Related papers: Locally coherent exact categories
For an internal category $\mathbb{C}$ in a cartesian category $\mathcal{C}$ we define, naturally in objects $X$ of $\mathcal{C}$, $Prin_{\mathbb{C}}(X)$. This is a category whose objects are principal $c \mathbb{C}$-bundles over $X$ and…
We mainly investigate abelian quotients of the categories of short exact sequences. The natural framework to consider the question is via identifying quotients of morphism categories as modules categories. These ideas not only can be used…
We study the spectrum of closed subcategories in a quasi-scheme, i.e. a Grothendieck category $X$. The closed subcategories are the direct analogs of closed subschemes in the commutative case, in the sense that when $X$ is the category of…
For a noetherian scheme that has an ample family of invertible sheaves, we prove that direct products in the category of quasi-coherent sheaves are not exact unless the scheme is affine. This result can especially be applied to all…
We consider a notion of exact sequences in any -not necessarily exact- pointed category relative to a given (E;M)-factorization structure. We apply this notion to introduce and investigate a new notion of exact sequences of semimodules over…
We define the notion of exact completion with respect to an existential elementary doctrine. We observe that the forgetful functor from the 2-category exact categories to existential elementary doctrines has a left biadjoint that can be…
The paper contains general results on the uniqueness of a DG enhancement for triangulated categories. As a consequence we obtain such uniqueness for the unbounded categories of quasi-coherent sheaves, for the triangulated categories of…
We develop a theory of unbounded derived categories of quasi-coherent sheaves on algebraic stacks. In particular, we show that these categories are compactly generated by perfect complexes for stacks that either have finite stabilizers or…
In this paper, we study ideal approximation theory associated to almost $n$-exact structures in extension closed subcategories of $n$-angulated categories. For $n=3$, an $n$-angulated category is nothing but a classical triangulated…
The higher direct image complex of a coherent sheaf (or finite complex of coherent sheaves) under a projective morphism is a fundamental construction that can be defined via a Cech complex or an injective resolution, both inherently…
A new construction to associate an internal category to an enriched one is presented. The key concept is that of extensive ambient category, and the construction follows the one that associates a category whose idempotents split to a given…
Using a local analog of the Wiener-Levi theorem, we investigate the class of measures on Euclidean space with discrete support and spectrum. Also, we find a new sufficient conditions for a discrete set in Euclidean space to be a coherent…
In "On o-minimal homotopy groups", o-minimal homotopy was developed for the definable category, proving o-minimal versions of the Hurewicz theorems and the Whitehead theorem. Here, we extend these results to the category of locally…
We classify certain subcategories in quotients of exact categories. In particular, we classify the triangulated and thick subcategories of an algebraic triangulated category, i.e. the stable category of a Frobenius category.
In this paper we prove first a general theorem on semiorthogonal decompositions in derived categories of coherent sheaves for flat families over a smooth base. Based on the results of math.AG/0510670, we then show that the derived…
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 present a new coherence theorem for comprehension categories, providing strict models of dependent type theory with all standard constructors, including dependent products, dependent sums, identity types, and other inductive types.…
We give conditions on a finitary endofunctor of a finitely accessible category to admit a final coalgebra. Our conditions always apply to the case of a finitary endofunctor of a locally finitely presentable (l.f.p.) category and they bring…
Approximate lattices are aperiodic generalisations of lattices of locally compact groups that were first studied in seminal work of Yves Meyer. They are defined as those uniformly discrete approximate subgroups (symmetric subsets stable…
Given a particular collection of categorical axioms, aimed at capturing properties of the category of locales, we show that if $\mathcal{C}$ is a category that satisfies the axioms then so too is the category $[ G, \mathcal{C}]$ of…