(x_\epsilon) \to \text{Aut}_k(x_0)$
is surjective, then we can adjust the choice of $\beta$ by precomposing
with an automorphism of $x_\epsilon$ whose image is
$\gamma^{-1} \circ \gamma'$ to make things work.
\end{proof}
\begin{lemma}
\label{lemma-S1-S2-localize}
Let $\mathcal{F}$ be a category cofibered in groupoids over
$\mathcal{C}_\Lambda$. Let $x_0 \in \Ob(\mathcal{F}(k))$.
Let $\mathcal{F}_{x_0}$ be the category cofibred in groupoids over
$\mathcal{C}_\Lambda$ constructed in
Remark \ref{remark-localize-cofibered-groupoid}.
\begin{enumerate}
\item If $\mathcal{F}$ satisfies (S1), then so does $\mathcal{F}_{x_0}$.
\item If $\mathcal{F}$ satisfies (S2), then so does $\mathcal{F}_{x_0}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Any diagram as in
Definition \ref{definition-S1-S2}
in $\mathcal{F}_{x_0}$ gives rise to a diagram in $\mathcal{F}$
and the output of condition (S1) or (S2) for this diagram in $\mathcal{F}$
can be viewed as an output for $\mathcal{F}_{x_0}$ as well.
\end{proof}
\begin{lemma}
\label{lemma-lifting-section}
Let $p: \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in
groupoids. Consider a diagram of $\mathcal{F}$
$$
\vcenter{
\xymatrix{
y \ar[r] \ar[d]_a & x_\epsilon \ar[d]_e \\
x \ar[r]^d & x_0
}
}
\quad\text{lying over}\quad
\vcenter{
\xymatrix{
A \times_k k[\epsilon] \ar[r] \ar[d] & k[\epsilon] \ar[d] \\
A \ar[r] & k.
}
}
$$
in $\mathcal{C}_\Lambda$. Assume $\mathcal{F}$ satisfies (S2).
Then there exists a morphism $s : x \to y$ with $a \circ s = \text{id}_x$
if and only if there exists a morphism $s_\epsilon : x \to x_\epsilon$
with $e \circ s_\epsilon = d$.
\end{lemma}
\begin{proof}
The ``only if'' direction is clear. Conversely, assume there exists a
morphism $s_\epsilon : x \to x_\epsilon$ with $e \circ s_\epsilon = d$.
Note that $p(s_\epsilon) : A \to k[\epsilon]$ is a ring map compatible
with the map $A \to k$. Hence we obtain
$$
\sigma = (\text{id}_A, p(s_\epsilon)) : A \to A \times_k k[\epsilon].
$$
Choose a pushforward $x \to \sigma_*x$. By construction we can factor
$s_\epsilon$ as $x \to \sigma_*x \to x_\epsilon$. Moreover, as $\sigma$
is a section of $A \times_k k[\epsilon] \to A$, we get a morphism
$\sigma_*x \to x$ such that $x \to \sigma_*x \to x$ is $\text{id}_x$.
Because $e \circ s_\epsilon = d$ we find that the diagram
$$
\xymatrix{
\sigma_*x \ar[r] \ar[d] & x_\epsilon \ar[d]_e \\
x \ar[r]^d & x_0
}
$$
is commutative. Hence by (S2) we obtain a morphism $\sigma_*x \to y$
such that $\sigma_*x \to y \to x$ is the given map $\sigma_*x \to x$.
The solution to the problem is now to take $a : x \to y$ equal to
the composition $x \to \sigma_*x \to y$.
\end{proof}
\begin{lemma}
\label{lemma-lifting-along-small-extension}
Consider a commutative diagram in a predeformation category $\mathcal{F}$
$$
\vcenter{
\xymatrix{
y \ar[r] \ar[d] & x_2 \ar[d]^{a_2} \\
x_1 \ar[r]^{a_1} & x
}
}
\quad\text{lying over}
\vcenter{
\xymatrix{
A_1 \times_A A_2 \ar[r] \ar[d] & A_2 \ar[d]^{f_2} \\
A_1 \ar[r]^{f_1} & A
}
}
$$
in $\mathcal{C}_\Lambda$ where
$f_2 : A_2 \to A$ is a small extension.
