(This is an updated version; following an idea of Voevodsky, we have strengthened our results so all of them apply to one form of motivic homotopy theory).
We give two general constructions for the passage from unstable to stable homotopy that apply to the known example of topological spaces, but also to new situations, such as motivic homotopy theory of schemes. One is based on the standard notion of spectra originated by Boardman. Its input is a well-behaved model category C and an endofunctor G, generalizing the suspension. Its output is a model category on which G is a Quillen equivalence. Under strong hypotheses the weak equivalences in this model structure are the appropriate analogue of stable homotopy isomorphisms.
The second construction is based on symmetric spectra, and is of value only when C has some monoidal structure that G preserves. In this case, ordinary spectra generally will not have monoidal structure, but symmetric spectra will. Our abstract approach makes constructing the stable model category of symmetric spectra straightforward. We study properties of these stabilizations; most importantly, we show that the two different stabilizations are Quillen equivalent under some hypotheses (that also hold in the motivic example).
In this paper we construct a symmetric monoidal closed model category of coherently commutative monoidal categories. The main aim of this paper is to establish a Quillen equivalence between a model category of coherently commutative monoidal categories and a natural model category of Permutative (or strict symmetric monoidal) categories, $\mathbf{Perm}$, which is not a symmetric monoidal closed model category. The right adjoint of this Quillen equivalence is the classical Segal's Nerve functor.
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.
We prove that the homotopy theory of parsummable categories (as defined by Schwede) with respect to the underlying equivalences of categories is equivalent to the usual homotopy theory of symmetric monoidal categories. In particular, this yields a model of symmetric monoidal categories in terms of categories equipped with a strictly commutative, associative, and unital (but only partially defined) operation.
In this paper, we study conditions for extending Quillen model category properties , between two symmetric monoidal categories, to their associated category of symmetric sequences and of operads. Given a Quillen equivalence $λ: \mathcal{C}=Ch_{\Bbbk,t} \leftrightarrows \mathcal{D}: R,$ so that $\mathcal{D}$ is any symmetric monoidal category and the adjoint pair $(λ, R)$ is weak monoidal, we prove that the categories of connected operads $Op_\mathcal{C}$ and $Op_\mathcal{D}$ are Quillen equivalent. This expands an analogous result of Schwede-Shipley(\cite{SS03}) when we replace these categorie…
In here we define the concept of fibered symmetric bimonoidal categories. These are roughly speaking fibered categories D->C whose fibers are symmetric monoidal categories parametrized by C and such that both D and C have a further structure of a symmetric monoidal category that satisfy certain coherences that we describe. Our goal is to show that we can correspond to a fibered symmetric bimonoidal category an E_{\infty}-ring spectrum in a functorial way.
We construct a Quillen model structure on the category of spectral categories, where the weak equivalences are the symmetric spectra analogue of the notion of equivalence of categories.
We show that all coalgebras over the sphere spectrum are cocommutative in the category of symmetric spectra, orthogonal spectra, $Γ$-spaces, $\mathcal{W}$-spaces and EKMM $\mathbb{S}$-modules. Our result only applies to these strict monoidal categories of spectra and does not apply to the $\infty$-category setting.
We describe the multiplicative structures that arise on categories of equivariant modules over certain equivariant commutative ring spectra. Building on our previous work on N-infinity ring spectra, we construct categories of equivariant operadic modules over N-infinity rings that are structured by equivariant linear isometries operads. These categories of modules are endowed with equivariant symmetric monoidal structures, which amounts to the structure of an "incomplete Mackey functor in homotopical categories". In particular, we construct internal norms which satisfy the double coset formula…
We use Lurie's symmetric monoidal envelope functor to give two new descriptions of $\infty$-operads: as certain symmetric monoidal $\infty$-categories whose underlying symmetric monoidal $\infty$-groupoids are free, and as certain symmetric monoidal $\infty$-categories equipped with a symmetric monoidal functor to finite sets (with disjoint union as tensor product). The latter leads to a third description of $\infty$-operads, as a localization of a presheaf $\infty$-category, and we use this to give a simple proof of the equivalence between Lurie's and Barwick's models for $\infty$-operads.
