homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
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see also algebraic topology
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(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
Simplicial presheaves over some site $S$ are
or equivalently, using the Hom-adjunction and symmetry of the closed monoidal structure on Cat
Regarding $\Simp\Set$ as a model category using the standard model structure on simplicial sets and inducing from that a model structure on $[S^{op}, \Simp\Set]$ makes simplicial presheaves a model for $\infty$-stacks, as described at infinity-stack homotopically.
In more illustrative language this means that a simplicial presheaf on $S$ can be regarded as an $\infty$-groupoid (in particular a Kan complex) whose space of $n$-morphisms is modeled on the objects of $S$ in the sense described at space and quantity.
Notice that most definitions of $\infty$-category the $\infty$-category is itself defined to be a simplicial set with extra structure (in a geometric definition of higher category) or gives rise to a simplicial set under taking its nerve (in an algebraic definition of higher category). So most notions of presheaves of higher categories will naturally induce presheaves of simplicial sets.
In particular, regarding a group $G$ as a one object category $\mathbf{B}G$ and then taking the nerve $N(\mathbf{B}G) \in \Simp\Set$ of these (the “classifying simplicial set of the group whose geometric realization is the classifying space $\mathcal{B}G$), which is clearly a functorial operation, turns any presheaf with values in groups into a simplicial presheaf.
Here are some basic but useful facts about simplicial presheaves.
Every simplicial presheaf $X$ is a homotopy colimit over a diagram of Set-valued sheaves regarded as discrete simplicial sheaves.
More precisely, for $X : S^{op} \to SSet$ a simplicial presheaf, let $D_X : \Delta^{op} \to Set \hookrightarrow SSet$ be given by $D_X : [n] \mapsto X_n$. Then there is a weak equivalence
See for instance remark 2.1, p. 6
(which is otherwise about descent for simplicial presheaves).
Let $[-,-] : SSet^{S^{op}} \to SSet$ be the canonical $SSet$-enrichment of the category of simplicial presheaves (i.e. the assignment of SSet-enriched functor categories).
It follows in particular from the above that every such hom-object $[X,A]$ of simplical presheaves can be written as a homotopy limit (in SSet for instance realized as a weighted limit, as described there) over evaluations of $X$.
First the above yields
Next from the co-Yoneda lemma we know that the Set-valued presheaves $X_n$ are in turn colimits over representables in $S$, so that
And finally the Yoneda lemma reduces this to
Notice that these kinds of computations are in particular often used when checking/computing descent and codescent along a cover or hypercover. For more on that in the context of simplicial presheaves see descent for simplicial presheaves.
Applications appear for instance at
The theory of simplicial presheaves and of simplicial sheaves was developed by J. Jardine in a long series of articles, some of which are listed below. It’s usage as a model for infinity-stacks was developed by Töen as described at infinity-stack homotopically.
For their interpretation in the more general context of (infinity,1)-sheaves see section 6.5.2 of