By now it is well known that there are two useful (objectwise or local) families of model structures on presheaves: the injective and projective. In fact, there is at least one more: the flasque. For some purposes, both the projective and the injective structure run into technical and annoying (but surmountable) difficulties for different reasons. The flasque model structure, which possesses a combination of the convenient properties of both structures, sometimes avoids these difficulties.
It is shown that the category of presheaves of simplicial groupoids and the category of presheaves of 2-groupoids have Quillen closed model structures. Furthermore, their homotopy categories are equivalent to the homotopy categories of simplicial presheaves and homotopy 2-types, respectively.
The usual way of defining weak equivalences for simplicial presheaves is to require an isomorphism on all sheaves of homotopy groups. We unravel some of the machinery here, and give a more concrete description in terms of local homotopy lifting properties. This characterization is used to prove some basic facts about the local homotopy theory of simplicial presheaves.
The category of simplicial R-coalgebras over a presheaf of commutative unital rings on a small Grothendieck site is endowed with a left proper, simplicial, cofibrantly generated model category structure where the weak equivalences are the local weak equivalences of the underlying simplicial presheaves. This model category is naturally linked to the R-local homotopy theory of simplicial presheaves and the homotopy theory of simplicial R-modules by Quillen adjunctions. We study the comparison with the R-local homotopy category of simplicial presheaves in the special case where R is a presheaf of…
We show that weak equivalences in a (cofibrantly generated) left Bousfield localization of the projective model category of simplicial presheaves can be characterized by a local lifting property if and only if the localization is exact.
We show that the category of presheaves of simplicial groupoids on a site $\mathcal{C}$ is a right proper simplicial model category. We define $G$-torsor of presheaf of 2-groupoids $G$, presheaf of simplicial groups $G$ and presheaf of simplicial groupoids $G$ on a site $\mathcal{C}$ and classify $G$-torsors by the homotopy classes $[\ast, \bar{W}G]$.
We define a map of simplicial presheaves, the Chern character, that assigns to every sequence of composable non connection preserving isomorphisms of vector bundles with holomorphic connections an appropriate sequence of holomorphic forms. We apply this Chern character map to the Cech nerve of a good cover of a complex manifold and assemble the data by passing to the totalization to obtain a map of simplicial sets. In simplicial degree 0, this map gives a formula for the Chern character of a bundle in terms of the clutching functions. In simplicial degree 1, this map gives a formula for the Ch…
We consider the category of presheaves of Gamma-spaces, or equivalently, of Gamma-objects in simplicial presheaves. Our main result is the construction of stable model structures on this category parametrised by local model structures on simplicial presheaves. If a local model structure on simplicial presheaves is monoidal, the corresponding stable model structure on presheaves of Gamma-spaces is monoidal and satisfies the monoid axiom. This allows us to lift the stable model structures to categories of algebras and modules over commutative algebras.
We construct models for the motivic homotopy category based on simplicial functors from smooth schemes over a field to simplicial sets. These spaces are homotopy invariant and therefore one does not have to invert the affine line in order to get a model for the motivic homotopy category.
We prove that Jardine's model category of simplicial presheaves can be obtained by localizing the `discrete' version at the collection of all hypercovers. One consequence is that the fibrant objects can be explicitly identified in terms of a hypercover descent condition. Another is a very simple approach to change-of-site functors. In an appendix, we discuss how this hypercover localization compares to the more naive process of localizing at the Cech complexes; the two are not the same in general, but agree in some cases of interest.
We show that a category $\mathscr{M}$ equipped with a model structure defined by a proper, locally small class of orbits $\mathscr{O}$ is Quillen equivalent to the category of small relative presheaves $\mathcal{P}(\mathscr{M},\mathscr{O})$.
The usual notion of a site fibred over a stack is expanded to a definition of a site C/A fibred over a presheaf of categories A. Presheaves of simplicial sets on the site fibred over a presheaf of categories A are contravariant enriched diagrams defined on A, taking values in simplicial sets. The standard model structure for presheaves of simplicial sets induces a coarse equivariant structure for enriched contravariant A-diagrams. If the presheaf of categories is a presheaf of groupoids G, then the associated homotopy theory is Quillen equivalent to the homotopy theory of simplicial presheaves…
We describe the moduli space of extensions in the model category of simplicial presheaves. This article can be seen as a generalization of Blomgren-Chacholski results in the case of simplicial sets. Our description of the moduli space of extensions treat the equivariant and the nonequivariant case in the same setting. As a new result, we describe the moduli space of M-bundles over a fixed space X, when M is a simplicial monoid. Moreover, the moduli space of M-bundles is classified by the classifying space of the simplicial submonoid generated by homotopy invertible elements of M. We give a gen…
We utilise the theory of crossed simplicial groups to introduce a collection of local Quillen model structures on the category of simplicial presheaves with a compact planar Lie group action on a small Grothendieck site. As an application, we give a characterisation of equivariant cohomology theories on a site as derived mapping spaces in these model categories.
