Parametrised Presentability over Orbital Categories

TL;DR

This paper extends the concept of presentability to parametrised homotopy theory over orbital categories, providing characterisations, theorems, and localisations relevant for equivariant homotopy and algebraic K-theory.

Contribution

It introduces a new framework for parametrised presentability, characterises it via straightening, and develops foundational theorems applicable to equivariant homotopy theory.

Findings

Characterisation of parametrised presentable categories

Parametrised adjoint functor theorem

Localization results in parametrised setting

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.