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We prove that the integral closedness of any ideal of height at least two is compatible with specialization by a generic element. This opens the possibility for proofs using induction on the height of an ideal. Also, with additional…
We explore the category of internal categories in the usual category of (right) group-sets, whose objects are referred to as categorified group-sets. More precisely, we develop a new Burnside theory, where the equivalence relation between…
We construct an internal language for cartesian closed bicategories. Precisely, we introduce a type theory modelling the structure of a cartesian closed bicategory and show that its syntactic model satisfies an appropriate universal…
For a given finite class of finite graphs H, a graph G is called a realization of H if the neighbourhood of its any vertex induces the subgraph isomorphic to a graph of H. We consider the following problem known as the Generalized…
We analyze the classification problem for finitely generated orderable groups from the viewpoint of descriptive set theory. We analyze the standard Borel space of finitely generated left-orderable groups, and the subspace of finitely…
A double category is constructed from a `fattened' version of a given category, motivated in part by a context of parallel transport. We also study monoidal structures on the underlying category and on the fattened category.
This is essentially an illustration for the general technology of homotopical enhancements developed recently in arxiv:2409.17489. We take the derived category of an abelian category, and we look at the full subcategory spanned by complexes…
We show how the separability problem is dual to that of decomposing any given matrix into a conic combination of rank-one partial isometries, thus offering a duality approach different to the positive maps characterization problem. Several…
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…
In a bicategory of spans (an example of a 'generic bicategory') the factorization of a span (s,t) as the span (s,1) followed by (1,t) satisfies a simple universal property with respect to all factorizations in terms of the generic…
A new construction to associate an internal category to an enriched one is presented. The key concept is that of extensive ambient category, and the construction follows the one that associates a category whose idempotents split to a given…
Via an overparameterized linear model with Gaussian features, we provide conditions for good generalization for multiclass classification of minimum-norm interpolating solutions in an asymptotic setting where both the number of underlying…
We find that second order quantification is problematic when a quantified concept variable is supposed to function predicatively. This issue is analyzed and it is shown that a constructive interpretation of the falling under relation…
We present an intrinsic and concrete development of the subdivision of small categories, give some simple examples and derive its fundamental properties. As an application, we deduce an alternative way to compare the homotopy categories of…
This paper is a coalgebra version of arXiv:1703.04266 and a sequel to arXiv:1607.03066. We present the definition of a pseudo-dualizing complex of bicomodules over a pair of coassociative coalgebras $\mathcal C$ and $\mathcal D$. For any…
We solve a problem posed by Cardinali and Sastry [2] about factorization of $2$-covers of finite classical generalized quadrangles. To that end, we develop a general theory of cover factorization for generalized quadrangles, and in…
In the work, the property of the second-order subdifferential is studied and second-order optimality conditions are obtained for the minimization problem. We also obtained necessary and sufficient conditions for an extremum for the extremal…
Given a monoidal $\infty$-category $C$ equipped with a monoidal recollement, we give a simple criterion for an object in $C$ to be dualizable in terms of the dualizability of each of its factors and a projection formula relating them.…
We formulate problems of tight closure theory in terms of projective bundles and subbundles. This provides a geometric interpretation of such problems and allows us to apply intersection theory to them. This yields new results concerning…
We outline a definition of accessible and presentable objects in a 2-category $\mathcal K$ endowed with a "KZ context", that is to say a pair of lax-idempotent monads interacting in a prescribed way; this perspective suggests a unified…