The Karoubi envelope plays a fundamental role in Tilson's Delay theorem, an important result in the theory of pseudovarieties of semigroups. It is also known that two semigroups with local units are Morita equivalent if and only if they have equivalent Karoubi envelops.
We introduce a new category, the Schützenberger category of a semigroup, which stands in relation to the Karoubi envelop of the semigroup as the Schützenberger groups stand to maximal subgroups in a semigroup. We analyze some aspects of this relationship, with an emphasis on the case of semigroups with local units.
This is joint work with Benjamin Steinberg.