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Julia Ramos González

Publications and source records attributed to Julia Ramos González.

8 recordsLinked to original sources
arXiv2025

The snail lemma and the long homology sequence

In the first part of the paper, we establish an homotopical version of the snail lemma (which is a generalization of the classical snake lemma). In the second part, we introduce the category $\mathbf{Seq}(\mathcal A)$ of sequentiable families of arrows in a category $\mathcal A$ and we compare it with the category of chain complexes in $\mathcal A.$ We apply the homotopy snail lemma to a morphism in $\mathbf{Seq}(\mathcal A)$ obtaining first a six-term exact sequence in $\mathbf{Seq}(\mathcal A)$ and then, unrolling the sequence in $\mathbf{Seq}(\mathcal A),$ a long exact sequence in $\mathcal A.$ When $\mathcal A$ is abelian, this sequence subsumes the usual long homology sequence obtained from an extension of chain complexes.

math.CT↗
arXiv2022

Exponentiable Grothendieck categories in flat Algebraic Geometry

We introduce and describe the $2$-category $\mathsf{Grt}_{\flat}$ of Grothendieck categories and flat morphisms between them. First, we show that the tensor product of locally presentable linear categories $\boxtimes$ restricts nicely to $\mathsf{Grt}_{\flat}$. Then, we characterize exponentiable objects with respect to $\boxtimes$: these are continuous Grothendieck categories. In particular, locally finitely presentable Grothendieck categories are exponentiable. Consequently, we have that, for a quasi-compact quasi-separated scheme $X$, the category of quasi-coherent sheaves $\mathsf{Qcoh}(X)$ is exponentiable. Finally, we provide a family of examples and concrete computations of exponentials.

math.AG↗
arXiv2021

On the tensor product of well generated dg categories

We endow the homotopy category of well generated (pretriangulated) dg categories with a tensor product satisfying a universal property. The resulting monoidal structure is symmetric and closed with respect to the cocontinuous RHom of dg categories (in the sense of Toën [26]). We give a construction of the tensor product in terms of localisations of dg derived categories, making use of the enhanced derived Gabriel-Popescu theorem [21]. Given a regular cardinal alpha, we define and construct a tensor product of homotopically alpha-cocomplete dg categories and prove that the well generated tensor product of alpha-continuous derived dg categories (in the sense of [21]) is the alpha-continuous dg derived category of the homotopically alpha-cocomplete tensor product. In particular, this shows that the tensor product of well generated dg categories preserves alpha-compactness.

math.CT↗
arXiv2021

A derived Gabriel-Popescu theorem for t-structures via derived injectives

We prove a derived version of the Gabriel-Popescu theorem in the framework of dg-categories and t-structures. This exhibits any pretriangulated dg-category with a suitable t-structure (such that its heart is a Grothendieck abelian category) as a t-exact localization of a derived dg-category of dg-modules. We give an original proof based on a generalization of Mitchell's argument in "A quick proof of the Gabriel-Popesco theorem" and involving derived injective objects. As an application, we also give a short proof that derived categories of Grothendieck abelian categories have a unique dg-enhancement.

math.CT↗
arXiv2019

Gabriel-Ulmer duality for topoi and its relation with site presentations

Let $κ$ be a regular cardinal. We study Gabriel-Ulmer duality when one restricts the 2-category of locally $κ$-presentable categories with $κ$-accessible right adjoints to its locally full sub-2-category of $κ$-presentable Grothendieck topoi with geometric $κ$-accessible morphisms. In particular, we provide a full understanding of the locally full sub-2-category of the 2-category of $κ$-small cocomplete categories with $κ$-colimit preserving functors arising as the corresponding 2-category of presentations via the restriction. We analyse the relation of these presentations of Grothendieck topoi with site presentations and we show that the 2-category of locally $κ$-presentable Grothendieck topoi with geometric $κ$-accessible morphisms is a reflective sub-bicategory of the full sub-2-category of the 2-category of sites with morphisms of sites genearated by the weakly $κ$-ary sites in the sense of Shulman [37].

math.CT↗
arXiv2017

Grothendieck categories as a bilocalization of linear sites

We prove that the 2-category Grt of Grothendieck abelian categories with colimit preserving functors and natural transformations is a bicategory of fractions in the sense of Pronk of the 2-category Site of linear sites with continuous morphisms of sites and natural transformations. This result can potentially be used to make the tensor product of Grothendieck categories from earlier work by Lowen, Shoikhet and the author into a bi-monoidal structure on Grt.

math.CT↗
arXiv2017

On the tensor product of linear sites and Grothendieck categories

We define a tensor product of linear sites, and a resulting tensor product of Grothendieck categories based upon their representations as categories of linear sheaves. We show that our tensor product is a special case of the tensor product of locally presentable linear categories, and that the tensor product of locally coherent Grothendieck categories is locally coherent if and only if the Deligne tensor product of their abelian categories of finitely presented objects exists. We describe the tensor product of non-commutative projective schemes in terms of Z-algebras, and show that for projective schemes our tensor product corresponds to the usual product scheme.

math.CT↗