Functor Categories

Core Idea

Given categories C and D, the functor category [C, D] (also written D^C) has functors F: C → D as objects and natural transformations as morphisms. Composition of natural transformations is defined component-wise and satisfies all category axioms, making functors and natural transformations into a genuine category. Presheaf categories [C^op, Set] are particularly important in mathematics and logic, providing models for sheaf theory, topos theory, and Kripke semantics for intuitionistic logic.

How It's Best Learned

Verify that vertical composition of natural transformations (composing η: F ⇒ G and ε: G ⇒ H component-wise) is associative and has identity natural transformations. Then recognize that a diagram of shape J in a category C is simply a functor J → C, making limits and colimits into functors on functor categories.

Common Misconceptions

• The functor category [C, D] may be large (a proper class) even when C and D are small; size issues matter in foundations.
• Horizontal composition and vertical composition of natural transformations are different operations; both are needed for the full 2-categorical structure.

Explainer

You have seen that functors are structure-preserving maps between categories, and that natural transformations are maps between functors that respect the functorial structure. The key insight of functor categories is that functors and natural transformations themselves form a category — with functors as objects and natural transformations as morphisms.

Given two categories C and D, the functor category [C, D] (also written D^C) is defined as follows: its objects are all functors F: C → D, and for any two functors F, G: C → D, a morphism from F to G in [C, D] is a natural transformation η: F ⇒ G. Vertical composition is how morphisms compose in [C, D]: given η: F ⇒ G and ε: G ⇒ H, the composite (ε ∘ η): F ⇒ H is defined component-wise by (ε ∘ η)_c = ε_c ∘ η_c for each object c ∈ C. You can verify that this composite is again natural, and that composition is associative with identity natural transformations as units — all the category axioms hold.

One of the most useful reformulations in category theory is that a diagram of shape J in C is exactly a functor J → C. The index category J specifies the "shape" of the diagram — which nodes and arrows should appear. A functor from J to C picks out objects and morphisms in C of that shape. This means that asking whether a limit of a diagram exists is asking whether a certain functor has a terminal object in a comma category, and constructions like limits and colimits become functors on functor categories. This uniformity of language is what makes category theory so efficient.

Presheaf categories [C^op, Set] are particularly important. A presheaf on C is a contravariant functor from C to sets — you can think of it as assigning a "set of sections" to each object of C, with restriction maps going backward along morphisms. The Yoneda lemma (which you will study next) shows that every category C embeds fully and faithfully into its presheaf category via the Yoneda embedding c ↦ Hom(−, c): C^op → Set. This means [C^op, Set] contains a faithful copy of C, but is much richer — it always has all limits and colimits, and supports an internal logic. Presheaf categories provide models for sheaf theory, for variable sets in topos theory, and for Kripke frames in intuitionistic logic, making them one of the central constructions in modern mathematics.