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The cartesian closed bicategory of generalised species of structures
Author(s) -
Fiore M.,
Gambino N.,
Hyland M.,
Winskel G.
Publication year - 2008
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdm096
Subject(s) - cartesian coordinate system , substitution (logic) , functor , mathematics , cartesian closed category , pure mathematics , algebra over a field , computer science , programming language , geometry
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.