Flat functors and free exact categories

2022

Let Σ be a small category and A be a Σ-co-complete (resp. Σcomplete) abelian category. It is a well-known fact that the category Fun(Σ, A) of functors of Σ in A is an abelian category, and that the functor colim Σ (−) : Fun(Σ, A) → A (resp. lim Σ (−) : Fun(Σ, A) → A) is left (resp. right) adjoint to κ Σ : A → Fun(Σ, A), where κ Σ is the associated constant diagram functor. In this paper we will show that the functor colim Σ (−) (resp. lim Σ (−)) is exact if and only if the pair of functors colim Σ (−), κ Σ (resp. κ Σ , lim Σ (−)) is Ext-adjoint. As an application of our findings, we will give new proofs of known results on the exactness of limits and colimits in abelian categories. Contents 1. Introduction 1 2. Preliminaries 3 3. A one-point extension of a small category 4 4. Main results 6 References 13

2021

In this note we show that under very mild conditions on a functor between exact categories F ∶ D → E it is possible to derive F at the level of unbounded complexes. We also give applications to deriving functors between Proand Indcategories.

Applied Categorical Structures, 2013

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 obtained as a composite of two others. Finally, we conclude how this notion encompasses both that of the exact completion of a regular category as well as that of the exact completion of a cartesian category with weak pullbacks.

2007

We identify two categories of locally compact objects on an exact category A. They correspond to the well-known constructions of the Beilinson category lim A and the Kato category k(A). We study their mutual relations and compare the two constructions. We prove that lim A is an exact category, which gives to this category a very convenient feature when dealing with K-theoretical invariants. It is natural therefore to consider the Beilinson category lim A as the most convenient candidate to the role of the category of locally compact objects over an exact category. We also show that the categories Ind_{aleph_0}(C), Pro_{aleph_0}(C) of countably indexed ind/pro-objects over any category C can be described as localizations of categories of diagrams over C.

Journal of Pure and Applied Algebra, 1981

arXiv (Cornell University), 2018

Finite modules, finitely presented modules and Mittag-Leffler modules are characterized by their behaviour by tensoring with direct products of modules. In this paper, we study and characterize the functors of modules that preserve direct products and direct limits.

Communications in Algebra, 2005

We characterize those small categories with the property that flat (contravariant) functors on them are coherently axiomatized in the language of presheaves on them. They are exactly the categories with the property that every finite diagram into them has a finite set of (weakly) initial cocones.

1966

In Chapter I (Sections 1-9) of [1], we list several properties of diagrammatic categories in direct comparison with those valid for the category Set [4], and by analogy with abelian categories .

Electronic Proceedings in Theoretical Computer Science

Lenses are an important tool in applied category theory. While individual lenses have been widely used in applications, many of the mathematical properties of the corresponding categories of lenses have remained unknown. In this paper, we study the category of small categories and asymmetric delta lenses, and prove that it has several good exactness properties. These properties include the existence of certain limits and colimits, as well as so-called imported limits, such as imported products and imported pullbacks, which have arisen previously in applications. The category is also shown to be extensive, and it has an image factorisation system.

In this paper we develop the 2-dimensional theory of flat functors. We define a 2-functor A