Parametrised Presentability over Orbital Categories
Kaif Hilman
TL;DR
The paper extends the theory of presentable $\infty$-categories to parametrised settings over orbital base categories ${\mathcal T}$, providing a straightening-based characterisation of ${\mathcal T}$-presentable categories and deriving a parametrised adjoint functor theorem. It develops foundational tools for parametrised colimits, mapping spaces, Ind-completions, and (co)limits, and then establishes a robust framework for ${\mathcal T}$-presentability, including Dwyer–Kan localisations, localisation–cocompletions, and a tight correspondence between presentables and idempotent-complete categories in the parametrised context. The results enable controlled constructions in equivariant and parametrised homotopy theory, with planned applications to parametrised noncommutative motives for equivariant algebraic K-theory. Overall, the work integrates straightening techniques, Beck–Chevalley conditions, and multiplicative aspects to produce a comprehensive theory of parametrised presentability and its categorical consequences.
Abstract
In this paper, we develop the notion of presentability in the parametrised homotopy theory framework of Barwick-Dotto-Glasman-Nardin-Shah over orbital categories. We formulate and prove a characterisation of parametrised presentable categories in terms of its associated straightening. From this we deduce a parametrised adjoint functor theorem from the unparametrised version, prove various localisation results, and we record the interactions of the notion of presentability here with multiplicative matters. Such a theory is of interest for example in equivariant homotopy theory, and we will apply it in a companion work to construct the category of parametrised noncommutative motives for equivariant algebraic K-theory.