Pieter Hofstra, Philip Scott
We present a survey of some developments in the general area of category-theoretic approaches to the theory of computation, with a focus on topics and ideas particularly close to the interests of Jim Lambek.
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A new categorical setting is defined in order to characterize the subrecursive classes belonging to complexity hierarchies. This is achieved by means of coercion functors over a symmetric monoidal category endowed with certain recursion schemes that imitate the bounded recursion scheme. This gives a categorical counterpart of generalized safe composition and safe recursion.
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。
Categorical aspects of congruence distributivity
本文研究了关系上的一个范畴条件，作为Jónsson对同余分配簇刻画的范畴化表述，并证明满足该条件的范畴不必是簇，拓扑空间等对偶范畴即为实例。
We give to the categorical theory PR of Primitive Recursion a logically simple, algebraic presentation, via equations between maps, plus one genuine Horner type schema, namely Freyd's uniqueness of the initialised iterated. Free Variables are introduced - formally - as another names for projections. Predicates χ: A -> 2 admit interpretation as (formal) Objects {A|χ} of a surrounding Theory PRA = PR + (abstr) : schema (abstr) formalises this predicate abstraction into additional Objects. Categorical Theory P\hat{R}_A \sqsupset PR_A \sqsupset PR then is the Theory of formally partial PR-maps, ha…
We give a definition of an operad with general groups of equivariance suitable for use in any symmetric monoidal category with appropriate colimits. We then apply this notion to study the 2-category of algebras over an operad in Cat. We show that any operad is finitary, that an operad is cartesian if and only if the group actions are nearly free (in a precise fashion), and that the existence of a pseudo-commutative structure largely depends on the groups of equivariance. We conclude by showing that the operad for strict braided monoidal categories has two canonical pseudo-commutative structure…
Logical Aspects of Virtual Double Categories
本论文提出一种符合范畴逻辑原则的虚拟双范畴方法，在语义侧将其作为谓词逻辑模型以包含现有框架，在语法侧开发名为FVDblTT的类型理论作为其内部语言。
There is a construction which lies at the heart of descent theory. The combinatorial aspects of this paper concern the description of the construction in all dimensions. The description is achieved precisely for strict n-categories and outlined for weak n-categories. The categorical aspects concern the development of descent theory in low dimensions in order to provide a template for a theory in all dimensions. The theory involves non-abelian cohomology, stacks, torsors, homotopy, and higher-dimensional categories. Many of the ideas are scattered through the literature or are folklore; a few a…
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。
We discuss relations between some category-theoretical notions for a finite tensor category and cointegrals on a quasi-Hopf algebra. Specifically, for a finite-dimensional quasi-Hopf algebra $H$, we give an explicit description of categorical cointegrals of the category ${}_H \mathscr{M}$ of left $H$-modules in terms of cointegrals on $H$. Provided that $H$ is unimodular, we also express the Frobenius structure of the `adjoint algebra' in the Yetter-Drinfeld category ${}^H_H \mathscr{YD}$ by using an integral in $H$ and a cointegral on $H$. Finally, we give a description of the twisted module …
These are extended notes for a series of lectures on Hall algebras given at the CRM Barcelona in February 2015. The basic idea of the theory of Hall algebras is that the collection of flags in an exact category encodes an associative multiplication law. While introduced by Steinitz and Hall for the category of abelian p-groups, it has since become clear that the original construction can be applied in much greater generality and admits numerous useful variations. These notes focus on higher categorical aspects based on the relation between Hall algebras and Waldhausen's S-construction.
We show that either of the two reasonable choices for the category of compact quantum groups is nice enough to allow for a plethora of universal constructions, all obtained "by abstract nonsense" via the adjoint functor theorem. This approach both recovers constructions which have appeared in the literature, such as the quantum Bohr compactification of a locally compact semigroup, and provides new ones, such as the coproduct of a family of compact quantum groups, and the compact quantum group freely generated by a locally compact quantum space. In addition, we characterize epimorphisms and mon…
Categorical-algebraic aspects of Heyting semilattices
本文研究Heyting半格范畴的范畴代数性质，纠正文献错误认知，证明其不具代数相干性、不满足并集正规性等结论，明确相关蕴含关系的严格性。
The system of intuitionistic modal logic ${\bf IEL}^{-}$ was proposed by S. Artemov and T. Protopopescu as the intuitionistic version of belief logic \cite{Artemov}. We construct the modal lambda calculus which is Curry-Howard isomorphic to ${\bf IEL}^{-}$ as the type-theoretical representation of applicative computation widely known in functional programming. We also provide a categorical interpretation of this modal lambda calculus considering coalgebras associated with a monoidal functor on a cartesian closed category. Finally, we study Heyting algebras and locales with corresponding operat…
A monoidal category of dependently sorted algebraic theories II: categorical aspects
本文从语境范畴的角度研究广义代数理论的闭对称幺半结构，定义指数函子和多重态射，并证明张量积的存在性和函子性。
We analyse some aspects of the notion of algebraic exponentiation introduced by the second author [16] and satisfied by the category of groups. We show how this notion provides a new approach to the categorical-algebraic question of the centralization. We explore, in the category of groups, the unusual universal properties and constructions determined by this notion, and we show how it is the origin of various properties of this category.
