The symmetric monoidal 2-category of permutative categories
Nick Gurski, Niles Johnson, Angélica M. Osorno
本文为置换范畴定义张量积，证明其使置换范畴的2-范畴成为闭对称幺半双范畴，奠定了相关结构的基础。
我们定义了置换范畴的张量积，并证明了若干关键性质。我们表明，该积使得置换范畴的2-范畴作为双范畴成为闭对称幺半的。
We define a tensor product for permutative categories and prove a number of key properties. We show that this product makes the 2-category of permutative categories closed symmetric monoidal as a bicategory.
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We offer two proofs that categories weakly enriched over symmetric monoidal categories can be strictified to categories enriched in permutative categories. This is a "many 0-cells" version of the strictification of bimonoidal categories to strict ones.
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 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.
Enriched Morita theory of monoids in a closed symmetric monoidal category
本文在丰富范畴论框架下，针对闭对称幺半范畴中的幺半群建立了Morita理论，拓展了经典代数中的Morita等价研究方法。
In that paper, we prove that the collection of all FRBSU monoidal categories and the collection of all crossed modules form a 2 category.
We construct a compact closed category out of any symmetric monoidal category by freely adding adjoints to its objects. The morphisms of the completion are defined as string diagrams annotated by objects and morphisms from the original category. The symmetric monoidal category embeds via a faithful monoidal functor into its completion, but in contrast to the non-symmetric case, this embedding is not full. Our construction factors through the Int construction, which yields another free construction: the free traced monoidal category on a symmetric monoidal category.
A universal property of the monoidal 2-category of cospans of finite linear orders and surjections
证明了有限线性级和满射的余扩张的monoidal 2-范畴是具有半群和余半群结构的对象X的泛幺半范畴，其中这两种结构满足一定的二维可分代数条件.
We introduce the notion of the quotient of a category $C$ by the action $A : M \longrightarrow C \times C$ of a unital symmetric monoidal category $M$. The quotient $C/M$ is a 2-category. We prove its existence and uniqueness by first showing that every small 2-category has a presentation in terms of generators and relations and then describing the generators and relations needed for the quotient $C/M$.
We give a 3-categorical, purely formal argument explaining why on the category of Kleisli algebras for a lax monoidal monad, and dually on the category of Eilenberg-Moore algebras for an oplax monoidal monad, we always have a natural monoidal structures. The key observation is that the 2-category of lax monoidal monads in any 2-category D with finite products is isomorphic to the 2-category of monoidal objects with oplax morphisms in the 2-category of monads with lax morphisms in D. As we explain at the end of the paper a similar phenomenon occurs in many other situations.
In this paper we construct a symmetric monoidal closed model category of coherently commutative Picard groupoids. We construct another model category structure on the category of (small) permutative categories whose fibrant objects are (permutative) Picard groupoids. The main result is that the Segal's nerve functor induces a Quillen equivalence between the two aforementioned model categories. Our main result implies the classical result that Picard groupoids model stable homotopy one-types.
Let $B$ be a bialgebra, and $A$ a left $B$-comodule algebra in a braided monoidal category $\Cc$, and assume that $A$ is also a coalgebra, with a not-necessarily associative or unital left $B$-action. Then we can define a right $A$-action on the tensor product of two relative Hopf modules, and this defines a monoidal structure on the category of relative Hopf modules if and only if $A$ is a bialgebra in the category of left Yetter-Drinfeld modules over $B$. Some examples are given.
We define the Drinfeld center of a monoidal category enriched over a braided monoidal category, and show that every modular tensor category can be realized in a canonical way as the Drinfeld center of a self-enriched monoidal category. We also give a generalization of this result for important applications in physics.
