In the monoidal category (Vect_k, ⊗_k, k), what is the unit object I?
AThe zero vector space {0}
BThe ground field k itself, viewed as a one-dimensional vector space
CThe direct sum of all finite-dimensional spaces
DThe space of linear maps from k to k
The unit object in (Vect_k, ⊗_k, k) is the field k viewed as a one-dimensional vector space. For any vector space V, there are natural isomorphisms k ⊗_k V ≅ V ≅ V ⊗_k k, capturing the left and right unit laws. The zero space {0} is the unit for direct sum ⊕, a different monoidal structure on Vect.
Question 2 True / False
In a monoidal category, the tensor product ⊗ is generally the same as the categorical product (the object satisfying the universal property of products).
TTrue
FFalse
Answer: False
These are different constructions that happen to coincide in some categories but not others. In (Set, ×, {*}), the tensor product is the categorical product. But in (Vect_k, ⊗_k, k), the tensor product is not the categorical product — the categorical product in Vect is the direct product (direct sum for finite families), not the tensor product. A monoidal category's tensor product only needs to be a bifunctor with a unit and coherence isomorphisms, not a categorical product.
Question 3 Short Answer
What does Mac Lane's coherence theorem for monoidal categories allow you to do in practice, and why is it non-trivial?
Think about your answer, then reveal below.
Model answer: The coherence theorem guarantees that every diagram built from the associator and unitor natural isomorphisms commutes. In practice, this means you can treat the tensor product as if it were strictly associative and unital — you can drop parentheses and suppress unit objects without any calculation, because all possible reassociations give the same result.
The theorem is non-trivial because the associator α_{A,B,C}: A⊗(B⊗C) → (A⊗B)⊗C is only an isomorphism, not the identity. Without coherence, different sequences of reassociation might yield different morphisms. The pentagon and triangle axioms are exactly the conditions that prevent inconsistency; coherence then extends this to all diagrams, not just the basic ones.