Eugenia Cheng
We address the fact that composition in an opetopic weak n-category is in general not unique and hence is not a well-defined operation. We define composition with a given k-cell in an n-category by a span of (n-k)-categories. We characterise such a cell as universal if its composition span gives an equivalence of (n-k)-categories.
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We define a family of structures called "opetopic algebras", which are algebraic structures with an underlying opetopic set. Examples of such are categories, planar operads, and Loday's combinads over planar trees. Opetopic algebras can be defined in two ways, either as the algebras of a "free pasting diagram" parametric right adjoint monad, or as models of a small projective sketch over the category of opetopes. We define an opetopic nerve functor that fully embeds each category of opetopic algebras into the category of opetopic sets. In particular, we obtain fully faithful opetopic nerve fun…
We define the category of tidy symmetric multicategories. We construct for each tidy symmetric multicategory Q a cartesian monad (E_Q,T_Q) and extend this assignation to a functor. We exhibit a relationship between the slice construction on symmetric multicategories, and the `free operad' monad construction on suitable monads. We use this to give an explicit description of the relationship between Baez-Dolan and Leinster opetopes.
We generalise the concepts introduced by Baez and Dolan to define opetopes constructed from symmetric operads with a category, rather than a set, of objects. We describe the category of 1-level generalised multicategories, a special case of the concept introduced by Hermida, Makkai and Power, and exhibit a full embedding of this category in the category of symmetric operads with a category of objects. As an analogy to the Baez-Dolan slice construction, we exhibit a certain multicategory of function replacement as a slice construction in the multitopic setting, and use it to construct multitope…
We continue our previous modifications of the Baez-Dolan theory of opetopes to modify the Baez-Dolan definition of universality, and thereby the category of opetopic n-categories and lax functors. For the case n=2 we exhibit an equivalence between this category and the category of bicategories and lax functors. We examine notions of strictness in the opetopic theory.
Properties and Characterisations of Cofree Cartesian Differential Categories
本文研究余自由笛卡尔微分范畴的刻画问题，提出在不指定基范畴的情况下，利用导数为零的微分常数，通过完备超度量空间和单子代数两种方式对其进行刻画，并探讨了其基本性质。
Magmal characterisations of cocartesian categories
本文综述了用幺半群范畴（含幺半群范畴）刻画余笛卡尔范畴的方法，证明了幺半群范畴$(\mathcal M, \otimes)$是余笛卡尔幺半群、对象有幺半群结构且态射为同态、张量积函子有右伴随三者等价，细化了经典刻画。
We provide various ways to characterise $Σ$-pure-injective objects in a compactly generated triangulated category. These characterisations mimic analogous well-known results from the model theory of modules. The proof involves two approaches. In the first approach we adapt arguments from the module-theoretic setting. Here the one-sorted language of modules over a fixed ring is replaced with a canonical multi-sorted language, whose sorts are given by compact objects. Throughout we use a variation of the Yoneda embedding, called the resticted Yoneda functor, which associates a multi-sorted struc…
We give a characterisation of the extriangulated categories which admit the structure of a triangulated category. We show that these are the extriangulated categories where for every object $X$ in the extriangulated category, the morphism $0 \rightarrow X$ is a deflation and the morphism $X \rightarrow 0$ is an inflation.
A characterisation for the category of Hilbert spaces
针对现有Hilbert空间范畴刻画依赖monoidal结构而排除四元数的问题，本文提出一种无需该假设的替代刻画方法，成功统一了实数、复数与四元数Hilbert空间范畴。
Double categories of relations relative to factorisation systems
本文通过稳定正交因子分解系统相对化关系双范畴，提出两种特征刻画方法，并系统关联多种双范畴类别与因子分解系统，涵盖正则范畴上的跨距和关系双范畴，与现有工作进行了比较。
An algebraically exact category in one that admits all of the limits and colimits which every variety of algebras possesses and every forgetful functor between varieties preserves, and which verifies the same interactions between these limits and colimits as hold in any variety. Such categories were studied by Adámek, Lawvere and Rosický: they characterised them as the categories with small limits and sifted colimits for which the functor taking sifted colimits is continuous. They conjectured that a complete and sifted-cocomplete category should be algebraically exact just when it is Barr-exac…
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.
