The linear-non-linear substitution 2-monad

TL;DR

This paper develops a new construction on 2-monads, creating a linear-non-linear 2-monad by combining free symmetric monoidal and cartesian 2-monads, with detailed theoretical analysis and characterizations.

Contribution

It introduces a colimit construction on 2-monads, characterizes its algebras, and applies it to combine symmetric monoidal and cartesian structures into a novel linear-non-linear 2-monad.

Findings

Constructed a colimit-based 2-monad from maps of 2-monads.

Provided a characterization of algebras for the new 2-monad.

Demonstrated the combination of symmetric monoidal and cartesian 2-monads.

Abstract

We introduce a general construction on 2-monads. We develop background on maps of 2-monads, their left semi-algebras, and colimits in 2-category. Then, we introduce the construction of a colimit induced by a map of 2-monads, show that we obtain the structure of a 2-monad and give a characterisation of its algebras. Finally, we apply the construction to the map of 2-monads between free symmetric monoidal and the free cartesian 2-monads and combine them into a linear-non-linear 2-monad.