Connected monads weakly preserve products

TL;DR

This paper investigates conditions under which connected monads weakly preserve products, establishing a criterion involving the monad’s action on the singleton set, and generalizes a known observation in algebraic structures.

Contribution

It provides a characterization of when a connected monad weakly preserves products based on its behavior on the singleton set, extending previous algebraic results.

Findings

Surjectivity of $F(A\times B) \to F(A) \times F(B)$ iff $F(\mathbf{1})=\mathbf{1}$

Generalization of Dent, Kearnes, and Szendrei's observation to non-associative monads

Clarification of the relationship between monad properties and product preservation

Abstract

If $F$ is a (not necessarily associative) monad on $Set$, then the natural transformation $F(A\times B)\to F(A)\times F(B)$ is surjective if and only if $F(\boldsymbol{1})=\boldsymbol{1}$. Specializing $F$ to $F_{\mathcal{V}}$, the free algebra functor for a variety $\mathcal{V}$, this result generalizes and clarifies an observation by Dent, Kearnes and Szendrei.