范畴论
Convergent rewriting systems on algebraic structures give methods to solve decision problems, to prove coherence results, and to compute homological invariants. These methods are based on higher-dimensional extensions of the critical…
Bondarko's (strong) weight complex functor is a triangulated functor from Voevodsky's triangulated category of motives to the homotopy category of chain complexes of classical Chow motives. Its construction is valid for any dg enhanced…
To the best of our knowledge, there are very few results on how Heyting-valued models are affected by the morphisms on the complete Heyting algebras that determine them: the only cases found in the literature are concerning automorphisms of…
Classical definitions of weak higher-dimensional categories are given inductively; for example, a bicategory has a set of objects and hom categories, and a tricategory has a set of objects and hom bicategories. However, more recent…
In this paper we introduce a notion of $\mathbf{O}$-monoidal $\infty$-categories for a finite sequence $\mathbf{O}^{\otimes}$ of $\infty$-operads, which is a generalization of the notion of higher monoidal categories in the setting of…
In view of the fact that many of the most familiar examples of category bases are equivalent to some topology, it is natural to ask whether category bases are always topological in nature. The answer is in the negative. In this paper, we…
We study right exact tensor products on the category of finitely presented functors. As our main technical tool, we use a multilinear version of the universal property of so-called Freyd categories. Furthermore, we compare our constructions…
The aim of this work is to point out a strong structural phenomenon hidden behind the existence of normalizers through the investigation of this property in the non-pointed context: given any category E, a certain property of the fibration…
We introduce a notion of split extension of (non-associative) bialgebras which generalizes the notion of split extension of magmas introduced by M. Gran, G. Janelidze and M. Sobral. We show that this definition is equivalent to the notion…
There are different meanings of the term "compositionality" within science: what one researcher would call compositional, is not at all compositional for another researcher. The most established conception is usually attributed to Frege,…
It is proved that if a ring is left hereditary, left perfect and right coherent, then the stable category has cokernels. Moreover, we show that the condition for a ring to be left perfect and right coherent is also necessary for the stable…
This paper studies presentations of the Sierpinski gasket as a final coalgebra for functors on several categories of metric spaces with additional designated points. The three categories which we study differ on their morphisms: one uses…
A reflective subuniverse in homotopy type theory is an internal version of the notion of a localization in topology or in the theory of $\infty$-categories. Working in homotopy type theory, we give new characterizations of the following…
An $\infty$-cosmos is a setting in which to develop the formal category theory of $(\infty,1)$-categories. In this paper, we explore a few atypical examples of $\infty$-cosmoi whose objects are 2-categories or bicategories rather than…
A fusion category is called transparent if the associator involving any invertible object is the identity map. For the Haagerup-Izumi fusion rings with $G = \mathbb{Z}_{2n+1}$ (the $\mathbb{Z}_3$ case is the Haagerup fusion ring with six…
We show that for any quantale $\mathcal{Q}$, a $\mathcal{Q}$-category is skeletal and complete if and only if it is injective with respect to fully faithful $\mathcal{Q}$-functors. This is a special case of known theorems due to Hofmann and…
The first part of this dissertation defines "dependently typed algebraic theories", which are a strict subclass of the generalised algebraic theories (GATs) of Cartmell. We characterise dependently typed algebraic theories as finitary…
We construct a category that classifies compact Hausdorff spaces by their shape and finite topological spaces by their weak homotopy type.
The purpose of this short and elementary note is to identify some classes of exact categories introduced in L. Previdi's thesis. Among other things we show: (1) An exact category is partially abelian exact if and only if it is abelian. (2)…
A coherent presentation of an n-category is a presentation by generators, relations and relations among relations. Confluent and terminating rewriting systems generate coherent presentations, whose relations among relations are defined by…