范畴论

Artificial intelligence (AI) systems are routinely modified after deployment through retraining and changes in their environments. These transformations raise a metaphysical question: under what conditions does an AI system remain the same…

Signed measures are traditionally introduced as countably additive set functions that may take both positive and negative values. The classical Jordan decomposition theorem shows that every finite signed measure can be expressed uniquely as…

We extend the notion of computads for weak \(\omega\)-categories to allow marking certain generators as invertible, and describe inductively the free \(\omega\)-categories they generate. This gives a simple, finite description of the…

We introduce and develop the theory of braided cogroupoids, a class of algebraic structures generalizing cogroupoids in a braided setting. We show that braided cogroupoids induce monoidal equivalences between the associated comodule…

We give a single compositional setting in which gradient-based learning and Hamiltonian-style mechanics appear as functorial semantics. The syntax is an operad Arr whose objects are input-output interfaces (pairs of manifolds) and whose…

We analyse the space of points of the canonical extension of a coherent doctrine. We first give a full characterisation of doctrine morphisms that are extensible, and relate it to the existing notion of p-model of a coherent category.…

We introduce a general framework for studying natural contravariant adjunctions that refine the adjunction between frames and spaces so that the fixpoints are $T_0$-spaces. Our objects of study are \textit{spatializable…

We introduce a new model for $(\infty,n)$-categories as Segal sheaves on lax grids, which are pasting diagrams of lax cubes. This model allows for a direct construction of the Gray tensor product via Day convolution. We show that this…

In the generality of a rigidly-compactly generated tensor triangulated category, we introduce semi-Bousfield classes in terms of the vanishing of the tensor product in positive degrees with respect to a fixed reasonable $t$-structure. We…

In this paper, we answer negatively to a question posed in the context of the 2025 Oberwolfach Mini-Workshop ``The Yang-Baxter Equation and Representations of Braid Groups'' regarding the existence of split extensions classifiers in the…

This article is a generalization of a result in Quillen's note ``Module theory over non-unital rings'' giving a one-to-one correspondence between bilocalization of abelian categories of modules and idempotent ideals of the base ring.…

K. S. S. Nambooripad introduced an interesting class of categories known as normal categories, which are categories with subobjects, morphisms admitting factorization and having sufficiently many cones. These normal categories plays…

We establish the equivalence between models of enhanced $2$-sketches and algebras over monads, including the (co)lax morphisms. More precisely, for any enhanced limit $2$-sketch $\mathbb{T}$ with tight cones, the enhanced $2$-category…

We investigate a new notion of regularity for tensor triangulated categories, called residual regularity. We show that residual regularity descends and ascends via finite separable extensions and we classify all finite groups whose derived…

Drawing on well-known results from the theory of canonical extensions and the theory of categories enriched over a quantale, we define canonical extensions of quantale-enriched categories and establish their basic properties.

Lezama \cite{LezamaLatorre2017} introduced the notion of semi-graded ring with the aim of generalizing $\mathbb{Z}$-graded rings and several families of noncommutative rings of polynomial type non-$\mathbb{N}$-graded such as the skew…

In the philosophical tradition of `analytic pragmatism', which attempts to account for linguistic meanings in terms of their practices of use, logical expressivism is a theory which offers a distinct perspective on logic. We shed light on…

A limit of a (small) diagram $d : J \to E$ in a complete category $E$ can be thought of as specifying a set of equations involving the objects of $E$. To motivate this intuitively, one can think of each object $d(j)$ as a "variable" and…

We make a systematic study of duality phenomena in tensor-triangular geometry, generalising and complementing previous results of Balmer--Dell'Ambrogio--Sanders and Dwyer--Greenlees--Iyengar. A key feature of our approach is the use of…

In this paper we construct classifying localic categories and groupoids for various bundles equipped with logical structure. When these bundles are local homeomorphisms, we recover the localic groupoids that classify geometric theories,…