Simple free star-autonomous categories and full coherence
Dominic Hughes
本文基于Eilenberg-Kelly-MacLane图和Trimble重排给出了范畴上的自由星自治范畴的一个简单表示，并得到了一个完全凝聚定理：正则映射的图的交换性是可判定的.
This paper gives a simple presentation of the free star-autonomous category over a category, based on Eilenberg-Kelly-MacLane graphs and Trimble rewiring, yielding a full coherence theorem: the commutativity of diagrams of canonical maps is decidable.
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Coherence for Star-Autonomous Categories
本文给出了星自治范畴的一个凝聚定理，它与Kelly和Mac Lane关于对称monoidal闭范畴的凝聚定理完全相似。这个定理的证明是基于范畴割消的结果，这是在一些细节。
Coherence of Proof-Net Categories
本文定义的证明网范畴的概念与线性逻辑中无常数命题的乘法片段的证明网中隐含的图密切相关。类似的图形出现在凯利和麦克莱恩的相干定理对称monoidal闭范畴。证明了证明网范畴的相干定理。这种一致性定理也证明了相应的线性逻辑的混合原则的箭头的存在。证明网范畴的概念抓住了星自治范畴概念的单位自由片段，星自治范畴是一类特殊的对称monoidal闭范畴。
Coherence in Linear Predicate Logic
相关的Kelly-Mac巷图证明的类别对应的乘法片段没有常数命题的经典线性一阶谓词逻辑没有或混合。为了获得这一结果，一致性首先建立的类别对应的乘法合取-析取片段与一阶量词的经典线性逻辑，一个片段缺乏否定。这些结果推广了作者在前两本书中发表的结果，其中对命题经典线性逻辑的相应片段的范畴建立了一致性，这些片段与证明网有关，并且可以被描述为没有单位对象的星自治范畴。
A coherence result for symmetric monoidal closed categories with biproducts is shown in this paper. It is explained how to prove, by using the same technique, coherence for compact closed categories with biproducts and for dagger compact closed categories with dagger biproducts.
Coherence in monoidal track categories
我们引入同伦的方法重写的基础上更高的维度类别证明的一致性结果的代数结构的类别。我们将（对称）幺半范畴的连贯性问题表示为迹范畴的非球面性问题，并利用多道生理记录仪上的重写方法来解决，将其推广到更一般的连贯性问题，即迹范畴中的三维字问题，包括辫状幺半范畴的情况.
Coherence in Substructural Categories
证明了MacLane关于monoidal范畴和对称幺半范畴的相干性结果可以推广到其它一些有乘法的范畴，即相关范畴、仿射范畴和Carnival范畴.所有的结果都是用带有"图“的自然变换（g-自然变换）表示的，并给出相应的态射定理.利用这些结果，得到了这些类的自由范畴之间的一些基本关系。
Coherent Bicartesian and Sesquicartesian Categories
相干性在这里被证明为sesquicartesian范畴，这是范畴与非空有限产品和任意有限的总和，包括空的总和，而且从产品的始对象与自身的第一和第二个投影是相同的。（每一个双artesian闭范畴，特别是范畴Set，就是这样一个范畴。）这种一致性等于存在一个忠实函子，它从由对象集合自由生成的这类范畴到有限序数上的关系范畴，并且它产生了一个非常容易的判定箭头相等的过程。对于双artesian范畴，限制相干性也成立，在这里，除了投影的这种相等性之外，我们还知道，对终对象与自身之和的第一次和第二次注入是相同的。本文的印刷版（在：R。Kahle et al. eds，证明论in Computer Science，Lecture Notes in Computer Science，vol. 2183，Springer，Berlin，2001，pp. 78-92）和版本以前张贴在这里声称证明不受限制的连贯性为上述bicartesian类别。这些版本中的引理5.1，也是sesquicartesian范畴的相干性证明所依赖的引理，然而是不正确的。本文件在术语上也与以前的版本不同。
Bicartesian Coherence
连贯性证明的类别与二进制产品和总和，但没有终端和始对象，没有分布。这种一致性相当于存在一个忠实函子从一个自由范畴的二元产品和总和的范畴的关系有限序数。这一结果是得到的证明理论规范化技术的帮助下。当存在终对象时，如果二进制和保持其双函子性质，则仍然可以证明相干性。人们发现，最简单的理解的连贯性，这是最好的一个可以希望在bicartesian类别。具有二进制乘积和的范畴的相干性提供了一个简单的判定过程。它也被用来证明所讨论的范畴是最大的，在这个意义上，在任何这样的范畴，这不是一个前序之间的所有方程箭头只涉及二进制产品和总和是相同的。这表明，在非分配合取-析取逻辑中证明的等价性的通常概念是最优定义的：进一步的假设会使这个概念崩溃为平凡。（对于具有比本文中所提出的更简单的二进制乘积和和的范畴的相干性的证明，可以在{ \it Proof-Theoretical Coherence}的第9.4节中找到，2007年9月的修订版，http：//www.mi.sanu.ac.yu/~kosta/coh.pdf。）
