Related papers: Grothendieck categories and support conditions

Lenses have a rich history and have recently received a great deal of attention from applied category theorists. We generalize the notion of lens by defining a category $\mathsf{Lens}_F$ for any category $\mathcal{C}$ and functor $F\colon…

In this article, we develop a notion of Quillen bifibration which combines the two notions of Grothendieck bifibration and of Quillen model structure. In particular, given a bifibration $p:\mathcal E\to\mathcal B$, we describe when a family…

The classical Grothendieck inequality is viewed as a statement about representations of functions of two variables over discrete domains by integrals of two-fold products of functions of one variable. An analogous statement is proved,…

For a polynomial in several variables depending on some parameters, we discuss some results to the effect that for almost all values of the parameters the polynomial is irreducible. In particular we recast in this perspective some results…

Some aspects of differential and integral calculi on generalized grassmann (paragrassmann) algebras are considered. The integration over paragrassmann variables is applied to evaluate the partition function for the $Z_{p+1}$ Potts model on…

Starting with a k-linear or DG category admitting a (homotopy) Serre functor, we construct a k-linear or DG 2-category categorifying the Heisenberg algebra of the numerical K-group of the original category. We also define a 2-categorical…

In this article, I classify the totally geodesic submanifolds in the complex 2-Grassmannians and in the quaternionic 2-Grassmannians. It turns out that for both of these spaces, the earlier classification of maximal totally geodesic…

This paper introduces the notion of complete connectedness of a Grothendieck topos, defined as the existence of a left adjoint to a left adjoint to a left adjoint to the global sections functor, and provides many examples. Typical examples…

For an arbitrary countable group G = <A|R> given by its generators A and defining relations R we discuss a specific method for embedding of G into a certain 2-generator group T. Our embedding explicitly lists the images of generators from A…

Let $U_\varepsilon^{\mathrm{res}}(L\mathfrak{sl}_2)$ be the restricted integral form of the quantum loop algebra $U_q(L\mathfrak{sl}_2)$ specialised at a root of unity $\varepsilon$. We prove that the Grothendieck ring of a tensor…

A procedure for constructing bivariant theories by means of Grothendieck duality is developed. This produces, in particular, a bivariant theory of Hochschild (co)homology on the category of schemes that are flat, separated and essentially…

Grothendieck's standard conjecture of Lefschetz type has two main forms: the weak form $C$ and the strong form $B$. The weak form is known for varieties over finite fields as a consequence of the proof of the Weil conjectures. This suggests…

Let G be an algebraic group over an algebraically closed field of positive characteristic such that its neutral connected component is a unipotent group. We consider a certain class of closed idempotents in the braided monoidal category…

We give a classification of all exact structures on a given idempotent complete additive category. Using this, we investigate the structure of an exact category with finitely many indecomposables. We show that the relation of the…

To an inclusion topological groups H->G, we associate a naive G-spectrum. The special case when H=G gives the dualizing spectrum D_G introduced by the author in the first paper of this series. The main application will be to give a purely…

If G is a finite group and k is a field, there is a natural construction of a Hopf algebra over k associated to G, the Drinfel'd double D(G). We prove that if G is any finite real reflection group with Drinfel'd double D(G) over an…

Cohomology of a topological space with coefficients in stacks of abelian 2-groups is considered. A 2-categorical analog of the theorem of Grothendieck is proved, relating cohomology of the space with coefficients in a 2-stage spectrum and…

The complete flag variety admits a natural action by both the orthogonal group and the symplectic group. Wyser and Yong defined orthogonal Grothendieck polynomials $\mathfrak{G}^{\mathsf{O}}_z$ and symplectic Grothendieck polynomials…

We prove that every Grothendieck topology induces a hereditary torsion pair in the category of presheaves of modules on a ringed site, and obtain a homological characterization of sheaves of modules: a presheaf of modules is a sheaf of…

Every Serre subcategory of an abelian category is assigned a unique type. The type of a Serre subcategory of a Grothendieck category is in the list: $$(0, 0), \ (0, -1), \ (1, -1), \ (0, -2), \ (1, -2), \ (2, -1), \ (+\infty, -\infty);$$…