Assume there is a map $h : A_1 \to A_2$ such that $f_2 = f_1 \circ h$.
Let $I = \Ker(f_2)$. Consider the ring map
$$
g : A_1 \times_A A_2 \longrightarrow k[I] = k \oplus I, \quad
(u, v) \longmapsto \overline{u} \oplus (v - h(u))
$$
Choose a pushforward $y \to g_*y$. Assume $\mathcal{F}$ satisfies (S2).
If there exists a morphism $x_1 \to g_*y$, then there exists a
morphism $b: x_1 \to x_2$ such that $a_1 = a_2 \circ b$.
\end{lemma}
\begin{proof}
Note that
$\text{id}_{A_1} \times g : A_1 \times_A A_2 \to A_1 \times_k k[I]$
is an isomorphism and that $k[I] \cong k[\epsilon]$. Hence we have a diagram
$$
\vcenter{
\xymatrix{
y \ar[r] \ar[d] & g_*y \ar[d] \\
x_1 \ar[r] & x_0
}
}
\quad\text{lying over}\quad
\vcenter{
\xymatrix{
A_1 \times_k k[\epsilon] \ar[r] \ar[d] & k[\epsilon] \ar[d] \\
A_1 \ar[r] & k.
}
}
$$
where $x_0$ is an object of $\mathcal{F}$ lying over $k$ (every object
of $\mathcal{F}$ has a unique morphism to $x_0$, see
discussion following Definition \ref{definition-predeformation-category}).
If we have a morphism $x_1 \to g_*y$ then
Lemma \ref{lemma-lifting-section}
provides us with a section $s : x_1 \to y$ of the map $y \to x_1$.
Composing this with the map $y \to x_2$ we obtain $b : x_1 \to x_2$
which has the property that $a_1 = a_2 \circ b$ because
the diagram of the lemma commutes and because $s$ is a section.
\end{proof}
\section{Tangent spaces of functors}
\label{section-tangent-spaces-functors}
\noindent
Let $R$ be a ring. We write $\text{Mod}_R$ for the category of
$R$-modules and $\text{Mod}^{fg}_R$ for the category of finitely
generated $R$-modules.
\begin{definition}
\label{definition-linear}
Let $L: \text{Mod}^{fg}_R \to \text{Mod}_R$,
resp.\ $L: \text{Mod}_R \to \text{Mod}_R$
be a functor. We say that $L$ is {\it $R$-linear} if for every
pair of objects $M, N$ of $\text{Mod}^{fg}_R$, resp.\ $\text{Mod}_R$
the map
$$
L : \Hom_R(M, N) \longrightarrow \Hom_R(L(M), L(N))
$$
is a map of $R$-modules.
\end{definition}
\begin{remark}
\label{remark-linear-enriched-over-modules}
One can define the notion of an $R$-linearity for any functor between
categories enriched over $\text{Mod}_R$. We made the definition
specifically for functors $L: \text{Mod}^{fg}_R \to \text{Mod}_R$ and
$L: \text{Mod}_R \to \text{Mod}_R$
because these are the cases that we have needed so far.
\end{remark}
\begin{remark}
\label{remark-linear-functor}
If $L: \text{Mod}^{fg}_R \to \text{Mod}_R$ is an $R$-linear functor,
then $L$ preserves finite products and sends the zero module to the zero
module, see
Homology, Lemma \ref{homology-lemma-additive-additive}.
On the other hand, if a functor $\text{Mod}^{fg}_R \to \textit{Sets}$
preserves finite products and sends the zero module to a one element set,
then it has a unique lift to a $R$-linear functor, see
Lemma \ref{lemma-linear-functor}.