Bicartesian Coherence Revisited
综述了关于有限积范畴和有限余积范畴的凝聚性的研究结果。对于这些结果，这是以前发表的作者在几个地方，一些配方和证明在这里更正，并更新事项。本文研究的范畴形式化了经典和直觉合取-析取逻辑中的证明的等价性，而没有合取在析取上的分布。
A note on Frobenius monoidal functors on autonomous categories
Frobenius monoidal函子保持π。相反，我们表明，（co）monoidal函子之间的自治范畴，保持Frobenius monoidal。我们将这个结果应用于自治范畴之间的线性分配函子。
We study the question when a *-autonomous (Mix-)category has a representation as a $*$-autonomous category of a compact one. We prove that necessary and sufficient condition is that weak distributivity maps are monic (or, equivalently epic). For a Mix-category, this condition is, in turn, equivalent to the requirement that Mix-maps be monic (or epic). We call categories satisfying this property torsion-free. An important side result is that torsion-free categories have canonical partial traces.
The Coherence Theorem for Ann-Categories
本文给出了Ann-范畴的凝聚定理的证明，其公理集和最初的基本性质在[9]中给出。设$$\A=(\A,{\Ah},c,(0,g,d),a,(1,l,r),{\Lh},{\Rh})$$是Ann-范畴。相干定理指出，在范畴$ \A$中，任何由上述同构和复合识别以及两个运算$\tx$，$\ts$构建的态射只依赖于它的源和它的目标。Mac Lane [7]建立了关于monoidal范畴和对称幺半范畴的第一个凝聚定理。在此之后，如参考文献所示，有许多结果涉及到某些类别的范畴的连贯性问题。对于Ann-范畴，应用Hoang Xuan Sinh在[2]中关于Gr-范畴的思想，通过忠实地将任意Ann-范畴“嵌入”到一个相当严格的Ann-范畴中，构造了凝聚定理的证明.这里，一个{ \it quite strict} Ann-范畴是一个Ann-范畴，它的所有约束都是严格的，除了交换性和左分配性。本文是文[9]的继续。如果没有解释，本文中的术语和符号与[9]相同。
Over the recent years, the theory of rewriting has been used and extended in order to provide systematic techniques to show coherence results for strict higher categories. Here, we investigate a further generalization to Gray categories, which are known to be equivalent to tricategories. This requires us to develop the theory of rewriting in the setting of precategories, which include Gray categories as particular cases, and are adapted to mechanized computations. We show that a finite rewriting system in precategories admits a finite number of critical pairs, which can be efficiently computed…
A symmetric monoidal category is a category equipped with an associative and commutative (binary) product and an object which is the unit for the product. In fact, those properties only hold up to natural isomorphisms which satisfy some coherence conditions. The coherence theorem asserts the commutativity of all linear diagrams involving the left and right unitors, the associator and the braiding. We prove the coherence for symmetric monoidal categories using a homotopical method based on rewriting. For that scope, we detail the con vergence proof of Lafont's string diagram rewriting system wh…
