How to invert well-pointed endofunctors
Matt Booth
TL;DR
The paper addresses inverting well-pointed endofunctors in enriched categories by extending Kelly's transfinite free-algebra construction to enriched ind-categories, producing a localisation $\Omega^\infty$ that inverts the given natural transformation $\theta: \mathrm{id}\to\Omega$. It shows that functors inverting $\theta$ factor uniquely through the localisation $L_\Omega(\mathcal{C})$, and recovers important dg-quotient and orbit-category constructions (Seidel, Chen–Wang, Keller) as instances of this localisation. It then develops spectra and stabilisation notions (spectrification and stabilisation) via $\mathrm{Sp}_{\Omega}(\mathcal{C})$ and $\mathcal{S}_\Omega\mathcal{C}$, with precise comparisons that identify these frameworks with the same core localisation $L_\Omega\mathcal{C}$, and discusses their connections to localisation by spectra, Heller's stabilisation, and known orbit/quotient phenomena. Overall, the work provides a unified, high-level categorical approach that ties together algebraic and homotopical localisations across dg-categories, ind-completions, and spectra, with implications for constructing and comparing localisations such as Keller's orbit category and the CWKW dg quotient.
Abstract
In this short note we observe that Kelly's transfinite construction of free algebras yields a way to invert well-pointed endofunctors. In enriched settings, this recovers constructions of Keller, Seidel, and Chen-Wang. We also relate this procedure to localisation by spectra and to Heller's stabilisation.