Tensor products of finitely presented functors

Abstract

We provide explicit constructions for various ingredients of right exact monoidal structures on the category of finitely presented functors. As our main tool, we prove a multilinear version of the universal property of so-called Freyd categories, which in turn is used in the proof of correctness of our constructions. Furthermore, we compare our construction with the Day convolution of arbitrary additive functors. Day convolution always yields a closed monoidal structure on the category of all additive functors. In contrast, right exact monoidal structures for finitely presented functor categories are not necessarily closed. We provide a necessary criterion for being closed that relies on the underlying category having weak kernels and a so-called finitely presented prointernal hom structure. Our results are stated in a constructive way and thus serve as a unified approach for the implementation of tensor products in various contexts.

Tensor products of finitely presented functors

“…Our results yield a unified approach to the implementation of monoidal structures for f. p. modules, f. p. graded modules, f. p. functors, that has in parts been realized in [BP19]. Let us therefore explicitly present the underlying constructions.…”

“…In particular, this provides a computationally unified approach to tensor products of f. p. modules and f. p. graded modules, that can both be interpreted as special instances of categories of f. p. functors [Pos17]. A computer implementation of the special case where the promonoidal structure on A is actually monoidal is realized within the CAP-project, a software project for constructive category theory [GSP18], [GP19], [GPS18], as an own package for Freyd categories [BP19].…”