Related papers: On presheaf submonads of quantale enriched categor…

We expand \v{C}ech cohomology of a topological space $X$ with values in a presheaf on $X$ to \v{C}ech cohomology of a commutative ring with unity $R$ with values in a presheaf on $R$. The strategy is to observe that both the set of open…

This paper provides a comprehensive overview of some of the foundational properties of categories enriched over quantaloids, along with several new results. We demonstrate that the category whose objects are quantaloid-enriched categories…

This is the first paper in the sequence devoted to derived category of moduli spaces of curves of genus $0$ with marked points. We develop several approaches to describe it equivariantly with respect to the action of the symmetric group…

In this paper, we deal with quantum theories on presheaves and sheaves on context categories consisting of commutative von Neumann algebras of bounded operators on a Hilbert space, from two viewpoints. One is to reduce presheaf-based topos…

An $n$-sesquicategory is an $n$-globular set with strictly associative and unital composition and whiskering operations, which are however not required to satisfy the Godement interchange laws which hold in $n$-categories. In…

Many structured categories of interest are most naturally described as algebras for a relative monad, but turn out nonetheless to be algebras for an ordinary monad. We show that, under suitable hypotheses, the left oplax Kan extension of a…

We construct a `weak' version EM^w(K) of Lack & Street's 2-category of monads in a 2-category K, by replacing their compatibility constraint of 1-cells with the units of monads by an additional condition on the 2-cells. A relation between…

We define a quantum analogue of the Grothendieck ring of finite dimensional modules of a quantum affine algebra of simply laced type. The construction is based on perverse sheaves on a variety related to quivers. We get also a new geometric…

Convex algebras, also called (semi)convex sets, are at the heart of modelling probabilistic systems including probabilistic automata. Abstractly, they are the Eilenberg-Moore algebras of the finitely supported distribution monad.…

A monoid structure on families of representations of a quiver is introduced by taking extensions of representations in families, i.e. subvarieties of the varieties of representations. The study of this monoid leads to interesting…

We study finite quasi-quantum groups in their quiver setting developed recently by the first author in arXiv:0902.1620 and arXiv:0903.1472. We obtain a classification of finite-dimensional pointed Majid algebras of finite corepresentation…

For a quantale ${\sf{V}}$, the category $\sf V$-${\bf Top}$ of ${\sf{V}}$-valued topological spaces may be introduced as a full subcategory of those ${\sf{V}}$-valued closure spaces whose closure operation preserves finite joins. In…

A fundamental result in the theory of monads is the characterisation of the category of algebras for a monad in terms of a pullback of the category of presheaves on the category of free algebras: intuitively, this expresses that every…

We give an elementary exposition of some fundamental facts about fibered (or rather opfibered) categories, in terms of monads and 2-categories. The account avoids any mention of category-valued functors and pseudofunctors.

A categorical model of the multiplicative and exponential fragments of intuitionistic linear logic ($\mathsf{MELL}$), known as a \emph{linear category}, is a symmetric monoidal closed category with a monoidal coalgebra modality (also known…

We give a detailed account of the theory of enrichment over a bicategory and show that it establishes a two-fold generalization of enrichment over both quantaloids and monoidal categories. We define complete B-categories, a generalization…

The Medvedev degree of a subshift is a dynamical invariant of computable origin that can be used to compare the complexity of subshifts that contain only uncomputable configurations. We develop theory to describe how these degrees can be…

We develop the theory of relative monads and relative adjunctions in a virtual equipment, extending the theory of monads and adjunctions in a 2-category. The theory of relative comonads and relative coadjunctions follows by duality. While…

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…

Let S(n) be the category of invariant subspaces of nilpotent operators with nilpotency index at most n. Such submodule categories have been studied already in 1936 by Birkhoff, they have attracted a lot of attention in recent years, for…