Functors: Maps Between Categories

Whenever we conceive of a collection of objects, we should decide at the same time how those objects are to be compared — Simmons states this as the operating principle of the subject.¹ Categories are themselves mathematical objects, so the principle applies to them, and the comparison gadget is called a functor. A functor translates one category into another (objects to objects, arrows to arrows) in a way that respects the only structure a category has: composition and identities. The standard constructions of mathematics (take the underlying set, form the free group, form the fundamental group, take the dual space) are all functors, and seeing them as such is one of the first practical uses of the subject.

The definition

The two axioms supply what the following bookkeeping fact needs: from any string $A_0\stackrel{f_1}{\to }\cdots \stackrel{f_n}{\to }{A}_n$ in $\mathcal{A}$, all ways of applying $F$ and composing yield the same arrow $F(A_0)\to F(A_n)$.² A functor is a picture of the source category drawn inside the target category, with all commuting diagrams preserved: if $gf=h$ in $\mathcal{A}$, then $F(g)F(f)=F(h)$ in $\mathcal{B}$. In particular, functors preserve isomorphisms: if $gf=1$ and $fg=1$ then $F(g)F(f)=1$ and $F(f)F(g)=1$, so $A\cong A^{\prime }$ implies $F(A)\cong F(A^{\prime })$.

Functors compose in the evident way (apply one, then the other), and every category has an identity functor $1_{\mathcal{A}}$. So categories and functors themselves form a category, written $\mathbf{CAT}$ (with $\mathbf{Cat}$ reserved for the category of small categories, a distinction taken up in the size lesson).³ The pattern structures and their structure-preserving maps form a category applies one level up.

Forgetful and free functors

The easiest functors discard structure.

• $U:\mathbf{Grp}\to \mathbf{Set}$ sends a group to its underlying set and a homomorphism to itself-as-function. It forgets the group structure.
• $\mathbf{Ring}\to \mathbf{Ab}$ forgets multiplication, keeping the additive group; $\mathbf{Ring}\to \mathbf{Mon}$ forgets addition, keeping the multiplicative monoid. Forgetting can be partial.
• $\mathbf{Ab}\to \mathbf{Grp}$ changes nothing at all: it forgets the property of being abelian rather than any structure.

Forgetful has no precise definition; it is a name for a family of examples, but the family is coherent enough to reason about.⁴ Simmons tabulates a dozen of these; the pattern is always a functor to a category of less-furnished objects that is the identity on underlying functions.

Free functors run the other way, adding structure as freely as possible.

• $F:\mathbf{Set}\to \mathbf{Grp}$ sends $S$ to the free group on $S$: words like $x^{-4}y{x}^{2}z{y}^{-3}$ in the elements of $S$, multiplied by concatenation, with only the cancellation forced by the group axioms. A function $f:S\to S^{\prime }$ induces $F(f)$ by renaming letters.
• $F:\mathbf{Set}\to {\mathbf{Vect}}_k$ sends $S$ to the vector space with basis $S$: formal finite $k$-linear combinations ${\sum }_s{\lambda }_{s}s$.
• $F:\mathbf{Set}\to \mathbf{CRing}$ sends $S$ to the polynomial ring $\mathbb{Z}[x_s:s\in S]$; a polynomial is a formal expression in the generators, so this is the free commutative ring.
• $F:\mathbf{Set}\to \mathbf{Mon}$ sends $S$ to the Kleene closure $S^{\ast }$, the free monoid on $S$.

The pairing is not an accident. Free and forgetful functors are partners in a precise sense — each free functor is left adjoint to its forgetful companion — made precise in the adjunctions module.⁵

Historically, some of the first functors came from algebraic topology, where the strategy is to convert a hard question about spaces into a tractable one about algebra. The fundamental group is a functor ${\pi }_1:{\mathbf{Top}}_{\ast }\to \mathbf{Grp}$ from based spaces to groups: a based continuous map $f$ induces a homomorphism ${\pi }_1(f)=f_{\ast }$, and functoriality, $(g\circ f{)}_{\ast }=g_{\ast }\circ f_{\ast }$, is what lets topological conclusions be drawn from algebraic computations.

Functors between small categories

When the source and target are one-object or thin categories, functoriality specializes to familiar notions.

Expand the $G$-set row. A functor $F:G\to \mathbf{Set}$ from a one-object category picks a single set $S=F(\ast )$ and, for each $g\in G$, a function $F(g):S\to S$; functoriality says $F(g^{\prime }g)=F(g^{\prime })\circ F(g)$ and $F(1)=1_S$. Writing $g\cdot s$ for $F(g)(s)$, these are precisely the axioms of a left action.⁶ Representation theory, from this angle, is the study of functors out of a one-object category — a first instance of a definition that yields new theorems once recast functorially.

