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  • Cited by 2
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    This chapter has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Batanin, Michael and Street, Ross 2000. The universal property of the multitude of trees. Journal of Pure and Applied Algebra, Vol. 154, Issue. 1-3, p. 3.


    Carboni, Aurelio and Johnstone, Peter 1995. Connected limits, familial representability and Artin glueing. Mathematical Structures in Computer Science, Vol. 5, Issue. 04, p. 441.


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  • Print publication year: 1992
  • Online publication date: September 2009

On clubs and data-type constructors

Summary

Introduction

The diagrams for symmetric monoidal closed categories proved commutative by Mac Lane and the author in [19] were diagrams of (generalized) natural transformations. In order to understand the connexion between these results and free models for the structure, the author introduced in [13] and [14] the notion of club, which was further developed in [15] and applied later to other coherence problems in [16] and elsewhere.

The club idea seemed to apply to several diverse kinds of structure on a category, but still to only a restricted number of kinds. In an attempt to understand its natural limits, the author worked out a general notion of “club”, as a monad with certain properties, not necessarily on Cat now, but on any category with finite limits. A brief account of this was included in the 1978 Seminar Report [17], but was never published; the author doubted that there were enough examples to make it of general interest.

During 1990 and 1991, however, we were fortunate to have with our research team at Sydney Robin Cockett, who was engaged in applying category theory to computer science. In lectures to our seminar he called attention to certain kinds of monads involved with data types, which have special properties : he was calling them shape monads, but in fact they are precisely examples of clubs in the abstract sense above.

In these circumstances it seems appropriate to set down those old ideas after all, and to complete them in various ways, in particular as regards enriched monads.

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Applications of Categories in Computer Science
  • Online ISBN: 9780511525902
  • Book DOI: https://doi.org/10.1017/CBO9780511525902
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