We study symmetric powers in the homotopy categories of abstract closed symmetric monoidal model categories, in both unstable and stable settings. As an outcome, we prove that symmetric powers preserve the Nisnevich and etale homotopy type in the unstable and stable motivic homotopy theories of schemes over a base. More precisely, if f is a weak equivalence of motivic spaces, or a weak equivalence between positively cofibrant motivic spectra, with respect to the Nisnevich or etale topology on schemes, then all symmetric powers Sym^n(f) are weak equivalences too. This gives left derived symmetr…
We introduce a tensor product for symmetric monoidal categories with the following properties. Let SMC denote the 2-category with objects small symmetric monoidal categories, arrows symmetric monoidal functors and 2-cells monoidal natural transformations. Our tensor product together with a suitable unit is part of a structure on SMC that is a 2-categorical version of the symmetric monoidal closed categories. This structure is surprisingly simple. In particular the arrows involved in the associativity and symmetry laws for the tensor and in the unit cancellation laws are 2-natural and satisfy c…
We give an operadic definition of a genuine symmetric monoidal G-category, and we prove that its classifying space is a genuine E_\infty G-space. We do this by developing some very general categorical coherence theory. We combine results of Corner and Gurski, Power, and Lack, to develop a strictification theory for pseudoalgebras over operads and monads. It specializes to strictify genuine symmetric monoidal G-categories to genuine permutative G-categories. All of our work takes place in a general internal categorical framework that has many quite different specializations. When G is a finite …
In this paper we give an expository account of quasistrict symmetric monoidal 2-categories, as introduced by Schommer-Pries. We reformulate the definition using a graphical calculus called wire diagrams, which facilitates computations and emphasizes the central role played by the interchangor coherence isomorphisms.
A new, self-contained, proof of a coherence result for categories equipped with two symmetric monoidal structures bridged by a natural transformation is given. It is shown that this coherence result is sufficient for $ω\timesω$-indexed family of iterated reduced bar constructions based on such a category.
Given a small category $I$ and a closed symmetric monoidal category $\mm$, we show that the diagram category $\mm^I$ with the objectwise product is a closed symmetric monoidal category. We then prove that if $I$ is a Reedy category and $\mm$ has a model structure compatible with its product, then so is the Reedy model structure on $\mm^I$ provided that $\mm$ is cofibrantly generated.
We establish an equivalence of homotopy theories between symmetric monoidal bicategories and connective spectra. For this, we develop the theory of $Γ$-objects in 2-categories. In the course of the proof we establish strictfication results of independent interest for symmetric monoidal bicategories and for diagrams of 2-categories.
We endow categories of non-symmetric operads with natural model structures. We work with no restriction on our operads and only assume the usual hypotheses for model categories with a symmetric monoidal structure. We also study categories of algebras over these operads in enriched non-symmetric monoidal model categories.
We introduce categorical models of $N_\infty$ spaces, which we call normed symmetric monoidal categories (NSMCs). These are ordinary symmetric monoidal categories equipped with compatible families of norm maps, and when specialized to a particular class of examples, they reveal a connection between the equivariant symmetric monoidal categories of Guillou-May-Merling-Osorno and those of Hill-Hopkins. We also give an operadic interpretation of the Mac Lane coherence theorem and generalize it to include NSMCs. Among other things, this theorem ensures that the classifying space of a NSMC is a $N_\…
We construct model category structures for monoids and modules in symmetric monoidal model categories which satisfy an extra axiom, the monoidal axiom, with applications to symmetric spectra and $Γ$-spaces.
