Questions: Functor Categories

Question 1

In the functor category [C, D], what plays the role of morphisms?

• A. Functors from C to D
• B. Objects of D
• C. Natural transformations between functors F, G: C → D
• D. Morphisms of C composed with morphisms of D

Answer: (C)

In [C, D], the objects are functors F: C → D, and the morphisms between two such functors F and G are natural transformations η: F ⇒ G. Composition of morphisms in [C, D] is vertical composition of natural transformations, defined component-wise at each object of C.

Question 2

A diagram of shape J in a category C is a collection of objects and morphisms in C indexed by J — it is not necessarily a functor.

• T. True
• F. False

Answer: False

A diagram of shape J in C is precisely a functor J → C. The index category J specifies which objects and morphisms must appear, and the functor sends each object of J to an object of C and each morphism of J to a morphism of C, respecting composition. This is why limits and colimits can be treated as functors on functor categories.

Question 3

What is the significance of presheaf categories [C^op, Set], and why do they appear across mathematics and logic?

Model answer: A presheaf on C is a functor C^op → Set, assigning a set to each object of C contravariantly. Presheaf categories are important because every category C embeds fully and faithfully into [C^op, Set] via the Yoneda embedding, and because presheaves model variable sets over a base, appearing in sheaf theory, topos theory, and Kripke semantics for intuitionistic logic.

The Yoneda embedding sends each object c ∈ C to the representable presheaf Hom(−, c): C^op → Set. The embedding is full and faithful, meaning [C^op, Set] contains a faithful copy of C — any category can be studied by studying its presheaves. The topos-theoretic and logical applications follow from the fact that [C^op, Set] always has limits, colimits, and an internal logic.