An \algebraic left Kan extension" is a left Kan extension which interacts well with the alge- braic 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 un- derstand. 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 alge- braicness 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.
[1] M. Batanin. Monoidal globular categories as a natural environment for the theory of weak n-categories. Advances in Mathematics 136:39{103 1998.
[2] M. Batanin. The Eckmann-Hilton argument and higher operads. Advances in Mathematics 217:334{385 2008.
[3] M. Batanin and C. Berger. Homotopy theory for algebras over polynomial monads. ArXiv:1305.0086.
[4] M. Batanin J. Kock and M. Weber. Feynman categories are operads regular patterns are substitudes. In preparation.
[5] R. Blackwell G. M. Kelly and A. J. Power. Two-dimensional monad theory. J. Pure Appl. Algebra 59:1{41 1989.
[6] J. Bourke. Codescent objects in 2-dimensional universal algebra. PhD thesis University of Sydney 2010.
[7] K. Costello. The A1-operad and the moduli space of curves. ArXiv:0402015v2.
[8] E. Dubuc. Free monoids. Journal of Algebra 29:208{228 1974.
[9] R. Guitart. Relations et carrées exacts. Ann. Sc. Math. Quéebec IV(2):103{125 1980.
[10] A. Joyal and R. Street. The geometry of tensor calculus I. Advances in Mathematics 88:55{ 112 1991.
[11] R. M. Kaufmann and B. Ward. Feynman categories. ArXiv:1312.1269 2014.
[12] G.M. Kelly. On clubs and doctrines. Lecture Notes in Math. 420:181{256 1974.
[13] 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.
[14] G.M. Kelly and R. Street. Review of the elements of 2-categories. Lecture Notes in Math. 420:75{103 1974.
[15] J. Kock. Polynomial functors and trees. Int. Math. Res. Not. 3:609{673 2011.
[16] S. R. Koudenburg. Algebraic weighted colimits. PhD thesis University of Sheffield 2012.
[17] S. R. Koudenburg. Algebraic Kan extensions in double categories. Theory and applications of categories 30:86{146 2015.
[18] S. Lack. Codescent objects and coherence. J. Pure Appl. Algebra 175:223{241 2002.
[19] S. Lack. Homotopy-theoretic aspects of 2-monads. Journal of Homotopy and Related Structures 2(2):229{260 2007.
[20] S. Lack. Icons. Applied Categorical Structures 18:289{307 2010.
[21] S. Lack and M. Shulman. Enhanced 2-categories and limits for lax morphisms. Advances in Mathematics 229(1):294{356 2012.
[22] F.W. Lawvere. Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. USA 50 1963.
[23] P-A. Mellièes and N. Tabareau. Free models of T-algebraic theories computed as Kan extensions. Unpublished article accompanying a talk given at CT08 in Calais available here.
[24] A. J. Power. A general coherence result. J. Pure Appl. Algebra 57:165{173 1989.
[25] R. Street. The formal theory of monads. J. Pure Appl. Algebra 2:149{168 1972.
[26] R. Street. Fibrations and Yoneda's lemma in a 2-category. Lecture Notes in Math. 420:104{ 133 1974.
[27] R. Street. Fibrations in bicategories. Cahiers Topologie Géeom. Differentielle 21:111{160 1980.
[28] R. Street and R.F.C. Walters. Yoneda structures on 2-categories. J.Algebra 50:350{379 1978.
[29] S. Szawiel and M. Zawadowski. Theories of analytic monads. ArXiv:1204.2703 2012.
[30] M. Weber. Familial 2-functors and parametric right adjoints. Theory and applications of categories 18:665{732 2007.
[31] M. Weber. Yoneda structures from 2-toposes. Applied Categorical Structures 15:259{323 2007.
[32] M. Weber. Internal algebra classifiers as codescent objects of crossed internal categories. Theory and applications of categories 30:1713{1792 2015.
[33] M. Weber. Operads as polynomial 2-monads. Theory and applications of categories 30:1659{ 1712 2015.
[34] M. Weber. Polynomials in categories with pullbacks. Theory and applications of categories 30:533{598 2015.
[35] R. J. Wood. Abstract pro arrows I. Cahiers Topologie Géeom. Difféerentielle Catéegoriques 23(3):279{290 1982.
[36] R. J. Wood. Abstract pro arrows II. Cahiers Topologie Géeom. Difféerentielle Catéegoriques 26(2):135{168 1985.