Definition
A simplicial presheaf on a category đ is a functor
$$
F\colon \mathcal{C}^{\mathrm{op}} \longrightarrow \mathbf{sSet},
$$
where $\mathbf{sSet}$ denotes the category of simplicial sets. Equivalently, it is a presheaf of simplicial sets, assigning to each object $U$ of đ a simplicial set $F(U)$ and to each morphism $V\to U$ a morphism of simplicial sets $F(U)\to F(V)$ functorially.
Overview
Simplicial presheaves provide a convenient framework for doing homotopyâtheoretic geometry on sites that may lack a classical topological space structure. They generalize ordinary presheaves of sets (the case where each $F(U)$ is a discrete simplicial set) and play a central role in the theory of ââtopoi, model categories of stacks, and derived algebraic geometry. By endowing the category of simplicial presheaves with a suitable model structureâmost commonly the Jardine (or local injective) model structureâone obtains a homotopical category in which weak equivalences are detected stalkwise after sheafification. This enables the definition of sheaves of spaces, higher stacks, and derived moduli problems.
Etymology/Origin
The term combines âsimplicial,â referring to simplicial setsâcombinatorial models for topological spaces introduced by Eilenberg and MacâŻLane (1940s)âwith âpresheaf,â a functorial assignment of data to the objects of a category, a concept formalized by Grothendieck in the 1950s. The systematic study of simplicial presheaves emerged in the 1980s, notably through work of Jardine, Joyal, and BousfieldâFriedlander, who introduced model structures on such functor categories.
Characteristics
| Feature | Description |
|---|---|
| Underlying category | Functor category $[\mathcal{C}^{\mathrm{op}},\mathbf{sSet}]$. |
| Model structures | ⢠Projective model structure: cofibrations are objectwise monomorphisms; fibrations are objectwise Kan fibrations. ⢠Injective model structure: cofibrations are objectwise cofibrations in $\mathbf{sSet}$; fibrations are objectwise fibrations. ⢠Local (Jardine) model structure: weak equivalences are stalkwise (or sheafâwise) weak equivalences after sheafification; fibrations are defined via a lifting property against local trivial cofibrations. |
| Sheaf condition | A simplicial presheaf $F$ is a simplicial sheaf (or sheaf of spaces) if it satisfies descent for hypercovers: for any hypercover $U_\bullet\to U$ in the site, the canonical map $F(U)\to \operatorname{holim}_{[n]\in\Delta}F(U_n)$ is a weak equivalence of simplicial sets. |
| Homotopyâtheoretic operations | ⢠Homotopy limits/colimits computed objectwise and then sheafified. ⢠Mapping spaces: given $F,G$, the internal hom $ \underline{\mathrm{Hom}}(F,G) $ is defined objectwise as the simplicial mapping set $\mathrm{Map}(F(U),G(U))$. |
| Relation to ââcategories | Via the homotopy coherent nerve, the ââcategory of simplicial presheaves presents the ââtopos $\operatorname{Sh}_\infty(\mathcal{C})$. |
| Applications | ⢠Construction of higher stacks and derived moduli spaces. ⢠Ătale and Nisnevich homotopy theory of schemes (e.g., motivic homotopy theory). ⢠Model for ââsheaves in higher topos theory. |
Related Topics
- Presheaf â a functor $\mathcal{C}^{\mathrm{op}}\to \mathbf{Set}$.
- Sheaf â a presheaf satisfying a gluing condition for coverings.
- Simplicial set â a combinatorial model for topological spaces.
- Model category â an abstract framework for homotopy theory; see the Jardine model structure.
- ââtopos â a higherâcategorical analogue of a Grothendieck topos, often presented by simplicial presheaves.
- Higher stack â a sheaf of ââgroupoids; simplicial sheaves are a concrete model.
- Derived algebraic geometry â uses simplicial (or differential graded) sheaves to encode derived geometric objects.
- Motivic homotopy theory â employs simplicial presheaves on the site of smooth schemes to construct a homotopy theory of algebraic varieties.