Categories

Summary

A category consists of objects and morphisms with composition. This note covers the formal definition, elementary examples spanning mathematics, and the size distinctions (small, locally small, large) needed to avoid set-theoretic paradoxes.

Overview

Category theory begins with a single structure: the category. Rather than studying mathematical objects in isolation, category theory studies objects together with the maps between them. The insight is that maps (morphisms) carry as much—often more—information than the objects themselves.

Main Content

Definition 1.1.1: Category

A category $\mathcal{A}$ consists of:

• A collection $\mathrm{ob}(\mathcal{A})$ of objects
• For each $A,B\in \mathrm{ob}(\mathcal{A})$, a collection $\mathcal{A}(A,B)$ of maps (morphisms, arrows) from $A$ to $B$
• For each $A,B,C$, a composition function $\mathcal{A}(B,C)\times \mathcal{A}(A,B)\to \mathcal{A}(A,C)$, written $(g,f)\mapsto g\circ f$
• For each $A$, an identity map $1_A\in \mathcal{A}(A,A)$

satisfying:

1. Associativity: $h\circ (g\circ f)=(h\circ g)\circ f$
2. Unit laws: $f\circ 1_A=f$ and $1_B\circ f=f$ for all $f:A\to B$

Notation: Maps $f:A\to B$ may also be written $A\stackrel{f}{\to }B$. Collections $\mathcal{A}(A,B)$ are called hom-sets (when they are sets).

Core Examples

Category	Objects	Maps	Composition
Set	sets	functions	function composition
Grp	groups	group homomorphisms	homomorphism composition
Top	topological spaces	continuous maps	composition
Vect${}_k$	$k$-vector spaces	linear maps	composition
Ring	rings	ring homomorphisms	composition
Poset $(P,\le )$	elements of $P$	unique map $a\to b$ iff $a\le b$	transitivity
Discrete $S$	elements of $S$	only identity maps	trivial
Monoid $M$	single object $\bullet$	elements of $M$	multiplication in $M$

Example: A Monoid as a Category (BCT, Ch. 1.1)

Any monoid $(M,\cdot ,e)$ is a category with one object $\bullet$. Maps $\bullet \to \bullet$ correspond to elements of $M$, composition is multiplication, and the identity map is $e$. A functor between one-object categories is exactly a monoid homomorphism.

Example: A Preorder as a Category (BCT, Ch. 1.1)

A preorder $(P,\le )$ is a category where $\mathcal{A}(a,b)$ has exactly one element if $a\le b$ and is empty otherwise. Composition encodes transitivity; identity maps encode reflexivity. Functors between preorder categories are exactly order-preserving maps.

Isomorphisms

Definition: Isomorphism

A map $f:A\to B$ is an isomorphism if there exists $g:B\to A$ such that $g\circ f=1_A$ and $f\circ g=1_B$. We write $A\cong B$ and call $A$ and $B$ isomorphic.

The inverse $g$ is unique when it exists. In Set, isomorphisms are bijections; in Top, they are homeomorphisms; in Grp, they are group isomorphisms.

Opposite Category

Definition: Opposite Category

The opposite (or dual) category ${\mathcal{A}}^{\mathrm{op}}$ has the same objects as $\mathcal{A}$, with ${\mathcal{A}}^{\mathrm{op}}(A,B)=\mathcal{A}(B,A)$ and composition reversed. A map in ${\mathcal{A}}^{\mathrm{op}}$ from $A$ to $B$ is a map $B\to A$ in $\mathcal{A}$.

Duality is a powerful principle: any theorem about all categories has a dual theorem obtained by reversing all arrows.

Size: Small, Large, Locally Small

Definition: Size Distinctions (BCT, Ch. 3.1–3.2)

• A category is small if $\mathrm{ob}(\mathcal{A})$ is a set (not a proper class) and each $\mathcal{A}(A,B)$ is a set.
• A category is locally small if each hom-collection $\mathcal{A}(A,B)$ is a set (but $\mathrm{ob}(\mathcal{A})$ may be a proper class).
• Set, Grp, Top are locally small but not small.
• CAT (category of all categories) is not even locally small — the collection of functors between two large categories need not be a set.

Russell’s paradox prevents us from forming the category of all sets (or all categories) naively. The Yoneda lemma requires local smallness (so that representable functors land in Set).

Connections

• Functors (Functors) are the maps between categories — they must preserve composition and identities.
• Natural transformations (Natural Transformations) are maps between functors.
• Limits and colimits (General Limits) are defined purely in terms of the categorical structure.
• The opposite category construction underlies all duality in category theory; colimits are limits in the opposite category.

See Also

• Functors — Maps between categories
• Natural Transformations — Maps between functors
• Adjoint Functors — Special pairs of functors between categories
• General Limits — Universal cones