Manuel Cortés-Izurdiaga
University of Almería
Title
Flatness in Finitely Accessible Additive Categories
Abstract
The notion of κ-accessible category, where κ is an infinite regular cardinal, was introduced by Lair in [5] under the name of catégorie modelable, as an extension of the κ-presentable categories previously studied by Gabriel and Ulmer in [4]. The term accessible was first used by Makkai and Paré in [6].
In the talk, we will be interested in finitely accessible (i. e., ℵ0-accessible) additive categories C. These categories have all directed colimits and a set of finitely presented objects such that every object in the category is a directed colimit of a directed system of such objects.
Even in the non-additive case, there is a notion of purity in such categories, that allows, in the additive setting, to define an exact structure in the sense of Quillen [8]. The admissible monomorphisms and epimorphisms (also called inflations and deflations, respectively) in this exact category are the pure monomorphisms and pure epimorphisms, respectively. Of course, not every epimorphism f : A → F is a pure epimorphism, but there might exist objects F satisfying the property that every epimorphism ending at F is pure. These objects are called flat. If C is a module category (over a non-commutative ring) or, more generally, a finitely accessible Grothendieck category, this notion agrees with the classical notion of flat object first considered by Stenström [9] it this categorical setting.
In the recent work [2], Cuadra and Simson studied flat objects in the category of comodules over a coalgebra and they stated some open problems for a finitely accessible Grothendieck category G: (a) Give a characterization of those categories G that have enough flat objects (i. e., every object in the category is an epimorphic image of a flat object); (b) If there are enough flat objects in G, are there enough projectives?
In the talk, which is based on the paper [1], we will show how (a) can be solved using the representation theorem of finitely accessible additive categories via functor categories given by Crawley-Boevey [3]. Regarding (b), we will give a criterion to determine when the finitely accessible additive category C has enough projective objects, provided that it has enough flat objects. This will allow us to give an affirmative answer to (b) in some particular situations.
Recently, Martini, Parra, Saorín and Virili [7] have shown that question (b) has a negative answer. Let us point out that our results are valid for finitely accessible additive categories satisfying certain mild conditions, which are much weaker than being Grothendieck.
[1] M. Cortés-Izurdiaga, Flatness in finitely accessible additive categories, preprint arXiv:2505.07742, 2025.
[2] J. Cuadra and D. Simson., Flat comodules and perfect coalgebras, Comm. Algebra, 35 (2007), 3164–3194.
[3] W. Crawley-Boevey, Locally finitely presented additive categories, Comm. Algebra 22 (1994), 1641–1674.
[4] P. Gabriel and F. Ulmer. Lokal präsentierbare Kategorien, Lecture Notes in Math., no. 221, Springer-Verlag, 1971.
[5] C. Lair, Catégories modelables et catégories esquissables, Diagrammes 6 (1981).
[6] M. Makkai, R. Paré, Accessible categories: the foundations of categorical model theory, Contemp. Math., no. 104, American Mathematical Society, 1989.
[7] L. Martini, C. Parra, M. Saorín, S. Virili., Locally finitely presented Grothendieck categories with a flat generator, preprint, arXiv:2508.00670.
[8] D. Quillen, Higher algebraic K -theory. I, in: Algebraic K -theory, I: Higher K -theories, Proceedings of the Conference, Battelle Memorial Institute, Seattle, Washington, 1972, Lecture Notes in Mathematics, vol. 341, Springer, Berlin, 1973, pp. 85–147.
[9] B. Stenström. Purity in functor categories, J. Algebra, 8 (1968), 352–361
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