Notions of Cauchy completeness for normed categories
该研究将Lawvere提出的两种柯西完备性刻画推广到任意交换quantale对应的\\(\mathcal{V}\\)-赋范范畴，同时改进了两类重要\\(\mathcal{V}\\)-赋范范畴中柯西列赋范收敛性的相关结果。
The category of extensions and a characterisation of $n$-exangulated functors
本文通过扩张范畴研究n-外角范畴，刻画了n-外角函子与自然变换，并在小n-外角范畴与小正合范畴的2-范畴之间建立了2-函子，同时构造了多种实例。
The paper studies analytic functors between presheaf categories. Generalising results of A. Joyal and of R. Hasegawa for analytic endofunctors on the category of sets, we give two characterisations of analytic functors between presheaf categories over groupoids: (i) as functors preserving filtered colimits, quasi-pullbacks, and cofiltered limits; and (ii) as functors preserving filtered colimits and wide quasi-pullbacks. The development establishes that small groupoids, analytic functors between their presheaf categories, and quasi-cartesian natural transformations between them form a 2-catego…
We determine internal characterisations for when a tensor category is (super) tannakian, for fields of positive characteristic. This generalises the corresponding characterisations in characteristic zero by P. Deligne. We also explore notions of Frobenius twists in tensor categories in positive characteristic.
We give new characterisations of regular Mal'tsev categories with distributive lattice of equivalence relations through variations of the so-called Triangular Lemma and Trapezoid Lemma in universal algebra. We then give new characterisations of equivalence distributive Goursat categories (which extend $3$-permutable varieties) through variations of the Triangular and Trapezoid Lemmas involving reflexive and positive relations.
We prove that a regular category $\mathcal C$ is a Mal'tsev category if and only if a strong form of the denormalised $3 \times 3$ Lemma holds true in $\mathcal C$. In this version of the $3 \times 3$ Lemma, the vertical exact forks are replaced by pullbacks of regular epimorphisms along arbitrary morphisms. The shape of the diagram it determines suggests to call it the Cuboid Lemma. This new characterisation of regular categories that are Mal'tsev categories (= $2$-permutable) is similar to the one previously obtained for Goursat categories (= $3$-permutable). We also analyse the "relative" v…
Characterisation of Lawvere-Tierney Topologies on Simplicial Sets, Bicolored Graphs, and Fuzzy Sets
本文完整刻画了单纯集、双色图与模糊集上的Lawvere-Tierney拓扑，并证明部分简单单纯集与图构成拟拓扑os。
A universal characterisation of the Bochner integral
本文通过双函子范畴中的初始对象，对所有$p \in [1,\infty)$的$L^p$ Bochner可积函数等价类空间给出了普适刻画，并由此唯一刻画了Bochner积分。
A semi-localization of a category is a full reflective subcategory with the property that the reflector is semi-left-exact. In this article we first determine an abstract characterization of the categories which are semi-localizations of an exact Mal'tsev category, by specializing a result due to S. Mantovani. We then turn our attention to semi-abelian categories, where a special type of semi-localizations are known to coincide with torsion-free subcategories. A new characterisation of protomodular categories in terms of binary relations is obtained, inspired by the one discovered in the point…
We prove that Mal'tsev and Goursat categories may be characterised through stronger variations of the Shifting Lemma, that is classically expressed in terms of three congruences $R$, $S$ and $T$, and characterises congruence modular varieties. We first show that a regular category $\mathcal C$ is a Mal'tsev category if and only if the Shifting Lemma holds for reflexive relations on the same object in $\mathcal C$. Moreover, we prove that a regular category $\mathcal C$ is a Goursat category if and only if the Shifting Lemma holds for a reflexive relation $S$ and reflexive and positive relation…
Dagger categories and the complex numbers: Axioms for the category of finite-dimensional Hilbert spaces and linear contractions
本研究揭示实数极限与范畴论极限的深层关联，基于已有全希尔伯特空间范畴刻画成果，用不含范数、实数等概念的范畴论结构完成有限维希尔伯特空间与线性压缩映射范畴的公理化刻画。
Prenormal categories
本文提出预范范畴的概念，通过替换等化子和核对子为余核和核，扩展经典代数结果，并展示交换半群作为核心例子。
We provide a characterisation of the category KH of compact Hausdorff spaces and continuous maps by means of categorical properties only. To this aim we introduce a notion of filtrality for coherent categories, relating certain lattices of subobjects to their Boolean centers. Our main result reads as follows: Up to equivalence, KH is the unique non-trivial well-pointed pretopos which is filtral and admits all set-indexed copowers of its terminal object.
A note on coextensivity of bounded hoops
该注记刻画了有界 hoop 范畴的共扩展态射，证明了有界 Wajsberg hoop 等相关范畴的共扩展性，还导出了$\boldsymbol{Heyt}$与$\boldsymbol{MV}$范畴的共扩展性。
We investigate how the concepts of intersection and sums of subobjects carry to exact categories. We obtain a new characterisation of quasi-abelian categories in terms of admitting admissible intersections in the sense of Hassoun and Roy. There are also many alternative characterisations of abelian categories as those that additionally admit admissible sums and in terms of properties of admissible morphisms. We then define a generalised notion of intersection and sum which every exact category admits. Using these new notions, we define and study classes of exact categories that satisfy the Jor…
We give a characterisation of functors whose induced functor on the level of localisations is an equivalence and where the isomorphism inverse is induced by some kind of replacements such as projective resolutions or cofibrant replacements.
