范畴论

Univalence, originally a type theoretical notion at the heart of Voevodsky's Univalent Foundations Program, has found general importance as a higher categorical property that characterizes descent and hence classifying maps in…

In this paper we introduce a generalisation of a covariant Grothendieck construction to the setting of sites. We study the basic properties of defined site structures on Grothendieck constructions as well as we treat the cohomological…

For a category $\mathcal E$ with finite limits and well-behaved countable coproducts, we construct a model structure, called the effective model structure, on the category of simplicial objects in $\mathcal E$, generalising the Kan--Quillen…

We study the Grothendieck monoid (a monoid version of the Grothendieck group) of an extriangulated category, and give some results which are new even for abelian categories. First, we classify Serre subcategories and dense 2-out-of-3…

We answer Mundici's problem number 3 (D. Mundici. Advanced {\L}ukasiewicz calculus. Trends in Logic Vol. 35. Springer 2011, p. 235): Is the category of locally finite MV-algebras equivalent to an equational class? We prove: (i) The category…

We show that coherent topoi are right Kan injective with respect to flat embeddings of topoi. We recover the ultrastructure on their category of points as a consequence of this result. We speculate on possible notions of ultracategory in…

We study questions motivated by results in the classical theory of dynamical systems in the context of triangulated and A-infinity categories. First, entropy is defined for exact endofunctors and computed in a variety of examples. In…

Lenses encode protocols for synchronising systems. We continue the work begun by Chollet et al. at the Applied Category Theory Adjoint School in 2020 to study the properties of the category of small categories and asymmetric delta lenses.…

We prove a fixpoint theorem for contractions on Cauchy-complete quantale-enriched categories. It holds for any quantale whose underlying lattice is continuous, and applies to contractions whose control function is sequentially…

We define a symmetric monoidal category modelling fuzzy concepts and fuzzy conceptual reasoning within G\"ardenfors' framework of conceptual (convex) spaces. We propose log-concave functions as models of fuzzy concepts, showing that these…

The constraint satisfaction problem (CSP) is a computational problem that includes a range of important problems in computer science. We point out that fundamental concepts of the CSP, such as the solution set of an instance and…

The real unit interval is the fundamental building block for many branches of mathematics like probability theory, measure theory, convex sets and homotopy theory. However, a priori the unit interval could be considered an arbitrary choice…

Dynamical systems are ubiquitous in science and engineering as models of phenomena that evolve over time. Although complex dynamical systems tend to have important modular structure, conventional modeling approaches suppress this structure.…

String diagrams are an increasingly popular algebraic language for the analysis of graphical models of computations across different research fields. Whereas string diagrams have been thoroughly studied as semantic structures, much less…

We propose a categorical framework for processes which interact bidirectionally with both an environment and a 'controller'. Examples include open learners, in which the controller is an optimiser such as gradient descent, and an approach…

Lenses are an important tool in applied category theory. While individual lenses have been widely used in applications, many of the mathematical properties of the corresponding categories of lenses have remained unknown. In this paper, we…

We construct a monoidal category of open transition systems that generate material history as transitions unfold, which we call situated transition systems. The material history generated by a composite system is composed of the material…

We show that the word problem for braided monoidal categories is at least as hard as the unknotting problem. As a corollary, so is the word problem for Gray categories. We conjecture that the word problem for Gray categories is decidable.

The exponential modalities of linear logic have been used by various authors to model infinite-dimensional quantum systems. This paper explains how these modalities can also give rise to the complementarity principle of quantum mechanics.…

We provide a categorical semantics for bounded Petri nets, both in the collective- and individual-token philosophy. In both cases, we describe the process of bounding a net internally, by just constructing new categories of executions of a…