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
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
The Joyal- model structure on simplicial sheaves over a site is a model category presentation of the hypercomplete (∞,1)-topos over that site.
It is Quillen equivalent to the Jardine-local model structure on simplicial presheaves.
There is also a Čech model structure on simplicial sheaves (see there) modelling not the hypercompletion but just the topological localization of the (∞,1)-category of (∞,1)-presheaves.
Let $C$ be a small site. Write $Sh(C)$ for the category of sheaves on $C$ and $Sh(C)^{\Delta^{op}}$ for the category of simplicial objects in $Sh(C)$: the category of simplicial sheaves over $C$.
There is a left proper simplicially enriched model category structure on $Sh(C)^{\Delta^{op}}$ such that
cofibrations are precisely the objectwise cofibrations (monomorphisms) of simplicial sets;
weak equivalences are the local weak equivalences of the underlying simplicial presheaves as defined at model structure on simplicial presheaves.
This is (Jardine, theorem 5).
Call this the local injective model structure on simplicial sheaves.
The geometric embedding of a category of sheaves into its category of presheaves
with $L$ given by sheafification extends to a Quillen equivalence
between the above local model structure on simplicial sheaves and the injective hyperlocal Jardine-model structure on simplicial presheaves.
This is (Jardine, theorem 5).
The simplicial combinatorial model category $Sh(C)^{\Delta^{op}}_{loc}$ is a presentation for the hypercompletion $\hat Sh_{(\infty,1)}(c)$ of the (∞,1)-category of (∞,1)-sheaves on $C$:
The proof is spelled out at hypercomplete (∞,1)-topos.
For $D$ a dense sub-site of $C$ we have an equivalence of (∞,1)-categories
By the comparison lemma at dense sub-site we have already an equivalence of categories
This implies the claim with the above proposition.
The local model structure on simplicial sheaves was proposed in
This is, with BrownAHT, among the first proposals for models for ∞-stacks which eventually came to be used in the theory of (∞,1)-toposes.
Discussion of the model structure:
Jardine’s lectures
discuss the Quillen equivalence between the model structure on simplicial sheaves and the model structure on simplicial presheaves.
Wendt discusses “the construction of classifying spaces of fibre sequences in model categories of simplicial sheaves” in
Last revised on June 19, 2022 at 13:55:02. See the history of this page for a list of all contributions to it.