By Guiseppe Longo (auth.), David H. Pitt, David E. Rydeheard, Peter Dybjer, Andrew M. Pitts, Axel Poigné (eds.)
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Additional info for Category Theory and Computer Science: Manchester, UK, September 5–8, 1989 Proceedings
This monoidal c a t e g o r y forms the basis for ultrametric spaces. T h e more useful monoidal s t r u c t u r e takes ® to be addition, a s y m m e t r i c 4For purists~ Seti is any maximal subcategory of Set which is a poser and consists only of monics of Set. 34 closed but not cartesian closed structure. This structure forms the basis for ordinary metric spaces. Note the distinction drawn here between the additive structure as a nonclosed monoidal structure vs. the underlying category as cartesian closed by virtue of the product supplying a closed monoidal structure.
5) Any monoid automatically forms a monoidal category having a discrete underlying category (all morphisms identities). Among these we will find groups particularly useful as metrics for "rigid" spaces. This is the one category in these examples t h a t is not bicomplete. 3 Functors Just as strict monoidal categories motivate monoidal categories, so do strict monoidat functors motivate monoidal functors. A strict monoidal functor F : P ~ P~ is a functor F : D --* D r between their underlying categories satisfying F x ®' F y = F ( x ® y) and I v = F I for objects and similarly for morphisms.
T h a t is, homsets become objects of P while functions between homsets and other sets become morphisms of P. The set or class of objects of C do not participate in this passage: the objects of C remain a set or class. A functor f : A ~ B participates partially in this passage: when we view it as the P-functor f : A ~ B, its object part remains unchanged but its action on morphisms, viewed for each pair uv of objects of A as a function r ( u , v ) : 5A(u,v) --+ 5B(f(u), f(v)), becomes a morphism of P between homobjects hA(u, v) and 5B(f(u), f(v)) of A, B respectively.
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.)