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We introduce a new categorical framework for studying derived functors, and in particular for comparing composites of left and right derived functors. Our central observation is that model categories are the objects of a double category…
The aim of this paper is to study, by using the mathematical tools developed by Chalupnik, Touze, and Van der Kallen, the effect of the Frobenius twist on Ext-group in the category of strict polynomial functors. As an application, we obtain…
We study group graded extensions of fusion 2-categories. As an application, we obtain a homotopy theoretic classification of fermionic strongly fusion 2-categories. We examine various examples in detail.
Two adjoint functors can be seen as generalisations of the two functions within a Galois connection. If instead the adjoints are not generalised from functions, but from relations, then analogously the object of study becomes a more general…
We present a logical and algebraic description of right adjoint functors between generalized quasi-varieties, inspired by the work of McKenzie on category equivalence. This result is achieved by developing a correspondence between the…
The question "What is category theory" is approached by focusing on universal mapping properties and adjoint functors. Category theory organizes mathematics using morphisms that transmit structure and determination. Structures of…
For any additive functor from modules (or, more generally, from an abelian category with enough projectives or injectives), we construct long sequences tying up together the derived functors, the satellites, and the stabilizations of the…
We consider limits over categories of extensions and show how certain well-known functors on the category of groups turn out as such limits. We also discuss higher (or derived) limits over categories of extensions.
We introduce the notion of numerical functors to generalise Eilenberg & MacLane's polynomial functors to modules over a binomial base ring. After shewing how these functors are encoded by modules over a certain ring, we record a precise…
Graduated locally finitely presentable categories are introduced, examples include categories of sets, vector spaces, posets, presheaves and Boolean algebras. A finitary functor between graduated locally finitely presentable categories is…
We show that the left and right adjoint of the Schur functor can be expressed in terms of the monoidal structure of strict polynomial functors. Using this result we give a necessary and sufficient condition for when the tensor product of…
In this mainly expository note, we state a criterion for when a left Kan extension of a lax monoidal functor along a strong monoidal functor can itself be equipped with a lax monoidal structure, in a way that results in a left Kan extension…
We identify the Atkin polynomials in terms of associated Jacobi polynomials. Our identificationthen takes advantage of the theory of orthogonal polynomials and their asymptotics to establish many new properties of the Atkin polynomials.…
We use tools of additive combinatorics for the study of subvarieties defined by {\it high rank} families of polynomials in high dimensional $\mathbb{F} _q$-vector spaces. In the first, analytic part of the paper we prove a number properties…
We compute the derived functors of (the functors associated to) the ideal of compact operators in Banach spaces and obtain new results about the extension and lifting of compact operators.
We investigate the link between rectangular Jack polynomials and Hankel hyperdeterminants. As an application we give an expression of the even power of the Vandermonde in term of Jack polynomials.
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
In this paper, we generalize the principle of the Long-Moody construction for representations of braid groups to other groups, such as mapping class groups of surfaces. Namely, we introduce endofunctors over a functor category that encodes…
We study the relationship between cartesian bicategories and a specialisation of Lawvere's hyperdoctrines, namely elementary existential doctrines. Both provide different ways of abstracting the structural properties of logical systems: the…
We investigate correspondence functors, namely the functors from the category of finite sets and correspondences to the category of $k$-modules, where $k$ is a commutative ring.They have various specific properties which do not hold for…