From: amnell@cc.helsinki.fi (Marko Amnell) Date: Wed, 3 Mar 1993 17:08:05 +0200 (EET) Can anyone recommend good introductions to enriched category theory? I have Kelly's "Basic concepts of enriched category theory", but would like to know about work after 1981 too. The article referenced below [CCMP], also available by anonymous ftp as boole.stanford.edu:/pub/man.tex, applies a monoidal category D to a notion of time having two dimensions, logical strength (having to finish in 1 minute is a stronger requirement than 2) and temporal extent (taking 1 minute and then 2 sums to 3). The category structure expresses strength while the monoidal structure captures temporal accumulation. A behavior, whether physical or computational, is defined to be a generalized metric space of events whose distances, construed as delays, are objects of D, with the space satisfying a triangle inequality. Following Lawvere, a "generalized metric space" over D is interpreted to mean a category enriched in D. The abstractness of this framework is instantiated to express a variety of concepts associated with time: precedence, simultaneity, strict delay, causality, upper and lower bounds on integer or real waiting time, rigid vs. flexible scheduling, etc. An algebra of behaviors is defined and its interpretation in these various notions of time considered. A brief history of enriched categories and its counterpart in computer science, the regular algebra of semirings, is included. Both started in the 1960's and developed independently for two decades prior to converging in the current work unifying the two perspectives. The article includes a more or less self-contained treatment of enriched categories from the perspective of this application, up to the theorem that if D is complete and closed then D-Cat is closed. Whereas Kelly's treatment of these concepts is more formal and complete, it was our intent that our treatment make the subject more accessible to engineering-oriented readers who would prefer to view such abstractions in the setting of a concrete application such as temporal structures. We are currently investigating the incorporation of Winskel's time-information duality into this model, in which every such behavior or schedule as a metric space of *events* dualizes via Stone or Pontrjagin duality to an automaton formalized as a complete metric space of *states* whose distances denote correlations between those states, with the correlations being of the standard statistical kind in the case that the monoid of distances forms a locally compact abelian group. Inner product spaces provide a convenient way of achieving such duality. We are also investigating extensions to non-abelian groups and monoids. In the natural process-algebra-cum-logic of such dynamic structures, static conjunction does not distribute over static disjunction, but dynamic conjunction does [Pr92g]. Vaughan Pratt @Article( CCMP, Author="Casley, R.T and Crew, R.F. and Meseguer, J. and Pratt, V.R.", Title="Temporal Structures", Journal="Math. Structures in Comp. Sci.", Volume=1, Number=2, Pages="179-213", Month=Jul, Year=1991) @InProceedings( Pr92g, Author="Pratt, V.R.", Title="Linear Logic for Generalized Quantum Mechanics", Booktitle="Proc. of Computation of Physics workshop", Address="Dallas", Month=Oct, Year=1992, Publisher="IEEE", Note="Also available as boole.stanford.edu:/pub/ql.tex") ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++