2019/03/26 by Ruben Henrard, Henrard, Ruben, Adam-Christiaan van Roosmalen +1 · 3 citations
Mathematics · #18E05 #18E10 #18E35 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1903.10861
openalex publication_date 2019/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we introduce quotients of exact categories by percolating subcategories. This approach extends earlier localization theories by Cardenas and Schlichting for exact categories, allowing new examples. Let A be a percolating subcategory of an exact category E, the quotient E /\mkern-6mu/ A is constructed in two steps. In the first step, we associate a set SA ⊆ Mor(E) to A and consider the localization E[S-1A]. In general, E[SA-1] need not be an exact category, but will be a one-sided exact category. In the second step, we take the exact hull E /\mkern-6mu/ A of E[SE-1]. The composition E → E[SA-1] → E /\mkern-6mu/ A satisfies the 2-universal property of a quotient in the 2-category of exact categories. We formulate our results in slightly more generality, allowing to start from a one-sided exact category. Additionally, we consider a type of percolating subcategories which guarantee that the morphisms of the set SA are admissible. In upcoming work, we show that these localizations induce Verdier localizations on the level of the bounded derived category.