Simplicial Model Category Research Articles

Homotopy Sheaves on Generalised Spaces

We study the homotopy right Kan extension of homotopy sheaves on a category to its free cocompletion, i.e. to its category of presheaves. Any pretopology on the original category induces a canonical pretopology of generalised coverings on the free cocompletion. We show that with respect to these pretopologies the homotopy right Kan extension along the Yoneda embedding preserves homotopy sheaves valued in (sufficiently nice) simplicial model categories. Moreover, we show that this induces an equivalence between sheaves of spaces on the original category and colimit-preserving sheaves of spaces on its free cocompletion. We present three applications in geometry and topology: first, we prove that diffeological vector bundles descend along subductions of diffeological spaces. Second, we deduce that various flavours of bundle gerbes with connection satisfy (∞,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\infty ,2)$$\\end{document}-categorical descent. Finally, we investigate smooth diffeomorphism actions in smooth bordism-type field theories on a manifold. We show how these smooth actions allow us to extract the values of a field theory on any object coherently from its values on generating objects of the bordism category.

Admissible replacements for simplicial monoidal model categories

Using Dugger's construction of universal model categories, we produce replacements for simplicial and combinatorial symmetric monoidal model categories with better operadic properties. Namely, these replacements admit a model structure on algebras over any given colored operad. As an application, we show that in the stable case, such symmetric monoidal model categories are classified by commutative ring spectra when the monoidal unit is a compact generator. In other words, they are strong monoidally Quillen equivalent to modules over a uniquely determined commutative ring spectrum.

2-Cartesian Fibrations I: A Model for $$\infty $$-Bicategories Fibred in $$\infty $$-Bicategories

In this paper, we provide a notion of \(\infty \)-bicategories fibred in \(\infty \)-bicategories which we call 2-Cartesian fibrations. Our definition is formulated using the language of marked biscaled simplicial sets: Those are scaled simplicial sets equipped with an additional collection of triangles containing the scaled 2-simplices, which we call lean triangles, in addition to a collection of edges containing all degenerate 1-simplices. We prove the existence of a left proper combinatorial simplicial model category whose fibrant objects are precisely the 2-Cartesian fibrations over a chosen scaled simplicial set S. Over the terminal scaled simplicial set, this provides a new model structure modeling \(\infty \)-bicategories, which we show is Quillen equivalent to Lurie’s scaled simplicial set model. We conclude by providing a characterization of 2-Cartesian fibrations over an \(\infty \)-bicategory. This characterization then allows us to identify those 2-Cartesian fibrations arising as the coherent nerve of a fibration of \({\text {Set}}^+_{\Delta }\)-enriched categories, thus showing that our definition recovers the preexisting notions of fibred 2-categories.

A Shadow Perspective on Equivariant Hochschild Homologies

Abstract Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category $\mathsf V$, as well as for small $\mathsf V$-categories. We show that each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. We also define a generalized theory of Hochschild homology twisted by an automorphism and show that it is Morita invariant. Hochschild homology of Green functors and $C_n$-twisted topological Hochschild homology fit into this framework, which allows us to conclude that these theories are Morita invariant. We also study linearization maps relating the topological and algebraic theories, proving that the linearization map for topological Hochschild homology arises as a lax shadow functor, and constructing a new linearization map relating topological restriction homology and algebraic restriction homology. Finally, we construct a twisted Dennis trace map from the fixed points of equivariant algebraic $K$-theory to twisted topological Hochschild homology.

Higher order Toda brackets

We describe two ways to define higher order Toda brackets in a pointed simplicial model category $${\mathcal {D}}$$ : one is a recursive definition using model categorical constructions, and the second uses the associated simplicial enrichment. We show that these two definitions agree, by providing a third, diagrammatic, description of the Toda bracket, and explain how it serves as the obstruction to rectifying a certain homotopy-commutative diagram in $${\mathcal {D}}$$ .

A simplicial approach to stratified homotopy theory

In this article we consider the homotopy theory of stratified spaces through a simplicial point of view. We first consider a model category of filtered simplicial sets over some fixed poset $P$, and show that it is a simplicial combinatorial model category. We then define a generalization of the homotopy groups for any fibrant filtered simplicial set $X$ : the filtered homotopy groups $s\pi_n(X)$. They are filtered simplicial sets built from the homotopy groups of the different pieces of $X$. We then show that the weak equivalences are exactly the morphisms that induce isomorphisms on those filtered homotopy groups. Then, using filtered versions of the topological realisation of a simplicial set and of the simplicial set of singular simplices, we transfer those results to a category whose objects are topological spaces stratified over $P$. In particular, we get a stratified version of Whitehead's theorem. Specializing to the case of conically stratified spaces, a wide class of topological stratified spaces, we recover a theorem of Miller saying that to understand the homotopy type of conically stratified spaces, one only has to understand the homotopy type of strata and holinks. We then provide a family of examples of conically stratified spaces and of computations of their filtered homotopy groups.

