Introduction to Categories

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 14 September 2012

Introduction

A category is

1. a collection of objects $𝒞$,
2. for each $X,Y\in 𝒞$ a set $\mathrm{Hom}(X,Y)$ of morphisms,
3. for each $X,Y,Z\in 𝒞$ a function

$$\begin{array}{ccc}\mathrm{Hom}(X,Y)\times \mathrm{Hom}(Y,Z)& ⟶& \mathrm{Hom}(X,Z)\\ (f,g)& ⟼& g\circ f\end{array}$$ such that

1. if $f\in \mathrm{Hom}(X,Y),g\in \mathrm{Hom}(Y,Z)$ and $h\in \mathrm{Hom}(Z,W)$ then $h\circ (g\circ f)=(h\circ g)\circ f,$
2. If $X\in 𝒞$ then there exists ${\text{id}}_X\in \mathrm{Hom}(X,X)$ such that
   1. if $f\in \mathrm{Hom}(X,Y)$ then $f\circ {\text{id}}_X=f$ and
   2. if $f\in \mathrm{Hom}(W,X)$ then ${\text{id}}_X\circ f=f$.

A functor from $𝒞$ to $𝒟$ is a collection of functions $$F:𝒞⟶𝒟\text{and}F:\mathrm{Hom}(X,Y)⟶\mathrm{Hom}(FX,FY),$$ for $X,Y\in 𝒞$, such that

1. If $f\in \mathrm{Hom}(X,Y)$ and $g\in \mathrm{Hom}(Y,Z)$ then $F(g\circ f)=F\left(g\right)\circ F\left(f\right)$.
2. If $X\in 𝒞$ then $F\left({\text{id}}_X\right)={\text{id}}_{F\left(X\right)}$.

A contravariant functor from $𝒞$ to $𝒟$ is a collection of functions $$F:𝒞⟶𝒟\text{and}F:\mathrm{Hom}(X,Y)⟶\mathrm{Hom}(FY,FX)$$ for $X,Y\in 𝒞$ such that

1. If $f\in \mathrm{Hom}(X,Y)$ and $g\in \mathrm{Hom}(Y,Z)$ then $F(g\circ f)=F\left(f\right)\circ F\left(g\right)$,
2. If $X\in 𝒞$ then $F\left({\text{id}}_X\right)={\text{id}}_{F\left(X\right)}$.

A full functor is a functor $F:𝒞⟶𝒟$ such that $$\text{if}X,Y\in 𝒞\text{then}F:\mathrm{Hom}(X,Y)⟶\mathrm{Hom}(FX,FY)\text{is surjective.}$$

A faithful functor is a functor $F:𝒞⟶𝒟$ such that $$\text{if}X,Y\in 𝒞\text{then}F:\mathrm{Hom}(X,Y)⟶\mathrm{Hom}(FX,FY)\text{is injective.}$$

An equivalence of categories is a pair of functors $F:𝒞⟶𝒟$ and $G:𝒟⟶𝒞$ with $$\text{natural isomorphisms}{\text{id}}_{𝒞}\simeq G\circ F\text{and}{\text{id}}_{𝒟}\simeq F\circ G.$$

A full subcategory of $𝒞$ is a subcategory $𝒮\subseteq 𝒞$ such that the inclusion $𝒮\hookrightarrow 𝒞$ is full.

A terminal object is $Y\in 𝒞$ such that $$\text{if}X\in 𝒞\text{then there exists a unique}f\in \mathrm{Hom}(X,Y).$$

An initial object is $X\in 𝒞$ such that $$\text{if}Y\in 𝒞\text{then there exists a unique}f\in \mathrm{Hom}(X,Y).$$

A null object is $0\in 𝒞$ such that $$0\text{is both initial and terminal.}$$

A split monic is a monic $\iota :A⟶B$ such that there exists a left inverse $r:B⟶A$.

A split epi is an epi $E\stackrel{p}{⟶}B$ such that there exists a right inverse $s:B⟶E$.

Categories

A category $𝒞$ is a collection of objects and morphisms, with composition maps $$\begin{array}{rcl}{\mathrm{Hom}}_{𝒞}(X,Y)\times {\mathrm{Hom}}_{𝒞}(Y,Z)& ⟶& {\mathrm{Hom}}_{𝒞}(X,Z)\\ (f,g)& ⟼& g\circ f\end{array}$$ for which associativity holds and identities exist (if $X,Y$ and $Z$ are objects of $𝒞$ then there exists an identity morphism ${\mathrm{id}}_X:X\to X$ such that ${\mathrm{id}}_X\circ f=f$ for all $f:Y\to X$ and $g\circ {\mathrm{id}}_X=g$ for all $g:X\to Z$).

Examples.

