Hongliang Lai, Lili Shen
Based on Garner's discovery that topological categories are total categories enriched in a quantaloid, this paper presents a series of results related to initial and final density in categorical topology via (co)density in quantaloid-enriched categories, focusing on (co-)Sierpiński objects, Galois correspondences and their fixed points.
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We formulate an elementary condition on an involutive quantaloid Q under which there is a distributive law from the Cauchy completion monad over the symmetrisation comonad on the category of Q-enriched categories. For such quantaloids, which we call Cauchy-bilateral quantaloids, it follows that the Cauchy completion of any symmetric Q-enriched category is again symmetric. Examples include Lawvere's quantale of non-negative real numbers and Walters' small quantaloids of closed cribles.
Applying (enriched) categorical structures we define the notion of ordered sheaf on a quantaloid Q, which we call `Q-order'. This requires a theory of semicategories enriched in the quantaloid Q, that admit a suitable Cauchy completion. There is a quantaloid Idl(Q) of Q-orders and ideal relations, and a locally ordered category Ord(Q) of Q-orders and monotone maps; actually, Ord(Q)=Map(Idl(Q)). In particular is Ord(Omega), with Omega a locale, the category of ordered objects in the topos of sheaves on Omega. In general Q-orders can equivalently be described as Cauchy complete categories enrich…
Quantaloidal Completions of Order-enriched Categories and Their Applications
本文提出序丰富范畴的量词范畴完备化概念，刻画其为幂集完备化的相容商，并证明特殊内射包为MacNeille完备化、自由量词范畴为下集完备化。
We thoroughly treat several familiar and less familiar definitions and results concerning categories, functors and distributors enriched in a base quantaloid Q. In analogy with V-category theory we discuss such things as adjoint functors, (pointwise) left Kan extensions, weighted (co)limits, presheaves and free (co)completion, Cauchy completion and Morita equivalence. With an appendix on the universality of the quantaloid Dist(Q) of Q-enriched categories and distributors.
Quantaloid-enriched categories: Factorization, weak classifiers, and symmetry
本文研究了quantaloid-enriched范畴的基础性质，证明了其左伴随分配子范畴存在(epi, extremal mono)-分解系统，并在稳定性条件下满足弱子对象分类子公理，进而将结构结果推广至quantale值集。
Injective symmetric quantaloid-enriched categories
本文研究小型、整且对合 quantaloid 上的对称强化范畴，刻画其内射对象、内射包和本质嵌入，并给出内射偏度量空间的精确表述。
It is shown that, for a small quantaloid Q, the category of small Q-categories and Q-functors is total and cototal, and so is the category of Q-distributors and Q-Chu transforms.
In fairly elementary terms this paper presents, and expands upon, a recent result by Garner by which the notion of topologicity of a concrete functor is subsumed under the concept of total cocompleteness of enriched category theory. Motivated by some key results of the 1970s, the paper develops all needed ingredients from the theory of quantaloids in order to place essential results of categorical topology into the context of quantaloid-enriched category theory, a field that previously drew its motivation and applications from other domains, such as quantum logic and sheaf theory.
Our subject is that of categories, functors and distributors enriched in a base quantaloid Q. We show how cocomplete Q-categories are precisely those which are tensored and conically cocomplete, or alternatively, those which are tensored, cotensored and order-cocomplete. Bearing this in mind, we analyze how Sup-valued homomorphisms on Q are related to Q-categories. With an appendix on action, representation and variation.
We study presheaves on semicategories enriched in a quantaloid: this gives rise to the notion of regular presheaf. A semicategory is regular when its representable presheaves are regular, and its regular presheaves then constitute an essential (co)localization of the category of all of its presheaves. The notion of regular semidistributor allows to establish the Morita equivalence of regular semicategories. Continuous orders and Omega-sets provide examples.
For any small quantaloid $\Q$, there is a new quantaloid $\D(\Q)$ of diagonals in $\Q$. If $\Q$ is divisible then so is $\D(\Q)$ (and vice versa), and then it is particularly interesting to compare categories enriched in $\Q$ with categories enriched in $\D(\Q)$. Taking Lawvere's quantale of extended positive real numbers as base quantale, $\Q$-categories are generalised metric spaces, and $\D(\Q)$-categories are generalised partial metric spaces, i.e.\ metric spaces in which self-distance need not be zero and with a suitably modified triangular inequality. We show how every small quantaloid-e…
With quantaloids carefully constructed from multi-adjoint frames, it is shown that multi-adjoint concept lattices, multi-adjoint property-oriented concept lattices and multi-adjoint object-oriented concept lattices are derivable from Isbell adjunctions, Kan adjunctions and dual Kan adjunctions between quantaloid-enriched categories, respectively.