Semantics of higher inductive types

AbstractHigher inductive typesare a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the “synthetic” development of homotopy theory within type theory, as well as in formalising ordinary set-level mathematics in type theory. In this paper, we construct models of a wide range of higher inductive types in a fairly wide range of settings.We introduce the notion ofcell monad with parameters: a semantically-defined scheme for specifying homotopically well-behaved notions of structure. We then show that any suitable model category hasweakly stable typal initial algebrasfor any cell monad with parameters. When combined with the local universes construction to obtain strict stability, this specialises to give models of specific higher inductive types, including spheres, the torus, pushout types, truncations, the James construction and general localisations.Our results apply in any sufficiently nice Quillen model category, including any right proper, simplicially locally cartesian closed, simplicial Cisinski model category (such as simplicial sets) and any locally presentable locally cartesian closed category (such as sets) with its trivial model structure. In particular, any locally presentable locally cartesian closed (∞, 1)-category is presented by some model category to which our results apply.

Polynomial functors in manifold calculus

Let M be a smooth manifold, and let O(M) be the poset of open subsets of M. Manifold calculus, due to Goodwillie and Weiss, is a calculus of functors suitable for studying contravariant functors (cofunctors) F:O(M)⟶Spaces from O(M) to the category of spaces. Weiss showed that polynomial cofunctors of degree ≤k are determined by their values on Ok(M), where Ok(M) is the full subposet of O(M) whose objects are open subsets diffeomorphic to the disjoint union of at most k balls. Afterwards Pryor showed that one can replace Ok(M) by more general subposets and still recover the same notion of polynomial cofunctor. In this paper, we generalize these results to cofunctors from O(M) to any simplicial model category M. If Fk(M) stands for the unordered configuration space of k points in M, we also show that the category of homogeneous cofunctors O(M)⟶M of degree k is weakly equivalent to the category of linear cofunctors O(Fk(M))⟶M provided that M has a zero object. Using a new approach, we also show that if M is a general model category and F:Ok(M)⟶M is an isotopy cofunctor, then the homotopy right Kan extension of F along the inclusion Ok(M)↪O(M) is also an isotopy cofunctor.

Presentably symmetric monoidal ∞–categories are represented by symmetric monoidal model categories

We prove the theorem stated in the title. More precisely, we show the stronger statement that every symmetric monoidal left adjoint functor between presentably symmetric monoidal infinity-categories is represented by a strong symmetric monoidal left Quillen functor between simplicial, combinatorial and left proper symmetric monoidal model categories.

Bousfield Localisations along Quillen Bifunctors

Consider a Quillen adjunction of two variables between combinatorial model categories from $\mathcal{C}\times\mathcal{D}$ to $\mathcal{E}$, and a set $\mathcal{S}$ of morphisms in $\mathcal{C}$. We prove that there is a localised model structure $L_{\mathcal{S}}\mathcal{E}$ on $\mathcal{E}$, where the local objects are the $\mathcal{S}$-local objects in $\mathcal{E}$ described via the right adjoint. These localised model structures generalise Bousfield localisations of simplicial model categories, Barnes and Roitzheim's familiar model structures, and Barwick's enriched Bousfield localisations. In particular, we can use these model structures to define Postnikov sections in more general left proper combinatorial model categories.

Calculus of functors and model categories, II

This is a continuation, completion, and generalization of our previous joint work with B. Chorny. We supply model structures and Quillen equivalences underlying Goodwillie's constructions on the homotopy level for functors between simplicial model categories satisfying mild hypotheses.

Cross effects and calculus in an unbased setting

We study functors F from C_f to D where C and D are simplicial model categories and C_f is the full subcategory of C consisting of objects that factor a fixed morphism f from A to B. We define the analogs of Eilenberg and Mac Lane's cross effects functors in this context, and identify explicit adjoint pairs of functors whose associated cotriples are the diagonals of the cross effects. With this, we generalize the cotriple Taylor tower construction of [10] from the setting of functors from pointed categories to abelian categories to that of functors from C_f to D to produce a tower of functors whose n-th term is a degree n functor. We compare this tower to Goodwillie's tower of n-excisive approximations to F found in [8]. When D is a good category of spectra, and F is a functor that commutes with realizations, the towers agree. More generally, for functors that do not commute with realizations, we show that the terms of the towers agree when evaluated at the initial object of C_f.

