Bruno Vallette
We give an explicit construction of the free monoid in monoidal abelian categories when the monoidal product does not necessarily preserve coproducts. We apply it to several new monoidal categories that appeared recently in the theory of Koszul duality for operads and props. This gives a conceptual explanation of the form of the free operad, free dioperad, free properad and free prop.
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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…
Skew-monoidal categories arise when the associator and the left and right units of a monoidal category are, in a specific way, not invertible. We prove that the closed skew-monoidal structures on the category of right R-modules are precisely the right bialgebroids over the ring R. These skew-monoidal structures induce quotient skew-monoidal structures on the category of R-R-bimodules and this leads to the following generalization: Opmonoidal monads on a monoidal category correspond to skew-monoidal structures with the same unit object which are compatible with the ordinary monoidal structure b…
Kornel Szlachányi recently used the term skew-monoidal category for a particular laxified version of monoidal category. He showed that bialgebroids $H$ with base ring $R$ could be characterized in terms of skew-monoidal structures on the category of one-sided $R$-modules for which the lax unit was $R$ itself. We define skew monoidales (or skew pseudo-monoids) in any monoidal bicategory $\mathscr M$. These are skew-monoidal categories when $\mathscr M$ is $\mathrm{Cat}$. Our main results are presented at the level of monoidal bicategories. However, a consequence is that quantum categories in th…
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…
We prove Heyneman-Radford Theorem in the framework of Monoidal Categories.
We construct an equivalence between the 2-categories VMonCat of rigid V-monoidal categories for a braided monoidal category V and VModTens of oplax braided functors from V into the Drinfeld centers of ordinary rigid monoidal categories. The 1-cells in each are the respective lax monoidal functors, and the 2-cells are the respective monoidal natural transformations. Our proof also gives an equivalence in the case that we consider only strong monoidal 1-cells on both sides. The 2-categories VMonCat and VModTens have G-graded analogues. We also get an equivalence of 2-categories between G-extensi…
We develop the theory of weak bimonoids in braided monoidal categories and show them to be quantum categories in a certain sense. Weak Hopf monoids are shown to be quantum groupoids. Each separable Frobenius monoid R leads to a weak Hopf monoid R \otimes R.
This paper is about skew monoidal tensored V-categories (= skew monoidal hommed V-actegories) and their categories of modules. A module over <M,*,R> is an algebra for the monad T = R * _ on M. We study in detail the skew monoidal structure of M^T and construct a skew monoidal forgetful functor from M^T to the category of E-objects in M where E=M(R,R) is the endomorphism monoid of the unit object R. Then we give conditions for the forgetful functor to be strong monoidal and for the category M^T of modules to be monoidal. In formulating these conditions a notion of `self-cocomplete' subcategorie…
The analogy between Yetter's deformation theory form (lax) monoidal functors and Gerstenahaber's deformation theory for associative algebras is solidified by shown that under reasonable conditions the category of functors with an action of a lax monoidal functor is abelian, that an analogue of the Hochschild cohomology of an algebra with coefficients in a bimodule exists for monoidal functors, and is given by right derived functors. The deformation cohmology of a monoidal natural transformation is shown to be a special case.
Skew monoidal categories are monoidal categories with non-invertible `coherence' morphisms. As shown in a previous paper bialgebroids over a ring R can be characterized as the closed skew monoidal structures on the category Mod R in which the unit object is R. This offers a new approach to bialgebroids and Hopf algebroids. Little is known about skew monoidal structures on general categories. In the present paper we study the one-object case: skew monoidal monoids (SMM). We show that they possess a dual pair of bialgebroids describing the symmetries of the (co)module categories of the SMM. Thes…
This article represents a preliminary attempt to link Kan extensions, and some of their further developments, to Fourier theory and quantum algebra through *-autonomous monoidal categories and related structures.
Division algebras in monoidal categories
针对幺半范畴中的除代数定义问题，将域上除代数的等价定义适配到幺半范畴场景，构建多类除代数定义并给出相关示例与推论。
Multiplier bimonoids (or bialgebras) in arbitrary braided monoidal categories are defined. They are shown to possess monoidal categories of comodules and modules. These facts are explained by the structures carried by their induced functors.
We study the basic monoidal properties of the category of Hopf modules for a coquasi Hopf algebra. In particular we discuss the so called fundamental theorem that establishes a monoidal equivalence between the category of comodules and the category of Hopf modules. We present a categorical proof of Radford's $S^4$ formula for the case of a finite dimensional coquasi Hopf algebra, by establishing a monoidal isomorphism between certain double dual functors.