In this paper we prove that for any simplicial set $B$, there is a Quillen equivalence between the covariant model structure on $\mathbf{S}/B$ and a certain localization of the projective model structure on the category of simplicial presheaves on the simplex category $Δ/B$ of $B$. We extend this result to give a new Quillen equivalence between this covariant model structure and the projective model structure on the category of simplicial presheaves on the simplicial category $\mathfrak{C}[B]$. We study the relationship with Lurie's straightening theorem. Along the way we prove some results on…
We show that Voevodsky's univalence axiom for intensional type theory is valid in categories of simplicial presheaves on elegant Reedy categories. In addition to diagrams on inverse categories, as considered in previous work of the author, this includes bisimplicial sets and $Θ_n$-spaces. This has potential applications to the study of homotopical models for higher categories.
We introduce, for \(\C\) a regular Cartesian Reedy category a model category whose fibrant objects are an analogue of quasicategories enriched in simplicial presheaves on \(C\). We then develop a coherent realization and nerve for this model structure and demonstrate using an enriched version of the necklaces of Dugger and Spivak that our model category is Quillen-equivalent to the category of categories enriched in simplicial presheaves on \(\C\). We then show that for any Cartesian-closed left-Bousfield localization of the category of simplicial presheaves on \(\C\), the coherent nerve and r…
We prove the existence of a model structure on the category of stratified simplicial sets whose fibrant objects are precisely $n$-complicial sets, which are a proposed model for $(\infty,n)$-categories, based on previous work of Verity and Riehl. We then construct a Quillen equivalent model based on simplicial presheaves over a category that can facilitate the comparison with other established models.
This is a sequel to a previous paper, developing an intrinsic, combinatorial homotopy theory for simplicial complexes; the latter form the cartesian closed subcategory of 'simple presheaves' in !Smp, the topos of symmetric simplicial sets, or presheaves on the category of finite, positive cardinals. We show here how this homotopy theory can be extended to the topos itself, !Smp. As a crucial advantage, the fundamental groupoid functor !Smp --> Gpd is left adjoint to a natural functor Gpd --> !Smp, the symmetric nerve of a groupoid, and preserves all colimits - a strong van Kampen property. Sim…
An n-truncated model structure on simplicial (pre-)sheaves is described having as weak equivalences maps that induce isomorphisms on certain homotopy sheaves only up to degree n. Starting from one of Jardine's intermediate model structures we construct such an n-type model structure via Bousfield-Friedlander localization and exhibit useful generating sets of trivial cofibrations. Injectively fibrant objects in these categories are called n-hyperstacks. The whole setup can consequently be viewed as a description of the homotopy theory of higher hyperstacks. More importantly, we construct analog…
If C is the model category of simplicial presheaves on a site with enough points, with fibrations equal to the global fibrations, then it is well-known that the fibrant objects are, in general, mysterious. Thus, it is not surprising that, when G is a profinite group, the fibrant objects in the model category of discrete G-spectra are also difficult to get a handle on. However, with simplicial presheaves, it is possible to construct an explicit fibrant model for an object in C, under certain finiteness conditions. Similarly, in this paper, we show that if G has finite virtual cohomological dime…
This paper presents a model structure for natural transformations of diagrams of simplicial presheaves of a fixed shape, in which the weak equivalences are defined by analogy with pro-equivalences between pro-objects.
We prove the existence of Morita model structures on the categories of small simplicial categories, simplicial sets, simplicial operads and dendroidal sets, modelling the Morita homotopy theory of $(\infty,1)$-categories and $\infty$-operads. We give a characterization of the weak equivalences in terms of simplicial presheaves, simplicial algebras and slice categories. In the case of the Morita model structure for simplicial categories and simplicial operads, we also show that each of these model structures can be obtained as an explicit left Bousfield localization of the Bergner model structu…
The most natural notion of a simplicial nerve for a (weak) bicategory was given by Duskin, who showed that a simplicial set is isomorphic to the nerve of a $(2,1)$-category (i.e. a bicategory with invertible $2$-morphisms) if and only if it is a quasicategory which has unique fillers for inner horns of dimension $3$ and greater. Using Duskin's technique, we show how his nerve applies to $(2,1)$-category functors, making it a fully faithful inclusion of $(2,1)$-categories into simplicial sets. Then we consider analogues of this extension of Duskin's result for several different two-dimensional …
We introduce the symmetricity notions of symmetric h-monoidality, symmetroidality, and symmetric flatness. As shown in our paper arXiv:1410.5675, these properties lie at the heart of the homotopy theory of colored symmetric operads and their algebras. In particular, the former property can be seen as the analog of Schwede and Shipley's monoid axiom for algebras over symmetric operads and allows one to equip categories of such algebras with model structures, whereas the latter ensures that weak equivalences of operads induce Quillen equivalences of categories of algebras. We discuss these prope…
For a category $\mathcal E$ with finite limits and well-behaved countable coproducts, we construct a model structure, called the effective model structure, on the category of simplicial objects in $\mathcal E$, generalising the Kan--Quillen model structure on simplicial sets. We then prove that the effective model structure is left and right proper and satisfies descent in the sense of Rezk. As a consequence, we obtain that the associated $\infty$-category has finite limits, colimits satisfying descent, and is locally Cartesian closed when $\mathcal E$ is, but is not a higher topos in general.…
