COMPLETELY REDUCIBLE SETS

We study the family of rational sets of words, called completely reducible and which are such that the syntactic representation of their characteristic series is completely reducible. This family contains, by a result of Reutenauer, the submonoids generated by bifix codes and, by a result of Berstel and Reutenauer, the cyclic sets. We study the closure properties of this family. We prove a result on linear representations of monoids which gives a generalization of the result concerning the complete reducibility of the submonoid generated by a bifix code to sets called birecurrent. We also give a new proof of the result concerning cyclic sets.

Homotopy theory of cyclic presheaves

We generalize the homotopy theory of cyclic sets to cyclic presheaves on small Grothendieck sites. This is achieved by constructing pointwise and local model structures reminiscent of the homotopy theory of simplicial presheaves.