  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.

Algebraic Topology;Category Theory