For any small involutive quantaloid Q we define, in terms of symmetric quantaloid-enriched categories, an involutive quantaloid Rel(Q) of Q-sheaves and relations, and a category Sh(Q) of Q-sheaves and functions; the latter is equivalent to the category of symmetric maps in the former. We prove that Rel(Q) is the category of relations in a topos if and only if Q is a modular, locally localic and weakly semi-simple quantaloid; in this case we call Q a Grothendieck quantaloid. It follows that Sh(Q) is a Grothendieck topos whenever Q is a Grothendieck quantaloid. Any locale L is a Grothendieck qua…
Exponentiable functors between quantaloid-enriched categories are characterized in elementary terms. The proof goes as follows: the elementary conditions on a given functor translate into existence statements for certain adjoints that obey some lax commutativity; this, in turn, is precisely what is needed to prove the existence of partial products with that functor; so that the functor's exponentiability follows from the works of Niefield [1980] and Dyckhoff and Tholen [1987].
Each distributor between categories enriched over a small quantaloid Q gives rise to two adjunctions between the categories of contravariant and covariant presheaves, and hence to two monads. These two adjunctions are respectively generalizations of Isbell adjunctions and Kan extensions in category theory. It is proved that these two processes are functorial with infomorphisms playing as morphisms between distributors; and that the free cocompletion functor of Q-categories factors through both of these functors.
Completeness of quantaloid-enriched categories up to Morita equivalence
本文研究了在莫里塔等价下量化loid富集范畴的完备性，通过预设单子和张量性特征化 Q-范畴的 (co) 完备性。
Ideals and continuity for quantaloid-enriched categories
本文将论域理论推广到多值、多类型的 quantaloid 强化范畴，定义 Q-范畴的 Phi-连续性，并考察平坦、不可约与锥形三类饱和理想。
Representation theorems are established for fixed points of adjoint functors between categories enriched in a small quantaloid. In a very general setting these results set up a common framework for representation theorems of concept lattices in formal concept analysis (FCA) and rough set theory (RST), which not only extend the realm of formal contexts to multi-typed and multi-valued ones, but also provide a general approach to construct various kinds of representation theorems. Besides incorporating several well-known representation theorems in FCA and RST as well as formulating new ones, it i…
This dissertation is devoted to a study of adjunctions concerning categories enriched over a quantaloid Q (or Q-categories for short), with the following types of adjunctions involved: (1) adjoint functors between Q-categories; (2) adjoint distributors between Q-categories; (3) adjoint functors between categories consisting of Q-categories. For a small quantaloid Q and a distributor between Q-categories, two adjunctions between the Q-categories of contravariant and covariant presheaves are presented. These adjunctions respectively extend the fundamental construction of Isbell adjunctions and K…
In the context of quantaloid-enriched categories, we explain how each saturated class of weights defines, and is defined by, an essentially unique full sub-KZ-doctrine of the free cocompletion KZ-doctrine. The KZ-doctrines which arise as full sub-KZ-doctrines of the free cocompletion, are characterised by two simple "fully faithfulness" conditions. Conical weights form a saturated class, and the corresponding KZ-doctrine is precisely (the generalisation to quantaloid-enriched categories of) the Hausdorff doctrine of [Akhvlediani et al., 2009].