\end{remark}
\begin{lemma}
\label{lemma-linear-functor}
Let $L: \text{Mod}^{fg}_R \to \textit{Sets}$,
resp.\ $L: \text{Mod}_R \to \textit{Sets}$ be a
functor. Suppose $L(0)$ is a one element set and $L$ preserves finite
products. Then there exists a unique $R$-linear functor
$\widetilde{L} : \text{Mod}^{fg}_R \to \text{Mod}_R$,
resp.\ $\widetilde{L} : \text{Mod}^{fg}_R \to \text{Mod}_R$,
such that
$$
\vcenter{
\xymatrix{
& \text{Mod}_R \ar[dr]^{\text{forget}} & \\
\text{Mod}^{fg}_R \ar[ur]^{\widetilde{L}} \ar[rr]^{L} & &
\textit{Sets}
}
}
\quad\text{resp.}\quad
\vcenter{
\xymatrix{
& \text{Mod}_R \ar[dr]^{\text{forget}} & \\
\text{Mod}_R \ar[ur]^{\widetilde{L}} \ar[rr]^{L} & &
\textit{Sets}
}
}
$$
commutes.
\end{lemma}
\begin{proof}
We only prove this in case $L: \text{Mod}^{fg}_R \to \textit{Sets}$.
Let $M$ be a finitely generated $R$-module. We define $\widetilde{L}(M)$ to be
the set $L(M)$ with the following $R$-module structure.
\medskip\noindent
Multiplication: If $r \in R$, multiplication by $r$ on $L(M)$ is defined to be
the map $L(M) \to L(M)$ induced by the multiplication map
$r \cdot : M \to M$.
\medskip\noindent
Addition: The sum map $M \times M \to M: (m_1, m_2) \mapsto m_1 + m_2$
induces a map $L(M \times M) \to L(M)$. By assumption $L(M \times M)$
is canonically isomorphic to $L(M) \times L(M)$. Addition on $L(M)$ is defined
by the map $L(M) \times L(M) \cong L(M \times M) \to L(M)$.
\medskip\noindent
Zero: There is a unique map $0 \to M$. The zero element of $L(M)$ is
the image of $L(0) \to L(M)$.
\medskip\noindent
We omit the verification that this defines an $R$-module $\widetilde{L}(M)$,
the unique such that is $R$-linearly functorial in $M$.
\end{proof}
\begin{lemma}
\label{lemma-morphism-linear-functors}
Let $L_1, L_2: \text{Mod}^{fg}_R \to \textit{Sets}$ be
functors that take $0$ to a one element set and preserve finite products.
Let $t : L_1 \to L_2$ be a morphism of functors. Then $t$ induces a morphism
$\widetilde{t} : \widetilde{L}_1 \to \widetilde{L}_2$ between the
functors guaranteed by Lemma \ref{lemma-linear-functor}, which is given simply
by $\widetilde{t}_M = t_M: \widetilde{L}_1(M) \to \widetilde{L}_2(M)$
for each $M \in \Ob(\text{Mod}^{fg}_R)$. In other words,
$t_M: \widetilde{L}_1(M) \to \widetilde{L}_2(M)$ is a map of $R$-modules.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\noindent
In the case $R = K$ is a field, a $K$-linear functor
$L : \text{Mod}^{fg}_K \to \text{Mod}_K$ is determined by its value $L(K)$.
\begin{lemma}
\label{lemma-linear-functor-over-field}
Let $K$ be a field. Let $L: \text{Mod}^{fg}_K \to
\text{Mod}_K$ be a $K$-linear functor. Then $L$ is isomorphic to the
functor $L(K) \otimes_K - : \text{Mod}^{fg}_K \to
\text{Mod}_K$.
\end{lemma}
\begin{proof}
For $V \in \Ob(\text{Mod}^{fg}_K)$, the isomorphism
$L(K) \otimes_K V \to L(V)$ is given on pure tensors by
$x \otimes v \mapsto L(f_v)(x)$, where $f_v: K \to V$ is the $K$-linear map
sending $1 \mapsto v$. When $V = K$, this is the isomorphism
$L(K) \otimes_K K \to L(K)$ given by multiplication by $K$.
For general $V$, it is an isomorphism by the case $V = K$ and the
fact that $L$ commutes with finite
products (Remark \ref{remark-linear-functor}).
\end{proof}
\noindent
For a ring $R$ and an $R$-module $M$, let $R[M]$ be the $R$-algebra whose
underlying $R$-module is $R \oplus M$ and whose multiplication is given by
$(r, m) \cdot (r', m') = (rr', rm' + r'm)$. When $M = R$ this is the ring of
dual numbers over $R$, which we denote by $R[\epsilon]$.