Heaps of modules: Categorical aspects
本文建立模的堆与环上仿射模之间的范畴等价，并利用这些对应关系显式描述（余）极限及比较短正合序列。
Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation
该研究从增强型2-范畴视角解释二维范畴论中结构的内在严格性，引入增强型2-范畴极限草图以弥补现有理论不足，并建立了这类结构的内化对称性。
We argue for the addition of category theory to the toolkit of toric topology, by surveying recent examples and applications. Our case is made in terms of toric spaces X_K, such as moment-angle complexes Z_K, quasitoric manifolds M, and Davis-Januszkiewicz spaces DJ(K). We first exhibit X_K as the homotopy colimit of a diagram of spaces over the small category cat(K), whose objects are the faces of a finite simplicial complex K and morphisms their inclusions. Then we study the corresponding cat(K)-diagrams in various algebraic Quillen model categories, and interpret their homotopy colimits as …
Some aspects of descent theory and applications
本文阐述作者在广义范畴结构有效下降态射的贡献，含下降理论基础描述、关键定义结果及博士期间四篇相关文章内容。
This thesis provides an introduction to the various category theory ideas employed in topological quantum field theory. These theories are viewed as symmetric monoidal functors from topological cobordism categories into the category of vector spaces. In two dimensions, they are classified by Frobenius algebras. In three dimensions, and under certain conditions, they are classified by modular categories. These are special kinds of categories in which topological notions such as braidings and twists play a prominent role. There is a powerful graphical calculus available for working in such categ…
Here are considered some categorical aspects of "Differential calculus" archetype of local approximation of arbitrary morphisms by "linear" ones.
We study 2-monads and their algebras using a Cat-enriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2-categorical points of view. Every 2-category with finite limits and colimits has a canonical model structure in which the weak equivalences are the equivalences; we use these to construct more interesting model structures on 2-categories, including a model structure on the 2-category of algebras for a 2-monad T, and a model structure on a 2-category of 2-monads on a fixed 2-category K.
This is the third installment in a series of papers on algebraic set theory. In it, we develop a uniform approach to sheaf models of constructive set theories based on ideas from categorical logic. The key notion is that of a "predicative category with small maps" which axiomatises the idea of a category of classes and class morphisms, together with a selected class of maps whose fibres are sets (in some axiomatic set theory). The main result of the present paper is that such predicative categories with small maps are stable under internal sheaves. We discuss the sheaf models of constructive s…
Infinite products and zero-one laws in categorical probability
本文在马尔可夫范畴框架下发展无穷张量积与柯尔莫哥洛夫扩张定理，并据此证明柯尔莫哥洛夫及休伊特-萨维奇零一律的一般范畴版本，从而将经典测度论结果推广到更广泛的概率语境。
This presentation discusses a new, modular, more mature architecture for the Information Flow Framework (IFF). The IFF uses institution theory as a foundation for the semantic integration of ontologies. It represents metalogic, and as such operates at the structural level of ontologies. The content, form and experience of the IFF could contribute to the development of a standard ontology for category theory. The foundational aspect of the IFF helps to explain the relationship between the fundamental concepts of set theory and category theory. The development of the IFF follows two design princ…
Implementing the biset category of finite groups
本文提出了一种通过标准范畴构造的塔结构来实现有限群的biset范畴，并展示了该实现如何利用Kleisli范畴中的biadjunction monad来描述biset的复合，从而为Schreier-Sims轨道算法提供了优雅的范畴解释。
Several results on exact sequences in categories of modules over trusses
本文在truss上模范畴中建立了蛇引理和九引理，推进了truss上模理论的范畴论研究。
Categorical aspects of the theory of modules over trusses are studied. Tensor product of modules over trusses is defined and its existence established. In particular, it is shown that bimodules over trusses form a monoidal category. Truss versions of the Eilenberg-Watts theorem and Morita equivalence are formulated. Projective and small-projective modules over trusses are defined and their properties studied.
This note is a survey on the basic aspects of moduli theory along with some examples. In that respect, one of the purposes of this current document is to understand how the introduction of stacks circumvents the non-representability problem of the corresponding moduli functor $\mathcal{F}$ by using the 2-category of stacks. To this end, we shall briefly revisit the basics of 2-category theory and present a 2-categorical version of Yoneda's lemma.
Probabilistic morphisms and Bayesian supervised learning
本文通过发展马尔可夫核的范畴论研究贝叶斯反演，提出涵盖贝叶斯密度估计的统一监督学习模型，并用高斯过程回归加以说明。
Reasoning about weak higher categorical structures constitutes a challenging task, even to the experts. One principal reason is that the language of set theory is not invariant under the weaker notions of equivalence at play, such as homotopy equivalence. From this point of view, it is natural to ask for a different foundational setting which more natively supports these notions. Our work takes up on suggestions in the original article arXiv:1705.07442 by Riehl--Shulman to further develop synthetic $(\infty,1)$-category theory in simplicial homotopy type theory, including in particular the stu…
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