For a quantaloid $\mathcal{Q}$, considered as a bicategory, Walters introduced categories enriched in $\mathcal{Q}$. Here we extend the study of monad-quantale-enriched categories of the past fifteen years by introducing monad-quantaloid-enriched categories. We do so by making lax distributive laws of a monad $\mathbb{T}$ over the discrete presheaf monad of the small quantaloid $\mathcal{Q}$ the primary data of the theory, rather than the lax monad extensions of $\mathbb{T}$ to the category of $\mathcal{Q}$-relations that they equivalently describe. The central piece of the paper establishes a…
Sheaves on a bicategory
本文建立双范畴上的富集理论，定义完备 B-范畴，并构造其与 Map(B) 上 2-预层的伴随，推广并统一 quantaloid 与幺半范畴中的层论结果。
Enriched categorical aspects of homological perturbation theory
本文用充实范畴语言重新阐述同调摄动理论，给出同调摄动引理的表述、证明与函子性性质，为该理论提供范畴化框架。
It is common practice in both theoretical computer science and theoretical physics to describe the (static) logic of a system by means of a complete lattice. When formalizing the dynamics of such a system, the updates of that system organize themselves quite naturally in a quantale, or more generally, a quantaloid. In fact, we are lead to consider cocomplete quantaloid-enriched categories as fundamental mathematical structure for a dynamic logic common to both computer science and physics. Here we explain the theory of totally continuous cocomplete categories as generalization of the well-know…
When is Cat(Q) cartesian closed?
本文研究了哪些量词oids Q 使得Cat(Q)为笛卡儿闭合，并统一了已知结果并提供了新例子。
For a small quantaloid $\mathcal{Q}$ we consider four fundamental 2-monads $\mathbb{T}$ on $\mathcal{Q}\text{-}{\bf Cat}$, given by the presheaf 2-monad $\mathbb{P}$ and the copresheaf 2-monad $\mathbb{P}^{\dagger}$, as well as by their two composite 2-monads, and establish that they all laxly distribute over $\mathbb{P}$. These four 2-monads therefore admit lax extensions to the category $\mathcal{Q}\text{-}{\bf Dist}$ of $\mathcal{Q}$-categories and their distributors. We characterize the corresponding $(\mathbb{T},\mathcal{Q})$-categories in each of the four cases, leading us to both known …
On enriched terms and 2-categorical universal algebra
本文引入递归生成的富化项与项可解释性概念，将其应用于2-范畴情形，提出2维项与2维等式理论，给出自由结构显式描述并证明2维Birkhoff簇定理。
Naturality of the $\infty$-Categorical Enriched Yoneda Embedding
本文通过将 Hinich 的 ∞-范畴丰富 Yoneda 嵌入构造为丰富预层函子与 Heine 函子之间部分伴随的单位，使其自然化，并进一步证明该部分伴随在原子对象上限制为真正的伴随。
Categorical Diffusion of Weighted Lattices
本文提出加权格的范畴扩散框架，通过Lawvere拉普拉斯和Hodge-Lawvere定理，提供了一种离散时间谐波流用于信息聚合的构造方法。
Tensor Enriched Categorical Generalization of the Eilenberg-Watts Theorem
本文在Bénabou宇宙中推广Eilenberg-Watts定理，证明交换幺半群范畴与余连续lax幺半丰富函子范畴的等价性，受六函子形式主义启发。
Smyth complete real-enriched categories
本文研究实富集范畴的 Smyth 完备性，证明其等价于分离且所有理想可表示，也等价于其 Alexandroff 实值拓扑为 sober。
In this work, we explore a double categorical framework for categories of enriched graphs, categories and the newly introduced notion of cocategories. A fundamental goal is to establish an enrichment of V-categories in V-cocategories, which generalizes the so-called Sweedler theory relatively to an enrichment of algebras in coalgebras. The language employed is that of V-matrices, and an interplay between the double categorical and bicategorical perspective provides a high-level flexibility for demonstrating essential features of these dual structures. Furthermore, we investigate an enriched fi…
We investigate an enriched-categorical approach to a field of discrete mathematics. The main result is a duality theorem between a class of enriched categories (called $\overline{\mathbb{Z}}$- or $\overline{\mathbb{R}}$-categories) and that of what we call ($\overline{\mathbb{Z}}$- or $\overline{\mathbb{R}}$-) extended L-convex sets. We introduce extended L-convex sets as variants of certain discrete structures called L-convex sets and L-convex polyhedra, studied in the field of discrete convex analysis. We also introduce homomorphisms between extended L-convex sets. The theorem claims that th…
A Biset-Enriched Categorical Model for Proto-Quipper with Dynamic Lifting
本文提出一种双集丰富范畴结构，用于建模量子编程语言Proto-Quipper中动态提升操作，实现电路生成与执行两个运行时的正确交互语义。
Generalising Nachbin's theory of "topology and order", in this paper we continue the study of quantale-enriched categories equipped with a compact Hausdorff topology. We compare these $\mathcal{V}$-categorical compact Hausdorff spaces with ultrafilter-quantale-enriched categories, and show that the presence of a compact Hausdorff topology guarantees Cauchy completeness and (suitably defined) codirected completeness of the underlying quantale enriched category.
