Re: Re: An elementary question

[Vaughan Pratt <pratt@cs.stanford.edu>]

> 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.


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