The Karoubi envelope of a semigroup is a small category which plays a distinguished role in finite semigroup theory thanks to the Delay theorem of Tilson. In this talk, an application to the classification of symbolic dynamical systems is presented: up to natural equivalence, the Karoubi envelope of the syntactic semigroup defined by a shift space is a flow equivalence invariant. From this invariant, other algebraic invariants  are deduced.
This is joint work with Benjamin Steinberg, from Carleton University, Ottawa.