Open AccessA Higher-Order Calculus for Categories
Author(s) -
Mario Cáccamo,
Glynn Winskel
Publication year - 2001
Publication title -
brics report series
Language(s) - English
Resource type - Journals
eISSN - 1601-5355
pISSN - 0909-0878
DOI - 10.7146/brics.v8i27.21687
Subject(s) - functor , mathematics , categorical variable , calculus (dental) , category theory , algebra over a field , diagrammatic reasoning , duality (order theory) , algebraic number , pure mathematics , computer science , programming language , medicine , mathematical analysis , statistics , dentistry
A calculus for a fragment of category theory is presented. The types in the language denote categories and the expressions functors. The judgements of the calculus systematise categorical arguments such as: an expression is functorial in its free variables; two expressions are naturally isomorphic in their free variables. There are special binders for limits and more general ends. The rules for limits and ends support an algebraic manipulation of universal constructions as opposed to a more traditional diagrammatic approach. Duality within the calculus and applications in proving continuity are discussed with examples. The calculus gives a basis for mechanising a theory of categories in a generic theorem prover like Isabelle.