nLab model structure on sSet-enriched presheaves



Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos

Enriched category theory



The model structure sPSh(C)sPSh(C) on sSet-enriched presheaves is supposed to be a presentation of the (∞,1)-category of (∞,1)-sheaves on an sSet-site CC.

It generalizes the model structure on simplicial presheaves which is the special case obtained when CC 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.

Notation and conventions

Regard the closed monoidal category sSet as a simplicially enriched category in the canonical way.

For CC any sSetsSet-category, write sk 1Csk_1 C for its underlying ordinary category.

Write [C op,sSet][C^{op}, sSet] for the enriched functor category.

Hence the ordinary category sk 1[C op,sSet]sk_1 [C^{op}, sSet] has as objects enriched functors C opsSetC^{op} \to sSet and a morphism f:ABf : A \to B in sk 1[C op,sSet]sk_1 [C^{op}, sSet] is a natural transformation given by a collection of morphisms f c:A(c)B(c)f_c : A(c) \to B(c) in sSet, for each object cCc \in C.


(global model structure)

Let CC be a simplicially enriched category.

The global projective model structure sk 1[C op,sSet] projsk_1[C^{op},sSet]_{proj} on sk 1[C op,sSet]sk_1[C^{op}, sSet]


The global projective model structure on sk 1[C op,sSet]sk_1 [C^{op}, sSet] makes the sSetsSet-category [C op,sSet][C^{op}, sSet] a combinatorial simplicial model category.


For CC an sSet-site, the local projective model structure on sk 1[C op,sSet]sk_1 [C^{op}, sSet] is the left Bousfield localization of sk 1[C op,sSet] projsk_1 [C^{op}, sSet]_{proj} at…

This appears on (ToënVezzosi, page 14).


Over an unenriched site

It seems that the claim is that, indeed, in the special case that CC happens to be an ordinary category, the model structure on sPSh(C) proj llocsPSh(C)_{proj}^{l loc} reproduces the projective local model structure on simplicial presheaves.

Presentation of (,1)(\infty,1)-toposes


For an sSet-site CC regarded as an (∞,1)-site, the local model structure on [C op,sSet][C^{op}, sSet] is a presentation of the (∞,1)-category of (∞,1)-sheaves on CC, in that there is an equivalence of (∞,1)-categories

([C op,sSet] loc) Sh (,1)(C). ([C^{op}, sSet]_{loc})^\circ \simeq Sh_{(\infty,1)}(C) \,.

This is Lurie, prop., remark


Derived geometry

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 sSetsSet-presheaves serves to model contexts of derived geometry. For instance over the etale (∞,1)-site.



The theory of model structures on sSetsSet-enriched presheaf categories was developed in

  • Bertrand Toën, Gabriele Vezzosi, Segal topoi and stacks over Segal categories Proceedings of the Program Stacks, Intersection theory and Non-abelian Hodge Theory , MSRI, Berkeley, January-May

    2002 (arXiv:0212330)

The relation to intrinsically defined (∞,1)-topos theory is around remark 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.