Functors

Summary

A functor is a structure-preserving map between categories, sending objects to objects and morphisms to morphisms while respecting composition and identities. Functors that are both full and faithful are the categorical notion of “embedding.” Contravariant functors go into the opposite category.

Overview

If categories are mathematical universes, functors are the translations between them. A functor from $\mathcal{A}$ to $\mathcal{B}$ carries every object and morphism of $\mathcal{A}$ over to $\mathcal{B}$ in a way that preserves all compositional structure. This makes functors the maps in the category CAT of categories.

Main Content

Definition 1.2.1: Functor

A functor $F:\mathcal{A}\to \mathcal{B}$ consists of:

• A function $F:\mathrm{ob}(\mathcal{A})\to \mathrm{ob}(\mathcal{B})$
• For each $A,A^{\prime }\in \mathcal{A}$, a function $F:\mathcal{A}(A,A^{\prime })\to \mathcal{B}(FA,F{A}^{\prime })$

satisfying:

1. $F(g\circ f)=F(g)\circ F(f)$ (preserves composition)
2. $F(1_A)=1_{FA}$ (preserves identities)

Notation: The image of a map $f:A\to A^{\prime }$ under $F$ is written $Ff$ or $F(f)$.

Key Examples

Functor	Domain → Codomain	Action
Forgetful $U:\mathbf{Grp}\to \mathbf{Set}$	groups → sets	Forgets group structure
Free $F:\mathbf{Set}\to \mathbf{Grp}$	sets → free groups	Generates free group
Power set $\mathcal{P}:\mathbf{Set}\to \mathbf{Set}$	sets → sets	$\mathcal{P}(f)(S)=f(S)$ (direct image)
Abelianization $(-)/[-,-]:\mathbf{Grp}\to \mathbf{Ab}$	groups → abelian groups	Quotients by commutator
Fundamental group ${\pi }_1:{\mathbf{Top}}_{\ast }\to \mathbf{Grp}$	pointed spaces → groups	Assigns ${\pi }_1$
Identity $1_{\mathcal{A}}:\mathcal{A}\to \mathcal{A}$	any → same	Does nothing
Constant $\Delta B:\mathcal{A}\to \mathcal{B}$	any → any	Sends everything to $B$

Example: Power Set as a Functor (BCT, Ch. 1.2)

The direct image power set functor $\mathcal{P}:\mathbf{Set}\to \mathbf{Set}$ sends a set $X$ to $\mathcal{P}(X)$ (its set of subsets) and a function $f:X\to Y$ to $f_{\ast }:\mathcal{P}(X)\to \mathcal{P}(Y)$ where $f_{\ast }(S)=\{f(s):s\in S\}$.

The inverse image version $f^{-1}:\mathcal{P}(Y)\to \mathcal{P}(X)$ is contravariant — it reverses arrows.

Contravariant Functors

Definition: Contravariant Functor

A contravariant functor from $\mathcal{A}$ to $\mathcal{B}$ is a (covariant) functor ${\mathcal{A}}^{\mathrm{op}}\to \mathcal{B}$. It sends $f:A\to A^{\prime }$ to $Ff:F{A}^{\prime }\to FA$ (arrow reversed).

The key example is the hom-functor: for fixed $B\in \mathcal{A}$, the functor $\mathcal{A}(-,B):{\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$ sends $A\mapsto \mathcal{A}(A,B)$ and $f:A\to A^{\prime }$ to the precomposition map $f^{\ast }:\mathcal{A}(A^{\prime },B)\to \mathcal{A}(A,B)$.

Full and Faithful Functors

Definition 1.2.15: Full, Faithful, Fully Faithful

A functor $F:\mathcal{A}\to \mathcal{B}$ is:

• Faithful if for all $A,A^{\prime }$, the function $F:\mathcal{A}(A,A^{\prime })\to \mathcal{B}(FA,F{A}^{\prime })$ is injective
• Full if for all $A,A^{\prime }$, the function $F:\mathcal{A}(A,A^{\prime })\to \mathcal{B}(FA,F{A}^{\prime })$ is surjective
• Fully faithful if it is both full and faithful (i.e., each such function is a bijection)

Fully faithful functors are the categorical embedding: they identify $\mathcal{A}$ as a full subcategory of $\mathcal{B}$. The Yoneda embedding $H^{\bullet }:\mathcal{A}\to [{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$ is a key example (Yoneda Embedding and Consequences).

Proposition: If $F$ is fully faithful and $FA\cong F{A}^{\prime }$ in $\mathcal{B}$, then $A\cong A^{\prime }$ in $\mathcal{A}$.

Equivalence of Categories

Definition: Equivalence of Categories

A functor $F:\mathcal{A}\to \mathcal{B}$ is an equivalence of categories if there exists $G:\mathcal{B}\to \mathcal{A}$ such that $GF\cong 1_{\mathcal{A}}$ and $FG\cong 1_{\mathcal{B}}$ (natural isomorphisms). Write $\mathcal{A}\simeq \mathcal{B}$.

$F$ is an equivalence if and only if $F$ is fully faithful and essentially surjective (every $B\in \mathcal{B}$ is isomorphic to $FA$ for some $A$).

Equivalence is weaker than isomorphism of categories (which requires $GF=1$ and $FG=1$ on the nose) but is the correct notion of “sameness” in category theory.

The Diagonal (Constant) Functor

Definition: Diagonal Functor

For categories $\mathcal{A}$ and $\mathcal{I}$, the diagonal functor $\Delta :\mathcal{A}\to [\mathcal{I},\mathcal{A}]$ sends each $A\in \mathcal{A}$ to the constant functor at $A$ (mapping every object of $\mathcal{I}$ to $A$ and every map to $1_A$). Used to define cones and limits.

Connections

• A natural transformation (Natural Transformations) is a “map between functors” — making functors into objects of a category.
• Functor categories $[\mathcal{A},\mathcal{B}]$ (Functor Categories) are categories whose objects are functors.
• Representable functors (Representable Functors) are functors $\mathcal{A}\to \mathbf{Set}$ isomorphic to a hom-functor.
• Adjoints (Adjoint Functors) are pairs of functors between categories with a special natural bijection.

See Also

• Categories — What functors map between
• Natural Transformations — Maps between functors
• Functor Categories — Categories of functors
• Representable Functors — Functors isomorphic to hom-functors