Takeshi Torii
We give another proof of the fact that there is a dual equivalence between the $\infty$-category of monoidal $\infty$-categories with left adjoint oplax monoidal functors and that with right adjoint lax monoidal functors by constructing a perfect pairing between them.
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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.
Uniqueness of monoidal adjunctions
本文研究了以右伴随lax和左伴随oplax $\mathcal{O}$-幺半函子构成的$\infty$-范畴间的对偶等价，证明了配备相容$\mathcal{O}$-幺半预层函子的两种等价具有典范等价性。
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…
Let V be a cofibrantly generated monoidal model category and let M be a monoidal V-model category. Given a cofibrant C-coloured operad P in V, we give sufficient conditions for the fibrant replacement and cofibrant replacement functors in M^C to preserve P-algebra structures. In particular, we show how P-algebra structures can be transferred along derived Quillen adjunctions of monoidal V-model categories, and we apply this result to the Quillen adjunctions defined by enriched Bousfield localizations and colocalizations on M. As an application, we prove that in the category of symmetric spectr…
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. …
Monoidal adjunctions and abelian envelopes
本文通过单子伴随证明伪张量范畴的单子阿贝尔包络存在性，提出通用准则并用组合方法证明超八面体和修改对称群插值范畴的新阿贝尔包络。
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。
Let $U$ be a strong monoidal functor between monoidal categories. If it has both a left adjoint $L$ and a right adjoint $R$, we show that the pair $(R,L)$ is a linearly distributive functor and $(U,U)\dashv (R,L)$ is a linearly distributive adjunction, if and only if $L\dashv U$ is a Hopf adjunction and $U\dashv R$ is a coHopf adjunction. We give sufficient conditions for a strong monoidal $U$ which is part of a (left) Hopf adjunction $L\dashv U$, to have as right adjoint a twisted version of the left adjoint $L$. In particular, the resulting adjunction will be (left) coHopf. One step further,…
Let $U:\mathcal{C}\rightarrow\mathcal{D}$ be a strong monoidal functor between abelian monoidal categories admitting a right adjoint $R$, such that $R$ is exact, faithful and the adjunction $U\dashv R$ is coHopf. Building on the work of Balan, we show that $R$ is separable (resp., special) Frobenius monoidal if and only if $R(\mathbb{1}_{\mathcal{D}})$ is a separable (resp., special) Frobenius algebra in $\mathcal{C}$. If further, $\mathcal{C},\mathcal{D}$ are pivotal (resp., ribbon) categories and $U$ is a pivotal (resp., braided pivotal) functor, then $R$ is a pivotal (resp., ribbon) functor…
The Grouplike Functor and Monoidal Adjunctions
本文在辫子幺半范畴中范畴化Hopf代数的群样元素函子，通过有限极限构造群对象，证明其函子性并给出自由Hopf幺半群函子的右伴随，推广经典对应。
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。
Projection formulas and induced functors on centers of monoidal categories
本文研究了单子adjunction诱导的braided lax单子函子，通过投射公式 morphisms 的可逆性，探讨了Hopf代数的限制与诱导沿态射的单子adjunction，并应用这些函子到 affine代数群、量子群和Radford-Majid双积Hopf代数的例子中。
We prove that the category of dg-coalgebras is symmetric monoidal closed and that the category of dg-algebras is enriched, tensored, cotensored and strongly monoidal over that of coalgebras. We apply this formalism to reconstruct several known adjunctions, notably the bar-cobar adjunction.
