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 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 — a functor — 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 , which is
(in particular cofibrantly generated),
(induced under the simplicial nerve by the canonical enrichment over itself),
The model structure 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 is the simplicial nerve with values in the category sSet of simplicial sets.
One readily checks that:
With the canonical model structure on (from Prop. ) and the classical model structure on simplicial sets, (?)NerveAdjunction is a Quillen adjunction
In fact:
is the transferred model structure obtained from under (1).
canonical model structure on
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: -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 (a model structure for (2,1)-sheaves):
A model structure on 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.