First-year graduateCategory TheoryLesson available

Enriched categories

Categories enriched in a monoidal category V, with hom-objects in place of hom-sets: additive categories as Ab-enriched, strict 2-categories as Cat-enriched, Lawvere's metric spaces as [0, ∞]-enriched, and enriched functors and natural transformations.

math.cat.enriched-categories

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Monoidal categories

Required prerequisites, all the way down: 21

You will learn to

Define V-enriched categories, functors and natural transformations.
Show that Lawvere metric spaces are categories enriched in ([0, ∞], ≥, +, 0), with the triangle inequality as composition.
Explain why additive categories are Ab-enriched and strict 2-categories are Cat-enriched.

Definition

For a monoidal category (V, ⊗, I), a V-category has hom-objects C(A, B) in V, composition morphisms C(B, C) ⊗ C(A, B) → C(A, C) and identities I → C(A, A), satisfying associativity and unit axioms.
A Lawvere metric space is a set with distances d(x, y) ∈ [0, ∞] such that d(x, x) = 0 and d(x, z) ≤ d(x, y) + d(y, z); it is exactly a category enriched in ([0, ∞], ≥, +, 0).

Theorem

Locally small categories are exactly the categories enriched in (Set, ×, 1).
Preorders are exactly the categories enriched in the two-element poset of truth values with conjunction.
Ab-enriched categories with one object are exactly rings.

Common misconceptions

Thinking enrichment adds arbitrary extra structure to hom-sets; composition must itself be a morphism of V.
Assuming Lawvere metric spaces are symmetric; enrichment does not force d(x, y) = d(y, x).

Optional discoveries

Higher categories Strict 2-categories are categories enriched in Cat; weakening their laws gives bicategories.