Every small monoidal category with universal finite joins of central idempotents is monoidally equivalent to the category of global sections of a sheaf of local monoidal categories on a topological space. Every small stiff monoidal category monoidally embeds into such a category of global sections. An infinitary version of these theorems also holds in the spatial case. These representation results are functorial and subsume the Lambek-Moerdijk-Awodey sheaf representation for toposes, the Stone representation of Boolean algebras, and the Takahashi representation of Hilbert modules as continuous…
A (not so) short note: the equivalence of various notions of symmetric monoidal category
本文证明了经典偏向对称幺半范畴、无偏对称幺半范畴和同伦对称幺半范畴在精确意义下等价。
Let $\mathcal{B}$ be a subcategory of a given category $\mathcal{D}$. Let $\mathcal{B}$ has monoidal structure. In this article, we discuss when can one extend the monoidal structure of $\mathcal{B}$ to $\mathcal{D}$ such that $\mathcal{B}$ becomes a sub monoidal category of monoidal category $\mathcal{D}$. Examples are discussed, and in particular, in an example of loop space, we elaborated all results discussed in this article.
In this paper, we show another proof of the problem by constructing a strict monoidal category M(C) consisting of M-functors and M-morphisms of a category C and we prove C is equivalent to it. The proof is based on a basic character of monoidal equivalences. Ideas and techniques of these proofs can been used to prove the equivalence between an Ann-category and an almost strict Ann-category.
We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely weak monoidal pseudofunctors to the 2-category of categories. In doing so, we investigate the relation between this `global' monoidal structure where the total category is monoidal and the fibration strictly preserves the structure, and a `fibrewise' one where the fibres are monoidal and the reindexing functors strongly preserve the structure, first hinted by Shulman. In particular, when the domain is cocartesian monoidal, lax monoidal…
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。
Coherence for Categorified Operadic Theories
我们早就知道，每个弱monoidal范畴A通过monoidal函子和monoidal自然变换等价于严格的幺半范畴st（A）。推广了弱幺半范畴范畴的定义，给出了强正则（运算）理论P的弱P-范畴的定义，并证明了每个弱P-范畴通过P-函子和P-变换等价于严格P-范畴.这个严格函子，然后被证明有一个有趣的泛性质。
Discrete vertices in simplicial objects internal to a monoidal category
本文在幺半范畴中引入单纯对象，比较三种离散顶点定义，证明基于点函子与templicial对象等价，且与丰富Segal预范畴在笛卡尔情形等价。
The monoidal centre as a limit
幺半范畴的中心是辫状幺半范畴。幺半范畴是范畴的monoidal双范畴中的monoidal对象（或伪对象）。本文提供了一个通用的构造在一个编织monoidal双范畴，产生一个编织monoidal对象从任何monoidal对象。研究了该构造的一些性质和存在的充分条件。
The category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category $\mathbb A$, the corresponding internal hom functor $|[ \mathbb A,-]|$ sends a double category $\mathbb B$ to the double category whose 0-cells are the double functors $\mathbb A \to \mathbb B$, whose horizontal and vertical 1-cells are the horizontal and vertical pseudotransformations, respectively, and whose 2-cells are the modifications. Some well-known functors of practical significance are checked to be compatible with this monoidal structure.
Tannaka Duality describes the relationship between algebraic objects in a given category and their representations; an important case is that of Hopf algebras and their categories of representations; these have strong monoidal forgetful "fibre functors" to the category of vector spaces. We simultaneously generalize the theory of Tannaka duality in two ways: first, we replace Hopf algebras with weak Hopf algebras and strong monoidal functors with separable Frobenius monoidal functors; second, we replace the category of vector spaces with an arbitrary braided monoidal category. To accomplish thi…
This article shows that the units of a skew monoidal category are unique up to a unique isomorphism, and internalises this fact to skew monoidales. Some benefits of certain extra structure on the unit maps are also discussed before the axioms of a skew monoidal category are shown to be independent.
Grothendieck prelopologies: towards a closed monoidal sheaf category
本文推广 Grothendieck 预拓扑至半笛卡尔范畴，通过不保持有限极限的层化过程，构造闭幺半群层范畴，并受量子格上层概念的启发。
We study the monoidal structure of the standard strictification functor $\textrm{st}:\mathbf{Bicat} \rightarrow \mathbf{2Cat}$. In doing so, we construct monoidal structures on the 2-category whose objects are bicategories and on the 2-category whose objects are 2-categories.
We show that each rigid monoidal category A over a field defines a family of universal tensor categories, which together classify all faithful monoidal functors from A to tensor categories. Each of the universal tensor categories classifies monoidal functors of a given 'homological kernel' and can be realised as a sheaf category, not necessarily on A. This yields a theory of 'local abelian envelope' which completes the notion of monoidal abelian envelopes.
