##
**Algebraic Kan extensions along morphisms of internal algebra classifiers.**
*(English)*
Zbl 1342.18017

In his treatment of algebraic theories and structure, an important fact Lawvere proved was that, for any functor \(f : A\to B\) between small categories \(A\) and \(B\) with finite products, the left Kan extension \(\mathrm{Lan}_f(m) : B\to \mathrm{Set}\) of any finite-product-preserving functor \(m : A\to \mathrm{Set}\) is also finite product preserving. The special properties of finite product and \(\mathrm{Set}\) which make this work are: (a) there is a canonical comparison \(f(a_1\times \dots \times a_n)\to f(a_1)\times \dots \times f(a_n)\); (b) for any set \(x\), the endofunctors \(x\times -\) and \(-\times x\) of \(\mathrm{Set}\) preserve colimits; and (c) the left Kan extension is constructible pointwise by the usual colimit formula.

After the Introduction, in Section 2 the present paper centres attention on showing how the above result about Kan extensions can be lifted to the context where finite products are replaced by pseudo-algebra structure for a 2-monad \(T\) on a 2-category \(\mathcal{K}\). Item (a) is now replaced by the condition that \(f : A\to B\) should be a colax morphism of pseudo-\(T\)-algebras. Items (b) and (c) require the introduction of a number of new pseudo-\(T\)-algebra concepts: algebraic left extension for pseudomorphisms, algebraic cocompleteness with respect to a morphism, exactness of a square containing a 2-cell, and exactness for a colax morphism.

The main theorem of the paper is stated in Section 3, with examples, while the proof is postponed until Section 5.7. This theorem involves three 2-monads \(T\), \(R\) and \(S\) on respective 2-categories \(\mathcal{K}\), \(\mathcal{M}\) and \(\mathcal{L}\). The idea is that \(T\) describes the ambient structure, while \(R\) and \(S\) describe types of structure which can be defined internal to a pseudo-\(T\)-algebra, where \(S\) structure is richer than \(R\) structure as expressed using an adjunction \(G: (\mathcal{M},R)\to (\mathcal{L}, S)\). There is a forgetful functor \(U^G_A: S\text{-}\mathrm{Alg}(A)\to R\text{-}\mathrm{Alg}(A)\) between the categories of algebras internal to each pseudo-\(T\)-algebra \(A\). Conditions are developed under which a left adjoint for \(U^G_A\) can be constructed via algebraic left extension. The main theorem states that these conditions are met when dealing with polynomial monads in \(\mathrm{Cat}\). For background, see the author’s paper [Theory Appl. Categ. 30, 1659–1712 (2015; Zbl 1375.18058)] and his other two papers in that volume.

The paper contains a wealth of new results for working within a 2-category. The merit of working at this level is reinforced by the diversity of the applications. For example, further insight is gained on situations studied by R. M. Kaufmann and B. C. Ward [“Feynman categories”, arXiv:1312.1269] and by M. Batanin and C. Berger [“Homotopy theory for algebras over polynomial monads”, arXiv:1305.0086].

After the Introduction, in Section 2 the present paper centres attention on showing how the above result about Kan extensions can be lifted to the context where finite products are replaced by pseudo-algebra structure for a 2-monad \(T\) on a 2-category \(\mathcal{K}\). Item (a) is now replaced by the condition that \(f : A\to B\) should be a colax morphism of pseudo-\(T\)-algebras. Items (b) and (c) require the introduction of a number of new pseudo-\(T\)-algebra concepts: algebraic left extension for pseudomorphisms, algebraic cocompleteness with respect to a morphism, exactness of a square containing a 2-cell, and exactness for a colax morphism.

The main theorem of the paper is stated in Section 3, with examples, while the proof is postponed until Section 5.7. This theorem involves three 2-monads \(T\), \(R\) and \(S\) on respective 2-categories \(\mathcal{K}\), \(\mathcal{M}\) and \(\mathcal{L}\). The idea is that \(T\) describes the ambient structure, while \(R\) and \(S\) describe types of structure which can be defined internal to a pseudo-\(T\)-algebra, where \(S\) structure is richer than \(R\) structure as expressed using an adjunction \(G: (\mathcal{M},R)\to (\mathcal{L}, S)\). There is a forgetful functor \(U^G_A: S\text{-}\mathrm{Alg}(A)\to R\text{-}\mathrm{Alg}(A)\) between the categories of algebras internal to each pseudo-\(T\)-algebra \(A\). Conditions are developed under which a left adjoint for \(U^G_A\) can be constructed via algebraic left extension. The main theorem states that these conditions are met when dealing with polynomial monads in \(\mathrm{Cat}\). For background, see the author’s paper [Theory Appl. Categ. 30, 1659–1712 (2015; Zbl 1375.18058)] and his other two papers in that volume.

