范畴论如何运作：集合中态射、对偶性与泛构造的元素与区分分析

拓扑斯的Gabriel-Ulmer对偶及其与场表示的关系

Gabriel-Ulmer duality for topoi and its relation with site presentations

该文研究正则基数下限制到κ-可展示Grothendieck拓扑斯时的Gabriel-Ulmer对偶，刻画了对应表示2-范畴的结构，分析其与场表示的关联并证明了相关反射子双范畴结论。

Categories of contexts

Morphisms between (formal) contexts are certain pairs of maps, one between objects and one between attributes of the contexts in question. We study several classes of such morphisms and the connections between them. Among other things, we show that the category CLc of complete lattices with complete homomorphisms is (up to a natural isomorphism) a full reflective subcategory of the category of contexts with so-called conceptual morphisms; the reflector associates with each context its concept lattice. On the other hand, we obtain a dual adjunction between CLc and the category of contexts with …

将紧致 $T_1$-空间反射到有界分配格

Reflecting compact $T_1$-spaces into bounded distributive lattices

本文构造了紧致 $T_1$-空间到有界分配格的反变反射，用闭 subfit 态射推广 Stone 对偶与 Isbell 对偶，并在完备紧致 subfit 格上得到对偶，进而恢复 Cornish 的经典结果。

Representing Semilattices as Relations

We represent finite join-semilattices and join-preserving morphisms as a category whose objects and morphisms are binary relations. It is a quotient category of $\mathsf{Rel}_f$'s arrow category, where self-duality arises by taking the relational converse on both objects and morphisms. We investigate the tensor product and the "tight" tensor product using both categories. We also refine the categorical equivalence i.e. finite De Morgan algebras with bounded lattice morphisms are equivalent to a category whose objects are the symmetric relations.

预层与算子代数态射的最小模型：基于盒算子Koszul对偶

A minimal model for prestacks and morphisms of operadic algebras via Koszul duality for box operads

本文通过盒算子的Koszul对偶理论，为编码$P$-代数态射和松弛预层的盒算子建立了最小模型，解决了Markl的开放问题。

Remarks on Morphisms of Spectral Geometries

Having in view the study of a version of Gel'fand-Neumark duality adapted to the context of Alain Connes' spectral triples, in this very preliminary review, we first present a description of the relevant categories of geometrical spaces, namely compact Hausdorff smooth finite-dimensional orientable Riemannian manifolds (or more generally Hermitian bundles of Clifford modules over them); we give some tentative definitions of the relevant categories of algebraic structures, namely "propagators" and "spectral correspondences" of commutative Riemannian spectral triples; and we provide a constructi…

Contravariance through enrichment

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.

Algebraic duality for partially ordered sets

For an arbitrary partially ordered set $P$ its {\em dual} $P^*$ is built as the collection of all monotone mappings $P\to\2$ where $\2=\{0,1\}$ with $0<1$. The set of mappings $P^*$ is proved to be a complete lattice with respect to the pointwise partial order. The {\em second dual} $P^{**}$ is built as the collection of all morphisms of complete lattices $P^*\to\2$ preserving universal bounds. Then it is proved that the partially ordered sets $P$ and $P^{**}$ are isomorphic.

Some notes on Esakia spaces

Under Stone/Priestley duality for distributive lattices, Esakia spaces correspond to Heyting algebras which leads to the well-known dual equivalence between the category of Esakia spaces and morphisms on one side and the category of Heyting algebras and Heyting morphisms on the other. Based on the technique of idempotent split completion, we give a simple proof of a more general result involving certain relations rather then functions as morphisms. We also extend the notion of Esakia space to all stably locally compact spaces and show that these spaces define the idempotent split completion of…

A Remark on Gelfand Duality for Spectral Triples

We present a duality between the category of compact Riemannian spin manifolds (equipped with a given spin bundle and charge conjugation) with isometries as morphisms and a suitable "metric" category of spectral triples over commutative pre-C*-algebras. We also construct an embedding of a "quotient" of the category of spectral triples introduced in arXiv:math/0502583v1 into the latter metric category. Finally we discuss a further related duality in the case of orientation and spin-preserving maps between manifolds of fixed dimension.

A De Vries-type Duality Theorem for Locally Compact Spaces -- III

In this paper we prove some new Stone-type duality theorems for some subcategories of the category $\ZLC$ of locally compact zero-dimensional Hausdorff spaces and continuous maps. These theorems are new even in the compact case. They concern the cofull subcategories $\SkeZLC$, $\QPZLC$, $\OZLC$ and $\OPZLC$ of the category $\ZLC$ determined, respectively, by the skeletal maps, by the quasi-open perfect maps, by the open maps and by the open perfect maps. In this way, the zero-dimensional analogues of Fedorchuk Duality Theorem and its generalization are obtained. Further, we characterize the in…

矢列式代数的对偶理论

Duality Theory for Sequent Algebras

本文研究由D. Vakarelov提出的矢列式代数，建立其与矢列式空间范畴的对偶等价，刻画矢列式同余，证明该范畴在布尔代数范畴上是拓扑的并得到相关完备性结果。

Division, adjoints, and dualities of bilinear maps

The distributive property can be studied through bilinear maps and various morphisms between these maps. The adjoint-morphisms between bilinear maps establish a complete abelian category with projectives and admits a duality. Thus the adjoint category is not a module category but nevertheless it is suitably familiar. The universal properties have geometric perspectives. For example, products are orthogonal sums. The bilinear division maps are the simple bimaps with respect to nondegenerate adjoint-morphisms. That formalizes the understanding that the atoms of linear geometries are algebraic ob…

Categorical Extension of Dualities: From Stone to de Vries and Beyond, II

Under a general categorical procedure for the extension of dual equivalences as presented in this paper's predecessor, a new algebraically defined category is established that is dually equivalent to the category $\bf LKHaus$ of locally compact Hausdorff spaces and continuous maps, with the dual equivalence extending a Stone-type duality for the category of extremally disconnected locally compact Hausdorff spaces and continuous maps. The new category is then shown to be isomorphic to the category $\bf CLCA$ of complete local contact algebras and suitable morphisms. Thereby, a new proof is pres…

Fibrations and Koszul duality in locally Cartesian localisations

I show that any locally Cartesian left localisation of a presentable infinity-category admits a right proper model structure in which all morphisms are cofibrations, and obtain a Koszul duality classification of its fibrations. By a simple criterion in terms of generators for a localisation to be locally Cartesian, this applies to any nullification functor. In particular, it includes examples with non-trivial "homotopical content." I further describe, and provide examples from, the set of fibrations in three contexts: the higher categorical Thomason model structure of Mazel-Gee, where fibratio…

单调子的石对偶性

Stone Duality for Monads

本文研究了单调子的石对偶性，通过构造一个反变的幂等伴随，将有等级的单调子范畴与内部范畴和内部反函子范畴联系起来，揭示了单调子的行为范畴与局部范畴之间的对应关系。