Free Profinite Categories and Semigroupoids and how Symbolic Dynamics can help us to understand them
(Work with Jorge Almeida)
Bret Tilson proposed in the 1987 paper
"Categories as algebra: an essential ingredient in the theory of monoids"
to see small categories and semigroupoids
as partial algebras generalizing the concepts of monoid and semigroup, respectively.
The results in Tilson's paper
were the first of many proving that its title was far from being exaggerated.
Free semigroups, free profinite semigroups and relatively free profinite semigroups play a central role in finite semigroup theory (the same for monoids).
Independently, Peter Jones
on the one hand, and Jorge Almeida and Pascal Weil on the other hand,
introduced the foundations of a theory of
free profinite categories and semigroupoids. Jones only considered
categories generated by finite-vertex graphs, while Almeida and
Weil considered also generating infinite-vertex profinite graphs.
With an example based on a very simple
symbolic dynamical system, we prove that their definition has flaws
if the generating profinite graph is infinite-vertex.
We fix these flaws, and again with the help of symbolic dynamics we
present some interesting examples of free profinite semigroupoids
infinite-vertex profinite graphs where some of the assumptions made
by Almeida and Weil remain valid.
Areas of interest
Semigroups, Category Theory
Alfredo Costa (CMUC/Mat. FCTUC)
March 13, 2007