The paper contains a wealth of new results for working within a 2-category. The merit of working at this level is reinforced by the diversity of the applications. For example, further insight is gained on situations studied by R. M. Kaufmann and B. C. Ward [“Feynman categories”, arXiv:1312.1269] and by M. Batanin and C. Berger [“Homotopy theory for algebras over polynomial monads”, arXiv:1305.0086].

Reviewer: Ross H. Street (North Ryde)

### MSC:

18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |

18D20 | Enriched categories (over closed or monoidal categories) |

55P48 | Loop space machines and operads in algebraic topology |

18D50 | Operads (MSC2010) |

### Citations:

Zbl 1375.18058### References:

[1] | M. Batanin. Monoidal globular categories as a natural environment for the theory of weak n-categories. Advances in Mathematics, 136:39{103, 1998.} · Zbl 0912.18006 |

[2] | M. Batanin. The Eckmann-Hilton argument and higher operads. Advances in Mathematics, 217:334{385, 2008.} · Zbl 1138.18003 |

[3] | M. Batanin and C. Berger. Homotopy theory for algebras over polynomial monads. ArXiv:1305.0086. · Zbl 1368.18006 |

[4] | M. Batanin, J. Kock and M. Weber. Feynman categories are operads, regular patterns are substitudes. In preparation. · Zbl 1398.18008 |

[5] | R. Blackwell, G. M. Kelly and A. J. Power. Two-dimensional monad theory. J. Pure Appl. Algebra, 59:1{41, 1989.} · Zbl 0675.18006 |

[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.} · Zbl 0291.18010 |

[9] | R. Guitart. Relations et carrées exacts. Ann. Sc. Math. Quéebec, IV(2):103{125, 1980.} · Zbl 0495.18008 |

[10] | A. Joyal and R. Street. The geometry of tensor calculus I. Advances in Mathematics, 88:55{ 112, 1991.} · Zbl 0738.18005 |

[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.} · Zbl 0334.18018 |

[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.} · Zbl 0334.18016 |

[15] | J. Kock. Polynomial functors and trees. Int. Math. Res. Not., 3:609{673, 2011.} · Zbl 1235.18007 |

[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.} · Zbl 1351.18004 |

[18] | S. Lack. Codescent objects and coherence. J. Pure Appl. Algebra, 175:223{241, 2002.} · Zbl 1142.18301 |

[19] | S. Lack. Homotopy-theoretic aspects of 2-monads. Journal of Homotopy and Related Structures, 2(2):229{260, 2007.} · Zbl 1184.18005 |

[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.} · Zbl 1236.18006 |

[22] | F.W. Lawvere. Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. USA, 50, 1963. · Zbl 0119.25901 |

[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.} · Zbl 0668.18010 |

[25] | R. Street. The formal theory of monads. J. Pure Appl. Algebra, 2:149{168, 1972.} · Zbl 0241.18003 |

[26] | R. Street. Fibrations and Yoneda’s lemma in a 2-category. Lecture Notes in Math., 420:104{ 133, 1974.} · Zbl 0327.18006 |

[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.} · Zbl 0401.18004 |

[29] | S. Szawiel and M. Zawadowski. Theories of analytic monads. ArXiv:1204.2703, 2012. · Zbl 1342.18006 |

[30] | M. Weber. Familial 2-functors and parametric right adjoints. Theory and applications of categories, 18:665{732, 2007.} · Zbl 1152.18005 |

[31] | M. Weber. Yoneda structures from 2-toposes. Applied Categorical Structures, 15:259{323, 2007.} · Zbl 1125.18001 |

[32] | M. Weber. Internal algebra classifiers as codescent objects of crossed internal categories. Theory and applications of categories, 30:1713{1792, 2015.} · Zbl 1351.18002 |

[33] | M. Weber. Operads as polynomial 2-monads. Theory and applications of categories, 30:1659{ 1712, 2015.} · Zbl 1375.18058 |

[34] | M. Weber. Polynomials in categories with pullbacks. Theory and applications of categories, 30:533{598, 2015.} · Zbl 1330.18002 |

[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.} |

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