Objects	Morphisms
Sets	Functions
Groups	Group homomorphisms
Rings	Ring homomorphisms
Vector spaces	Linear transformations
$A$-modules	$A$-modulehomomorphisms
Abelian groups	$\mathbb{Z}$-modulehomomorphisms
Topological spaces	Continuous functions
Manifolds	Smooth maps
Complex manifolds	Holomorphic maps
Algebras	Homomorphisms of algebras
Lie algebras	Lie algebra homomorphisms
Varieties	Morphisms of varieties
Affine varieties	Regular functions
Schemes	Morphisms of schemes
Affine schemes	Morphisms of schemes
Sheaves	Morphisms of sheaves
Vector bundles	Morphisms of vector bundles
Principal bundles	Morphisms of principal bundles
Categories	Functors
Functors	Natural transformations
Complexes	Chain maps
Homotopy category	Chain maps
Derived category	Morphisms

The category of categories

The category of categories has

• Objects: Categories
• Morphisms: Functors

Let $𝒜$ and $ℬ$ be sets of objects. A functor is a map $ℱ$ which takes objects to objects and morphisms to morphisms $$\begin{array}{rcll}ℱ:& 𝒜& ⟶& ℬ\\ & M& ⟼& ℱ\left(M\right)\end{array}\text{and}\begin{array}{rcll}ℱ:& {\mathrm{Hom}}_{𝒜}(M,N)& ⟶& {\mathrm{Hom}}_{ℬ}(ℱ\left(M\right),ℱ\left(N\right))\\ & f& ⟼& ℱ\left(f\right)\end{array}$$ such that $$ℱ\left({\mathrm{id}}_M\right)={\mathrm{id}}_{ℱ\left(M\right)}\text{and}ℱ(f_1\circ f_2)=ℱ(f_1)\circ ℱ(f_2).$$

Example. Let $A$ and $B$ be algebras with $A\subseteq B$ (e.g. $A=ℂS_3$ and $B=ℂS_4$). Let $𝒜$ be the category of $A$-modules and $ℬ$ be the category of $B$-modules. Then induction is a functor $$\begin{array}{rcll}{\mathrm{Ind}}_A^B:& 𝒜\mathrm{-mod}& ⟶& ℬ\mathrm{-mod}\\ & M& ⟼& B{\otimes }_{A}M\end{array}\text{with}\begin{array}{rcll}{\mathrm{Ind}}_A^B\left(f\right):& B{\otimes }_{A}M& ⟶& B{\otimes }_{A}N\\ & b\otimes m& ⟼& b\otimes f\left(m\right)\end{array}$$ if $f:M\to N$ is an $A$-module homomorphism.

The category of functors

Let $𝒜$ and $ℬ$ be categories. The category of functors from $𝒜$ to $ℬ$ has

• Objects: functors $ℱ:𝒜\to ℬ$, and
• Morphisms: natural transformations.

A natural transformation $\varphi :ℱ\to 𝒢$ is a collection of morphisms $$\{{\varphi }_M:ℱ(M)\to 𝒢(M\left)\right|M\in 𝒜\}$$ such that if $f:M\to N$ then the following diagram commutes. $$\begin{array}{ccc}ℱ\left(M\right)& \underset{ℱ\left(f\right)}{⟶}& ℱ\left(N\right)\\ \downarrow {\varphi }_M& & {\varphi }_N\downarrow \\ 𝒢\left(M\right)& \stackrel{𝒢\left(f\right)}{⟶}& 𝒢\left(N\right)\end{array}$$

Example. An additive category is a category $𝒜$ such that $${\mathrm{Hom}}_{𝒜}(M,N)\text{is an abelian group}$$ and there is a $0$ object in $𝒜$ and direct sums $M\oplus N$ exist in $𝒜$.

A 2-category is a category $𝒜$ such that $${\mathrm{Hom}}_{𝒜}(M,N)\text{is a category}$$ and there is a $0$ object in $𝒜$ and direct sums $M\oplus N$ exist in $𝒜$.

The category of categories is an example of a 2-category.

Example of a category of functors

Let $X$ be a topological space with topology $𝒯$. The topology $𝒯$ is a category with

• Objects: open sets $U$, and
• Morphisms: inclusions $U_1\hookrightarrow U_2$

Let $𝒞$ be the category of commutative rings with identity. A presheaf (of commutative rings) on $X$ is a contravariant functor $ℱ:𝒯\to 𝒞$. A morphism of presheaves is a morphism of functors $ℱ$ to $𝒢$.

The category of presheaves is the category of functors ${𝒯}^{\mathrm{op}}\to 𝒞$.

