Njia za utafitiCategory TheoryImepangwa

∞-categories

Optional research direction: (∞,1)-categories as quasi-categories and other models, ∞-groupoids and the homotopy hypothesis, limits, adjunctions and the Yoneda lemma in the ∞-setting, and applications in homotopy theory and algebraic geometry.

math.cat.infinity-categories

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Higher categories Algebraic topology

Masharti ya awali ya lazima 54 (hadi chini kabisa)

Utajifunza

Define simplicial sets and quasi-categories, and recognise nerves of categories and Kan complexes as examples.
Explain why composition in a quasi-category is defined only up to a contractible space of choices.
Survey the models of (∞,1)-categories (quasi-categories, complete Segal spaces, simplicial categories) and how they are compared.

Fasili

A simplicial set is a functor Δᵒᵖ → Set.
A quasi-category is a simplicial set in which every inner horn Λᵏ[n] → X with 0 < k < n extends to the simplex Δ[n].
A Kan complex is a simplicial set in which every horn extends; Kan complexes model ∞-groupoids.

Nadharia

The nerve of a category is a quasi-category, and a simplicial set is isomorphic to the nerve of a category if and only if its inner horns have unique fillers.
Joyal and Lurie developed category theory inside quasi-categories, including limits, adjunctions and a Yoneda lemma.
The main models of (∞,1)-categories are related by Quillen equivalences of model categories (Bergner, Joyal–Tierney, Lurie).

Mawazo potofu ya kawaida

Thinking ∞-categories are just categories with more arrows; their composition and coherence are homotopical.
Treating the homotopy hypothesis as a single theorem in every formulation; it is a guiding principle, established for particular definitions.

∞-categories

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Utajifunza

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Nadharia

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