---
date: 2022-03-25 20:38
modification date: Friday 25th March 2022 20:38:56
title: group actions on categories
aliases: [group actions on categories]

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Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #homotopy/stable-homotopy/equivariant 
- Refs:
	- #todo/add-references
- Links:
	- [equivariant functor](equivariant%20functor.md)
	- [RAPC](Unsorted/RAPC.md)
	- [Yoneda](Unsorted/Yoneda%20lemma.md)

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# group actions on categories

Categorifying a group action: regard $G$ as a groupoid with one object, then a functor $F\in [G, \Top]$ is precisely the data of a group action, $\colim F\cong \Orb_G(X)$ is the orbit space, and $\lim F = \Fix_G(X)$a are the fixed points.
A functor $F\in [G, \Vect\slice k]$ is a $k\dash$representation of $G$, i.e. a $kG\dash$module.

![](attachments/Pasted%20image%2020220325203858.png)

Equivalently, a [categorical fibration](categorical%20fibration) over $\B G$.