model category, model -category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of -categories
Model structures
for -groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant -groupoids
for rational -groupoids
for rational equivariant -groupoids
for -groupoids
for -groups
for -algebras
general -algebras
specific -algebras
for stable/spectrum objects
for -categories
for stable -categories
for -operads
for -categories
for -sheaves / -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 — a functor — 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 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 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 is the nerve of groupoids with values in sSet.
With the natural model structure on and the standard model structure on simplicial sets this is a Quillen adjunction
and is the transferred model structure obtained from under this adjunction.
canonical model structure on
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 (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.