The category of complexes

Let $𝒜$ be a abelian category (e.g. $𝒜$ is the category of $A$-modules). The category $\mathrm{Kom}\left(𝒜\right)$ of complexes over $𝒜$ has

• Objects: complexes over $𝒜$, and
• Morphisms: chain maps

A complex $M$ over $𝒜$ is a sequence of morphisms $$\cdots \to M^i\stackrel{d^i}{\to }{M}^{i+1}\stackrel{d^{i+1}}{\to }\cdots \text{with}{d}^{i+1}\circ d^i=0.$$ i.e. a $\mathbb{Z}$-graded $A$-module $M$ with a map $d:M\to M$ with $\mathrm{deg}d=1$ and $\mathrm{deg}{d}^2=0$. A chain map $f:M\to N$ is a collection of morphisms $$\{f^i:M^i\to N^i|i\in \mathbb{Z}\}$$ such that

commutes.

Totalization

Elements of the category $\mathrm{Kom}\left(\mathrm{Kom}\right(𝒜\left)\right)$ look like

Let $𝒜$ be an abelian category (i.e. ${\mathrm{Hom}}_{𝒜}(X,Y)$ are abelian groups, there exists a $0$ object and direct sums $X\oplus Y$).

The totalization functor $\mathrm{tot}:\mathrm{Kom}\left(\mathrm{Kom}\right(𝒜\left)\right)\to \mathrm{Kom}\left(𝒜\right)$ is given by

Cohomology

An abelian category is a category $𝒜$ such that ${\mathrm{Hom}}_{𝒜}(M,N)$ are abelian groups, there exists a $0$ object and direct sums $M\oplus N$ and kernels and cokernels exist.

Let $𝒜$ be an abelian category and let $\mathrm{Kom}\left(𝒜\right)$ be the category of complexes over $𝒜$ $$M=(\cdots \to M^i\stackrel{d^i}{\to }{M}^{i+1}\to \cdots )\text{with}{d}_{i+1}\circ d_i=0.$$ The cohomology of a complex $\left(M,d\right)$ is $$H\left(M\right)=\frac{Z\left(M\right)}{B\left(M\right)},\text{where}Z\left(M\right)=\ker d\text{and}B\left(M\right)=\mathrm{im}d$$ and $$\begin{array}{rcll}H\left(f\right):& H\left(M\right)& ⟶& H\left(M\right)\\ & \left[c\right]& ⟼& \left[f\right(c\left)\right]\end{array},\text{if}f:M\to N\text{is a morphism in}\mathrm{Kom}\left(𝒜\right).$$

The derived category $D\left(𝒜\right)$

Let $𝒜$ be an abelian category, let $\mathrm{Kom}\left(𝒜\right)$ be the category of complexes over $𝒜$, and let $H\left(M\right)$ be the cohomology of a complex $M$.

A quasiisomorphism is a morphism $f:M\to N$ in $\mathrm{Kom}(𝒜)$ such that $H(f:H(M)\to H(N\left)\right)$ is an isomorphism.

The derived category of $𝒜$ is the category $D\left(𝒜\right)$ with a functor $Q:\mathrm{Kom}\left(𝒜\right)\to D\left(𝒜\right)$ such that

1. if $f$ is a quasiisomorphism then $Q\left(f\right)$ is an isomorphism, and
2. if $F:\mathrm{Kom}\left(𝒜\right)\to 𝒞$ is a functor that takes quasiisomorphisms to isomorphisms then there exists a unique functor $\stackrel{\sim }{F}:D\left(𝒜\right)\to 𝒞$ such that

The homotopy category

Let $𝒜$ be an abelian category and $\mathrm{Kom}\left(𝒜\right)$ the category of complexes over $𝒜$. Let $M$ and $N$ be objects of $\mathrm{Kom}\left(𝒜\right)$. A homotopy between morphisms $f:M\to N$ and $g:M\to N$ is a collection of morphisms $$\{h^i:M^i\to N^{i+1}|i\in \mathbb{Z}\}\text{such that}{f}^i-g^i=h^{i+1}\circ d^i+d^{i-1}\circ h^i.$$

HW: Show that if $f$ and$ g$are\; homotopic\; then$ H\left(f\right)=H\left(g\right)$ .$$$$

The homotopy category $\mathrm{Ho}\left(𝒜\right)$ has

• Objects: complexes over $𝒜$, and
• Morphisms: chain maps modulo homotopy equivalence.

Notes and References

These notes originated from a presentation in a working seminar at University of Melbourne, 12 November 2009.

References

[KL] M. Khovanov and A. Lauda, A diagrammatic approach to categorification of quantum groups I, Representation Theory 13 (2009), 309–347. MR2525917

[Ro] R. Rouquier, 2 Kac-Moody algebras, arXiv:08125023

[MH] A. Mathas and J. Hu, arXiv:0907.2985

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