*-Autonomous categories in quantum theory
*-自治范畴最初是由M定义的。巴尔描述了一种类型的对偶进行了许多monoidal封闭范畴。后来，它们被现在的作者推广到包括 *-自治promonoidal范畴。总之，这些在“卷积”乘积下的结构清楚地表明了 *-自治在量子数学和相关领域的有用性。
Coherence for Skew-Monoidal Categories
我激发了一个变化（由于K。Szlachányi）的monoidal范畴称为skew-monoidal范畴，其中单位律和结合律不需要是同构，只需要自然变换。相干性必须以不同于众所周知的monoidal情况的方式来表述。在我的（据我所知的新）版本中，它成为规范化重写的独特性的声明。我提出了这个一致性定理的证明，并正式完全依赖类型的程序设计语言Agda。
On Cyclic Star-Autonomous Categories
我们讨论了循环星自治范畴，也就是说，无辫的星自治范畴，其中每个对象p的左和右顶点由相干的自然同构连接。我们解决的连贯性问题已经出现了这样的循环同构，我们表明，这样的循环结构是自然的设置中，考虑丰富的profunctors。具体地说，如果V是一个循环星自治范畴，则V-富集的原函子的集合具有一个正则循环结构。在辫星自治范畴的情况下，我们讨论了循环结构和平衡或挠结构之间的对应关系。最后，我们证明了每一个循环星自治范畴都等价于其中的循环同构是恒等式的范畴。
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。
Coherence theorems are fundamental to how we think about monoidal categories and their generalizations. In this paper we revisit Mac Lane's original proof of coherence for monoidal categories using the Grothendieck construction. This perspective makes the approach of Mac Lane's proof very amenable to generalization. We use the technique to give efficient proofs of many standard coherence theorems and new coherence results for bicategories with shadow and for their functors.
Intermutation
本文证明了具有自然变换\emph{intermutation}的范畴的相干结果，该变换由从$(A\wedge B)\vee(C\wedge D)$到${(A\vee C)\wedge(B\vee D)}$的箭头组成，其中$\wedge$和$\vee$是两个双核函子.互换发生在迭代的或$n$ -折叠的幺半范畴中，这些范畴是结合$n$ -折叠闭路空间引入的，并且对于这些范畴，Balteanu、Fiedorowicz、Schwanzl和Vogt先前得到了一个相关但不同的相干结果。本文的结果在一定程度上加强了这一先前的结果，并表明二重闭路空间是以这些作者所设想的方式从一种更一般的范畴中产生的，这种范畴在它们的意义上不是二重monoidal的。特别地，某些具有有限乘积和上乘积的范畴就是这样。连贯性在这里首先证明的范畴，其中$\wedge$和$\vee$的一个假设只有互换，并在接下来的范畴，其中还假设自然的结合同构。当进一步假设自然交换同构时，证明了对称幺半范畴范畴在凝聚意义下的凝聚性。一个限制的连贯性的结果，涉及的但书中发现的对称monoidal封闭范畴的连贯性，证明了在存在两个nonisomorphic单位对象。互换和单位对象的相干条件是从一个统一原则导出的，粗略地说，这个原则是关于直到一个不是同构的自然变换为止，具有函子结构的范畴的保持。
We prove coherence theorems for bicategories, pseudofunctors and pseudonatural transformations. These theorems boil down to proving the coherence of some free $(4,2)$-categories. In the case of bicategories and pseudofunctors, existing rewriting techniques based on Squier's Theorem allow us to conclude. In the case of pseudonatural transformations this approach only proves the coherence of part of the structure, and we use a new rewriting result to conclude. To this end, we introduce the notions of white-categories and partial coherence.
