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 of groupoids and functors is a presentation of the (2,1)-category of groupoids, functors and natural isomorphisms.
This is one flavor of the various natural model structures on categories and higher categories.
Let Grpd be the 1-category of small groupoids and functors between them. Say thata morphism in $Grpd$ — a functor $f \colon C \longrightarrow D$ — is
a weak equivalence if it is an equivalence of categories;
a fibration if it is an isofibration;
a cofibration if it is injective on objects.
Equipped with this structure $Grpd_{nat}$ is a model category which is
a simplicial model category with respect to the natural sSet-enriched category structure induced by the canonical enrichment over itself, under the nerve.
This is originally due to (Anderson 78) and (Bousfield 89). A detailed discussion is in (Strickland 00, section 6). In the context of the model structure for (2,1)-sheaves it appears as (Hollander 01, theorem 2.1).
The model structure $Grpd_{nat}$ is the restriction of the canonical model structure on Cat from categories to groupoids.
See natural model structure for more.
Let
be the pair of adjoint functors, where $N$ is the nerve of groupoids with values in sSet.
With the natural model structure on $Grpd$ and the standard model structure on simplicial sets this is a Quillen adjunction
and $Grpd_{nat}$ is the transferred model structure obtained from $sSet_{Quillen}$ under this adjunction.
canonical model structure on $Grpd$
Some aspects (like the pullback stability of fibrations of groupoids in its prop. 2.8) appeared in
Volume 15, Issue 1, May 1970, Pages 103-132
The full model category structure appears originally in
and
A detailed description is in section 6 of
The model structure on functors with values in $Grpd_{nat}$ (a model structure for (2,1)-sheaves) is discussed in
Last revised on May 1, 2023 at 09:37:20. See the history of this page for a list of all contributions to it.