Contravariance

Some natural operations reverse arrows. Simmons's dichotomy is sharp: a functor either preserves the direction of every arrow (covariant) or reverses the direction of every arrow (contravariant) — never a mixture.¹

Phrasing contravariance through the opposite category keeps every functor literally covariant and pushes the reversal into the source; this convention simplifies later work. The main examples:

• Dual spaces. Fix a field $k$. Sending $V\mapsto V^{\ast }=\mathrm{Hom}(V,k)$ and $f\mapsto f^{\ast }$ (precomposition) is a contravariant functor from ${\mathbf{Vect}}_k$ to itself.
• Function rings. Sending a space $X$ to the ring $C(X)$ of continuous real-valued functions on $X$, and a continuous $f:X\to Y$ to the homomorphism $C(f):C(Y)\to C(X)$, $q\mapsto q\circ f$, is a contravariant functor from $\mathbf{Top}$ to $\mathbf{Ring}$.⁷
• Open sets. Sending a space $S$ to its poset of open sets $\mathcal{O}S$ and a continuous $\phi$ to the inverse-image map ${\phi }^{\leftarrow }$ is contravariant from $\mathbf{Top}$ to $\mathbf{Pos}$.

Contravariant functors into $\mathbf{Set}$ are important enough to be named: a presheaf on $\mathcal{A}$ is a functor ${\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$. The name comes from topology: a presheaf on the poset of open sets of a space assigns to each open $U$ a set $F(U)$ (say, the continuous functions on $U$) and to each inclusion $U\subseteq U^{\prime }$ a restriction map $F(U^{\prime })\to F(U)$. Presheaves become central in the Yoneda module.

The hom-functors

For a fixed object $K$ of any category $\mathcal{C}$, composition itself gives two functors into $\mathbf{Set}$, one of each variance:

On objects, $A\mapsto \mathcal{C}(K,A)$ and $A\mapsto \mathcal{C}(A,K)$. On an arrow $f:A\to B$, the covariant functor post-composes, $r\mapsto f\circ r$, and the contravariant functor pre-composes, $l\mapsto l\circ f$. The variance check is a two-line computation each way: $(g\circ f)\circ r$ associates as $g\circ (f\circ r)$, while $l\circ (g\circ f)$ associates as $(l\circ g)\circ f$ — post-composition preserves order, pre-composition reverses it.⁸ The dash $-$ is a placeholder for the argument. These hom-functors are central to the subject; the representables module is entirely about them.

Three powerset functors

Different functors can share an object assignment. On objects, send every set $A$ to its powerset $\mathcal{P}A$; there are three standard ways to extend this to arrows $f:A\to B$:⁹

• Direct image (covariant): $\exists (f)(X)=f[X]$, so $b\in \exists (f)(X)$ iff $b=f(a)$ for some $a\in X$.
• Inverse image (contravariant): $I(f)(Y)=f^{\leftarrow }(Y)$, so $a\in I(f)(Y)$ iff $f(a)\in Y$.
• Universal image (covariant): $\forall (f)(X)=(f[X^{\prime }]{)}^{\prime }$ (complement of the image of the complement), so $b\in \forall (f)(X)$ iff every $a$ with $f(a)=b$ lies in $X$.

In a more general setting $\exists$ and $\forall$ are precisely the categorical semantics of the two quantifiers, and the three functors line up into adjunctions $\exists ⊣I⊣\forall$. The functor equations to check are $\exists (g\circ f)=\exists (g)\circ \exists (f)$ and its two companions, with the middle one reversed: $I(g\circ f)=I(f)\circ I(g)$.

Faithful, full, essentially surjective

Functions between sets are classified by injectivity and surjectivity; functors admit a finer classification because they act on two levels, objects and arrows. The useful notions live at the hom-set level.

Faithfulness is a hom-set-by-hom-set condition, not global injectivity on arrows. A faithful functor can send two arrows with different sources to the same arrow: the inclusion of a two-object discrete category into $1$ collapses everything yet is vacuously faithful on each hom-set.¹⁰

The stock examples calibrate the definitions:

A subcategory $\mathcal{S}$ of $\mathcal{A}$ selects some objects and some arrows among them, closed under composition and identities; it is full when it keeps all arrows between selected objects, so a full subcategory is specified by its objects alone ($\mathbf{Ab}$ inside $\mathbf{Grp}$). Inclusion functors of subcategories are automatically faithful, and full exactly for full subcategories. One caution: the image of a functor need not be a subcategory, since composable image-arrows can have a composite that is not itself an image — $F(g)\circ F(f)$ exists downstairs even when $g$ and $f$ do not compose upstairs.¹⁰

Full-and-faithful functors are the categorical notion of embedding: they reflect isomorphism ($F(A)\cong F(A^{\prime })$ implies $A\cong A^{\prime }$), and adding essential surjectivity yields the functors that are equivalences of categories, the notion of the same category. Defining equivalence requires comparing functors to identity functors, which is what natural transformations do; the Yoneda embedding being full and faithful rests on the same vocabulary.