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.
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