Questions: Equivalence of Categories

Question 1

A functor F: C → D is an equivalence of categories if and only if it satisfies which three properties?

• A. Full, faithful, and surjective on objects
• B. Full, faithful, and essentially surjective
• C. Injective on objects, surjective on morphisms, and faithful
• D. Bijective on objects and morphisms

Answer: (B)

Essentially surjective means every object of D is *isomorphic* to some F(c) — not necessarily equal. Surjective on objects would be too strict (closer to isomorphism). All three conditions are necessary: faithful ensures F reflects structural distinctions, full ensures no morphisms are missed, and essentially surjective ensures every object of D is 'reached' up to isomorphism.

Question 2

If two categories are equivalent, they is expected to also be isomorphic as categories.

• T. True
• F. False

Answer: False

Equivalence is strictly weaker than isomorphism. Isomorphism requires GF = Id_C and FG = Id_D exactly (functors are inverses on the nose). Equivalence only requires natural isomorphisms GF ≅ Id_C and FG ≅ Id_D, so objects may be sent to isomorphic rather than identical objects. The inclusion of a skeleton into FinSet is an equivalence but not an isomorphism.

Question 3

What distinguishes 'essentially surjective' from 'surjective on objects,' and why does equivalence use the weaker condition?

Model answer: Essentially surjective means every object d in D is isomorphic to F(c) for some c in C, while surjective on objects means every d equals F(c) for some c. Category theory treats isomorphic objects as 'the same,' so requiring exact equality would impose set-theoretic rigidity incompatible with categorical reasoning.

Category theory operates 'up to isomorphism' — objects are defined by their relationships to other objects, not as labeled points. Insisting on equality of objects would ignore the categorical notion of sameness. Essentially surjective captures the correct notion: if D contains an object isomorphic to something in the image of F, then categorically speaking, F 'covers' that object.