Philippe Gaucher
The category of flows is not cartesian closed. We construct a closed symmetric monoidal structure which has moreover a satisfactory behavior from the computer scientific viewpoint.
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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.
This paper proves three different coherence theorems for symmetric monoidal bicategories. First, we show that in a free symmetric monoidal bicategory every diagram of 2-cells commutes. Second, we show that this implies that the free symmetric monoidal bicategory on one object is equivalent, as a symmetric monoidal bicategory, to the discrete symmetric monoidal bicategory given by the disjoint union of the symmetric groups. Third, we show that every symmetric monoidal bicategory is equivalent to a strict one. We give two topological applications of these coherence results. First, we show that t…
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 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…
Building on structure observed in equivariant homotopy theory, we define an equivariant generalization of a symmetric monoidal category: a $G$-symmetric monoidal category. These record not only the symmetric monoidal products but also symmetric monoidal powers indexed by arbitrary finite $G$-sets. We then define $G$-commutative monoids to be the natural extension of ordinary commutative monoids to this new context. Using this machinery, we then describe when Bousfield localization in equivariant spectra preserves certain operadic algebra structures, and we explore the consequences of our defin…
Monoidal Relative Categories Model Monoidal $\infty$-Categories
本文证明单调相对范畴的同调理论等价于单调∞-范畴的同调理论，并在对称单调情况下同样成立。通过应用，证明了每个可呈现的单调或可呈现的对称单调∞-范畴都由单调或对称单调模型范畴呈现。
Monoidal Kleisli Bicategories and the Arithmetic Product of Coloured Symmetric Sequences
将算术积推广至着色对称序列，证明其构成正规oplax幺半结构，利用幺半双范畴高效验证相干条件。
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.
Coherence for indexed symmetric monoidal categories
本文针对索引对称幺半范畴这一双范畴细化结构，拓展现有相干定理，证明由其定义的双范畴上一大类运算具有典范同构，并引入判定同构存在性的严格图形演算。
The 2-monoidal structure of symmetric sequences
该研究证明Dwyer与Hess提出的对称序列范畴的2-幺半结构猜想，通过分析两种幺半结构的相互作用，得出每个operad都具有Day-operad结构的结论。
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 $N_\…
Symmetric monoidal extensions and graph cobordisms between finite sets
本文针对对称幺半$(\u221e,n)$-范畴，发展了一套工具以显式描述自由对称幺半扩张，并应用于构造有限集间图配边的对称幺半$(\u221e,2)$-范畴，作为有限集范畴的扩张，恢复了Barkan--Steinebrunner和Galatius的结果。
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.
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.
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 general notion of operad is given, which includes as instances, the operads originally conceived to study loop spaces, as well as the higher operads that arise in the globular approach to higher dimensional algebra. In the framework of this paper, one can also describe symmetric and braided analogues of higher operads, likely to be important to the study of weakly symmetric, higher dimensional monoidal structures.
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…
Admissible replacements for simplicial monoidal model categories
本文利用Dugger万有模型范畴构造，为单纯与组合对称幺半群模型范畴构造了具有更好operadic性质的替换，并证明稳定情形下当幺半群单位是紧生成元时，此类范畴可由交换环谱分类，且与唯一确定的交换环谱上的模强幺半群Quillen等价。
On the Kelly monoidal structure of $Λ$-sequences and unital operads
本文研究了Λ序列的凯利单体结构，证明了其与对称序列右模的同构，并建立了普遍的正常弱单体结构，将单位运算子识别为单体。
We study Quillen model categories equipped with a monoidal skew closed structure that descends to a genuine monoidal closed structure on the homotopy category. Our examples are 2-categorical and include permutative categories and bicategories. Using the skew framework, we adapt Eilenberg and Kelly's theorem relating monoidal and closed structure to the homotopical setting. This is applied to the construction of monoidal bicategories arising from the pseudo-commutative 2-monads of Hyland and Power.
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…
We consider the composition product of symmetric sequences in the case where the underlying symmetric monoidal structure does not commute with coproducts. Even though this composition product is not a monoidal structure on symmetric sequences, it has enough structure, namely that of a `normal oplax' monoidal product, to be able to define monoids (which are then operads on the underlying category) and make a bar construction. The main benefit of this work is in the dual setting, where it allows us to define a cobar construction for cooperads.
We verify that a certain functor $D\colon\text{Sp}^Σ(\text{Ch}^+)\to\text{Ch}$ is symmetric monoidal. This functor is used elsewhere in developing the model category theory of symmetric spectra and of chain complexes graded over $\mathbb{N}$ or $\mathbb{Z}$.
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.
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.
