homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
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equivalences in/of $(\infty,1)$-categories
A homotopical category is a structure used in homotopy theory, related to but more flexible than a model category.
A homotopical category is a category with weak equivalences where on top of the 2-out-of-3-property the morphisms satisfy the 2-out-of-6-property:
The 2-out-of-6-property implies the 2-out-of-3 property, hence every homotopical category is a category with weak equivalences.
Every model category yields a homotopical category.
A functor $F : C \to D$ between homotopical categories which preserves weak equivalences is a homotopical functor.
Every homotopical category $C$ “presents” or “models” an (infinity,1)-category $L C$, a simplicially enriched category called the simplicial localization of $C$, which is in some sense the universal solution to inverting the weak equivalence up to higher categorical morphisms.
category with a calculus of fractions
This definition is in paragraph 33 of