Coherence of Associativity in Categories with Multiplication
MacLane关于乘法范畴的通常的凝聚定理假设某个五边形图交换，以得出结合性同构在某种实际意义上是良好定义的结论。实践方面包括创建结合同构从一个给定的张量与单位的权利或左。我们表明，通过重新检查麦克莱恩的原始参数，如果张量与身份被限制在一边，那么良好的定义构造同构遵循自然性只，没有必要的交换性的五边形图。这一观察结果是通过注意到通常的相干定理与称为汤普森群F的非对称群的某些性质的相似性而发现的。这份报纸将被视为这种联系的广告。
It is known that the quantale of sup-preserving maps from a complete lattice to itself is a Frobenius quantale if and only if the lattice is completely distributive. Since completely distributive lattices are the nuclear objects in the autonomous category of complete lattices and sup-preserving maps, we study the above statement in a categorical setting. We introduce the notion of Frobenius structure in an arbitrary autonomous category, generalizing that of Frobenius quantale. We prove that the monoid of endomorphisms of a nuclear object has a Frobenius structure. If the environment category i…
Local coherence and recollement of abelian categories
本文研究阿贝尔范畴局部相干性在粘合下的传递条件，引入拟射影伴随函子对，推广相干准则，将Stenström的刻画扩展到紧组装Grothendieck范畴，汇集环相干性的等价条件。
Coherence and Confluence
范畴论中的相干性证明，从麦克·莱恩最初对幺半范畴的相干性证明开始，有时是基于类似于lambda演算或一般的项重写系统中的汇合技术。这适用于断言一个范畴是一个前序的连贯性结果，即“所有图都是可交换的”。本文是关于这个类比的，特别注意证明了凝聚性的范畴不是广群的情况。
Given an additive equational category with a closed symmetric monoidal structure and a potential dualizing object, we find sufficient conditions that the category of topological objects over that category has a good notion of full subcategories of strong and weakly topologized objects and show that each is equivalent to the chu category of the original category with respect to the dualizing object.
Symmetric bimonoidal intermuting categories and $ω\timesω$ reduced bar constructions
本文给出了一个新的、完备的证明，证明了具有两个由自然变换桥接的对称monoidal结构的范畴的相干结果。结果表明，这个相干性结果是充分的$ω\timesω$ -指标族的迭代约化杆结构的基础上这样一个范畴。
Coherence for Categorified Operadic Theories
给定一个代数理论，它可以描述一个（可能是对称的）操作$P$，我们提出了一个定义的\emph{weakening}（或\emph{categorification}）的理论，其中严格为$P$ -代数的方程只持有到相干同构。推广了幺半范畴和对称幺半范畴的理论以及文献中定义的几个相关概念。利用这个定义，我们推广了所有幺半范畴都与严格幺半范畴等价的结果，并证明了严格化函子具有一个有趣的泛性质，即从严格$P$ -范畴到弱$P$ -范畴的遗忘函子的左伴随.我们进一步表明，得到的分类是独立的，我们选择的介绍$P$，并扩展我们的一些结果，许多排序的理论，使用multicategories。
We define a traced pseudomonoid as a pseudomonoid in a monoidal bicategory equipped with extra structure, giving a new characterisation of Cauchy complete traced monoidal categories as algebraic structures in Prof, the monoidal bicategory of profunctors. This enables reasoning about the trace using the graphical calculus for monoidal bicategories, which we illustrate in detail. We apply our techniques to study traced ∗-autonomous categories, proving a new equivalence result between the left ⊗-trace and the right ⅋-trace, and describing a new condition under which traced ͨ…
In this paper, we develop the proof theory of skew prounital closed categories. These are variants of the skew closed categories of Street where the unit is not represented. Skew closed categories in turn are a weakening of the closed categories of Eilenberg and Kelly where no structural law is required to be invertible. The presence of a monoidal structure in these categories is not required. We construct several equivalent presentations of the free skew prounital closed category on a given set of generating objects: a categorical calculus (Hilbert-style system), a cut-free sequent calculus a…
Categories of polymorphic lenses in computer science, and of open games in compositional game theory, have a curious structure that is reminiscent of compact closed categories, but differs in some crucial ways. Specifically they have a family of morphisms that behave like the counits of a compact closed category, but have no corresponding units; and they have a `partial' duality that behaves like transposition in a compact closed category when it is defined. We axiomatise this structure, which we refer to as a `teleological category'. We precisely define a diagrammatic language suitable for th…
