The forgetful functor U: Grp → Set sends each group (G, ·) to its underlying set G and each group homomorphism to the same function. Which functor law does this illustrate most directly?
AIt reverses the direction of morphisms, showing contravariance
BIt sends identity morphisms to identity morphisms and preserves composition, satisfying the functoriality axioms
CIt creates new morphisms in Set that did not exist in Grp
DIt requires the groups to be isomorphic to their image sets
The forgetful functor preserves the two defining laws: F(id_G) = id_{U(G)} (identities go to identities) and U(h∘g) = U(h)∘U(g) (composition is preserved). It 'forgets' the group structure but does not alter the morphisms themselves — a homomorphism is also a function, so no new morphisms are created. The functor is covariant (arrows go in the same direction), not contravariant.
Question 2 True / False
A contravariant functor from C to D is the same as a covariant functor from C to D^op, where D^op is D with most morphisms reversed.
TTrue
FFalse
Answer: False
Almost — but the direction is slightly off. A contravariant functor F: C → D is equivalently described as a covariant functor F: C^op → D (from the opposite of C to D), or equivalently C → D^op. The key point is that contravariance means arrows in C get reversed when mapped into D, which is captured by saying F is covariant on the opposite category C^op. This reframing allows all general theorems about covariant functors to apply automatically to contravariant ones.
Question 3 Short Answer
What distinguishes a functor from a mere object-level mapping between categories — that is, what additional structure must a functor preserve?
Think about your answer, then reveal below.
Model answer: A functor must also act on morphisms: for every morphism f: A → B in C, it must produce a morphism F(f): F(A) → F(B) in D. Moreover, it must preserve composition (F(g∘f) = F(g)∘F(f)) and identity morphisms (F(id_A) = id_{F(A)}).
A mapping that only assigns objects to objects ignores the relational structure that makes a category meaningful — the morphisms. Functoriality laws ensure the mapping is coherent with the categorical structure: it respects how morphisms compose and how identity morphisms behave. Without these laws, the mapping would be a function between the object-classes of two categories but would carry no structural information.