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
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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 a distinguished class of morphisms (called ‘weak equivalences’) satisfying the following conditions:
Every identity map is a weak equivalence.
It has the 2-out-of-6-property: if morphisms $h \circ g$ and $g \circ f$ are weak equivalences, then so are $f$, $g$, $h$ and $h \circ g \circ f$.
If the first condition in Def. 2 holds, then 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 page 23 of
with the main development of the concept starting in subsection 33 on page 96.
Last revised on March 15, 2020 at 11:22:18. See the history of this page for a list of all contributions to it.