We exhibit sufficient conditions for a monoidal monad T on a monoidal category C to induce a monoidal structure on the Eilenberg--Moore category C^T that represents bimorphisms. The category of actions in C^T is then shown to be monadic over the base category C.
A symmetric monoidal Comparison Lemma
本文证明比较引理的对称幺半版本，建立松弛与强对称幺半超层范畴间的等价，并用于获得紧支撑上同调理论结果的松弛对称幺半推广。
Free skew monoidal categories
本文显式刻画单个生成对象上的自由幺半范畴 Fsk，并推广到任意范畴上的自由斜幺半范畴，进而建立相关 operad 间的伴随及与 colax L-代数的对应。
The Strict 2-Category Structure of Distorted Monoidal Categories
本文提出扭曲单子范畴的严格2-范畴结构，通过构造性演算实现非可逆交换的正式推理，扩展单子范畴的表达能力以建模不可逆过程。
An $\mathcal{O}$-monoidal Grothendieck construction
本文在2-范畴中定义弱O-幺半群（O-伪幺半群），统一幺半与对称幺半范畴的无偏概念，并利用经典Grothendieck构造建立O-幺半版本，联系到Set的lax O-幺半函子与离散纤维化的严格O-幺半函子。
Many monoidal-type objects are known to be classified by maps from the Catalan simplicial set $\mathbb{C}$ to various nerves of categories and higher categories. There are, for example, three different nerves of the 2-category of categories $\mathsf{Cat}$ with the property that maps from $\mathbb{C}$ into them are in one to one correspondence with strict monoidal, monoidal, and skew monoidal categories. In this last case, the classification result can be understood as justifying the definition of skew monoidal categories intrinsically in terms of the structure of $\mathbb{C}$. In this paper, w…
We explain two related constructions on the data of two monoidal symmetric closed categories $\mathscr{A}$ and $\mathscr{E}$ and monoidal functors $F: \mathscr{E}\to \mathscr{A}$ and $G: \mathscr{A}\to \mathscr{E}$. In a first part, we recall and partly extend work of A. Kock: In case $F$ is left-adjoint to $G$, and this adjunction is monoidal, we can equip the Eilenberg-Moore category $\mathscr{E}^T$ for $T$ being the canonical monad associated to the adjunction, with the structure of symmetric monoidal closed category, provided $\mathscr{E}$ has equalizers and $\mathscr{E}^T$ co-equalizers. …
The monoid of multipliers of a semigroup object in a monoidal category is introduced, arising from an abstraction of the definition of the translational hull of an ordinary semigroup or of the multiplier algebra of a Banach algebra and dually, the monoid of comultipliers of a cosemigroup object is obtained. Its set-theoretic version, the classical translational hull, is shown to provide a functor from a subcategory of ordinary semigroups to monoids, similar to a left adjoint. The abstract multiplier monoid of a semigroup object is related to the concrete translational hull of its convolution s…
Lemme de Yoneda pour les foncteurs à valeurs monoidales
本研究针对闭对称幺半范畴$\u2716M$，证明小范畴$I$对应的函子范畴$\u2716M^I$是闭$\u2716M$-模，重写幺半值函子的米田引理，推导伴随函子定理并验证$\u2716M^I$为闭对称幺半范畴。
Closed categories, star-autonomy, and monoidal comonads
本文确定了幺半范畴范畴C中的内homs对于C上的幺半群余单子G提升到Eilenberg-Moore余代数范畴C^G所需的结构。我们应用这一点来提升星自治，以期重铸量子广群的定义。
We study semi-strict tricategories in which the only weakness is in vertical composition. We construct these as categories enriched in the category of bicategories with strict functors, with respect to the cartesian monoidal structure. As these are a form of tricategory it follows that doubly-degenerate ones are braided monoidal categories. We show that this form of semi-strict tricategory is weak enough to produce all braided monoidal categories. That is, given any braided monoidal category $B$ there is a doubly-degenerate ``vertically weak'' semi-strict tricategory whose associated braided m…
We study cocompleteness, co-wellpoweredness and generators in the centralizer category of an object or morphism in a monoidal category, and the center or the weak center of a monoidal category. We explicitly give some answers for when colimits, cocompleteness, co-wellpoweredness and generators in these monoidal categories can be inherited from their base monidal categories. Most importantly, we investigate cofree objects of comonoids in these monoidal categories.
