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
The canonical model structure on the 1-category Grpd of groupoids (with functors between them) is a presentation of the (2,1)-category of groupoids, functors and natural isomorphisms.
This is one flavor of the various canonical model structures on classes of categories and higher categories.
Let Grpd be the 1-category of small groupoids with functors between them. Say that a morphism in $Grpd$ — a functor $f \colon C \longrightarrow D$ — is:
a weak equivalence iff it is an equivalence of categories hence a weak homotopy equivalence of groupoids,
a fibration iff it is an isofibration,
a cofibration iff it is an isocofibration, hence injective on objects.
Equipped with the classes from Def. Grpd is a model category $Grpd_{can}$, which is
(in particular cofibrantly generated),
(induced under the simplicial nerve by the canonical enrichment over itself),
The model structure $Grpd_{nat}$ is the restriction of the canonical model structure on Cat from categories to groupoids.
See at canonical model structure for more.
Consider the pair of adjoint functors
where $N$ is the simplicial nerve with values in the category sSet of simplicial sets.
One readily checks that:
With the canonical model structure on $Grpd$ (from Prop. ) and the classical model structure on simplicial sets, (?)NerveAdjunction is a Quillen adjunction
In fact:
$Grpd_{can}$ is the transferred model structure obtained from $sSet_{Qu}$ under (1).
canonical model structure on $Grpd$
Some aspects (like the pullback stability of fibrations of groupoids in its prop. 2.8) appeared in
The existence of the model structure is stated (without proof) in:
and (by referencing Anderson, still without proof) in:
Proofs are spelled out in:
André Joyal, Myles Tierney, Thm. 2 (p. 221) Strong stacks and classifying spaces, in: Category Theory Lecture Notes in Mathematics 1488, Springer (1991) 213-236 [doi:10.1007/BFb0084222]
(in the generality of internal groupoids in a topos, hence of stacks)
Neil Strickland, §6.1 of: $K(n)$-local duality for finite groups and groupoids , Topology 39 4 (2000) [arXiv:math/0011109, doi:10.1016/S0040-9383(99)00031-2]
The model structure on functors with values in $Grpd_{nat}$ (a model structure for (2,1)-sheaves):
A model structure on $Cat$ but localized such as to make the fibrant objects be groupoids:
Last revised on November 2, 2023 at 07:42:11. See the history of this page for a list of all contributions to it.