A unified framework for generalized multicategories

Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the "lax algebras" or "Kleisli monoids" relative to a "monad" on a bicategory. However, the meanings of these words differ from author to author, as do the specific bicategories considered. We propose a unified framework: by working with monads on double categories and related structures (rather than bicategories), one can define generalized multicategories in a way that unifies all previous examples, while at the same time simplifying and clarifying much of the theory.

[1]  J. Lambek.  Deductive systems and categories II. Standard constructions and closed categories , 1969 .

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

[3]  R. Lowen.  Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad , 1997 .

[4]  Dominic R. Verity,et al.  Enriched Categories, Internal Categories and Change of Base , 2011 .

[5]  Ronald Brown,et al.  Double groupoids and crossed modules , 1976 .

[6]  Marco Grandis,et al.  Adjoint for double categories , 2004 .

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

[8]  Tom Leinster.  Operads in Higher-Dimensional Category Theory , 2000 .

[9]  Claudio Hermida.  From coherent structures to universal properties , 2000 .

[10]  Tom Leinster,et al.  Higher Operads, Higher Categories: Opetopes , 2004 .

[11]  Homotopy Algebras for Operads , 2000, math/0002180.

[12]  ON PROPERTY-LIKE STRUCTURES , 1997 .

[13]  Brian Day,et al.  Monoidal Bicategories and Hopf Algebroids , 1997 .

[14]  G. M. Kelly.  Applications of Categories in Computer Science: On clubs and data-type constructors , 1992 .

[15]  G. M. Kelly.  Many-variable functorial calculus. I. , 1972 .

[16]  Mark Weber,et al.  Operads within Monoidal Pseudo Algebras , 2004, Appl. Categorical Struct..

[17]  Walter Tholen,et al.  Ordered Topological Structures , 2009 .

[18]  I. Kríz,et al.  Operads, algebras, modules and motives , 2018, Astérisque.

[19]  P. Johnstone.  Sketches of an Elephant: A Topos Theory Compendium Volume 1 , 2002 .

[20]  The Span Construction , 2010 .

[21]  Ieke Moerdijk,et al.  Axiomatic homotopy theory for operads , 2002, math/0206094.

[22]  Glynn Winskel,et al.  The cartesian closed bicategory of generalised species of structures , 2008 .

[23]  R. J. Macg,et al.  PATHS IN DOUBLE CATEGORIES , 2006 .

[24]  F. William Lawvere,et al.  Metric spaces, generalized logic, and closed categories , 1973 .

[25]  Jeffrey Morton.  Categorified algebra and quantum mechanics. , 2006 .

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

[27]  Stephen Lack,et al.  Limits of small functors , 2007 .

[28]  Richard Garner.  Polycategories via pseudo-distributive laws , 2006, math/0606735.

[29]  G. M. Kelly,et al.  Structures defined by finite limits in the enriched context, I , 1982 .

[30]  Dirk Hofmann,et al.  One Setting for All: Metric, Topology, Uniformity, Approach Structure , 2004, Appl. Categorical Struct..

[31]  A. Joyal.  Foncteurs analytiques et espèces de structures , 1986 .

[32]  J. P. May,et al.  The geometry of iterated loop spaces , 1972 .

[33]  Marco Grandis,et al.  Limits in double categories , 1999 .

[34]  Michael Shulman,et al.  Framed bicategories and monoidal fibrations , 2007, 0706.1286.

[35]  Ross Street,et al.  Elementary cosmoi I , 1974 .

[36]  Ross Street,et al.  Limits indexed by category-valued 2-functors , 1976 .

[37]  Law Fw.  FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES. , 1963 .

[38]  Gavin J. Seal.  CANONICAL AND OP-CANONICAL LAX ALGEBRAS , 2005 .

[39]  Walter Tholen,et al.  Metric, topology and multicategory—a common approach , 2003 .

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

[41]  F. W. Lawvere,et al.  FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES. , 1963, Proceedings of the National Academy of Sciences of the United States of America.

[42]  G. M. Kelly.  An abstract approach to coherence , 1972 .

[43]  J. M. Boardman,et al.  Homotopy Invariant Algebraic Structures on Topological Spaces , 1973 .

[44]  Joachim Lambek,et al.  Deductive systems and categories III. Cartesian closed categories, intuitionist propositional calculus, and combinatory logic , 1972 .

[45]  Christoph Schubert,et al.  Extensions in the theory of lax algebras , 2010 .

[46]  James Dolan,et al.  Higher-Dimensional Algebra III: n-Categories and the Algebra of Opetopes , 1997 .

[47]  Walter Tholen,et al.  Lectures on Lax-Algebraic Methods in General Topology Summer School in Categorical Methods in Algebra and Topology , 2007 .

[48]  Jon P. May,et al.  THE UNIQUENESS OF INFINITE LOOP SPACE MACHINES , 1978 .

[49]  Michael Batanin,et al.  Monoidal Globular Categories As a Natural Environment for the Theory of Weakn-Categories☆ , 1998 .

[50]  Albert Burroni.  $T$-catégories (catégories dans un triple) , 1971 .

[51]  John Power.  Enriched Lawvere Theories , .

[52]  Ross Street,et al.  ENRICHED CATEGORIES AND COHOMOLOGY , 1983 .

[53]  Rings, modules, and algebras in infinite loop space theory , 2004, math/0403403.

[54]  James Dolan,et al.  From Finite Sets to Feynman Diagrams , 2001 .

[55]  A. Joyal.  Une théorie combinatoire des séries formelles , 1981 .

[56]  Michael Shulman,et al.  Comparing composites of left and right derived functors , 2007, 0706.2868.

[57]  Tom Leinster,et al.  Generalized enrichment of categories , 2002, math/0204279.

[58]  Eugenia Cheng.  Weak n -categories: opetopic and multitopic foundations , 2003 .

[59]  Brian Day,et al.  Lax monoids, pseudo-operads, and convolution , 2003 .

[60]  J. Benabou.  Introduction to bicategories , 1967 .