Obstructions to Coherence: Natural Noncoherent Associativity
我们研究当连贯性失效时会发生什么。范畴的张量积和自然的结合同构，不一定满足五边形的连贯性要求（称为结合范畴）被认为是。自然正方形交换和五边形不交换的结合面体的范畴版本被构造（称为加泰罗尼亚群胚，$\An$）。这些广群用于自由结合范畴的构造。它们也被用在结合范畴理论的构造中（以2-草图的形式给出）。生成元和关系的基本群，$π(\An)$，加泰罗尼亚广群-思想作为一个单纯复形。这些群体被证明不仅仅是自由群体。每个结合范畴$B$都有相关的基本群$π(\bf B_n)$和同态$π(P_n):π(\An) \longrightarrow π(\bf B_n)$。如果$π(P_n)$的图像是平凡的，即任何两个对象之间只有一条关联路径，那么这个范畴是连贯的。否则，$π(P_n)$的图像会妨碍相干性。在关联范畴的非相干分类方面取得了一些进展。
Relevant Categories and Partial Functions
一个相关范畴是一个具有对角自然变换且满足某些相干条件的对称么半群闭范畴。在这个意义上，每一个卡氏闭范畴都是一个相关范畴。“相关”这个名称来自于与相关逻辑的联系。证明了同构于点集范畴的部分函数的集范畴是一个相关范畴，但它不是一个闭范畴。
String diagrams for Strictification and Coherence
本文提出非严格幺半范畴弦图的生成元与关系表示，证明其正确性，并由此给出Mac Lane严格性定理与相干性定理的新图形证明，支持数字电路与编程语言等应用。
In this paper we give sufficient conditions for lifting an enhanced factorization system $ (\mathcal{E}, \mathcal{M}) $ on a $ 2 $-category $ \mathbf{D} $ to the functor $ 2 $-category $ \mathbf{D}^{\mathbf{C}} $, where $ \mathbf{C} $ is a small $ 2 $-category. Due to previous work of Lack, our work provides coherence results for $ 2 $-monads on functor $ 2 $-categories. These coherence results are of immediate interest to an ongoing collaboration of Guillou, May, Merling, and Osorno on equivariant infinite loop space theory.
Coherence, Homotopy and 2-Theories
2-理论是描述具有额外结构的范畴的规范方法。2-理论态射用于讨论一个结构如何被另一个结构替换。这是范畴连贯理论的核心。我们在2-理论和2-理论态射的范畴上放置一个Quillen模型范畴结构，其中弱等价是2-理论的双等价。2-理论的双等价（Morita等价）诱导2-代数范畴的双等价，并被2-代数范畴的双等价诱导。这个模型范畴结构允许我们讨论2-理论的同伦和相干性的泛性质。
Traced monoidal categories as algebraic structures in $\mathbf{Prof}$
本研究提出迹伪幺半群的定义，给出柯西完备迹幺半范畴在$\u003cmath\u003e\mathbf{Prof}\u003c/math\u003e$中的新代数刻画，可借助图形演算推理迹，并将其用于研究迹$*$-自守范畴，得到迹的等价结果与范畴自守的新条件。
Unbiasing symmetric monoidal categories in Lean
本文在Lean 4中形式化了对称单子范畴的去偏过程，通过扩展数据到伪函子并利用Kleisli双范畴实现高 arity张量积的一致性。
We give a self-contained introduction to accessible categories and how they shed light on both model- and set-theoretic questions. We survey for example recent developments on the study of presentability ranks, a notion of cardinality localized to a given category, as well as stable independence, a generalization of pushouts and model-theoretic forking that may interest mathematicians at large. We give many examples, including recently discovered connections with homotopy theory and homological algebra. We also discuss concrete versions of accessible categories (such as abstract elementary cla…
Relating toy models of quantum computation: comprehension, complementarity and dagger mix autonomous categories
玩具模型已经被用来从标准希尔伯特空间模型的丰富背景中分离量子计算的重要特征。另一方面，范畴论是一种将数学结构的组成部分分开的通用工具，并且一次分析一层。将这两种方法结合起来似乎很自然，一些作者已经在追求这一想法。我们探索 * 类别理解结构 * 作为一种工具，为玩具模型添加功能。我们用它来理解有限维希尔伯特空间的基本模型中的量子命题和概率。我们还分析了互补的量子可观的集合和关系的类别。这就引入了 * 测试空间 * 的领域，这是一个研究得很好的模型。我们提出了这个模型的许多可能的扩展之一，使理解建设。显然，以这种方式获得的所有模型都具有相同的范畴结构，用互补运算 * 扩展 * 了熟悉的匕首紧致框架。我们称得到的结构为 * 匕首混合自治 *，因为它扩展了计算机科学中流行的混合自治范畴，就像匕首紧凑结构扩展紧凑范畴一样。在量子计算中，只要互补性被视为全局结构的一部分，匕首混合自治范畴似乎就很自然地出现了。
Note on star-autonomous comonads
我们开发了一种替代的方法，通过线性分布的类别星自治comonads。它表明，在自治的情况下，星自治comonad和Hopf comonad的概念相吻合。
Rewriting techniques for relative coherence
该研究拓展代数结构相干性的重写证明方法，覆盖部分相干性与笛卡尔结构两类情形，构建抽象及项重写系统层面的重写技术与定理，可推导幺半范畴等的相干定理。
We define strict and weak duality involutions on 2-categories, and prove a coherence theorem that every bicategory with a weak duality involution is biequivalent to a 2-category with a strict duality involution. For this purpose we introduce "2-categories with contravariance", a sort of enhanced 2-category with a basic notion of "contravariant morphism", which can be regarded either as generalized multicategories or as enriched categories. This enables a universal characterization of duality involutions using absolute weighted colimits, leading to a conceptual proof of the coherence theorem.