Extranatural transformations, adjunctions of two variables and conjugation
本文将普通伴随的共轭关系推广到双变量伴随，借助曲面图记号在形式范畴论框架下研究共轭，并阐明其可用于理解 Grothendieck 六运算中 Fausk-Hu-May 的投影公式与内部伴随构造。
Symmetric monoidal closed categories may be related to one another not only by the functors between them but also by enrichment of one in another, and it was known to G. M. Kelly in the 1960s that there is a very close connection between these phenomena. In this first part of a two-part series on this subject, we show that the assignment to each symmetric monoidal closed category $V$ its associated $V$-enriched category $\underline{V}$ extends to a 2-functor valued in an op-2-fibred 2-category of symmetric monoidal closed categories enriched over various bases. For a fixed $V$, we show that th…
We review some of the endeavors in trying to connect Petri nets with free symmetric monoidal categories. We give a list of requirement such connections should respect if they are meant to be useful for practical/implementation purposes. We show how previous approaches do not satisfy them, and give compelling evidence that this depends on trying to make the correspondence functorial in the direction from nets to free symmetric monoidal categories, in order to produce an adjunction. We show that dropping this immediately honors our desiderata, and conclude by introducing an Idris library which i…
In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2-categories. In particular, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2-category D into which M fully and faithfully embeds. Since a 2D topological quantum field theory is equivalent to a commutative Frobenius algebra, this result also shows that every 2D TQFT is obtained from an ambijunction in some 2-category. Our theorem is proved by extending the theory of adjoint monads to the context of…
$A_{\infty}$-structures in monoidal DG categories and strong homotopy unitality
本文在任意幺半DG范畴中借助无界扭曲复形定义A∞代数、余代数、模与余模，引入强同伦幺性，构造自由-遗忘同伦伴随及导出范畴，并证明A∞代数与余代数的同伦伴随给出导出模-余模等价，应用于A∞单子与增强正合单子。
We define Hopf monads on an arbitrary monoidal category, extending the definition given previously for monoidal categories with duals. A Hopf monad is a bimonad (or opmonoidal monad) whose fusion operators are invertible. This definition can be formulated in terms of Hopf adjunctions, which are comonoidal adjunctions with an invertibility condition. On a monoidal category with internal Homs, a Hopf monad is a bimonad admitting a left and a right antipode. Hopf monads generalize Hopf algebras to the non-braided setting. They also generalize Hopf algebroids (which are linear Hopf monads on a cat…
We prove an abstract Fubini-type theorem in the context of monoidal and enriched category theory, and as a corollary we establish a Fubini theorem for integrals on arbitrary convergence spaces that generalizes (and entails) the classical Fubini theorem for Radon measures on compact Hausdorff spaces. Given a symmetric monoidal closed adjunction satisfying certain hypotheses, we show that an associated monad of natural distributions D is commutative. Applying this result to the monoidal adjunction between convergence spaces and convergence vector spaces, the commutativity of D amounts to a Fubin…
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。
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…
Identity in the presence of adjunction
本文在半群范畴中发展了伴随理论，证明刚性半群范畴是promonoidal的从而可附加单位对象，推广了先前对称与非对称情形的结果，并给出了有限张量范畴的刻画。
The main result of this paper is the construction of a trace and a trace pairing for endomorphisms satisfying suitable conditions in a monoidal category. This construction is a common generalization of the trace for endomorphisms of dualizable ob jects in a balanced monoidal category and the trace of nuclear operators on a locally convex topological vector space with the approximation property.
The sheaves--spectrum adjunction
本文研究稳定可展示对称幺半∞范畴的smashing谱，证明其函子是谱层函子的右伴随，建立非稳定推广，并给出Clausen-Scholze范畴化locale的范畴表示。
