Semigroup actions on sets and the burnside ring
Date
2018Source Title
Applied Categorical Structures
Print ISSN
0927-2852
Electronic ISSN
1572-9095
Publisher
Springer Science
Volume
26
Issue
1
Pages
7 - 28
Language
English
Type
ArticleItem Usage Stats
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Abstract
In this paper we discuss some enlargements of the category of sets with semigroup
actions and equivariant functions. We show that these enlarged categories possess
two idempotent endofunctors. In the case of groups these enlarged categories are equivalent
to the usual category of group actions and equivariant functions, and these idempotent
endofunctors reverse a given action. For a general semigroup we show that these enlarged
categories admit homotopical category structures defined by using these endofunctors and
show that up to homotopy these categories are equivalent to the usual category of sets with
semigroup actions. We finally construct the Burnside ring of a monoid by using homotopical
structure of these categories, so that when the monoid is a group this definition agrees
with the usual definition, and we show that when the monoid is commutative, its Burnside
ring is equivalent to the Burnside ring of its Gr¨othendieck group.
Keywords
Semigroup actionsMonoid actions
Reverse actions
Homotopical category
Burnside ring
16W22
20M20
20M35
55U35