研究生一年级Category Theory计划中
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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前置知识
21 项必需的前置知识（一直追溯到底）
你将学会
- 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.
定义
- 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).
定理
- 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.
常见误解
- 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).
可选发现
- Higher categories Strict 2-categories are categories enriched in Cat; weakening their laws gives bicategories.
延伸阅读
- G. M. Kelly, Basic Concepts of Enriched Category Theory (1982)
- F. William Lawvere, Metric spaces, generalized logic, and closed categories (1973)