Natural transformations of superpositions of selected functors in some closed categories

S. V. Solov'ev

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 8, Tome 160 (1987), pp. 257-261

Résumé

Under the condition that the distinguished object is a generator, one has obtained: a complete description of the natural transformations of the superpositions of the tensor product $\otimes$ of the interior $\mathrm{hom}$ functor $[\,,\,]$, in compact closed categories, including the category of finitely generated projective modules over a commutative ring with unit; the generalization of this result to a certain class of superpositions of $\otimes$, $[\,,\,]$, and of the biproduct $\oplus$; conditions for the commutativity of the diagram of the natural transformations for the mentioned superpositions, reducing to the verification of the commutativity of the diagrams of the components, when all the arguments of the functors are equal to the distinguished object.