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http://hdl.handle.net/1942/32603
Title: | Localizations of (one-sided) exact categories | Authors: | HENRARD, Ruben VAN ROOSMALEN, Adam-Christiaan |
Issue Date: | 2019 | Abstract: | 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 $\mathcal{A}$ be a percolating subcategory of an exact category $\mathcal{E}$, the quotient $\mathcal{E} {/\mkern-6mu/} \mathcal{A}$ is constructed in two steps. In the first step, we associate a set $S_\mathcal{A} \subseteq \operatorname{Mor}(\mathcal{E})$ to $\mathcal{A}$ and consider the localization $\mathcal{E}[S^{-1}_\mathcal{A}]$. In general, $\mathcal{E}[S_\mathcal{A}^{-1}]$ need not be an exact category, but will be a one-sided exact category. In the second step, we take the exact hull $\mathcal{E} {/\mkern-6mu/} \mathcal{A}$ of $\mathcal{E}[S_\mathcal{E}^{-1}]$. The composition $\mathcal{E} \rightarrow \mathcal{E}[S_\mathcal{A}^{-1}] \rightarrow \mathcal{E} {/\mkern-6mu/} \mathcal{A}$ 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 $S_\mathcal{A}$ are admissible. In upcoming work, we show that these localizations induce Verdier localizations on the level of the bounded derived category. | Keywords: | Mathematics - Category Theory;18E05, 18E10, 18E35 | Document URI: | http://hdl.handle.net/1942/32603 | Link to publication/dataset: | http://arxiv.org/abs/1903.10861v3 | Category: | O | Type: | Preprint |
Appears in Collections: | Research publications |
Files in This Item:
File | Description | Size | Format | |
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1903.10861v3.pdf | Non Peer-reviewed author version | 652.96 kB | Adobe PDF | View/Open |
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