Get Applications of Categories in Computer Science: Proceedings PDF

By M. P. Fourman, P. T. Johnstone, A. M. Pitts

ISBN-10: 0511525907

ISBN-13: 9780511525902

ISBN-10: 0521427266

ISBN-13: 9780521427265

Purposes of type idea and similar themes of arithmetic to desktop technological know-how were a becoming quarter in recent times. This booklet comprises chosen papers at the topic from the London Mathematical Society Symposium held on the collage of Durham in July 1991.

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Additional resources for Applications of Categories in Computer Science: Proceedings of the London Mathematical Society Symposium, Durham 1991

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These rules are to represent bijections stable under cutf so the following equations must hold: • g o /® = (g o /)® for g : Ai,C, A2 —> A 3 and f : Fi —> F 2 ,C,F 3 , and where T\ contains the sequence A,B. • /® og = (fog)® for g : Ax —-> A 2 ,C,A 3 and f : Fi,C,r 2 —> F 3 , and where one o/Fi,F 2 contains the sequence A,B. f = /®. om for f :Fi,A,£,F 2 — F 3 =% •gof® = {gof)® for g : Ai,C, A2 —> A 3 and f : Tx —> F 2 ,C,r 3 , and where one o/F 2 ,F 3 contains the sequence A,B. • f® og = (fog)® for g : Ai —> A 2 ,C,A 3 oncf / : Fi,C,F 2 —> F 3 , and where F 3 contains the sequence A, 2?.

We can also obtain some slightly stronger results by taking advantage of the ordering. We omit most of the proofs, which may be easily obtained from the results above, using monotonicity and continuity of composition. 1 Let C be an ordered ccc and let (T, 6,6,7) be a computational comonad over C with a computational pairing. • //"split o merge > id then curry o uncurry > id. • //split o merge < id then cuTfy o uncurry < id. • If merge0 split > id then unoTrry ocuTry > id. ry < id. Next we introduce a simple generalization of the notion of cartesian closed ordered category, obtained by relaxing the requirement that currying and uncurrying form an isomorphism.

Then: curfy(g 6 ( / x id)) = = < = = = < = = cilrry(# o T(f x e) o Tsplit 06) curry(# o T(f x e) o T split oS o merge) curry(# o T(f x e) o merge 0(6 x 6)) curry(# o merge o(Tf x Te) o (6 x 6)) curry(# o merge o((Tf o 6) x (Te o 6))) curry(# o mergeo((T/ o S) x id)) curry(# o merge) o (Tf o S) cuTry(^) o Tf o 6 curfy(g)bf. • To show that uncITrry is a lax natural transformation, suppose that / : A - A Ai a n d h . A> _+c uncurry(/i) 6 ( / x id) = > = = = > = = uncurry(/i) o split o T ( / x e) o Tsplit o£ uncurry(/i) o (Tf x Te) o split oT split o£ uncurry(Zi) o (Tf x Te) o (8 x 8) o split uncurry(/i) o ((Tf o 8) x (Te o 8)) o split uncuny(h) o ((Tf o 8) x \d) o sptit uncurry(/i o T / o 8) o split uncurry(/i 6 / ) o split uncurry(/i 6 / ) .

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