Tbilisi Mathematical Journal

Algebraic Kan extensions along morphisms of internal algebra classifiers

Mark Weber

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


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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  • M. Batanin. Monoidal globular categories as a natural environment for the theory of weak $n$-categories. Advances in Mathematics, 136:39–103, 1998.
  • M. Batanin. The Eckmann-Hilton argument and higher operads. Advances in Mathematics, 217:334–385, 2008.
  • M. Batanin and C. Berger. Homotopy theory for algebras over polynomial monads. ArXiv:1305.0086.
  • M. Batanin, J. Kock and M. Weber. Feynman categories are operads, regular patterns are substitudes. In preparation.
  • R. Blackwell, G. M. Kelly and A. J. Power. Two-dimensional monad theory. J. Pure Appl. Algebra, 59:1–41, 1989.
  • J. Bourke. Codescent objects in 2-dimensional universal algebra. PhD thesis, University of Sydney, 2010.
  • K. Costello. The $A_{\infty}$-operad and the moduli space of curves. ArXiv:0402015v2.
  • E. Dubuc. Free monoids. Journal of Algebra, 29:208–228, 1974.
  • R. Guitart. Relations et carrés exacts. Ann. Sc. Math. Québec, IV(2):103–125, 1980.
  • A. Joyal and R. Street. The geometry of tensor calculus I. Advances in Mathematics, 88:55–112, 1991.
  • R. M. Kaufmann and B. Ward. Feynman categories. ArXiv:1312.1269, 2014.
  • G.M. Kelly. On clubs and doctrines. Lecture Notes in Math., 420:181–256, 1974.
  • G.M. Kelly. Basic concepts of enriched category theory, LMS lecture note series, volume 64. Cambridge University Press, 1982. Available online as TAC reprint no. 10.
  • G.M. Kelly and R. Street. Review of the elements of 2-categories. Lecture Notes in Math., 420:75–103, 1974.
  • J. Kock. Polynomial functors and trees. Int. Math. Res. Not., 3:609–673, 2011.
  • S. R. Koudenburg. Algebraic weighted colimits. PhD thesis, University of Sheffield, 2012.
  • S. R. Koudenburg. Algebraic Kan extensions in double categories. Theory and applications of categories, 30:86–146, 2015.
  • S. Lack. Codescent objects and coherence. J. Pure Appl. Algebra, 175:223–241, 2002.
  • S. Lack. Homotopy-theoretic aspects of 2-monads. Journal of Homotopy and Related Structures, 2(2):229–260, 2007.
  • S. Lack. Icons. Applied Categorical Structures, 18:289–307, 2010.
  • S. Lack and M. Shulman. Enhanced 2-categories and limits for lax morphisms. Advances in Mathematics, 229(1):294–356, 2012.
  • F.W. Lawvere. Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. USA, 50, 1963.
  • P-A. Melliès and N. Tabareau. Free models of T-algebraic theories computed as Kan extensions. Unpublished article accompanying a talk given at CT08 in Calais available http://www.tabareau.fr/here.
  • A. J. Power. A general coherence result. J. Pure Appl. Algebra, 57:165–173, 1989.
  • R. Street. The formal theory of monads. J. Pure Appl. Algebra, 2:149–168, 1972.
  • R. Street. Fibrations and Yoneda's lemma in a $2$-category. Lecture Notes in Math., 420:104–133, 1974.
  • R. Street. Fibrations in bicategories. Cahiers Topologie Géom. Differentielle, 21:111–160, 1980.
  • R. Street and R.F.C. Walters. Yoneda structures on $2$-categories. J.Algebra, 50:350–379, 1978.
  • S. Szawiel and M. Zawadowski. Theories of analytic monads. ArXiv:1204.2703, 2012.
  • M. Weber. Familial 2-functors and parametric right adjoints. Theory and applications of categories, 18:665–732, 2007.
  • M. Weber. Yoneda structures from 2-toposes. Applied Categorical Structures, 15:259–323, 2007.
  • M. Weber. Internal algebra classifiers as codescent objects of crossed internal categories. Theory and applications of categories, 30:1713–1792, 2015.
  • M. Weber. Operads as polynomial 2-monads. Theory and applications of categories, 30:1659–1712, 2015.
  • M. Weber. Polynomials in categories with pullbacks. Theory and applications of categories, 30:533–598, 2015.
  • R. J. Wood. Abstract pro arrows I. Cahiers Topologie Géom. Différentielle Catégoriques, 23(3):279–290, 1982.
  • R. J. Wood. Abstract pro arrows II. Cahiers Topologie Géom. Différentielle Catégoriques, 26(2):135–168, 1985.