Functors

Maps between categories: objects and arrows carried across, composites intact, sometimes with the arrows turned around.

The idea

A functor is a map between categories. It sends objects to objects and arrows to arrows, and it respects the only structure a category carries: composition and identities.

Definition (Functor).

A functor $F \colon \mathcal{C} \to \mathcal{D}$ assigns to each object $A$ of $\mathcal{C}$ an object $F(A)$ of $\mathcal{D}$, and to each arrow $f \colon A \to A'$ an arrow $F(f) \colon F(A) \to F(A')$, subject to two axioms: $F(g \circ f) = F(g) \circ F(f), \qquad F(1_{A}) = 1_{F(A)},$ the first for every composable pair $f \colon A \to A'$, $g \colon A' \to A''$ and the second for every object $A$.

Composites go to composites and identities go to identities.

One example shows the axioms at work. Send a set $X$ to its set of subsets $P(X)$, and a function $k \colon X \to Y$ to the direct-image map carrying a subset $S$ to $\{k(s) : s \in S\}$. This is a functor from $\mathbf{Set}$ to itself: the image of a subset under a composite is the image of its image, and the image under an identity is the subset itself.

Some constructions reverse direction. Send the same $k$ to the preimage map instead and it runs from $P(Y)$ to $P(X)$, so a composite can only be undone outermost first and the factors compose in the opposite order. Such a construction is contravariant, always for the same reason: its value is built out of maps out of the object, so the only way to use an arrow is to precompose with it.

Ways to work on it

• Walkthrough. The functor axioms, with examples from power sets, dual spaces, and one-object categories.
• Proof. Why every functor preserves isomorphisms, and why that makes functorial constructions invariants.
• Practice. Call variance correctly, decide which axiom a proposed assignment breaks, and compute images and preimages.
• Hardest. Count the functors between two posets, and test whether a familiar group construction is functorial.

Not sure where to start? Take the ten-question placement test.