Coextension of scalars in operad theory

TL;DR

This paper characterizes when the restriction functor between operadic algebras admits a right adjoint, based on a factorization condition for operations in the codomain operad.

Contribution

It provides a necessary and sufficient condition for the existence of a right adjoint to the restriction functor in operad theory.

Findings

Identifies a factorization axiom for operad maps

Establishes criteria for adjoint functors in operad algebras

Advances understanding of scalar extension in operad theory

Abstract

The functor between operadic algebras given by restriction along an operad map generally has a left adjoint. We give a necessary and sufficient condition for the restriction functor to admit a right adjoint. The condition is a factorization axiom which roughly says that operations in the codomain operad can be written essentially uniquely as operations in arity one followed by operations in the domain operad.