Mwaka wa kwanza wa uzamiliCategory TheoryImepangwa
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
Somo la mada hii bado halijaandikwa. Muhtasari ulio hapa chini ni sehemu ya ramani ya mtaala na haujakaguliwa.
Inajengwa juu ya
Masharti ya awali ya lazima 21 (hadi chini kabisa)
Utajifunza
- 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.
Fasili
- 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).
Nadharia
- 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.
Mawazo potofu ya kawaida
- 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).
Ugunduzi wa hiari
- Higher categories Strict 2-categories are categories enriched in Cat; weakening their laws gives bicategories.
Usomaji zaidi
- G. M. Kelly, Basic Concepts of Enriched Category Theory (1982)
- F. William Lawvere, Metric spaces, generalized logic, and closed categories (1973)