Algebraic Kan extensions along morphisms of internal algebra classifiers

TL;DR

This paper investigates algebraic left Kan extensions along morphisms of internal algebra classifiers, providing conditions that ensure their algebraic nature and explaining their occurrence in operad theory and related fields.

Contribution

It identifies monad-theoretic conditions that produce algebraic Kan extensions between universal internal algebra classifiers, extending previous work and unifying various applications.

Findings

Conditions for Guitart-exactness are established.

Explains algebraic Kan extensions in operad and Feynman category contexts.

Includes generalizations for polynomial monads.

Abstract

An algebraic left Kan extension is a left Kan extension which interacts well with the algebraic structure present in the given situation, and these appear in various subjects such as the homotopy theory of operads and in the study of conformal field theories. In the most interesting examples, the functor along which we left Kan extend goes between categories that enjoy universal properties which express the meaning of the calculation we are trying to understand. These universal properties say that the categories in question are universal examples of some categorical structure possessing some kind of internal structure, and so fall within the theory of internal algebra classifiers described in earlier work of the author. In this article conditions of a monad-theoretic nature are identified which give rise to morphisms between such universal objects, which satisfy the key condition of Guitart-exactness, which guarantees the algebraicness of left Kan extending along them. The resulting setting explains the algebraicness of the left Kan extensions arising in operad theory, for instance from the theory of Feynman categories of Kaufmann and Ward, generalisations thereof, and also includes the situations considered by Batanin and Berger in their work on the homotopy theory of algebras of polynomial monads.