In this work, we use Ising chain and Kitaev chain to check the validity of an earlier proposal in arXiv:2011.02859 that enriched fusion (higher) categories provide a unified categorical description of all gapped/gapless quantum liquid phases, including symmetry-breaking phases, topological orders, SPT/SET orders and certain gapless quantum phases. In particular, we show explicitly that, in each gapped phase realized by these two models, the spacetime observables form a fusion category enriched in a braided fusion category. We also study the categorical descriptions of the boundaries of these m…
We provide a definition of enrichment that applies to a wide variety of categorical structures, generalizing Leinster's theory of enriched $T$-multicategories. As a sample of newly enrichable structures, we describe in detail the examples of enriched monoidal categories and enriched double categories, with a focus on monoidal double categories as broadly convenient bases of enrichment.
A Categorical Approach to Möbius Inversion via Derived Functors
本文通过丰富范畴中的导出函子定义Möbius上同调，证明其欧拉示性数恢复经典Möbius反演，并给出Rota的Galois联络的范畴版本，统一了组合学与同调代数。
We give a functorial construction of equivariant spectra from a generalized version of Mackey functors in categories. This construction relies on the recent description of the category of equivariant spectra due to Guillou and May. The key element of our construction is a spectrally-enriched functor from a spectrally-enriched version of permutative categories to the category of spectra that is built using an appropriate version of K-theory. As applications of our general construction, we produce a new functorial construction of equivariant Eilenberg--MacLane spectra for Mackey functors and for…
We propose a definition of involutive categorical bundle (Fell bundle) enriched in an involutive monoidal category and we argue that such a structure is a possible suitable environment for the formalization of different equivalent versions of spectral data for commutative C*-categories.
Formal weakly enriched category theory
本文构造了编码弱充实的形式范畴论，证明其与拟范畴的无穷宇宙等价，并导出充实版Quillen定理A。
Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the down-set monad becomes the filter monad, cocomplete ordered set translates to continuous lattice, distributivity means disconnectedness, and so on. Curiously, the dual(?) notion of completeness does not behave as the mirror image of the one of cocompleteness; and in this paper we have a closer look at complete spaces. In particular, we construct the "up-set monad" on representable…
The Classification of 3+1d Symmetry Enriched Topological Order
该研究利用2 - 范畴的（去）等变对有有限\(G\) - 对称性的（3 + 1）维拓扑序分类，指出含\(G\) - 对称性的（3 + 1）维费米子拓扑序对应特定范畴，且相关范畴数据与特定构造一致，还说明了此方法可对所有辫合融合2 - 范畴分类。
Properties of categories enriched over the category of metric spaces are investigated and applied to a study of constructions known from that category and the category of Banach spaces. For every class of morphisms satisfying a mild smallness condition we prove the corresponding approximate-injectivity class is weakly reflective, and we study the properties of the reflection morphisms. As an application we present a new categorical proof of the essential uniqueness of the Gurarii space.
Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory
本文通过富范畴理论为基于电路的量子函数式编程语言建立范畴模型，将电路语言嵌入宿主语言，并利用W*-代数范畴的dcpo-富结构扩展递归类型。
Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries
本研究针对带阿贝尔在位对称性的1+1维有能隙相，用富集融合范畴描述其时空可观测量，统一了对称相分类与格点对偶性，给出非平凡SPT相等实例，通过全息对偶关联二维拓扑序与一维有能隙量子液体，为量子相统一定义提供思路。
The 2-category V-Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. The exception is the case in which V is symmetric, which leads to V-Cat being symmetric as well. This paper describes how these facts are related to a categorical analogue of topological delooping. It seems that the analogy of loop spaces is a good guide for how to define the concept of enrichment over various types of monoidal objects, including k-fold monoidal categories and their higher dimensional counterparts. The main result is that for V a…
We investigate certain categorical aspects of Voevodsky's triangulated categories of motives. For this, various recollements for Grothendieck categories of enriched functors and their derived categories are established. In order to extend these recollements further with respect to Serre's localization, the concept of the (strict) Voevodsky property for Serre localizing subcategories is introduced. This concept is inspired by the celebrated Voevodsky theorem on homotopy invariant presheaves with transfers. As an application, it is shown that Voevodsky's triangulated categories of motives fit in…
Double categories of profunctors
本文通过引入双范畴余极限的概念，严格刻画了富集范畴、函子和联络函子的虚拟双范畴，强调在对象层面的等价性和富集在虚拟双范畴中的统一性。
Categorical Time-Reversal Symmetries
本文探讨了反单位ary对称性在分类相中的作用，通过实融合范畴模型揭示时间反演对称性等物理结构，并分类具有反线性对称性的能隙相。
We prove a rectification theorem for enriched infinity-categories: If V is a nice monoidal model category, we show that the homotopy theory of infinity-categories enriched in V is equivalent to the familiar homotopy theory of categories strictly enriched in V. It follows, for example, that infinity-categories enriched in spectra or chain complexes are equivalent to spectral categories and dg-categories. A similar method gives a comparison result for enriched Segal categories, which implies that the homotopy theories of n-categories and (infinity,n)-categories defined by iterated infinity-categ…
A notion of central importance in categorical topology is that of topological functor. A faithful functor E -> B is called topological if it admits cartesian liftings of all (possibly large) families of arrows; the basic example is the forgetful functor Top -> Set. A topological functor E -> 1 is the same thing as a (large) complete preorder, and the general topological functor E -> B is intuitively thought of as a complete preorder relative to B. We make this intuition precise by considering an enrichment base Q_B such that Q_B-enriched categories are faithful functors into B, and show that, …
Hausdorff and Gromov distances are introduced and treated in the context of categories enriched over a commutative unital quantale V. The Hausdorff functor which, for every V-category X, provides the powerset of X with a suitable V-category structure, is part of a monad on V-Cat whose Eilenberg-Moore algebras are order-complete. The Gromov construction may be pursued for any endofunctor K of V-Cat. In order to define the Gromov "distance" between V-categories X and Y we use V-modules between X and Y, rather than V-category structures on the disjoint union of X and Y. Hence, we first provide a …
Convergence of martingales via enriched dagger categories
本文通过拓扑丰富类别理论，首次证明了鞅和逆向鞅在均值下的收敛性，并展示了其在概率分析中的应用。
Categories enriched over a commutative unital quantale can be studied as generalized, or many-valued, ordered structures. Because many concepts, such as complete distributivity, in lattice theory can be characterized by existence of certain adjunctions, they can be reformulated in the many-valued setting in terms of categorical postulations. So, it is possible, by aid of categorical machineries, to establish theories of many-valued complete lattices, many-valued completely distributive lattices, and so on. This paper presents a systematical investigation of many-valued complete distributivity,…
We propose a categorical foundation for the connection between pure and mixed states in quantum information and quantum computation. The foundation is based on distributive monoidal categories. First, we prove that the category of all quantum channels is a canonical completion of the category of pure quantum operations (with ancilla preparations). More precisely, we prove that the category of completely positive trace-preserving maps between finite-dimensional C*-algebras is a canonical completion of the category of finite-dimensional vector spaces and isometries. Second, we extend our result …
Accessible aspects of 2-category theory
该研究针对范畴结构及其伪映射难构成局部可展示2-范畴的问题，证明足够弱的范畴结构对应的2-范畴具有可达性，并探究了这类2-范畴的柔性极限及其与滤过余极限的相互作用，丰富了2-范畴论的可达性相关结论。
On $2$-categorical $\infty$-cosmoi
本文研究作为 $2$-范畴的 $\infty$-宇宙（称为 $2$-宇宙），证明具有柔性极限的 $2$-范畴产生最小 Cauchy 完备的 $2$-宇宙，并探讨可达 $2$-宇宙的基本理论。
Sweedler theory for double categories
本文在幺半双范畴背景下建立对偶代数结构的张量与余张量充实，证明了module在monad上的纤维化在comonad上的comodule的opfibration中得到充实，并将其应用于V-矩阵以推广Sweedler的测量余代数理论。
Multifunctorial Equivariant Algebraic K-Theory
本文构造从 O-伪代数 G-范畴多范畴到正交 G-谱的富化多函子 K，证明等变 K-理论可保持多范畴参数化的代数结构。
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