A mathematician observes that in her category C, every hom-collection hom(A,B) is a real vector space, and composition hom(B,C) × hom(A,B) → hom(A,C) is bilinear (linear in each argument separately). The correct description of C is:
AAn ordinary category with extra notation — the vector space structure is incidental
BA category enriched over Vect, the monoidal category of real vector spaces
CA 2-category, since the linear structure introduces morphisms between morphisms
DA monoidal category, since the hom-sets carry a tensor product structure
The defining feature of enrichment is that hom-sets become hom-objects in a monoidal category V, and composition is expressed as a morphism in V. Here, hom-objects are vector spaces (objects in Vect) and composition is bilinear — precisely a morphism in Vect via the tensor product hom(B,C) ⊗ hom(A,B) → hom(A,C). This is Vect-enrichment (stronger than Ab-enrichment, which requires only abelian group structure). A 2-category is enriched over Cat, not Vect, and monoidal structure refers to a tensor product on the category's objects, not on its hom-sets.
Question 2 Multiple Choice
In the Lawvere metric space construction, a set X with a distance function d(A,B) ≥ 0 satisfying d(A,A) = 0 and d(A,C) ≤ d(A,B) + d(B,C) is viewed as a category enriched over ([0,∞], +, 0). Which pairing makes this interpretation work?
BObjects = points, hom-object hom(A,B) = d(A,B); the triangle inequality is exactly the composition axiom
CObjects = points, morphisms = paths; the distance is the cost of following a morphism
DThe metric space is monoidal, not enriched, because addition is commutative
In a category enriched over V = ([0,∞], +, 0), each hom-object hom(A,B) is a non-negative real number. Setting hom(A,B) = d(A,B), the composition axiom becomes: d(B,C) + d(A,B) ≥ d(A,C), exactly the triangle inequality. The identity axiom gives d(A,A) ≤ 0, hence d(A,A) = 0. Every metric space is thus an enriched category — a striking unification of geometry and category theory that shows these familiar structures are instances of the same abstract framework.
Question 3 True / False
In an enriched category, it is the objects (not the hom-sets) that gain additional structure from the ambient monoidal category V.
TTrue
FFalse
Answer: False
This is the central misconception about enrichment. The objects of an enriched category remain unstructured (or carry their own separate structure unrelated to V). What enrichment changes is the hom-collections: instead of plain sets, each hom(A,B) becomes an object in V. Composition is no longer a function between sets — it is a morphism in V: hom(B,C) ⊗ hom(A,B) → hom(A,C). The enrichment lives entirely in the morphism-structure, not in the object-structure.
Question 4 True / False
When V = Set with cartesian product as tensor product, a V-enriched category is exactly an ordinary category.
TTrue
FFalse
Answer: True
Set-enrichment is the base case. Hom-objects in Set are just sets, the tensor product (cartesian product) of hom-sets is the ordinary set-product, and composition is an ordinary function — reducing to the familiar definition of a category. All the enriched axioms collapse to the standard unit and associativity laws. This confirms that ordinary categories are 'Set-enriched categories,' and every choice of non-trivial V (Ab, Vect, Cat, metric spaces, etc.) generalizes this foundation.
Question 5 Short Answer
What is the key structural difference between an ordinary category and a V-enriched category, and why does the choice of V matter for composition?
Think about your answer, then reveal below.
Model answer: In an ordinary category, hom(A,B) is a plain set and composition is a function of sets. In a V-enriched category, hom(A,B) is an object in a monoidal category V, and composition is a morphism in V: hom(B,C) ⊗ hom(A,B) → hom(A,C), where ⊗ is V's tensor product. The choice of V determines the structure of the morphism-collections and how composition behaves: V = Ab gives bilinear composition (preadditive categories), V = [0,∞] gives the triangle inequality (metric spaces), V = Cat gives 2-categories. Different choices of V thus simultaneously generalize many distinct mathematical structures under a single framework.
The identity is also enriched: instead of selecting an element id_A from hom(A,A), it is a morphism I → hom(A,A) from the unit object I of V. All axioms (associativity, unitality) are expressed as commutative diagrams in V using V's associator and unitors. This means enrichment is a genuine generalization, not just notation — it forces us to think of composition and identity as internal operations in V rather than as set-level functions.