model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
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
Write
$Groups(sSet)$ for the category of simplicial groups
$Groups(sSet)_{proj}$ for the model structure on simplicial groups whose weak equivalences and fibrations are those of the underlying morphisms in the classical model structure on simplicial sets (hence the underlying simplicial weak equivalences and Kan fibrations, respectively);
$sSet_{\geq 1}$ for the category of reduced simplicial sets;
$(sSet_{\geq 1})$ for the model structure on reduced simplicial sets whose weak equivalences and cofibrations are those of the underlying morphisms in the classical model structure on simplicial sets.
(Quillen equivalence between simplicial groups and reduced simplicial sets)
The
simplicial classifying space-construction $\overline{W}(-)$ is the right adjoint
simplicial loop space-construction $\Omega(-)$ is the left adjoint
in a Quillen equivalence between the projective model structure on simplicial groups and the injective model structure on reduced simplicial sets.
(e.g. Goerss & Jardine 09, V Prop. 6.3)
Created on July 4, 2021 at 11:47:36. See the history of this page for a list of all contributions to it.