Abstract :

A concrete monoid over a category C is a subset of the endomorphisms of an object of C containing the identity and closed under composition To contrast an abstract monoid is just a one object category. There is a natural notion of division between concrete monoids distinct from the usual division of abstract monoids This concrete division is identied via two examples and then dened giving rise to a bicategory of concrete monoids over C whose arrows are concrete divisions The Poincare classes of the arrows of this bicategory are found to have a simple and appealing characterization allowing us to dene a category of concrete monoids over C . These denitions are illustrated with examples from the theories of semigroups, matrices, vines and automata With the aid of these denitions, we make functorial the well known constructions of the action monoid of an automaton, and the endomorphism monoid of an object of a category.