\medskip\noindent
Now let $S$ be a ring and assume $R$ is an $S$-algebra.
Then the assignment $M \mapsto R[M]$ determines a functor
$\text{Mod}_R \to S\text{-Alg}/R$, where $S\text{-Alg}/R$
denotes the category of $S$-algebras over $R$. Note that
$S\text{-Alg}/R$ admits finite products: if $A_1 \to R$ and
$A_2 \to R$ are two objects, then $A_1 \times_R A_2$ is a product.
\begin{lemma}
\label{lemma-preserves-products}
Let $R$ be an $S$-algebra. Then the functor
$\text{Mod}_R \to S\text{-Alg}/R$ described above preserves finite products.
\end{lemma}
\begin{proof}
This is merely the statement that if $M$ and $N$ are $R$-modules, then the map
$R[M \times N] \to R[M] \times_R R[N]$ is an isomorphism in
$S\text{-Alg}/R$.
\end{proof}
\begin{lemma}
\label{lemma-tangent-space-functor}
Let $R$ be an $S$-algebra, and let $\mathcal{C}$ be a strictly full
subcategory of $S\text{-Alg}/R$ containing $R[M]$ for all
$M \in \Ob(\text{Mod}^{fg}_R)$.
Let $F: \mathcal{C} \to \textit{Sets}$ be a functor. Suppose that
$F(R)$ is a one element set and that for any $M, N \in
\Ob(\text{Mod}^{fg}_R)$, the induced map
$$
F(R[M] \times_R R[N]) \to F(R[M]) \times F(R[N])
$$
is a bijection. Then $F(R[M])$ has a natural $R$-module structure for any $M
\in \Ob(\text{Mod}^{fg}_R)$.
\end{lemma}
\begin{proof}
Note that $R \cong R[0]$ and $R[M] \times_R R[N] \cong R[M \times N]$ hence
$R$ and $R[M] \times_R R[N]$ are objects of $\mathcal{C}$ by our assumptions on
$\mathcal{C}$. Thus the conditions on $F$ make sense.
The functor $\text{Mod}_R \to S\text{-Alg}/R$ of
Lemma \ref{lemma-preserves-products}
restricts to a functor $\text{Mod}^{fg}_R \to \mathcal{C}$
by the assumption on $\mathcal{C}$. Let $L$ be the composition
$\text{Mod}^{fg}_R \to \mathcal{C} \to \textit{Sets}$, i.e.,
$L(M) = F(R[M])$.
Then $L$ preserves finite products by
Lemma \ref{lemma-preserves-products}
and the assumption on $F$. Hence
Lemma \ref{lemma-linear-functor}
shows that $L(M) = F(R[M])$ has a natural $R$-module structure for any
$M \in \Ob(\text{Mod}^{fg}_R)$.
\end{proof}
\begin{definition}
\label{definition-tangent-space-over-R}
Let $\mathcal{C}$ be a category as in
Lemma \ref{lemma-tangent-space-functor}.
Let $F : \mathcal{C} \to \textit{Sets}$ be a functor such that
$F(R)$ is a one element set. The {\it tangent space $TF$ of $F$} is
$F(R[\epsilon])$.
\end{definition}
\noindent
When $F : \mathcal{C} \to \textit{Sets}$ satisfies the hypotheses of
Lemma \ref{lemma-tangent-space-functor},
the tangent space $TF$ has a natural $R$-module structure.
\begin{example}
\label{example-tangent-space-functor}
Since $\mathcal{C}_\Lambda$ contains all $k[V]$ for finite dimensional
vector spaces $V$ we see that
Definition \ref{definition-tangent-space-over-R} applies with
$S = \Lambda$, $R = k$, $\mathcal{C} = \mathcal{C}_\Lambda$, and
$F : \mathcal{C}_\Lambda \to \textit{Sets}$ a
predeformation functor. The tangent space is $TF = F(k[\epsilon])$.
\end{example}
\begin{example}
\label{example-tangent-space-prorepresentable-functor}
Let us work out the tangent space of
Example \ref{example-tangent-space-functor}
when