Quantum categories have been recently studied because of their relation to bialgebroids, small categories, and skew monoidales. This is the first of a series of papers based on the author's PhD thesis in which we examine the theory of quantum categories developed by Day, Lack, and Street. A quantum category is an opmonoidal monad on the monoidale associated to a biduality $R\dashv R^{\circ}$, or enveloping monoidale, in a monoidal bicategory of modules $\mathsf{Mod}(\mathcal{V})$ for a monoidal category $\mathcal{V}$. Lack and Street proved that quantum categories are in equivalence with right…
We introduce monoidal categories whose monoidal products of any positive number of factors are lax coherent and whose nullary products are oplax coherent. We call them $\mathsf{Lax}^+\mathsf{Oplax}^0$-monoidal. Dually, we consider $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal categories which are oplax coherent for positive numbers of factors and lax coherent for nullary monoidal products. We define $\mathsf{Lax}^+_0\mathsf{Oplax}^0_+$-duoidal categories with compatible $\mathsf{Lax}^+\mathsf{Oplax}^0$- and $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal structures. We introduce comonoids in $\mathsf{La…
In our recent paper [Sh1] a version of the "generalized Deligne conjecture" for abelian $n$-fold monoidal categories is proven. For $n=1$ this result says that, given an abelian monoidal $k$-linear category $\mathscr{A}$ with unit $e$, $k$ a field of characteristic 0, the dg vector space $\mathrm{RHom}_{\mathscr{A}}(e,e)$ is the first component of a Leinster 1-monoid in $\mathscr{A}lg(k)$ (provided a rather mild condition on the monoidal and the abelian structures in $\mathscr{A}$, called homotopy compatibility, is fulfilled). In the present paper, we introduce a new concept of a ${\it graded}…
Given a monoidal category C, an ordinary category M, and a monad T in M, the lifts in a strict sense of a fixed action of C on M to an action of C on the Eilenberg-Moore category of T-modules in M are in a bijective correspondence with certain families of natural transformations, indexed by the objects in the monoidal category C. These families are analogues of distributive laws between monads.
A Graded Schur Lemma and a graded-monoidal structure for induced modules over graded-commutative algebras
本文在有限阿贝尔群分次和辫化幺半范畴框架下推广舒尔引理，并证明当代数按分次群双特征标交换时，诱导模范畴具有由双特征标扭转的分次幺半结构，从而统一并推广了超代数情形。
We study integrals of Hopf monoids in duoidal endohom categories of naturally Frobenius map monoidales in monoidal bicategories. We prove two Maschke type theorems, relating the separability of the underlying monoid and comonoid, respectively, to the existence of normalized integrals. It covers the examples provided by Hopf monoids in braided monoidal categories, weak Hopf algebras, Hopf algebroids over central base algebras, Hopf monads on autonomous monoidal categories and Hopf categories.
Here we describe three straightforward examples of what was called a graphic Fourier transformation in [4]. At least two of these examples may be viewed simply as monoidal comonads on suitable monoidal closed functor categories, but the third example, which involves "centres" of monoidal closed functor categories, is generally not comonadic. For the first two examples (i.e., association schemes and RCFT's), a more elaborate "probicategory" set-up was envisaged in an earlier version of this note, but many readers missed the main point so it is simplified (hopefully) below.
Centers of Algebras in Monoidal 2-Categories
介绍幺半 2- 范畴中代数的左、右和完全中心并证明其莫里塔不变性，对戴维多夫理论进行范畴化，统一阐述融合范畴等的中心理论，还发展了 2- 伴随和模 2- 范畴的基变换构造。
We define a notion of category enriched over an oplax monoidal category $V$, extending the usual definition of category enriched over a monoidal category. Even though oplax monoidal structures involve infinitely many functors $V^n\to V$, defining categories enriched over $V$ only requires the lower arity maps $(n \leq 3)$, similarly to the monoidal case. The focal point of the enrichment theory shifts, in the oplax case, from the notion of $V$-category (given by collections of objects and hom-objects together with composition and unit maps) to the one of categories enriched over $V$ (genuine c…
The centre of a monoidal category is a braided monoidal category. Monoidal categories are monoidal objects (or pseudomonoids) in the monoidal bicategory of categories. This paper provides a universal construction in a braided monoidal bicategory that produces a braided monoidal object from any monoidal object. Some properties and sufficient conditions for existence of the construction are examined.
We denote the monoidal bicategory of two-sided modules (also called profunctors, bimodules and distributors) between categories by $\mathrm{Mod}$; the tensor product is cartesian product of categories. For a groupoid $\scr{G}$, we study the monoidal centre $\mathrm{ZPs}(\scr{G},\mathrm{Mod}^{\mathrm{op}})$ of the monoidal bicategory $\mathrm{Ps}(\scr{G},\mathrm{Mod}^{\mathrm{op}})$ of pseudofunctors and pseudonatural transformations; the tensor product is pointwise. Alexei Davydov defined the full centre of a monoid in a monoidal category. We define a higher dimensional version: the full monoi…
We introduce semidirect products of skew monoidal categories as a categorification of semidirect products of monoids (or, perhaps more familiarly, of groups). We also discuss how this construction interacts with monoidal, autonomous and closed monoidal structures. We end by producing some examples of monoidal categories which are left closed but not right closed.
