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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Homotopy groups
Basic facts
Theorems
A simplicial weak equivalence is a weak equivalence in the standard homotopy theory of simplicial sets, hence with respect to the classical model structure on simplicial sets.
A morphism between simplicial sets is a simplicial weak equivalence if any of the following equivalent conditions is satisfied (where denotes Kan fibrant replacement):
has the right homotopy lifting property with respect to .
induces isomorphisms on simplicial homotopy groups, i.e., induces an isomorphism on and all homotopy groups for any choice of basepoints.
is a simplicial homotopy equivalence for every Kan complex (where denotes the hom-functor).
has the right homotopy lifting property with respect to (allowing subdivisions for homotopies also).
belongs to the class of weak equivalences in the classical model structure on simplicial sets, whose cofibrations are monomorphisms and fibrant objects are Kan complexes.
The morphism is a composition of a trivial cofibration and a trivial fibration on the classical model structure on simplicial sets, both of which are defined using lifting properties.
Applying the category of elements functor produces a Thomason weak equivalence of categories. The class of Thomason weak equivalences forms the smallest basic localizer, i.e., the smallest class of functors between small categories that contains identities, is closed under retracts and the 2-out-of-3 property, contains all functors for which the category A has a terminal object, and is locally determined: if and are functors, with , and for any the induced functor of comma categories is a Thomason weak equivalence, then so is .
Last revised on July 10, 2021 at 19:55:09. See the history of this page for a list of all contributions to it.