Parametrized higher category theory

In this paper, we develop the notion of presentability in the parametrised homotopy theory framework of Barwick et al. (Parametrized higher category theory and higher algebra: a general introduction, 2016) 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 Hilman (Parametrised noncommutative motives and cubical descent in equivariant algebraic K-theory, 2022) to construct the category of parametrised noncommutative motives for equivariant algebraic K-theory.

show abstract

An Equivariant Generalisation of McDuff–Segal’s Group–Completion Theorem

Hilman

2023

International Mathematics Research Notices

View full text Add to dashboard Cite

In this short note, we prove a $G$–equivariant generalisation of McDuff–Segal’s group–completion theorem for finite groups $G$. A new complication regarding genuine equivariant localisations arises and we resolve this by isolating a simple condition on the homotopy groups of $\mathbb{E}_{\infty }$–rings in $G$–spectra. We check that this condition is satisfied when our inputs are a suitable variant of $\mathbb{E}_{\infty }$–monoids in $G$–spaces via the existence of multiplicative norm structures, thus giving a localisation formula for their associated $G$–spherical group rings.

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Parametrized higher category theory

Abstract: Parametrized higher category theory 511(1) The 1-category of Borel G-spectra, ie spectra with G-action. This iswhich is the stabilization of Fun.BG; Spc/.(2) The 1-category of "naive" G-spectra, ie spectral presheaves on O G . This isSp/; which is the stabilization of Spc G .

Cited by 8 publications

References 9 publications

The Adams isomorphism revisited

The Adams isomorphism revisited

Parametrised Presentability Over Orbital Categories

An Equivariant Generalisation of McDuff–Segal’s Group–Completion Theorem

Contact Info

Product

Resources

About