Abelian versus triangulated quotients
Henning Krause
TL;DR
This work examines how localisation in triangulated categories translates, via abelianisation, to abelian localisations and quotients. It establishes a key compatibility: for a triangulated $\mathcal{T}$ and subcategory $\mathcal{S}$, the abelian quotient $\operatorname{Ab}(\mathcal{T})/\operatorname{Ab}_{\mathcal{S}}(\mathcal{T})$ is equivalent to $\operatorname{Ab}(\mathcal{T}/\mathcal{S})$, and it demonstrates that abelian quotients can escape a given Grothendieck universe through explicit counterexamples involving derived categories. The paper then develops the notion of the abelian hull $\operatorname{Ab}(\mathcal{C})$ for an arbitrary category $\mathcal{C}$ and shows that localisation behaves well with respect to the hull via $\operatorname{Ab}(\operatorname{add}(\mathcal{C}))/\mathcal{S} \simeq \operatorname{Ab}(\operatorname{add}(\mathcal{C}[\Sigma^{-1}]))$, where $\mathcal{S}$ is generated by kernels and cokernels of the localisation morphisms. A concrete quiver example demonstrates that quotient and hull constructions can fail to preserve the ambient universe size, highlighting fundamental set-theoretic limitations in the theory of quotients and abelianisations across categorical contexts.
Abstract
It is shown that any localisation of triangulated categories induces (up to an equivalence) a localisation of abelian categories when one passes to their abelianisations. From this one obtains for any enlargement of Grothendieck universes an example of an abelian category and a Serre subcategory within the smaller universe such that the corresponding quotient does only exist within the bigger universe. The second part of this note provides an analogue for the abelian hull of an arbitrary category.