What Separable Frobenius Monoidal Functors Preserve

TL;DR

This paper precisely characterizes the equations in monoidal categories preserved by separable Frobenius monoidal functors, including various operators and structures like Yang-Baxter operators and weak bimonoids.

Contribution

It develops a detailed theory of what equations are preserved by these functors, including a characterization of stable monoidal expressions and the image of Yang-Baxter operators.

Findings

Separable Frobenius monoidal functors preserve lax Yang-Baxter operators.

They also preserve weak Yang-Baxter operators and weak bimonoids in braided cases.

Every weak Yang-Baxter operator is the image of a genuine one under such a functor.

Abstract

Separable Frobenius monoidal functors were defined and studied under that name by Szlachanyi and by Day and Pastro, and in a more general context by Cockett and Seely. Our purpose here is to develop their theory in a very precise sense. We determine what kinds of equations in monoidal categories they preserve. For example we show they preserve lax (meaning not necessarily invertible) Yang-Baxter operators, weak Yang-Baxter operators in the sense of Alonso Alvarez et al., and (in the braided case) weak bimonoids in the sense of Pastro and Street. In fact, we characterize which monoidal expressions are preserved (or rather, are stable under conjugation in a well-defined sense). We show that every weak Yang-Baxter operator is the image of a genuine Yang-Baxter operator under a separable Frobenius monoidal functor. Prebimonoidal functors are also defined and discussed.