∞-category
Overview
Categories where morphisms between morphisms exist up to higher homotopy. Joyal introduced quasi-categories; Lurie developed the theory in Higher Topos Theory and Higher Algebra.
Era: 2000s
Proposed at: Cambridge, USA (MIT)
Formal structure
Domain: Simplicial set S; n-simplices; morphisms = 1-simplices
Quasi-category: inner horn fillers ∃; Hom_C(x,y) = (∞-groupoid)
Constraints: Composition defined up to contractible choice; ∞-limits, ∞-adjunctions
Objective: Stable ∞-categories; derived algebraic geometry; factorization systems