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
Paths and cylinders
Homotopy groups
Basic facts
Theorems
In simplicial homotopy theory there is a standard construction (Kan 58), traditionally denoted $G$, for the simplicial analog of the homotopy type of the loop space $\Omega X$ of a connected topological space, now incarnated as a simplicial group: It is the left adjoint in an adjunction
between pointed simplicial sets and simplicial groups, whose right adjoint is the simplicial classifying space construction.
This adjunction is a Quillen adjunction for the Kan–Quillen model structure? on pointed simplicial sets and its transferred model structure on simplicial groups.
When restricted to reduced simplicial sets, this Quillen adjunction becomes a Quillen equivalence.
The generalization of this construction to non-reduced simplicial sets and simplicial groupoids is the Dwyer-Kan loop groupoid functor.
For a more complete list of references, see the article simplicial delooping functor.
Daniel Kan, On homotopy theory and c.s.s. groups, Ann. of Math. 68 (1958), 38-53 (jstor:1970042) (Sections 10–11.)
Peter May, chapter VI of Simplicial objects in algebraic topology, Van Nostrand, Princeton (1967) (ISBN:9780226511818, djvu)
More details and relation to décalage:
Last revised on June 5, 2021 at 19:28:09. See the history of this page for a list of all contributions to it.