We study internal structures in regular categories using monoidal methods. Groupoids in a regular Goursat category can equivalently be described as special dagger Frobenius monoids in its monoidal category of relations. Similarly, connectors can equivalently be described as Frobenius structures with a ternary multiplication. We study such ternary Frobenius structures and the relationship to binary ones, generalising that between connectors and groupoids.
We start from any small strict monoidal braided Ab-category and extend it to a monoidal nonstrict braided Ab-category which contains braided bialgebras. The objects of the original category turn out to be modules for these bialgebras
A higher-order Eckmann-Hilton argument
本文提出纯代数的高阶高维Eckmann-Hilton论证，证明三个适当互换的幺半结构迫使每个典范辫成为对称，并应用于n-退化半严格(n+1)-范畴。
Quotient Category of a Multiring Category
本文为阿贝尔幺半范畴的Serre商范畴引入张量结构，证明多环范畴（多张量范畴）关于双边Serre张量理想的商仍为多环范畴（多张量范畴），并指出张量范畴的双边Serre张量理想平凡。
Elementary remarks on units in monoidal categories
我们探讨了一种替代的定义单位在一个幺半范畴最初由于Saavedra：一个Saavedra单位是一个可消幂等（在1-范畴意义上）。这个概念在左右约束方面比通常的概念更经济，并且受到高阶范畴理论的启发。首先，我们描述了一个给定的半幺半范畴上所有可能的单位结构的半幺半范畴，并观察到它是可收缩的（如果非空）。然后证明了这两个单位概念在强函子意义下是等价的。其次，证明了（强）monoidal函子的单位相容性条件正是该函子提升到单位范畴的条件，并解释了Saavedra单位的概念如何自然地导致等价的非代数的公平幺半范畴的概念，其中可收缩的多个单位被视为一个整体而不是选择一个单位.最后，考虑单位比较的宽松版本。纸是独立的。所有的论点都是基本的，其中一些具有某种美感。
Freely adjoining monoidal duals
本文构造了自由邻接右对偶的幺半范畴，证明其满足双伴随且全忠实，并给出显式几何与组合模型，推广至对偶序列情形。
Cartesian differential categories as skew enriched categories
本文将笛卡尔微分范畴刻画为基于斜幺半范畴的充实范畴，并证明每个小笛卡尔微分范畴可完全保持结构地嵌入由微分模态诱导的范畴，从而解决了该领域的一个公开问题。
It is well-known that the category of Kleisli algebras for a monoidal monad carries a canonical monoidal structure. We define the notion of a commutative graded monad and present a strictly two-categorical proof that Kleisli algebras for such monads equally carry a canonical monoidal structure which reduce to the monoidal monad case when the commutative graded monad in question is graded over the trivial monoidal category.
