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June 2016 Algebraic Kan extensions along morphisms of internal algebra classifiers
Mark Weber
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Tbilisi Math. J. 9(1): 65-142 (June 2016). DOI: 10.1515/tmj-2016-0006


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.


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Mark Weber. "Algebraic Kan extensions along morphisms of internal algebra classifiers." Tbilisi Math. J. 9 (1) 65 - 142, June 2016.


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

zbMATH: 1342.18017
MathSciNet: MR3461767
Digital Object Identifier: 10.1515/tmj-2016-0006

Primary: 18D10
Secondary: 18D20, 18D50, 55P48

Rights: Copyright © 2016 Tbilisi Centre for Mathematical Sciences


Vol.9 • No. 1 • June 2016
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