Invertibility and parity in symmetric monoidal categories
本文引入了形式态之间可逆对象的奇偶性概念，并利用其证明了相应的协整定理。通过自由置换范畴的详细处理及超整数的骨骼模型，展示了两者之间的等价关系。
A symmetric monoidal fracture square
本文研究从K-局部与K-完备子范畴恢复刚性紧生成对称幺半稳定∞-范畴，并保持对称幺半结构，推广至G-谱情形，给出各向同性分离的对称幺半变体。
We show how to construct a Gamma-bicategory from a symmetric monoidal bicategory, and use that to show that the classifying space is an infinite loop space upon group completion. We also show a way to relate this construction to the classic Gamma-category construction for a bipermutative category. As an example, we use this machinery to construct a delooping of the K-theory of a bimonoidal category as defined by Baas-Dundas-Rognes.
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.
A monoidal Grothendieck construction for $\infty$-categories
本文提出了一种在∞-范畴上构造单子Grothendieck构造的方法，证明了coCartesian纤维的∞-范畴与Day卷积单子结构的等价性。
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.
What are symmetric monoidal categories?
研究对称幺半范畴，证明其2-范畴与$\sP$-伪代数及严格特殊$\sF$-伪代数的2-范畴等价，此等价为无限循环空间理论提供简化增强，有独立研究意义。
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 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 …
On the equivalence of two approaches to multiplicative homotopy theories
本文探讨了两种乘法同调理论框架的等价性，解决了Pavlov猜想并解答了Hovey第十问题的特殊情况，同时证明了多个主定理的变体。
In [MMO] (arXiv:1704.03413), we reworked and generalized equivariant infinite loop space theory, which shows how to construct $G$-spectra from $G$-spaces with suitable structure. In this paper, we construct a new variant of the equivariant Segal machine that starts from the category $\scr{F}$ of finite sets rather than from the category ${\scr{F}}_G$ of finite $G$-sets and which is equivalent to the machine studied by Shimakawa and in [MMO]. In contrast to the machine in [MMO], the new machine gives a lax symmetric monoidal functor from the symmetric monoidal category of $\scr{F}$-$G$-spaces t…
Symmetric Monoidal Bicategories and Biextensions
本文借助范畴扩张及 Picard 群胚系数的上同调数据研究幺半 2-范畴与双范畴，证明幺半结构生成双扩张，并揭示交换性条件的渐进稳定化与双扩张对称性及 MacLane 同调的立方 Q-构造的对应关系。
On products of skeleta
该研究在相容有限余极限的对称幺半∞-范畴中，证明骨架滤过函子是典范松弛对称幺半的，同时建立了∞-前幺半范畴的局部化等新结论，为同伦论中相关构造提供了范畴论基础。
Lax monoidal adjunctions, two-variable fibrations and the calculus of mates
本文建立面向 ∞-范畴函子的配偶演算，推广 unstraightening 等价，并通过二变量纤维化刻画 lax/oplax 幺半伴随及 Gray 张量积函子。
Constructions of spectra from symmetric monoidal categories are typically functorial with respect to strict structure-preserving maps, but often the maps of interest are merely lax monoidal. We describe conditions under which one can transport the weak equivalences from one category to another with the same objects and a broader class of maps. Under mild hypotheses this process produces an equivalence of homotopy theories. We describe examples including algebras over an operad, such as symmetric monoidal categories and $n$-fold monoidal categories; and diagram categories, such as $Γ$-categorie…
Genuine vs. naïve symmetric monoidal G-categories
本文证明在等变弱等价意义下，真正的对称幺半 G-范畴等价于带 G-作用的普通对称幺半范畴，并给出全局无限环路空间的运算子模型及与 G-可部分求和范畴的等价。
We define a tracelike transformation to be a natural family of conjugation invariant maps $T_{x,C}: hom_C(x,x) \to hom_C(1,1)$ for all dualisable objects $x$ in any symmetric monoidal infinity-category $C$. This generalises the trace from linear algebra that assigns a scalar $Tr(f) \in k$ to any endomorphism $f:V \to V$ of a finite-dimensional $k$-vector space. Our main theorem computes the moduli space of tracelike transformations using the one-dimensional cobordism hypothesis with singularities. As a consequence we show that the trace $Tr$ can be uniquely extended to a tracelike transformati…
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 mixed Hodge theory to show that the functor of singular chains with rational coefficients is formal as a lax symmetric monoidal functor, when restricted to complex schemes whose weight filtration in cohomology satisfies a certain purity property. This has direct applications to the formality of operads or, more generally, of algebraic structures encoded by a colored operad. We also prove a dual statement, with applications to formality in the context of rational homotopy theory. In the general case of complex schemes with non-pure weight filtration, we relate the singular chains functor…
The long hunt for a symmetric monoidal category of spectra finally ended in success with the simultaneous discovery of the third author's discovery of symmetric spectra and the Elmendorf-Kriz-Mandell-May category of S-modules. In this paper we define and study the model category of symmetric spectra, based on simplicial sets and topological spaces. We prove that the category of symmetric spectra is closed symmetric monoidal and that the symmetric monoidal structure is compatible with the model structure. We prove that the model category of symmetric spectra is Quillen equivalent to Bousfield a…
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.