The Syntax of Coherence
本文处理的范畴连贯性内的二维推广Lawvere的函语义。2-理论是一种用结构描述范畴的句法方法。从这里提供的角度来看，许多相干结果成为简单的陈述准Yoneda引理和2理论态射。给定两个2-理论和它们之间的一个2-理论-态射，研究了相应的2-代数范畴之间的诱导关系。导出拟伴随的强度由2-理论态射的强度分类。这些拟伴随反映了一个结构可以被另一个结构取代的程度。一个二维类似的克罗内克产品的定义和构造。这种操作允许人们从旧的一致性定律中产生新的一致性定律。
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_\…
An alternative description of symmetric monoidal categories, and symmetric 2-groups
本文提出了一种对称单子族的等价描述，通过结合-交换同构代替传统的结合子和交换子同构，并利用埃利恩伯格-麦克拉伦立方上同调对称2-群进行上同调分类。
Hypergraph categories have been rediscovered at least five times, under various names, including well-supported compact closed categories, dgs-monoidal categories, and dungeon categories. Perhaps the reason they keep being reinvented is two-fold: there are many applications---including to automata, databases, circuits, linear relations, graph rewriting, and belief propagation---and yet the standard definition is so involved and ornate as to be difficult to find in the literature. Indeed, a hypergraph category is, roughly speaking, a "symmetric monoidal category in which each object is equipped…
Differential categories are now an established abstract setting for differentiation. However not much attention has been given to the process which is inverse to differentiation: integration. This paper presents the parallel development for integration by axiomatizing an integral transformation, $s_A: !A \to !A \otimes A$, in a symmetric monoidal category with a coalgebra modality. When integration is combined with differentiation, the two fundamental theorems of calculus are expected to hold (in a suitable sense): a differential category with integration which satisfies these two theorem is c…
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 study the question when a $*$-autonomous Mix-category has a representation as a $*$-autonomous Mix-subcategory of a compact one. We define certain partial trace-like operation on morphisms of a Mix-category, which we call a mixed trace, and show that any structure preserving embedding of a Mix-category into a compact one induces a mixed trace on the former. We also show that, conversely, if a Mix-category ${\bf K}$ has a mixed trace, then we can construct a compact category and structure preserving embedding of ${\bf K}$ into it, which induces the same mixed trace. Finally, we find a specif…
Presentations for unbraided, braided and symmetric pseudomonoids are defined. Biequivalences characterising the semistrict bicategories generated by these presentations are proven. It is shown that these biequivalences categorify results in the theory of monoids and commutative monoids, and generalise standard coherence theorems for braided and symmetric monoidal categories.
