Tbilisi Mathematical Journal

Algebraic Kan extensions along morphisms of internal algebra classifiers

Mark Weber

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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.

Article information

Source
Tbilisi Math. J., Volume 9, Issue 1 (2016), 65-142.

Dates
Received: 15 November 2015
Accepted: 10 January 2016
First available in Project Euclid: 12 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1528769043

Digital Object Identifier
doi:10.1515/tmj-2016-0006

Mathematical Reviews number (MathSciNet)
MR3461767

Zentralblatt MATH identifier
1342.18017

Subjects
Primary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23]
Secondary: 18D20: Enriched categories (over closed or monoidal categories) 18D50: Operads [See also 55P48] 55P48: Loop space machines, operads [See also 18D50]

Keywords
Kan extensions internal algebras exact squares operads modular envelope 2-monads

Citation

Weber, Mark. Algebraic Kan extensions along morphisms of internal algebra classifiers. Tbilisi Math. J. 9 (2016), no. 1, 65--142. doi:10.1515/tmj-2016-0006. https://projecteuclid.org/euclid.tbilisi/1528769043


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