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Journal of the London Mathematical Society Advance Access originally published online on December 20, 2007
Journal of the London Mathematical Society 2008 77(1):203-220; doi:10.1112/jlms/jdm096
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© 2007 London Mathematical Society

The cartesian closed bicategory of generalised species of structures

M. Fiore

Computer Laboratory
University of Cambridge
15 JJ Thomson Avenue
Cambridge
CB3 0FD
United Kingdom

N. Gambino

Laboratoire de Combinatoire et Informatique Mathématique
Départements de Mathématiques et d’Informatique
Université du Québec à Montréal
Case Postale 8888
Succursale Centre-Ville
Montréal QC H3C 3P8
Canada
nicola.gambino{at}gmail.com

M. Hyland

DPMMS
University of Cambridge
Wilberforce Road
Cambridge
CB3 0WA
United Kingdom
M.Hyland{at}dpmms.cam.ac.uk

G. Winskel

Computer Laboratory
University of Cambridge
15 JJ Thomson Avenue
Cambridge
CB3 0FD
United Kingdom
Glynn.Winskel{at}cl.cam.ac.uk

The concept of generalised species of structures between small categories and, correspondingly, that of generalised analytic functor between presheaf categories are introduced. An operation of substitution for generalised species, which is the counterpart to the composition of generalised analytic functors, is also put forward. These definitions encompass most notions of combinatorial species considered in the literature — including of course Joyal's original notion — together with their associated substitution operation. Our first main result exhibits the substitution calculus of generalised species as arising from a Kleisli bicategory for a pseudo-comonad on profunctors. Our second main result establishes that the bicategory of generalised species of structures is cartesian closed.

2000 Mathematics Subject Classification 18D05 (primary), 18D15, 18D50, 18F20, 03F52, 05A99 (secondary).

Received January 22, 2007; published online October 20, 2007.


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