Examples of exact categories in representation theory are given by the category of ∆−filtered modules over quasi-hereditary algebras, but also by various categories related to matrix problems, such as poset representations or representations of bocses. Motivated by the matrix problem background, we study in this article the reduction of exact structures, and consider the poset (Ex(A), ⊂) of all exact structures on a fixed additive category A. This poset turns out to be a complete lattice, and under suitable conditions results of Enomoto's imply that it is boolean.We initiate in this article a detailed study of exact structures E by generalizing notions from abelian categories such as the length of an object relative to E and the quiver of an exact category (A, E). We investigate the Gabriel-Roiter measure for (A, E), and further study how these notions change when the exact structure varies. category of finite type, using Auslander algebra, or [INP18] where the more general concept of extriangulated structures is studied.While every exact category (A, E) can be embedded into a module category, notions like length or simple object cannot be borrowed from such an embedding. The first goal of this paper is to give an intrinsic definition relative to the class of morphisms in E, thus, in section 3, we call an object E−simple if it does not admit proper monomorphisms that belong to the class E. And we say that X is anThis change of definition requires to work out a number of notions and results that are granted in abelian categories, such as the notion of simple objects, artinian and finite objects or the length of an object. It turns out that in general not all the desired properties can be guaranteed. We also define, in section 3, the notion of the quiver of an exact category Q(A, E).The motivation for studying reductions of exact structures stems from the matrix reduction technique. The method of matrix reduction has been applied successfully by the Kiev school to solve various important problems in representation theory, like the Brauer-Thrall conjectures, or to show the tame-wild dichotomy. While the basic technique is elementary, the formalism of matrix reductions is somewhat complicated. Various models have been proposed to formalise matrix reductions: poset representations or bimodule problems cover only some cases. For the general case, one needs to study bocs representations, as introduced by Roiter in [Ro79], or iterated quotients of bimodule problems as in [Brü].This result allows to show that the length function l E of a finite essentially small exact category (A, E) is a measure for the poset ObjA in the sense of Gabriel [Gab]. We further show that most of the work of Krause [Kr11] on the Gabriel-Roiter measure for abelian length categories can be generalized to the context of exact categories: For the partially ordered set (indA, ⊂ E ) equipped with the length function l E , we define the Gabriel-Roiter measure as a morphism of partially ordered sets which refines the length function l E , see Theorem 7...

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Integral categories form a sub-class of pre-abelian categories whose systematic study was initiated by Rump in 2001. In the first part of this article we determine whether several categories of topological and bornological vector spaces are integral. Moreover, we establish that the class of integral categories is not contained in the class of quasi-abelian categories. In the last part of the article we show that a category is quasi-abelian if and only if it has admissible intersections, in the sense considered recently by Brüstle, Hassoun and Tattar. This exhibits that a rich class of non-abelian categories having this property arises naturally in functional analysis.A pre-abelian category is an additive category in which every morphism has a kernel and a cokernel. Within the class of pre-abelian categories, one defines the following notions. Semi-abelian categories are the ones in which the canonical morphism between the coimage and image is always both monic and epic, but not necessarily an isomorphism as one would expect in an abelian category. Quasi-abelian categories are the ones in which