For any site of definition $\mathcal C$ of a Grothendieck topos $\mathcal E$, we define a notion of a $\mathcal C$-ary Lawvere theory $τ: \mathscr C \to \mathscr T$ whose category of models is a stack over $\mathcal E$. Our definitions coincide with Lawvere's finitary theories when $\mathcal C=\aleph_0$ and $\mathcal E = \operatorname{\mathbf {Set}}$. We construct a fibered category $\operatorname{\mathbf {Mod}}^{\mathscr T}$ of models as a stack over $\mathcal E$ and prove that it is $\mathcal E$-complete and $\mathcal E$-cocomplete. We show that there is a free-forget adjunction $F \dashv U:…
In this paper we investigate the categories of braided objects, algebras and bialgebras in a given monoidal category, some pairs of adjoint functors between them and their relations. In particular we construct a braided primitive functor and its left adjoint, the braided tensor bialgebra functor, from the category of braided objects to the one of braided bialgebras. The latter is obtained by a specific elaborated construction introducing a braided tensor algebra functor as a left adjoint of the forgetful functor from the category of braided algebras to the one of braided objects. The behaviour…
A straightening-unstraightening equivalence for $\infty$-operads
本文为 $\infty$-operads 建立伸直-非伸直伴随等价，并证明 Hinich-Moerdijk 比较函子诱导 operadic 与 dendroidal 左纤维化等价，同时刻画幺半非伸直函子的本质像。
An adjunction theorem for Davydov-Yetter cohomology and infinitesimal braidings
本文针对域上线性幺半范畴间幺半函子的Davydov-Yetter上同调，证明了其与右伴随相关的伴随定理，将上同调表为中心中余端对象为系数的恒等函子上同调，并应用于积函子的无穷小辫子、Hopf代数R-矩阵切空间及限制函子形变等问题。
Strictification and non-strictification of monoidal categories
本文综述幺半范畴严格化与非严格化的多种构造方法，证明其给出自由（非）严格幺半范畴并构成自由-遗忘2-伴随，同时展示这些构造与Power融贯定理及双范畴Yoneda嵌入的联系。
In some bicategories, the 1-cells are `morphisms' between the 0-cells, such as functors between categories, but in others they are `objects' over the 0-cells, such as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, because the `morphisms between 0-cells', such as ring homomorphisms, are missing. We can include them by using a pseudo double category, but usually these morphisms also induce base change functors acting on the 1-cells. We avoid complicated coherence problems by describing base change `nonalgebraically', using cat…
Rigid Algebras and Cospans
本文引入刚性代数概念，将其推广到任意对称幺半$(\infty,2)$-范畴，并证明刚性代数的$(\infty,2)$-范畴实际上是$(\infty,1)$-范畴；通过余跨构造与刚性交换代数的$(\infty,1)$-范畴之间的伴随关系，建立了与给定$(\infty,1)$-范畴的典范等同。
How to build a Hopf algebra
构造了一个从满足附加性条件的 $(\infty,3)$-范畴收缩到辫子幺半 $(\infty,1)$-范畴中 Hopf 代数的函子，为 Tannaka 重建提供了高阶范畴论解释。
Let $\mathcal{A}$ and $\mathcal{B}$ be monoidal categories and let $R:\mathcal{A} \rightarrow \mathcal{B}$ be a lax monoidal functor. If $R$ has a left adjoint $L$, it is well-known that the two adjoints induce functors $\overline{R}={\sf Alg}(R):{\sf Alg}(\mathcal{A})\rightarrow {\sf Alg }(\mathcal{B})$ and $\underline{L}={\sf Coalg(L)}:{\sf Coalg}(\mathcal{B})\rightarrow {\sf Coalg}(\mathcal{A})$ respectively. The pair $(L,R)$ is called "liftable" if the functor $\overline{R}$ has a left adjoint and if the functor $\underline{L}$ has a right adjoint. A pleasing fact is that, when $\mathcal{A…
In the theory of coalgebras $C$ over a ring $R$, the rational functor relates the category of modules over the algebra $C^*$ (with convolution product) with the category of comodules over $C$. It is based on the pairing of the algebra $C^*$ with the coalgebra $C$ provided by the evaluation map $\ev:C^*\ot_R C\to R$. We generalise this situation by defining a {\em pairing} between endofunctors $T$ and $G$ on any category $\A$ as a map, natural in $a,b\in \A$, $$β_{a,b}:\A(a, G(b)) \to \A(T(a),b),$$ and we call it {\em rational} if these all are injective. In case $\bT=(T,m_T,e_T)$ is a monad an…
For functors $L:\A\to \B$ and $R:\B\to \A$ between any categories $\A$ and $\B$, a {\em pairing} is defined by maps, natural in $A\in \A$ and $B\in \B$, $$\xymatrix{\Mor_\B (L(A),B) \ar@<0.5ex>[r]^α & \Mor_\A (A,R(B))\ar@<0.5ex>[l]^β}.$$ $(L,R)$ is an {\em adjoint pair} provided $α$ (or $β$) is a bijection. In this case the composition $RL$ defines a monad on the category $\A$, $LR$ defines a comonad on the category $\B$, and there is a well-known correspondence between monads (or comonads) and adjoint pairs of functors. For various applications it was observed that the conditions for a unit o…
Central Hopf Monads and Braided Commutative Algebras
针对辫张量范畴与张量范畴的结构，构造相对中心相关作用函子的右伴随，证明其单子性并诱导Hopf单子，提供特定Hopf代数表示范畴中辫交换代数的计算方法，还给出Taft代数的例子。
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