We present a method of constructing monoidal, braided monoidal, and symmetric monoidal bicategories from corresponding types of monoidal double categories that satisfy a lifting condition. Many important monoidal bicategories arise naturally in this way, and applying our general method is much easier than explicitly verifying the coherence laws of a monoidal bicategory for each example. Abstracting from earlier work in this direction, we express the construction as a functor between locally cubical bicategories that preserves monoid objects; this ensures that it also preserves monoidal functor…
We show that the equivalence between several possible characterizations of Frobenius algebras, and of symmetric Frobenius algebras, carries over from the category of vector spaces to more general monoidal categories. For Frobenius algebras, the appropriate setting is the one of rigid monoidal categories, and for symmetric Frobenius algebras it is the one of sovereign monoidal categories. We also discuss some properties of Nakayama automorphisms.
We present a method of constructing symmetric monoidal bicategories from symmetric monoidal double categories that satisfy a lifting condition. Such symmetric monoidal double categories frequently occur in nature, so the method is widely applicable, though not universally so.
The Relative Monoidal Center and Tensor Products of Monoidal Categories
本文建立增广幺半范畴理论，定义平衡双模，构造相对张量积与相对幺半中心并证明其Morita对偶性，还在量子代数等实例中验证了相对中心的范畴等价性。
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。
Frobenius monoidal functors preserve duals. We show that conversely, (co)monoidal functors between autonomous categories which preserve duals are Frobenius monoidal. We apply this result to linearly distributive functors between autonomous categories.
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…
We study the bialgebra structures on quiver coalgebras and the monoidal structures on the categories of locally nilpotent and locally finite quiver representations. It is shown that the path coalgebra of an arbitrary quiver admits natural bialgebra structures. This endows the category of locally nilpotent and locally finite representations of an arbitrary quiver with natural monoidal structures from bialgebras. We also obtain theorems of Gabriel type for pointed bialgebras and hereditary finite pointed monoidal categories.
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。
We show that, with some technical conditions, an abelian category can be embedded into the category of bimodules over a ring. The case of semisimple rigid monoidal categories is studied in more detail.
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.
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.
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.
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.
On pushouts of strict symmetric monoidal categories
本文为严格对称幺半范畴的推出给出显式构造，证明辫子强幺半函子强保持中心幂等元，并推论其中心幂等元格与严格化版本同构，服务于计算并发建模。
Monoidal Ringel duality and monoidal highest weight envelopes
本文提出通过单子增强的半无限环el对偶，将非阿贝尔单子范畴嵌入具有最高权结构的阿贝尔单子范畴，并应用于李代数表示的单子结构研究。
We study the action of monads on categories equipped with several monoidal structures. We identify the structure and conditions that guarantee that the higher monoidal structure is inherited by the category of algebras over the monad. Monoidal monads and comonoidal monads appear as the base cases in this hierarchy. Monads acting on duoidal categories constitute the next case. We cover the general case of $n$-monoidal categories and discuss several naturally occurring examples in which $n\leq 3$.
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.
We prove that a category which is symmetric (relaxed) monoidal closed, (small) complete, well-powered and has a small cogenerating family, is cocomplete.
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 introduce monoidal width as a measure of complexity for morphisms in monoidal categories. Inspired by well-known structural width measures for graphs, like tree width and rank width, monoidal width is based on a notion of syntactic decomposition: a monoidal decomposition of a morphism is an expression in the language of monoidal categories, where operations are monoidal products and compositions, that specifies this morphism. Monoidal width penalises the composition operation along ``big'' objects, while it encourages the use of monoidal products. We show that, by choosing the correct categ…
We discuss generalised duality theory for monoidal categories and its applications to the categories of exact endofunctors, graded vector spaces, and topological vector spaces.
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 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.
The basic data for a skew-monoidal category are the same as for a monoidal category, except that the constraint morphisms are no longer required to be invertible. The constraints are given a specific orientation and satisfy Mac Lane's five axioms. Whilst recent applications justify the use of skew-monoidal structure, they do not give an intrinsic justification for the form the structure takes (the orientation of the constraints and the axioms that they satisfy). This paper provides a perspective on skew-monoidal structure which, amongst other things, makes it quite apparent why this particular…
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…
This is an account of some work of Markus Rost and his students Dominik Boos and Susanne Maurer. We adapt it to the braided monoidal setting.
We describe a perfect correspondence between skew monoidal categories and certain generalised multicategories, called skew multicategories, that arise in nature.
We present Trimble's definition of a tetracategory and prove that the spans in (strict) 2-categories with certain limits have the structure of a monoidal tricategory, defined as a one-object tetracategory. We recall some notions of limits in 2-categories for use in the construction of the monoidal tricategory of spans.
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。
Multicategories from Symmetric Monoidal Categories
本文研究对称幺半范畴的底层多范畴，证明其在典范融贯同构下唯一，定义了对应遗忘函子并证明其有弱左伴随，且伴随单子给出严格化构造。
Change of Enrichment for Monoidal Model Categories
研究幺半模型范畴的富集变化，将经典范畴论定理扩展到该范畴，在特定假设下改进先前结果。以链复形范畴为例，其两种幺半结构产生幺半模型范畴，还借此重现增广dg - 范畴的平方零扩张伴随。
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.
Coinductive Streams in Monoidal Categories
本文将因果流函数推广为任意对称幺半范畴上的幺半流，用于为过程理论数据流编程提供语义，并利用余归纳弦图推理反馈幺半范畴，最后以随机数据流语言为例说明其应用。
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