The Categorical Theory of Self-similarity

We demonstrate how the identity N N = N in a monoidal category allows us to construct a functor from the full subcategory generated by N and to the endomorphism monoid of the object N. This provides a categorical foundation for one-object analogues of the symmetric monoidal categories used by J.-Y. Girard in his Geometry of Interaction series of papers, and explicitly described in terms of inverse semigroup theory in 6, 11]. This functor also allows the construction of one-object analogues of other categorical structures. We give the example of one-object analogues of the categorical trace, and compact closedness. Finally, we demonstrate how the categorical theory of self-similarity can be related to the algebraic theory (as presented in 11]), and Girard's dynamical algebra, by considering one-object analogues of projections and inclusions.