Related papers: Smash Products for Non-cartesian Internal Prestack…

Let $\mathcal{S}$ be a small category admitting binary products. We show that the whole theory of monoidal $\mathcal{S}$-fibered categories, which is customarily formulated in terms of the usual internal tensor product, can be rephrased…

Let $R$ be a ring and Ch($R$) the category of chain complexes of $R$-modules. We put an abelian model structure on Ch($R$) whose homotopy category is equivalent to $K(Proj)$, the homotopy category of all complexes of projectives. However,…

We introduce the condensed fiber product of two $G$-crossed braided fusion categories, generalizing existing constructions in the literature. We show that this product is closely related to the cohomological construction known as zesting.…

We give new formulas for Grothendieck polynomials of two types. One type expresses any specialization of a Grothendieck polynomial in at least two sets of variables as a linear combination of products Grothendieck polynomials in each set of…

We put a monoidal model category structure on the category of chain complexes of quasi-coherent sheaves over a quasi-compact and semi-separated scheme X. The approach generalizes and simplifies methods used by the author to build monoidal…

This paper is dedicated to the study of smashing weight structures (one may say that these are weight structures "coherent with arbitrary coproducts"), and the application of their properties to $t$-structures. In particular, we prove that…

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…

We study derived categories of Gorenstein varieties X and X^+ connected by a flop. We assume that the flopping contractions f: X \to Y, f^+: X^+ \to Y have fibers of dimension bounded by 1 and Y has canonical hypersurface singularities of…

Let $\mathcal C$ be a category with finite colimits, writing its coproduct $+$, and let $(\mathcal D, \otimes)$ be a braided monoidal category. We describe a method of producing a symmetric monoidal category from a lax braided monoidal…

We generalize Fulton's determinantal construction of Schur modules to the skew setting, providing an explicit and functorial presentation using only elementary linear algebra and determinantal identities, in parallel with the partition…

We introduce an axiomatization of Grothendieck sites with additional structure, and we describe sheaves that reconstruct groupoids which are internal to the site structure. This setting applies to various concrete situations, where a Nash…

Definability is a key notion in the theory of Grothendieck fibrations that characterises when an external property of objects can be accessed from within the internal logic of the base of a fibration. In this paper we consider a…

We construct a functor that inputs a retract in an $(\infty,3)$-category satisfying some adjunctibility conditions and outputs a Hopf algebra in a braided monoidal $(\infty,1)$-category. Provided the braided monoidal category is…

We consider cohomology of small categories with coefficients in a natural system in the sense of Baues and Wirsching. For any funtor L: K -> CAT, we construct a spectral sequence abutting to the cohomology of the Grothendieck construction…

We introduce a new stable birational invariant, which takes the form of a functor sending a degenerating variety to the homotopy type of a chain complex. Our invariant is a categorification of the motivic volume of Nicaise and Shinder. From…

The aim of this note is to take benefit of the foam nature of the Khovanov-Kuperberg algebras to compute the Grothendieck groups of their categories of finitely generated projective modules. The computation relies on the Hattori-Stallings…

For every smooth projective variety, we construct an action of the Heisenberg algebra on the direct sum of the Grothendieck groups of all the symmetric quotient stacks which contains the Fock space as a subrepresentation. The action is…

We provide a uniform construction of "mixed versions" or "graded lifts" in the sense of Beilinson-Ginzburg-Soergel which works for arbitrary Artin stacks. In particular, we obtain a general construction of graded lifts of many categories…

Let $S$ be a locally Noetherian normal scheme and $\blacklozenge/S$ a set of properties of $S$-schemes. Then we shall write Sch$_{\blacklozenge/S}$ for the full subcategory of the category of $S$-schemes Sch$_{/S}$ determined by the objects…

In this paper we apply some tools developed in our previous work on Grothendieck $\infty$-groupoids to the finite-dimensional case of weak 3-groupoids. We obtain a semi-model structure on the category of Grothendieck 3-groupoids of suitable…