on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
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see also algebraic topology
Introductions
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Paths and cylinders
Homotopy groups
Basic facts
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equivalences in/of $(\infty,1)$-categories
The model structure on presheaves of simplicial groupoids is one of the models for ∞-stack (∞,1)-toposes. It is a slight variant on the model structure on simplicial presheaves. (At that link more general information is collected).
For various applications it is useful to
use simplicial groupoids instead of Kan complexes or simplicial sets;
use a model structure on (pre)sheaves with values in simplicial groupoids instead of the model structure on simplicial presheaves (see there for motivation on why to consider these models in the first place).
An example is the discussion of principal infinity-bundles in section 3 of (JardineLuo)
Write $(G \dashv \bar W) : Grpd^\Delta \leftrightarrow sSet$ for the Quillen adjunction discussed at model structure on simplicial groupoids. This directly prolongs to an adjunction on presheaves
The transferred model structure along $\bar W$ on $[C^{op}, Grpd^\Delta]$ of the global injective model structure on simplicial presheaves exists on $[C^{op}, sSet, Grpd^\Delta]$: fibrations and weak equivalences are those that become global injective fibrations and weak equivalences, respectively, under $\bar W$.
This appears as (LBK, theorem 3.10).
A model structure on sheaves with values in simplicial groupoids is due to
Andre Joyal and Myles Tierney, Strong stacks and classifying spaces, in: Category theory (Como, 1990), volume 1488 of Lecture Notes in Math., pages 213–236. Springer, Berlin (1991) (web)
Andre Joyal and Myles Tierney, On the homotopy theory of sheaves of simplicial groupoids. Math. Proc. Cambridge Philos. Soc., 120(2):263–290, 1996.
A model structure on presheaves with values in simplicial groupoids is due to
A model structure on simplicial sheaves of groupoids is discussed in
Sjoerd Crans, Quillen closed model structure for sheaves, J. Pure Appl. Algebra 101 (1995), 35-57 (web)
Rick Jardine, Luo, Higher order principal bundles (web)