Operadic categories as (pseudo)-simplicial groupoids
从任意操作范畴构造一个单纯群胚（操作神经），将局部终端、纤维函子和基数函子整合为单一结构，并给出操作范畴的新刻画。
Conservative functors to pointed categories
本文引入预指向范畴概念，研究指向范畴的保守函子，推广经典引理并建立普遍指向化构造。
Ideally regular categories
本文提出了一种扩展的理想正则范畴概念，用于分类正则商，并通过单调性过同调范畴进行特征化。
As shown in a previous paper by the same authors, the theory of Galois functors provides a categorical framework for the characterisation of bimonads on any category as Hopf monads and also for the characterisation of opmonoidal monads on monoidal categories as right Hopf monads in the sense of Bruguieres and Virelizier. Hereby the central part is to describe conditions under which a comparison functor between the base category and the category of Hopf modules becomes an equivalence (Fundamental Theorem). For monoidal categories, Aguiar and Chase extended the setting by replacing the base cate…
This article consists of an interesting characterisation of a skew monoidale in the monoidal bicategory $Span$. After discussing the shift or decalage functor on simplicial sets we characterise these skew monoidales as categories $\mathbb C$ together with a functor $R \colon Dec(\mathbb C) \longrightarrow \mathbb C$ which satisfies two conditions and give an example where the unit of the skew monoidale is not of a restricted type.
Exponentiable virtual double categories and presheaves for double categories
本文通过虚拟双范畴的可指数性，构造了伪双范畴间的 lax 函子虚拟双范畴，并揭示了其作为指数中单子与模的虚拟双范畴的泛性质，简化了伪双范畴的 Yoneda 理论。
We generalise Yoshino's definition of a degeneration of two Cohen Macaulay modules to a definition of degeneration between two objects in a triangulated category. We derive some natural properties for the triangulated category and the degeneration under which the Yoshino-style degeneration is equivalent to the degeneration defined by a specific distinguished triangle analogous to Zwara's characterisation of degeneration in module varieties.
Aspects of algebraic Algebras
本文研究Kock-Zöberlein单子下Kleisli与Eilenberg-Moore范畴之间的中间范畴，刻画自由代数、研究加权极限存在性，并应用于滤子单子的幂等完备化。
Stratifications of abelian categories
本文研究可通过迭代重摄分解的阿贝尔范畴即分层，给出其等价于有限维代数有限维模范畴及为ε-分层范畴的充要条件，推广了相关已有结论。
Given a 2-category $\twocat{K}$ admitting a calculus of bimodules, and a 2-monad T on it compatible with such calculus, we construct a 2-category $\twocat{L}$ with a 2-monad S on it such that: (1)S has the adjoint-pseudo-algebra property. (2)The 2-categories of pseudo-algebras of S and T are equivalent. Thus, coherent structures (pseudo-T-algebras) are transformed into universally characterised ones (adjoint-pseudo-S-algebras). The 2-category $\twocat{L}$ consists of lax algebras for the pseudo-monad induced by T on the bicategory of bimodules of $\twocat{K}$. We give an intrinsic characterisa…
The elementary theory of the 2-category of small categories
本文给出了内部范畴2-范畴的基本描述，扩展了Bourke和Lawvere的刻画，引入了2-良点性、全子对象分类器等概念，并展示了生成族与正交分解系统的提升。
Enriched aspects of calculus of relations and $2$-permutability
本文研究了Ord-范畴中关系演算的丰富特性与2-置换性，提出了Ord-Mal'tsev范畴的概念并给出了相关性质和示例。
Presentability and topoi in internal higher category theory
该研究发展了任意∞-topos ℬ内部的可表现范畴与topoi理论，给出相关经典结果的内部对应版本，建立结构性结果并比较内外概念，证明嵌入与等价关系，还给出若干范畴论应用。
We consider categories of relational structures that fully embed every category of universal algebras, and prove a partial characterisation of these in terms of an infinitary variant of the notion of nowhere density of Nešetřil and Ossona de Mendez. More precisely, we show that the Gaifman class of an algebraically universal category contains subdivided complete graphs of any infinite size, and establish that any monotone category satisfying this may be oriented to obtain an algebraically universal category. For the proof of the above, we also develop a categorical framework for relational gad…