Using the category of finite sets and injections, we construct a new model for the multilinearization of multifunctors between spaces that appears in the derivatives of Goodwillie calculus. We show that this model yields a lax monoidal functor from the category of symmetric functor sequences to the category of symmetric sequences of spaces after evaluating at the unit.
Equivariant algebraic $K$-theory of symmetric monoidal Mackey functors
本文统一了等变对称幺半范畴的代数 K 理论构造，并证明每个连通 genuine G-谱等价于 Bohmann-Osorno 范畴 Mackey 函子的等变代数 K 理论。
Infinity categories with duality and hermitian multiplicative infinite loop space machines
本文证明具有对偶的预加性无穷范畴可产生直和埃尔米特K-理论谱，并构造E-无穷环谱，同时给出典范对偶的存在性及对偶的分类与扭曲。
We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. As an application we show that with an added assumption about underlying model structures Margolis' axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence. Also, the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (S-modules, \W-spaces, orthogonal sp…
On the global homotopy theory of symmetric monoidal categories
本文证明可和范畴的整体同伦论可由对称幺半范畴建模，并等价于可和单纯集的同伦论，为整体相干交换幺半群提供桥梁。
The 2-category V-Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. The exception is the case in which V is symmetric, which leads to V-Cat being symmetric as well. This paper describes how these facts are related to a categorical analogue of topological delooping. It seems that the analogy of loop spaces is a good guide for how to define the concept of enrichment over various types of monoidal objects, including k-fold monoidal categories and their higher dimensional counterparts. The main result is that for V a…
We study the category of Reedy diagrams in a $\mm$-model category. Explicitly, we show that if K is a small category, V is a closed symmetric monoidal category and C is a closed V-module, then the diagram category V^K is a closed symmetric monoidal category and the diagram category C^K is a closed V^K-module. We then prove that if further K is a Reedy category, V is a monoidal model category and C is a V-model category, then with the Reedy model category structures, V^K is a monoidal model category and C^K$ is a $\mm^K-model category provided that either the unit 1 of V is cofibrant or V is co…
The higher algebra and geometry of monoidal bicategories
该研究通过分析小立方操作符的几何，揭示了辫状、硅勒克和对称单调双范畴作为$\mathsf{E}_k$-单子的结构特性。
Joyal and Street note in their paper on braided monoidal categories [Braided tensor categories, Advances in Math. 102(1993) 20-78] that the 2-category V-Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. The exception that they mention is the case in which V is symmetric, which leads to V-Cat being symmetric as well. The symmetry in V-Cat is based upon the symmetry of V. The motivation behind this paper is in part to describe how these facts relating V and V-Cat are in turn related to a categorical analogue of to…
A variant of the trace in a monoidal category is given in the setting of closed monoidal derivators, which is applicable to endomorphisms of fiberwise dualizable objects. Functoriality of this trace is established. As an application, an explicit formula for the trace of the homotopy colimit of endomorphisms over finite EI-categories is deduced. This result can be seen as a generalization of the additivity of traces in monoidal categories with a compatible triangulation.
Monoidal Structures in Orthogonal Calculus
本文研究正交微积分中带 lax 对称幺半结构函子的 Taylor 逼近与导数谱，证明其 Taylor 逼近保持 lax 对称幺半结构，并给出导数谱的 O(n)-等变结构映射。
Symmetric Monoidal Smash Products in Homotopy Type Theory
本文针对同伦类型论中迭代砸积引发的指数级相干验证难题，提出用于构造函数同伦的归纳启发式方法，证明带点类型宇宙上的砸积具有（1-相干）对称幺半性，并在 Cubical Agda 中形式化关键结果。
Continuing our project on noncommutative (stable) homotopy we construct symmetric monoidal $\infty$-categorical models for separable $C^*$-algebras $\mathtt{SC^*_\infty}$ and noncommutative spectra $\mathtt{NSp}$ using the framework of Higher Algebra due to Lurie. We study smashing (co)localizations of $\mathtt{SC^*_\infty}$ and $\mathtt{NSp}$ with respect to strongly self-absorbing $C^*$-algebras. We analyse the homotopy categories of the localizations of $\mathtt{SC^*_\infty}$ and give universal characterizations thereof. We construct a stable $\infty$-categorical model for bivariant connect…
In this paper we prove that a morphism between schemes or stacks naturally corresponds to a symmetric monoidal functor between stable infinity-categories of quasi-coherent complexes. It can be viewed as a derived analogue of Tannaka duality. As a consequence, we deduce that an algebraic stack satisfying a certain condition can be recovered from the stable infinity-category of quasi-coherent complexes with tensor operation.
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