Polynomials in categories with pullbacks

The theory developed by Gambino and Kock, of polynomials over a locally cartesian closed category E, is generalised for E just having pullbacks. The 2-categorical analogue of the theory of polynomials and polynomial functors is given, and its relationship with Street's theory of fibrations within 2-categories is explored. Johnstone's notion of "bagdomain data" is adapted to the present framework to make it easier to completely exhibit examples of polynomial monads.

[1]  N. Gambino,et al.  Polynomial functors and polynomial monads , 2009, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  G. M. Kelly,et al.  Two-dimensional monad theory , 1989 .

[3]  Thorsten Altenkirch,et al.  Foundations of Software Science and Computation Structures: 6th International Conference, FOSSACS 2003 Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2003 Warsaw, Poland, April 7–11, 2003 Proceedings , 2003, Lecture Notes in Computer Science.

[4]  Ross Street,et al.  Cosmoi of internal categories , 1980 .

[5]  D. Tambara,et al.  On multiplicative transfer , 1993 .

[6]  Witt vectors and Tambara functors , 2003, math/0304495.

[7]  F. William Lawvere,et al.  Ordinal sums and equational doctrines , 1969 .

[8]  G. M. Kelly.  On clubs and doctrines , 1974 .

[9]  G. M. Kelly,et al.  BASIC CONCEPTS OF ENRICHED CATEGORY THEORY , 2022, Elements of ∞-Category Theory.

[10]  Ieke Moerdijk,et al.  Algebraic set theory , 1995 .

[11]  Eduardo J. Dubuc,et al.  Kan Extensions in Enriched Category Theory , 1970 .

[12]  A. Kock.  Fibrations as Eilenberg-Moore algebras , 2013, 1312.1608.

[13]  A. Joyal,et al.  The geometry of tensor calculus, I , 1991 .

[14]  A. Kock.  Monads for which Structures are Adjoint to Units , 1995 .

[15]  M. Brun,et al.  Witt vectors and equivariant ring spectra applied to cobordism , 2007 .

[16]  Peter T. Johnstone,et al.  Fibrations and partial products in a 2-category , 1993, Appl. Categorical Struct..

[17]  P. T. Johnstone,et al.  Variations on the Bagdomain Theme , 1994, Theor. Comput. Sci..

[18]  Ross Street,et al.  Fibrations and Yoneda's lemma in a 2-category , 1974 .

[19]  S. Lack,et al.  Codescent objects and coherence , 2002 .

[20]  Mark Weber.  Familial 2-functors and parametric right adjoints , 2007 .

[21]  Ross Street,et al.  Fibrations in bicategories , 1980 .

[22]  R. J. Wood,et al.  Abstract pro arrows I , 1982 .

[23]  Jean Giraud,et al.  Méthode de la descente , 1964 .

[24]  R. Street,et al.  Review of the elements of 2-categories , 1974 .

[25]  Homotopy theory for algebras over polynomial monads , 2013, 1305.0086.

[26]  Mark Weber,et al.  Yoneda Structures from 2-toposes , 2007, Appl. Categorical Struct..

[27]  Joachim Kock,et al.  Polynomial functors and opetopes , 2007, 0706.1033.

[28]  S. Lack,et al.  The formal theory of monads II , 2002 .

[29]  ON PROPERTY-LIKE STRUCTURES , 1997 .

[30]  A. J. Power,et al.  A general coherence result , 1989 .