Some applications of the multiplicative tensor product of matrix factorizations

TL;DR

This paper introduces a new example of semi-unital semi-monoidal categories using the multiplicative tensor product of matrix factorizations, expanding the understanding of such structures in category theory.

Contribution

It provides the first simple example of a semi-unital semi-monoidal category using the multiplicative tensor product of matrix factorizations.

Findings

Identifies a one-step connected subcategory of matrix factorizations that is semi-unital semi-monoidal.

Defines the concept of right pseudo-monoidal category and shows $(MF(1),\widetilde{\otimes})$ as an example.

Establishes the applicability of the multiplicative tensor product in category theory structures.

Abstract

The notion of semi-unital semi-monoidal category was defined a couple of years ago using the so called "Takahashi tensor product" and so far, the only example of it in the literature is complex. In this paper, we use the recently defined "multiplicative tensor product of matrix factorizations" to give a simple example of this a notion. In fact, if $MF(1)$ denotes the category of matrix factorizations of the constant power series $1$, we define the concept of one-step connected category and prove that there is a one-step connected subcategory of $(MF(1),\widetilde{\otimes})$ which is semi-unital semi-monoidal. We also define the concept of right pseudo-monoidal category which generalizes the notion of monoidal category and we prove that $(MF(1),\widetilde{\otimes})$ is an example of this concept.