By Guiseppe Longo (auth.), David H. Pitt, David E. Rydeheard, Peter Dybjer, Andrew M. Pitts, Axel Poigné (eds.)

ISBN-10: 354051662X

ISBN-13: 9783540516620

**Read Online or Download Category Theory and Computer Science: Manchester, UK, September 5–8, 1989 Proceedings PDF**

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**Additional info for Category Theory and Computer Science: Manchester, UK, September 5–8, 1989 Proceedings**

**Example text**

Many of the principles developed in this paper transcend this choice of denominator; fixing it at S e t helps fix ideas. To complete the description of the functor I> we must describe its action on functors F, G of S S M . This is much more straightforward than the object part. This is facilitated by the strong condition U -- U~F in the definition of the morphisms F of S S M . The functor 47 F I> G : S S M 2 __. 47 (5)) a n d F 1> G takes each m o r p h i s m {f,s,g) of P 1> ~ to ( F f , s, Gg) as shown, where s denotes b o t h t h e inner a n d o u t e r square.

T h a t 1 ® 2 = 2 in 31 corresponds to the implication u < v < w --* u < w. This s t r u c t u r e was used to define "prossets" (preordered specification sets) [GP87], which were used to prove K a h n ' s principle relative to a p o m s e t - b a s e d semantics of nets [Pra86]. In the cartesian closed 3, the i n t e r p r e t a t i o n of 0 is no constraint as before. However 1 E 3 denotes accidental order, a n d 2 denotes causal order. T h e i d e n t i t y 1 ® 2 -- 1 in 3 corresponds to the notion t h a t if u accidentally precedes v whereas v c a u s e s (causally precedes) w, then one can infer from this t h a t u accidentally precedes w, b u t one cannot infer t h a t u causally precedes w.

Any kind m a y be used in any of these ways. Even if a kind does not a p p e a r to have been c o n s t i t u t e d as a category of metric spaces or labeled structures this will not stop us using it as though it were. 1In [Pra84] one of us proposed adapting semirings, as used in the Floyd-Warshallalgorithm [AHU74], for formalizing concurrent real-time processes. As pointed out in [Pra891, monoidal categories are to semirings as categories are to posets. 25 To obtain spaces with metrics we define a functor !

### Category Theory and Computer Science: Manchester, UK, September 5–8, 1989 Proceedings by Guiseppe Longo (auth.), David H. Pitt, David E. Rydeheard, Peter Dybjer, Andrew M. Pitts, Axel Poigné (eds.)

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