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
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
The model structure on sSet-enriched presheaves is supposed to be a presentation of the (∞,1)-category of (∞,1)-sheaves on an sSet-site .
It generalizes the model structure on simplicial presheaves which is the special case obtained when happens to be just an ordinary category.
This means that in as far as the model structure on simplicial presheaves models ∞-stacks, the model structure on sSet-enriched categories model derived stacks.
The construction of the model structure on sSet-enriched categories closely follows the discussion of the model structure on simplicial presheaves, only that everything now takes place in enriched category theory.
Regard the closed monoidal category sSet as a simplicially enriched category in the canonical way.
For any -category, write for its underlying ordinary category.
Write for the enriched functor category.
Hence the ordinary category has as objects enriched functors and a morphism in is a natural transformation given by a collection of morphisms in sSet, for each object .
(global model structure)
Let be a simplicially enriched category.
The global projective model structure on
The global projective model structure on makes the -category a combinatorial simplicial model category.
For an sSet-site, the local projective model structure on is the left Bousfield localization of at…
This appears on (ToënVezzosi, page 14).
It seems that the claim is that, indeed, in the special case that happens to be an ordinary category, the model structure on reproduces the projective local model structure on simplicial presheaves.
For an sSet-site regarded as an (∞,1)-site, the local model structure on is a presentation of the (∞,1)-category of (∞,1)-sheaves on , in that there is an equivalence of (∞,1)-categories
This is Lurie, prop. 6.5.2.14, remark 6.5.2.15.
Where a topos or (∞,1)-topos over an ordinary site encodes higher geometry, over a genuine sSet-site one speaks of derived geometry. An ∞-stack on such a higher site is also called a derived stack.
Therefore the model structure on -presheaves serves to model contexts of derived geometry. For instance over the etale (∞,1)-site.
(…)
The theory of model structures on -enriched presheaf categories was developed in
2002 (arXiv:0212330)
The relation to intrinsically defined (∞,1)-topos theory is around remark 6.5.2.15 of
Last revised on March 2, 2020 at 12:05:32. See the history of this page for a list of all contributions to it.