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…
We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal $\infty$-category $\mathcal{V}$. Our theory of enriched $\infty$-categories has many desirable properties; for instance, if the enriching $\infty$-category $\mathcal{V}$ is presentably symmetric monoidal then $\mathrm{Cat}^\mathcal{V}_\infty$ is as well. These features render the theory useful even when an $\infty$-category of enriched $\infty$-categories comes from a model category (as is often the case in examples of interest, e.g. dg-categories, spectral categories, and $(\infty,n)$-categories). This…
We prove a version of J.P. May's theorem on the additivity of traces, in symmetric monoidal stable $\infty$-categories. Our proof proceeds via a categorification, namely we use the additivity of topological Hochschild homology as an invariant of stable $\infty$-categories and construct a morphism of spectra $\mathrm{THH}(\mathbf C)\to \mathrm{End}(\mathbf 1_\mathbf C)$ for $\mathbf C$ a stably symmetric monoidal rigid $\infty$-category. We also explain how to get a more general statement involving traces of finite (homotopy) colimits.
We obtain combinatorial model categories of parametrised spectra, together with systems of base change Quillen adjunctions associated to maps of parameter spaces. We work with simplicial objects and use Hovey's sequential and symmetric stabilisation machines. By means of a Grothendieck construction for model categories, we produce combinatorial model categories controlling the totality of parametrised stable homotopy theory. The global model category of parametrised symmetric spectra is equipped with a symmetric monoidal model structure (the external smash product) inducing pairings in twisted…
We prove, under mild assumptions, that a Quillen equivalence between symmetric monoidal model categories gives rise to a Quillen equivalence between their model categories of (non-symmetric) operads, and also between model categories of algebras over operads. We also show left properness results on model categories of operads and algebras over operads. As an application, we prove homotopy invariance for (unital) associative operads.
We prove two results from Morita theory of stable model categories. Both can be regarded as topological versions of recent algebraic theorems. One is on recollements of triangulated categories, which have been studied in the algebraic case by Jørgensen. We give a criterion which answers the following question: When is there a recollement for the derived category of a given symmetric ring spectrum in terms of two other symmetric ring spectra? The other result is on well generated triangulated categories in the sense of Neeman. Porta characterizes the algebraic well generated categories as local…
We introduce a notion of fine Tannakian infinity-categories and prove Tannakian characterization results for symmetric monoidal stable infinity-categories over a field of characteristic zero. It connects derived quotient stacks with symmetric monoidal stable infinity-categories which satisfy a certain simple axiom. We also discuss several applications to examples.
The main objective of this paper is to construct a symmetric monoidal closed model category of coherently commutative monoidal quasi-categories. We construct another model category structure whose fibrant objects are (essentially) those coCartesian fibrations which represent objects that are known as symmetric monoidal quasi-categories in the literature. We go on to establish a Quillen equivalence between the two model categories.
Given symmetric monoidal infinity-categories C and D, subject to mild hypotheses on D, we define an infinity-categorical analog of the Day convolution symmetric monoidal structure on the functor category Fun(C, D). An E_infinity monoid for the Day convolution product is a lax monoidal functor from C to D.
We show that weak monoidal Quillen equivalences induce equivalences of symmetric monoidal $\infty$-categories with respect to the Dwyer-Kan localization of the symmetric monoidal model categories. The result will induce a Dold-Kan correspondence of coalgebras in $\infty$-categories. Moreover, it shows that Shipley's zig-zag of Quillen equivalences lifts to an explicit symmetric monoidal equivalence of $\infty$-categories for the stable Dold-Kan correspondence. We study homotopy coherent coalgebras associated to a monoidal monoidal category. We show examples when these coalgebras cannot be rigi…
In this paper we develop the theory of operads, algebras and modules in cofibrantly generated symmetric monoidal model categories. We give J-semi model strucures, which are a slightly weaker version of model structures, for operads and algebras and model structures for modules. In a second part we develop the thoery of S-modules of [EKMM]., which allows a general homotopy theory for commutative algebras and pseudo unital symmetric monoidal categories of modules over them. Finally we prove a base change and projection formula.
