On 2017-08-13 22:55, Dana Scott wrote:
Does (P (<) Q) have a nice categorical definition as a functor in the category of posets?
Yes [3]. On 08/13/17 9:42 PM, Patrik Eklund wrote:
What would be the practical applications of that construction?
Sequential composition, aka concatenation, aka ordinal addition [1,2]. Second diagram of Figure 1, /et seq,/ [3] answers Dana's question more generally for V-categories, with the category of preordered sets as the case V = 2. The simplified proof of Theorem 9 notwithstanding, our paper could benefit today from a more pedagogically sensitive treatment. Vaughan Pratt [1] G. Birkhoff. An extended arithmetic. Duke Mathematical Journal, 3(2), June 1937. [2] G. Birkhoff. Generalized arithmetic. Duke Mathematical Journal, 9(2), June 1942. [3] Casley, R.T., Crew, R.F., Meseguer, J., and Pratt, V.R., ``Temporal Structures'', Proc. Category Theory and Computer Science 1989, ed. D. Pitt et al, LNCS 389, 21-51, Springer-Verlag, 1989. Revised journal version in Mathematical Structures in Computer Science, Volume 1:2, 179-213, July 1991. A version missing some figures is downloadable as http://boole.stanford.edu/pub/man.pdf, the missing figures should be in the older version http://boole.stanford.edu/pub/man90.pdf. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]