I describe a generalization of the notion of operadic category due to Batanin and Markl. For each such operadic category I describe a skew monoidal category of collections, such that a monoid in this skew monoidal category is precisely an operad over the operadic category. In fact I describe two skew monoidal categories with this property. The first has the feature that the operadic category can be recovered from the skew monoidal category of collections; the second has the feature that the right unit constraint is invertible. In the case of the operadic category S of finite sets and functions…
Monoidal Context Theory
本文泛性质地刻画幺半透镜的 produoidal 范畴，并以对称幺半透镜描述多方消息传递协议，提出消息理论公理化及自由消息理论构造。
The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. The titular category has nice formal properties: it is bicomplete and is a symmetric monoidal category, with monoidal product closely related to the Boardman-Vogt tensor product of operads. Tools developed in this article, which is the first part of a larger work, include a generalized version of multilinearity of functors, a free prop construction defined on certain "generalized graphs", and the relationship between the category of props and the categories of per…
Doubles for monoidal categories
在最近的一篇论文中，Daisuke Tambara定义了monoidal V-范畴A的endodistributor（= endodistributor）上的双边作用。当A是自治的（=刚性=紧的）时，他证明了如此配备的内模（我们称之为Tambara模）的V范畴（我们称之为Tamb（A））等价于卷积monoidal V范畴的monoidal中心Z[A，V]范畴[A，V]。我们的论文在某种程度上扩展了这些想法。对于一般A，我们构造了一个拟原幺V范畴DA（我们建议称之为A的double），它具有等价[DA，V] \simeq Tamb（A）.当A是闭的时，我们定义了强（左强）Tambara模，并证明了它们构成一个V-范畴Tamb_s（A）（分别为Tamb_{ls}（A）），它等价于[A，V]的中心（分别为lax中心）.我们构造DA的局部化D_s A和D_{ls} A，使得存在等价Tamb_s（A）\simeq [D_s A，V]和Tamb_{ls}（A）\simeq [ D_{ls} A，V].当A是自治的时，每个Tambara模都是强的;这意味着等价Z[A，V] \simeq [DA，V]。
Operads for Symmetric Monoidal Categories
本文显式构造了以对称幺半范畴为代数的范畴operad，并将其嵌入四个operad的序列，从而得到四种逐级更严格的对称幺半范畴概念。
Module Categories As Spans
本文提出在一般单子2-范畴中模块与代数跨度之间的对应关系，并通过统一和推广模块函子的实现来展示其理论价值。
We introduce homotopical methods based on rewriting on higher-dimensional categories to prove coherence results in categories with an algebraic structure. We express the coherence problem for (symmetric) monoidal categories as an asphericity problem for a track category and we use rewriting methods on polygraphs to solve it. The setting is extended to more general coherence problems, seen as 3-dimensional word problems in a track category, including the case of braided monoidal categories.
Diagrammatics for lax and Frobenius monoidal functors and weak morphism classifiers
本文通过图示法构造了单子 functor 的图示结构，包括松、反松和弗罗贝尼乌斯松 functor 的变体，揭示了其与弱 morphism 分类器的关系。
In this paper we give an example of duoidal $\infty$-categories. We introduce map $\mathcal{O}$-monoidales in an $\mathcal{O}$-monoidal $(\infty,2)$-category for an $\infty$-operad $\mathcal{O}^{\otimes}$. We show that the endomorphism mapping $\infty$-category of a map $\mathcal{O}$-monoidale is a coCartesian $(Δ^{\rm op},\mathcal{O})$-duoidal $\infty$-category. After that, we introduce a convolution product on the mapping $\infty$-category from an $\mathcal{O}$-comonoidale to an $\mathcal{O}$-monoidale. We show that the $\mathcal{O}$-monoidal structure on the duoidal endomorphism mapping $\i…
Monoidal categories enriched in a braided monoidal category $\mathcal{V}$ are classified by braided oplax monoidal functors from $\mathcal{V}$ to the Drinfeld centers of ordinary monoidal categories. In this article, we prove that this classifying functor is strongly monoidal if and only if the original $\mathcal{V}$-monoidal category is tensored over $\mathcal{V}$. We then define a completion operation which produces a tensored $\mathcal{V}$-monoidal category $\overline{\mathcal{C}}$ from an arbitrary $\mathcal{V}$-monoidal category $\mathcal{C}$, and we determine many equivalent conditions w…
Closed symmetric monoidal structures on the category of graphs
本文研究了图分类的闭合对称单子结构，证明其存在盒积和范畴积两种闭合对称单子积。
We introduce the notion of a monoidal category enriched in a braided monoidal category $\mathcal V$. We set up the basic theory, and prove a classification result in terms of braided oplax monoidal functors to the Drinfeld center of some monoidal category $\mathcal T$. Even the basic theory is interesting; it shares many characteristics with the theory of monoidal categories enriched in a symmetric monoidal category, but lacks some features. Of particular note, there is no cartesian product of braided-enriched categories, and the natural transformations do not form a 2-category, but rather sat…
Using the symmetric monoidal closed category structure of the category of measurable spaces, in conjunction with the Giry monad which we show is a strong monad, we analyze Bayesian inference maps and their construction in relation to the tensor product probability. This perspective permits the inference maps to be seen as a pullback construction.
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