By Guiseppe Longo (auth.), David H. Pitt, David E. Rydeheard, Peter Dybjer, Andrew M. Pitts, Axel Poigné (eds.)
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Extra resources for Category Theory and Computer Science: Manchester, UK, September 5–8, 1989 Proceedings
P ! --~ P '! is a functor. T h e proof is a d i a g r a m chase verifying t h a t F ! A and F ! f really do satisfy t h e axioms for P '-categories a n d P '-functors. For t h e case when F is a strong s y m m e t r i c monoidal functor, the above proof shows in fact t h a t F ! is so too, a n d this establishes our desired C o r o l l a r y 3 The construction P ~-* D! _: S S M - - * S S M . 4 Examples Consider the case where P is discrete, having no m o r p h i s m s o t h e r t h a n identities.
The event map and translation are consistent with respect to the labelings). We can view P o m as the full subcategory of P r o m = 2! I> S e t that generalizes pomsets by allowing P to be a preorder. In this paper, since the construction t h a t we emphasize is P I> 8, we are interested only in labels drawn from sets. When more elaborately structured label sources are needed, as for the category P o s l> 2 of order ideals, the comma must be taken over the appropriate common denominator, here P o s (so that an order ideal is a triple (P, f, 2) for f : P --* 2 monotone).
P ! -* P '! as follows. D e f i n i t i o n Given a monoidal functor F : P --~ P ~, we define t h e functor F ! : D! --* P ~! induced by F as follows: For any P - c a t e g o r y A, we get the P '-category F ! A,u ~ VA = F(~A(U,V)) = FrnA,u~ w o n~A(U,v)#A(u,~) = FjA,u o k For any P - f u n c t o r f : A --* B , we get a P '-functor F ! f by using t h e same o b j e c t function a n d by t a k i n g ( F ! f ) ~ -- F ( r ~ ) Proposition 2 F! : P ! --~ P '! is a functor. T h e proof is a d i a g r a m chase verifying t h a t F !