In this paper we propose, firstly, a categorification of virtual braid groups and groupoids in terms of "locally" braided objects in a symmetric category (SC), and, secondly, a definition of self-distributive structures (SDS) in an arbitrary SC. SDS are shown to produce braided objects in a SC. As for examples, we interpret associativity and the Jacobi identity in a SC as generalized self-distributivity, thus endowing associative and Leibniz algebras with a "local" braiding. Our "double braiding" approach provides a natural interpretation for virtual racks and twisted Burau representation. Hom…
We establish a highly flexible condition that guarantees that all colored symmetric operads in a symmetric monoidal model category are admissible, i.e., the category of algebras over any operad admits a model structure transferred from the original model category. We also give a necessary and sufficient criterion that ensures that a given weak equivalence of admissible operads admits rectification, i.e., the corresponding Quillen adjunction between the categories of algebras is a Quillen equivalence. In addition, we show that Quillen equivalences of underlying symmetric monoidal model categori…
Motivated by algebraic quantum field theory, we study presheaves of symmetric tensor categories defined over the base of a space, intended as a spacetime. Any section of a presheaf (that is, any "superselection sector", in the applications that we have in mind) defines a holonomy representation whose triviality is measured by Cheeger-Chern-Simons characteristic classes, and a non-abelian unitary cocycle defining a Lie group gerbe. We show that, given an embedding in a presheaf of full subcategories of the one of Hilbert spaces, the section category of a presheaf is a Tannaka-type dual of a loc…
In previous work, we used an $\infty$-categorical version of ultraproducts to show that, for a fixed height $n$, the symmetric monoidal $\infty$-categories of $E_{n,p}$-local spectra are asymptotically algebraic in the prime $p$. In this paper, we prove the analogous result for the symmetric monoidal $\infty$-categories of $K_{p}(n)$-local spectra, where $K_{p}(n)$ is Morava $K$-theory at height $n$ and the prime $p$. This requires $\infty$-categorical tools suitable for working with compactly generated symmetric monoidal $\infty$-categories with non-compact unit. The equivalences that we prod…
We prove that various structures on model $\infty$-categories descend to corresponding structures on their localizations: (i) Quillen adjunctions; (ii) two-variable Quillen adjunctions; (iii) monoidal and symmetric monoidal model structures; and (iv) enriched model structures.
We prove that every stable, combinatorial model category has a natural enrichment by symmetric spectra (or more precisely, a natural equivalence class of enrichments). This in some sense generalizes the simplicial enrichments of model categories provided by the Dwyer-Kan hammock localization. As one particular application, we are able to associate to every object of a stable, combinatorial model category a "homotopy endomorphism ring spectrum". The homotopy types of these ring spectra are preserved by Quillen equivalences, and so are an invariant for stable model categories.
We prove that given $\mathcal{C}$ a presentably symmetric monoidal $\infty$-category, and any essentially small $\infty$-operad $\mathcal{O}$, the $\infty$-category of $\mathcal{O}$-algebras in $\mathcal{C}$ is enriched, tensored and cotensored over the presentably symmetric monoidal $\infty$-category of $\mathcal{O}$-coalgebras in $\mathcal{C}$. We provide a higher categorical analogue of the universal measuring coalgebra. For categories in the usual sense, the result was proved by Hyland, López Franco, and Vasilakopoulou.
Lyubashenko has described enriched 2-categories as categories enriched over V-Cat, the 2-category of categories enriched over a symmetric monoidal V. I have generalized this to the k-fold monoidal V. The symmetric case can easily be recovered. The introduction of this paper proposes a recursive definition of V-n-categories and their morphisms. Then I consider the special case of V-2-categories and give the details of the proof that with their morphisms these form the structure of a 3-category.
