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Category Theory Notes

Jargon

Object

Algebraic structure

Morphism

A structure-preserving map from one mathematical object to another.

In category theory, obey conditions specific to category theory itself.

Homomorphism

A structure-preserving map between two algebraic structures of the same type.

Isomorphism

A homomorphism or morphism that admits an inverse. Two objects are isomorphic if an isomorphism exists between them.

Automorphism

An isomorphism from a mathematical object to itself, i.e. an invertible endomorphism.

Endomorphism

A morphism (or homomorphism) from a mathematical object to itself.

Epimorphism

Also known as an epic morphism, or an epi. A morphism, $f: X → Y$ that is right-cancellative i.e. \[ ∀ g_1, g_2: Y → Z, \ g_1 ˆ f = g_2 ˆ f \implies g_1 = g_2 \]

Epimorphisms are categorical analogues of surjective functions.

Right-cancellative

Surjective

“onto” \[ f : X → Y, \ ∀ y ∈ Y,\ ∃ x ∈ X,\ f(x) = y \]

Monomorphism

An injective homomorphism, e.g. $X \hookrightarrow Y$. In category theory, a monomorphism (a.k.a monic morphism or mono) is a left-cancellative morphism, i.e. \[ ∀ g_1, g_2 : Z → X, \ f : X → Y\ s.t.\ f ˆ g_1 = f ˆ g_2 \implies g_1 = g_2 \]

Monomorphisms are a categorical generalization of injective functions

Left-cancellative

Injective

“one-to-one” \[ ∀ a, b ∈ X,\ f(a) = f(b) \implies a = b \] contrapositive: \[ ∀ a, b ∈ X,\ a ≠ b \implies f(a) ≠ f(b) \]

Category

An algebraic structure comprised of objects linked by arrows, e.g. the category of sets, which links sets with functions. Arrows can be composed associatively and there exists an identity arrow.

Set

Function

Associativity

Identity

Abelian category

A category in which morphisms and objects can be added and in which kernels and cokernels exist and have desireable properties, e.g. the category of abelian groups, Ab.

• has a zero object
• has all binary products and binary coproducts
• has all kernels and cokernels
• all monomorphisms and epimorphisms are normal

Group

Ab

The category of abelian groups.

Zero object

An object that is both intial and terminal. a.k.a. null object.

Pointed category

A category with a zero object.

Strict initial object

An initial object for which every morphism into $I$ is an isomorphism.

Initial object

$I ∈ C, ∀ X ∈ C ∃$ precisely one morphism $I → X$ a.k.a. coterminal or universal

Terminal object

$T$ is terminal if $∀ X ∈ C ∃$ a single morphism $X → T$. a.k.a terminal element

Product

The “most general” object which admits a morphism to each of the given objects.

Commutative diagram

Coproduct

a.k.a. categorical sum

Commutative diagram

Coproduct

Kernel

Cokernel

Normal

Module

Ring

Monoid

An algebraic structure with a single associative binary operation and an identity element. A monoid is a semigroup with an identity.

Image

Codomain

Domain

Homology

Cohomology

Combinatorial topology

Algebraic toplogy

Abstract algebra

Henri Poincaré

David Hilbert