ORBITS AND MONOIDS IN A TOPOS

The linguistic intuition shows that many facts which are usually expre ssed in mathematics by using the explicit existence of a ''set of inte gers'', should be expressible without it. We make this intuition preci se in the context of toposes (not necessarily) with a natural number o bject. This leads to the study of orbits of elements for endomaps X -- > X in a topos. These can be identified with the monoids with one gene rator. We get results related to finiteness and existence of cyclic po ints, and study the fibration of the orbits.