Tannaka Reconstruction of Weak Hopf Algebras in Arbitrary Monoidal Categories
我们介绍了一个变种的图形演算的Cockett和Seely的monoidal函子，并说明它与讨论Tannaka重建，其中一些是已知的，其中一些是新的。新的部分是：给定一个可分的Frobenius函子F：A --> B，从一个幺半范畴A到一个适当完备或余完备的辫状自治范畴B，Tannaka重构的通常公式给出B中的一个弱双代数;此外，如果A是自治的，这个弱双代数实际上是一个弱霍普夫代数。
Hopf monads
我们引入并研究了自治范畴上的Hopf单子（即，带的幺半范畴）。霍普夫单子将霍普夫代数推广到非辫子（和非线性）的情形。事实上，自治范畴之间的任何monoidal adjunction都会产生一个Hopf monad。我们推广了霍普夫代数理论的许多基本结果（如霍普夫模的分解、积分的存在性、Maschke半单性判据等.）到霍普夫单子我们还介绍和研究准三角和带状霍普夫单子（再次定义在非编织设置）。
From fibered symmetric bimonoidal categories to symmetric spectra
本文定义了纤维对称双单范畴的概念。粗略地说，这些是纤维范畴D-> C，其纤维是由C参数化的对称幺半范畴，并且使得D和C都具有满足我们所描述的某些凝聚性的对称幺半范畴的进一步结构。我们的目标是证明我们可以用函子的方式对应于纤维对称双单类范畴的E_{\infty}-环谱。
In this thesis I lift the Curry--Howard--Lambek correspondence between the simply-typed lambda calculus and cartesian closed categories to the bicategorical setting, then use the resulting type theory to prove a coherence result for cartesian closed bicategories. Cartesian closed bicategories---2-categories `up to isomorphism' equipped with similarly weak products and exponentials---arise in logic, categorical algebra, and game semantics. I show that there is at most one 2-cell between any parallel pair of 1-cells in the free cartesian closed bicategory on a set and hence---in terms of the dif…
Quadratic categories, Koszul resolutions and operads
一个二次代数是一个齐次代数生成的元素的次数为1。Manin赋予二次代数范畴两个张量积。这些结构已经被金斯伯格和卡普拉诺夫改编成歌剧。Berger为n-齐次代数定义了这样的张量积。本文定义了二次范畴的概念，即具有两个张量积的范畴。Manin和Ginsburg-Kapranov结构是二次范畴的例子。我们还定义了一个Koszul复杂的，n-齐次运算，并展示了如何将这一概念可以应用到研究的连贯关系的n-范畴。
We investigate monoidal categories of formal contexts, in which states correspond to formal concepts. In particular we examine the category of bonds or Chu correspondences between contexts, which is known to be equivalent to the *-autonomous category of complete sup-lattices. We show that a second monoidal structure exists on both categories, corresponding to the direct product of formal contexts defined by Ganter and Wille, and discuss the use of these categories as compositional models of meaning.
We prove 2-categorical conservativity for any {0,T}-free fragment of MALL over its corresponding intuitionistic version: that is, that the universal map from a closed symmetric monoidal category to the *-autonomous category that it freely generates is fully faithful, and similarly for other doctrines. This implies that linear logics and graphical calculi for *-autonomous categories can also be interpreted canonically in closed symmetric monoidal categories. In particular, every closed symmetric monoidal category can be fully embedded in a *-autonomous category, preserving both tensor products …
Tangent categories were introduced by Rosicky as a categorical setting for differential structures in algebra and geometry; in recent work of Cockett, Crutwell and others, they have also been applied to the study of differential structure in computer science. In this paper, we prove that every tangent category admits an embedding into a representable tangent category---one whose tangent structure is given by exponentiating by a free-standing tangent vector, as in, for example, any model of Kock and Lawvere's synthetic differential geometry. The key step in our proof uses a coherence theorem fo…
Centers and homotopy centers in enriched monoidal categories
本文研究了幺半范畴中幺半群的中心和同伦中心的理论，这些幺半群本身在duoidal范畴中是丰富的。Duoidal范畴（由Aguillar和Mahajan引入，名为2-幺半范畴）是具有两个monoidal结构的范畴，它们通过一些不一定可逆的凝聚态射联系在一起。在这个意义上的幺半群的中心包括许多不是“经典的”的例子。特别是，范畴的2-范畴是我们意义上的中心的一个例子。同伦中心（类似于经典的Hochschild复形）的例子包括2-范畴、2-函子和自然变换的Gray范畴Gray和dg-范畴、dg-函子和相干dg-变换的Tamarkin同伦2-范畴。
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