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Register a new userDerived categories of (one-sided) exact categories and their localizations
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Adam-Christiaan van Roosmalen
Publication date
2019
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English
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We consider the quotient of an exact or one-sided exact category $mathcal{E}$ by a so-called percolating subcategory $mathcal{A}$. For exact categories, such a quotient is constructed in two steps. Firstly, one localizes $mathcal{E}$ at a suitable class $S_mathcal{A} subseteq operatorname{Mor}(mathcal{E})$ of morphisms. The localization $mathcal{E}[S_mathcal{A}^{-1}]$ need not be an exact category, but will be a one-sided exact category. Secondly, one constructs the exact hull $mathcal{E}{/mkern-6mu/} mathcal{A}$ of $mathcal{E}[S_mathcal{A}^{-1}]$ and shows that this satisfies the 2-universal property of a quotient amongst exact categories. In this paper, we show that this quotient $mathcal{E} to mathcal{E} {/mkern-6mu/} mathcal{A}$ induces a Verdier localization $mathbf{D}^b(mathcal{E}) to mathbf{D}^b(mathcal{E} {/mkern-6mu/} mathcal{A})$ of bounded derived categories. Specifically, (i) we study the derived category of a one-sided exact category, (ii) we show that the localization $mathcal{E} to mathcal{E}[S_mathcal{A}^{-1}]$ induces a Verdier quotient $mathbf{D}^b(mathcal{E}) to mathbf{D}^b(mathcal{E}[S^{-1}_mathcal{A}])$, and (iii) we show that the natural embedding of a one-sided exact category $mathcal{F}$ into its exact hull $overline{mathcal{F}}$ lifts to a derived equivalence $mathbf{D}^b(mathcal{F}) to mathbf{D}^b(overline{mathcal{F}})$. We furthermore show that the Verdier localization is compatible with several enhancements of the bounded derived category, so that the above Verdier localization can be used in the study of localizing invariants, such as non-connective $K$-theory.
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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.
One-sided exact categories are obtained via a weakening of a Quillen exact category. Such one-sided exact categories are homologically similar to Quillen exact categories: a one-sided exact category $mathcal{E}$ can be (essentially uniquely) embedded into its exact hull ${mathcal{E}}^{textrm{ex}}$; this embedding induces a derived equivalence $textbf{D}^b(mathcal{E}) to textbf{D}^b({mathcal{E}}^{textrm{ex}})$. Whereas it is well known that Quillens obscure axioms are redundant for exact categories, some one-sided exact categories are known to not satisfy the corresponding obscure axiom. In fact, we show that the failure of the obscure axiom is controlled by the embedding of $mathcal{E}$ into its exact hull ${mathcal{E}}^{textrm{ex}}.$ In this paper, we introduce thr
It is well known that a resolving subcategory $mathcal{A}$ of an abelian subcategory $mathcal{E}$ induces several derived equivalences: a triangle equivalence $mathbf{D}^-(mathcal{A})to mathbf{D}^-(mathcal{E})$ exists in general and furthermore restricts to a triangle equivalence $mathbf{D}^{mathsf{b}}(mathcal{A})to mathbf{D}^{mathsf{b}}(mathcal{E})$ if $operatorname{res.dim}_{mathcal{A}}(E)<infty$ for any object $Ein mathcal{E}$. If the category $mathcal{E}$ is uniformly bounded, i.e. $operatorname{res.dim}_{mathcal{A}}(mathcal{E})<infty$, one obtains a triangle equivalence $mathbf{D}(mathcal{A})to mathbf{D}(mathcal{E})$. In this paper, we show that all of the above statements hold for preresolving subcategories of (one-sided) exact categories. By passing to a one-sided language, one can remove the assumption that $mathcal{A}subseteq mathcal{E}$ is extension-closed completely from the classical setting, yielding easier criteria and more examples. To illustrate this point, we consider the Isbell category $mathcal{I}$ and show that $mathcal{I}subseteq mathsf{Ab}$ is preresolving but $mathcal{I}$ cannot be realized as an extension-closed subcategory of an exact category. We also consider a criterion given by Keller to produce derived equivalences of fully exact subcategories. We show that this criterion fits into the framework of preresolving subcategories by considering the relative weak idempotent completion of said subcategory.
Let $n$ be an integer greater or equal than $3$. We give a simultaneous generalization of $(n-2)$-exact categories and $n$-angulated categories, and we call it one-sided $n$-suspended categories. One-sided $n$-angulated categories are also examples of one-sided $n$-suspended categories. We provide a general framework for passing from one-sided $n$-suspended categories to one-sided $n$-angulated categories. Besides, we give a method to construct $n$-angulated quotient categories from Frobenius $n$-prile categories. These results generalize their works by Jasso for $n$-exact categories, Lin for $(n+2)$-angulated categories and Li for one-sided suspended categories.
Restriction categories were introduced to provide an axiomatic setting for the study of partially defined mappings; they are categories equipped with an operation called restriction which assigns to every morphism an endomorphism of its domain, to be thought of as the partial identity that is defined to just the same degree as the original map. In this paper, we show that restriction categories can be identified with emph{enriched categories} in the sense of Kelly for a suitable enrichment base. By varying that base appropriately, we are also able to capture the notions of join and range restriction category in terms of enriched category theory.
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