  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 $E\in
\mathcal{E}$. 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.

Category Theory;K-Theory and Homology