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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.
We prove a version of the Deligne conjecture for $n$-fold monoidal abelian categories $A$ over a field $k$ of characteristic 0, assuming some compatibility and non-degeneracy conditions for $A$. The output of our construction is a weak Leinster $(n,1
Let $R$ be a commutative noetherian ring with a semi-dualizing module $C$. The Auslander categories with respect to $C$ are related through Foxby equivalence: $xymatrix@C=50pt{mathcal {A}_C(R) ar@<0.4ex>[r]^{Cotimes^{mathbf{L}}_{R} -} & mathcal {B}_C
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 cl
We give a characterisation of functors whose induced functor on the level of localisations is an equivalence and where the isomorphism inverse is induced by some kind of replacements such as projective resolutions or cofibrant replacements.
Given a bounded-above cochain complex of modules over a ring, it is standard to replace it by a projective resolution, and it is classical that doing so can be very useful. Recently, a modified version of this was introduced in triangulated categor