范畴论

Tambara modules are strong profunctors between monoidal categories. They've been defined by Tambara in the context of representation theory, but quickly found their way in applications when it was understood Tambara modules provide a useful…

Most texts on Category Theory are written in a very terse style, in which people pretend a) that all concepts are visualizable, and b) that the readers can reconstruct the diagrams that the authors had in mind based on only the most…

While any infimum in a poset can also be computed as a supremum, and vice versa, categorical limits and colimits do not always approximate each other. If I approach a point from below, and you approach it from above, then we will surely…

In this paper, the categorial property of compactness of an object, i. e. commuting of the corresponding $\Hom$ functor with coproducts, is studied in categories of $S$-acts and the corresponding structural properties of compact $S$-acts…

We develop a point-free approach for constructing the Nakano-Vashaw-Yakimov-Balmer spectrum of a noncommutative tensor triangulated category under some mild assumptions. In particular, we provide a conceptual way of classifying radical…

We construct a category of fibrant objects $\mathbb{C}\langle P\rangle$ in the sense of K. Brown from any indexed frame (a kind of indexed poset generalizing triposes) $P$, and show that its homotopy category is the Barr-exact category…

One interpretation of the Kleisli construction (given by Miranda and related to work of Par\'e) is as a nerve sending a monad $P$ to the Kleisli double category of $P$. In this paper we find more general nerve constructions on the…

We introduce a functor $\mathcal V\colon \mathrm{DblCat}_{h,nps}\to \mathrm{2Cat}_{h,nps}$ extracting from a double category a $2$-category whose objects and morphisms are the vertical morphisms and squares. We give a characterisation of…

We describe how to perform surgeries on Joyal-Street style diagrams and thereby construct free Markov categories. We also show that the construction is functorial over ordered directed acyclic graphs.

We define a notion of Koszul dual of a monoid object in a monoidal biclosed model category. Our construction generalizes the classic Yoneda algebra $Ext_A(k,k)$. We apply this general construction to define the Koszul dual of a category…

We introduce the notion of an EILC topos: a topos $\mathcal{E}$ such that every essential geometric morphism with codomain $\mathcal{E}$ is locally connected. We then show that the topos of sheaves on a topological space $X$ is EILC if $X$…

This paper focus on the presheaf monad and its submonads on the realm of $V$-categories, for a quantale $V$. First we present two characterisations of presheaf submonads, both using $V$-distributors: one based on admissible classes of…

In earlier work, Batanin has shown that an important class of definitions of higher categories could be apprehended together simply as monads over globular sets. This allowed him to generalize the notion of polygraph, initially introduced…

Homotopy type theory is a formal language for doing abstract homotopy theory -- the study of identifications. But in unmodified homotopy type theory, there is no way to say that these identifications come from identifying the path-connected…

It is well-known that the category of Kleisli algebras for a monoidal monad carries a canonical monoidal structure. We define the notion of a commutative graded monad and present a strictly two-categorical proof that Kleisli algebras for…

We define a notion of category enriched over an oplax monoidal category $V$, extending the usual definition of category enriched over a monoidal category. Even though oplax monoidal structures involve infinitely many functors $V^n\to V$,…

We exhibit a bridge between the theory of cellular categories, used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the combinatorial cellular categories…

We develop some basic concepts in the theory of higher categories internal to an arbitrary $\infty$-topos. We define internal left and right fibrations and prove a version of the Grothendieck construction and of Yoneda's lemma for internal…

We prove that for each locally $\alpha$-presentable category $\mathcal K$ there exists a regular cardinal $\gamma$ such that any $\alpha$-accessible functor out of $\mathcal K$ (into another locally $\alpha$-presentable category) is…

Let $\Sigma$ be a small category and $\mathcal{A}$ be a $\Sigma$-co-complete (resp. $\Sigma$-complete) abelian category. It is a well-known fact that the category $\operatorname{Fun}(\Sigma,\mathcal{A})$ of functors of $\Sigma$ in…