Related papers: A Categorical Perspective on Gluing

In analogy with the classical theory of topological groups, for finitely complete categories enriched with Grothendieck topologies, we provide the concepts of localized G-topological space, initial Grothendieck topologies and continuous…

We develop general foundations of topological algebra over a linearly topologized ring k in a format applicable to both formal schemes and analytic adic spaces. We are especially interested in determining exact closed tensor categories of…

Topos properties of the category of covering groupoids over a fixed groupoid are discussed. A classification result for connected covering groupoids over a fixed groupoid analogous to the fundamental theorem of Galois theory is given.

We give criteria for subcategories of a compactly generated algebraic triangulated category to be precovering or preenveloping. These criteria are formulated in terms of closure conditions involving products, coproducts, directed homotopy…

Function is defined as the ensemble of tasks that enable the product to complete the designed purpose. Functional tools, such as functional modeling, offer decision guidance in the early phase of product design, where explicit design…

A novel generalization of photon surfaces to the case of massive charged particles is given for spacetimes with at least one isometry, including stationary ones. A related notion of glued massive particle surfaces is also defined. These…

In this work we describe a class of subsets of the Euclidean plane which, with the induced length metric, are locally $CAT(0)$ spaces and we show that the gluing of two such subsets along a piece of their boundary is again a locally…

This paper has been withdrawn and replaced by arXiv:1309.5035. In this paper we describe some examples of so called spherical functors between triangulated categories, which generalize the notion of a spherical object. We also give…

In this paper, we define a generalization of indexed categories and contextual categories which we call contextually indexed (contextual) categories. While contextual categories are models of ordinary type theories, contextually indexed…

The paper introduces a novel framework based on category theory to enhance the explainability of artificial intelligence systems, particularly focusing on word embeddings. Key topics include the construction of categories $\mathcal{L}_T$…

We compare closed and rigid monoidal categories. Closedness is defined by the tensor product having a right adjoint: the internal hom functor. Rigidity, on the other hand, generalises the duality of finite-dimensional vector spaces. In the…

Invertibility is an important concept in category theory. In higher category theory, it becomes less obvious what the correct notion of invertibility is, as extra coherence conditions can become necessary for invertible structures to have…

In this paper we look at Grothendieck's work on classifying holomorphic bundles over the complex projective line. The paper is divided into $4$ parts. The first and second part we build up the necessary background to talk about vector…

Internal categories feature notions of limit and completeness, as originally proposed in the context of the effective topos. This paper sets out the theory of internal completeness in a general context, spelling out the details of the…

A notion of stratification is introduced for any compactly generated triangulated category T endowed with an action of a graded commutative noetherian ring R. The utility of this notion is demonstrated by establishing diverse consequences…

Classification questions are often about understanding components of a category. It is much more desirable however to be able to understand the entire homotopy type of this category and not just the set of its components. In this paper we…

We define the Grothendieck group of an n-angulated category and show that for odd n its properties are as in the special case of n=3, i.e. the triangulated case. In particular, its subgroups classify the dense and complete n-angulated…

We propose a new framework for integrating quantifiers with other logical connectives in a higher-categorical setting. Our method systematically incorporates key coherence conditions-including those akin to the Beck-Chevalley property-and…

We deal with the problem of semantic classification of challenging and highly-cluttered dataset. We present a novel, and yet a very simple classification technique by leveraging the ease of classifiability of any existing well separable…

A $\mathcal{C}$-set is a functor from the category $\mathcal{C}$ to the category of finite sets and functions. The category of $\mathcal{C}$-sets, $\mathcal{C} - \operatorname*{set}$, is defined as the category whose objects are…