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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 [11]), and Girard's dynamical algebra, by considering one-object analogues of projections and inclusions.
@article{TAC_1999_6_a2,
author = {Peter Hines},
title = {The categorical theory of self-similarity},
journal = {Theory and applications of categories},
pages = {33--46},
publisher = {mathdoc},
volume = {6},
year = {1999},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_1999_6_a2/}
}
Peter Hines. The categorical theory of self-similarity. Theory and applications of categories, The Lambek Festschrift, Tome 6 (1999), pp. 33-46. http://geodesic.mathdoc.fr/item/TAC_1999_6_a2/