We demonstrate how the identity $N\otimes N \cong N$ in a monoidal category allows us to construct a functor from the full subcategory generated by $N$ and $\otimes$ 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 ), and Girard's dynamical algebra, by considering one-object analogues of projections and inclusions.
Keywords: Monoidal Categories, Categorical Trace, Compact Closure, Linear Logic, Inverse Semigroups.
1991 MSC: 18D10, 20M18.
Theory and Applications of Categories, Vol. 6, 1999, No. 3, pp 33-46.