Reshaping limit diagrams and cofinality in higher category theory
该研究在高阶范畴论框架下，探讨∞-范畴与(n,1)-范畴中的（余）极限问题，提出余极限图的单纯形重塑方法、∞-范畴函子的n-共尾性刻画，并简要介绍n-筛选性基础内容。
We study categorical models for the unitless fragment of multiplicative linear logic. We find that the appropriate notion of model is a special kind of promonoidal category. Since the theory of promonoidal categories has not been developed very thoroughly, at least in the published literature, we need to develop it here. The most natural way to do this - and the simplest, once the (substantial) groundwork has been laid - is to consider promonoidal categories as an instance of the general theory of pseudomonoids in a monoidal bicategory. Accordingly, we describe and explain the notions of monoi…
Monoidal su-categories
本文提出单范畴su-范畴概念，证明余端光学范畴Optic[C]在其构成的2-范畴中是2-初始的，为余端光学提供了泛性质刻画。
Topological lax comma categories
本文研究拓扑空间与lax逗号范畴之间的关系，证明在特定条件下，忘却函子保持指数和有效降映射，并刻画了有效降映射。
We investigate the categories of weak maps associated to an algebraic weak factorisation system (AWFS) in the sense of Grandis-Tholen. For any AWFS on a category with an initial object, cofibrant replacement forms a comonad, and the category of (left) weak maps associated to the AWFS is by definition the Kleisli category of this comonad. We exhibit categories of weak maps as a kind of "homotopy category", that freely adjoins a section for every "acyclic fibration" (=right map) of the AWFS; and using this characterisation, we give an alternate description of categories of weak maps in terms of …
Cofibrant generation of pure monomorphisms in presheaf categories
研究预示范畴中纯单射的共纤生成条件，通过分类C的特性确定其生成方式，特别当C为单群S时，探讨S作用下的集合纯单射生成条件。
Duality in Monoidal Categories
研究了单向范畴中闭合与刚性的关系，通过sl2晶体范畴提供反例，并探讨了Grothendieck-Verdier对偶性、QF-2代数和有限群交叉模的应用。
Internal higher topos theory
本文建立内在于任意 ∞-拓扑斯的高阶拓扑斯理论，给出多种刻画与结构定理，并据此刻画 B 上 ∞-拓扑斯、推导拉回公式及联系光滑几何态射。
Module structures of an algebra on a fixed finite dimensional vector space form an algebraic variety. Isomorphism classes correspond to orbits of the action of an algebraic group on this variety and a module is a degeneration of another if it belongs to the Zariski closure of the orbit. Riedtmann and Zwara gave an algebraic characterisation of this concept in terms of the existence of short exact sequences. Jensen, Su and Zimmermann, as well as independently Yoshino, studied the natural generalisation of the Riedtmann-Zwara degeneration to triangulated categories. The definition has an intrins…
Linear Arboreal Categories
本文通过强化树状范畴公理排除分支行为，提出线性树状范畴，并借助伴随关系建立其与满足线性化条件的树状范畴的联系，进而统一卵石关系余单子与卵石余单子，获得模态逻辑的线性变体，并证明迹包含与迹等价的新保持和刻画定理。
Let $A$ be a ring and $\M_A$ the category of $A$-modules. It is well known in module theory that for any $A $-bimodule $B$, $B$ is an $A$-ring if and only if the functor $-\otimes_A B: \M_A\to \M_A$ is a monad (or triple). Similarly, an $A $-bimodule $\C$ is an $A$-coring provided the functor $-\otimes_A\C:\M_A\to \M_A$ is a comonad (or cotriple). The related categories of modules (or algebras) of $-\otimes_A B$ and comodules (or coalgebras) of $-\otimes_A\C$ are well studied in the literature. On the other hand, the right adjoint endofunctors $\Hom_A(B,-)$ and $\Hom_A(\C,-)$ are a comonad and…
The stable module category and model structures for hierarchically defined groups
本文为包括 Kropholler 层次在内的一大类无限群构造了紧生成的张量三角稳定范畴，证明其作为模型范畴的同伦范畴与其他已知模型范畴 Quillen 等价，并刻画了平凡表示为紧对象的群。
Magnitude of module categories
通过表示有限代数的奥斯特兰德代数量级定义模范畴不变量，利用平移箭图的奥斯特兰德 - 赖滕 - 欧拉特征辅助计算，确定了多种代数模范畴量级，得到关于表示有限双列代数的推测性刻画。
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…
Q-system Completion of 2-functors
本文证明在目标 C*-2-范畴 Q-system 完备时，*-2-函子构成的 C*-2-范畴也完备，并据此刻画完备范畴、证明酉融合范畴在 C*-代数上的作用 2-范畴完备。
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