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Explain duals and adjoints better

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aisejohan committed Nov 6, 2019
1 parent 9c12754 commit 93305405409b78dcbed68c757543a018b752a56e
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  1. +49 −11 categories.tex
@@ -8554,17 +8554,55 @@ \section{Monoidal categories}
\begin{remark}
\label{remark-left-dual-adjoint}
Lemma \ref{lemma-left-dual} says in particular that $Z \mapsto Z \otimes Y$
is the right adjoint of $Z' \mapsto Z' \otimes X$. Conversely, if this is
true, then we get $\eta : \mathbf{1} \to X \otimes Y$ by evaluating
the unit of the adjunction on $\mathbf{1}$
and $\epsilon : Y \otimes X \to \mathbf{1}$ by evaluating the counit
of the adjunction on $\mathbf{1}$
(Section \ref{section-adjoint}). Thus the
requirement that $Z \mapsto Z \otimes Y$ be the right adjoint of
$Z' \mapsto Z' \otimes X$ is an equivalent formulation of the
property of being a left dual. Uniqueness of adjoint functors
guarantees that a left dual of $X$, if it exists, is unique up
to unique isomorphism.
is the right adjoint of $Z' \mapsto Z' \otimes X$. In particular, uniqueness
of adjoint functors guarantees that a left dual of $X$, if it exists, is
unique up to unique isomorphism.
Conversely, assume the functor $Z \mapsto Z \otimes Y$ is a right adjoint of
the functor $Z' \mapsto Z' \otimes X$, i.e., we're given a bijection
$$
\Mor(Z' \otimes X, Z) \longrightarrow \Mor(Z', Z \otimes Y)
$$
functorial in both $Z$ and $Z'$. The unit of the adjunction produces
maps
$$
\eta_Z : Z \to Z \otimes X \otimes Y
$$
functorial in $Z$ and the counit of the adjoint produces maps
$$
\epsilon_{Z'} : Z' \otimes Y \otimes X \to Z'
$$
functorial in $Z'$. In particular, we find
$\eta = \eta_\mathbf{1} : \mathbf{1} \to X \otimes Y$ and
$\epsilon = \epsilon_\mathbf{1} : Y \otimes X \to \mathbf{1}$.
As an exercise in the relationship between units, counits, and
the adjunction isomorphism, the reader can show that we have
$$
(\epsilon \otimes \text{id}_Y) \circ \eta_Y = \text{id}_Y
\quad\text{and}\quad
\epsilon_X \circ (\eta \otimes \text{id}_X) = \text{id}_X
$$
However, this isn't enough to show that
$(\epsilon \otimes \text{id}_Y) \circ (\text{id}_Y \otimes \eta) =
\text{id}_Y$ and
$(\text{id}_X \otimes \epsilon) \circ (\eta \otimes \text{id}_X) =
\text{id}_X$, because we don't know in general that
$\eta_Y = \text{id}_Y \otimes \eta$ and we don't know that
$\epsilon_X = \epsilon \otimes \text{id}_X$. For this it would suffice
to know that our adjunction isomorphism has the following property:
for every $W, Z, Z'$ the diagram
$$
\xymatrix{
\Mor(Z' \otimes X, Z) \ar[r] \ar[d]_{\text{id}_W \otimes -} &
\Mor(Z', Z \otimes Y) \ar[d]^{\text{id}_W \otimes -} \\
\Mor(W \otimes Z' \otimes X, W \otimes Z) \ar[r] &
\Mor(W \otimes Z', W \otimes Z \otimes Y)
}
$$
If this holds, we will say {\it the adjunction is compatible with
the given tensor structure}. Thus the requirement that
$Z \mapsto Z \otimes Y$ be the right adjoint of $Z' \mapsto Z' \otimes X$
compatible with the given tensor structure is an equivalent formulation of the
property of being a left dual.
\end{remark}

\begin{lemma}

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