Related papers: Formalising Yoneda Ext in Univalent Foundations
Motivated by quotient algorithms, such as the well-known $p$-quotient or solvable quotient algorithms, we describe how to compute extensions $\tilde H$ of a finite group $H$ by a direct sum of isomorphic simple $\mathbb{Z}_p H$-modules such…
This paper aims to study Ext-groups between certain functors defined on the category of finitely generated free groups. Rational Ext-groups between the abelianization functor and its symmetric powers are known, and are almost always equal…
We introduce a hierarchy for unital Kirchberg algebras with finitely generated K-groups by which the first and second homotopy groups of the automorphism groups serve as a complete invariant of classification. We also introduce an invariant…
Let $(\mathcal{A},\mathcal{E})$ be an exact category. We establish basic results that allow one to identify sub(bi)functors of $\operatorname{Ext}_{\mathcal{E}}(-,-)$ using additivity of numerical functions and restriction to subcategories.…
We prove the existence of a sequence of commutative diagrams generalizing existing results on the cohomology of the Borel-Serre boundary and well-rounded retract to the context of the well-tempered complex. Our main theorem provides a…
The notion of a duality between two derived functors as well as an extension theorem for derived functors to larger categories in which they need not be defined is introduced. These ideas are then applied to extend and study the coext…
In this paper, we first define the equivariant infinitesimal $\eta$-form, then we compare it with the equivariant $\eta$-form, modulo exact forms, by a locally computable form. As a consequence, we obtain the singular behavior of the…
We develop a homotopy theory for additive categories endowed with endofunctors, analogous to the concept of a model structure. We use it to construct the homotopy theory of a Hovey triple (which consists of two compatible complete cotorsion…
We combine Homotopy Type Theory with axiomatic cohesion, expressing the latter internally with a version of "adjoint logic" in which the discretization and codiscretization modalities are characterized using a judgmental formalism of "crisp…
Algebraic $kk$-theory, introduced by Corti\~nas and Thom, is a bivariant $K$-theory defined on the category $\mathrm{Alg}$ of algebras over a commutative unital ring $\ell$. It consists of a triangulated category $kk$ endowed with a functor…
Escard\'o and Simpson defined a notion of interval object by a universal property in any category with binary products. The Homotopy Type Theory book defines a higher-inductive notion of reals, and suggests that the interval may satisfy…
This article studies the algebraic structure of homology theories defined by a left Hopf algebroid U over a possibly noncommutative base algebra A, such as for example Hochschild, Lie algebroid (in particular Lie algebra and Poisson), or…
We present the type theory CaTT, originally introduced by Finster and Mimram to describe globular weak $\omega$-categories, and we formalise this theory in the language of homotopy type theory. Most of the studies about this type theory…
Drinfeld twists, and the twists of Giaquinto and Zhang, allow for algebras and their modules to be deformed by a cocycle. We prove general results about cocycle twists of algebra factorisations and induced representations and apply them to…
In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension…
A cohomology theory of the adjoint of Hopf algebras, via deformations, is presented by means of diagrammatic techniques. Explicit calculations are provided in the cases of group algebras, function algebras on groups, and the bosonization of…
These notes give a short introduction to finite Coxeter groups, their classification, and some parts of their representation theory, with a focus on the infinite families. They are based on lectures delivered by the author at the…
Homotopy type theory is an interpretation of Martin-L\"of's constructive type theory into abstract homotopy theory. There results a link between constructive mathematics and algebraic topology, providing topological semantics for…
The semigroups of unital extensions of separable $C^\ast$-algebras come in two flavours: a strong and a weak version. By the unital $\mathrm{Ext}$-groups, we mean the groups of invertible elements in these semigroups. We use the unital…
We consider a theory of noncommutative Gr\"obner bases on decreasingly filtered algebras whose associated graded algebras are commutative. We transfer many algorithms that use commutative Gr\"obner bases to this context. As an important…