Lifting homotopy T-algebra maps to strict maps

The settings for homotopical algebra—categories such as simplicial groups, simplicial rings, A∞ spaces, E∞ ring spectra, etc.—are often equivalent to categories of algebras over some monad or triple T. In such cases, T is acting on a nice simplicial model category in such a way that T descends to a monad on the homotopy category and defines a category of homotopy T-algebras. In this setting there is a forgetful functor from the homotopy category of T-algebras to the category of homotopy T-algebras.Under suitable hypotheses we provide an obstruction theory, in the form of a Bousfield–Kan spectral sequence, for lifting a homotopy T-algebra map to a strict map of T-algebras. Once we have a map of T-algebras to serve as a basepoint, the spectral sequence computes the homotopy groups of the space of T-algebra maps and the edge homomorphism on π0 is the aforementioned forgetful functor. We discuss a variety of settings in which the required hypotheses are satisfied, including monads arising from algebraic theories and operads. We also give sufficient conditions for the E2-term to be calculable in terms of Quillen cohomology groups.We provide worked examples in G-spaces, G-spectra, rational E∞ algebras, and A∞ algebras. Explicit calculations, connected to rational unstable homotopy theory, show that the forgetful functor from the homotopy category of E∞ ring spectra to the category of H∞ ring spectra is generally neither full nor faithful. We also apply a result of the second named author and Nick Kuhn to compute the homotopy type of the space E∞(Σ+∞CokerJ,LK(2)R).

Commutative ring objects in pro-categories and generalized Moore spectra

We develop a rigidity criterion to show that in simplicial model categories with a compatible symmetric monoidal structure, operad structures can be automatically lifted along certain maps. This is applied to obtain an unpublished result of M. J. Hopkins that certain towers of generalized Moore spectra, closely related to the K(n)-local sphere, are E-infinity algebras in the category of pro-spectra. In addition, we show that Adams resolutions automatically satisfy the above rigidity criterion. In order to carry this out we develop the concept of an operadic model category, whose objects have homotopically tractable endomorphism operads.

Recognizing mapping spaces

Given a fixed object A in a suitable pointed simplicial model category C, we study the problem of recovering the target Y from the pointed mapping space map∗(A,Y) (up to A-equivalence). We describe a recognition principle, modeled on the classical ones for loop spaces, but using the more general notion of an A-mapping algebra. It has an associated transfinite procedure for recovering CWAY from map∗(A,Y), inspired by Dror-Farjoun’s construction of CWA-approximations.

A model categorical approach to group completion of E n -algebras

A group completion functor Q is constructed in the category of algebras in simplicial sets over a cofibrant En-operad \({\mathcal{M}}\) . It is shown that Q defines a Bousfield–Friedlander simplicial model category on \({\mathcal{M}}\) -algebras.

Simplicial descent categories

Let D be a category and E a class of morphisms in D . In this paper we study the question of how to transfer homotopic structure from the category of simplicial objects in D , Δ ∘ D , to D through a ‘good’ functor s : Δ ∘ D → D , which we call simple functor. For instance, the Bousfield–Kan homotopy colimit in a Quillen simplicial model category is a good simple functor. As a remarkable example outside the setting of Quillen models we include Deligne simple of mixed Hodge complexes. We prove here that the simple functor induces an equivalence on the corresponding localized categories. We also describe a natural structure of Brown category of cofibrant objects on Δ ∘ D . We use these facts to produce cofiber sequences on the localized category of D by E , which give rise to a natural Verdier triangulated structure in the stable case.

Representability theorems for presheaves of spectra

Suppose that M is a simplicial model category and that F is a contravariant simplicial functor defined on M which takes values in pointed simplicial sets. This note displays conditions on the simplicial model category M and the functor F such that F is representable up to weak equivalence. The conditions on F are homotopy coherent versions of the classical conditions for Brown representability, while M should have the fundamental properties of the stable model structure for presheaves of spectra on a Grothendieck site.

Complete Segal spaces arising from simplicial categories

In this paper, we compare several functors which take simplicial categories or model categories to complete Segal spaces, which are particularly nice simplicial spaces which, like simplicial categories, can be considered to be models for homotopy theories. We then give a characterization, up to weak equivalence, of complete Segal spaces arising from these functors.

Simplicial model category structures on the category of chain functors

The model structure on the category of chain functors Ch, developed in [4], has the main features of a simplicial model category structure, taking into account the lack of arbitrary (co-)limits in Ch. After an appropriate tensor and cotensor structure in Ch is established (§1, §3), Quillen’s axiom SM7 is verified in §5 and §6. Moreover, it turns out that in the definition of a simplicial model structure, the category of simplicial sets can be replaced by the category of simplicial spectra endowing Ch with the structure of an approximate simplicial stable model structure (= approximate ss-model structure) (§7). In §8 the model structure on Ch is shown to be proper.