Dear Vaughan, I will under the CATLIST and at this point not reply more in detail, but let me just now say that your "categories are not algebraic in sets, at least they are algebraic in graphs, which in turn are algebraic in sets" is within the realm of "algebraic categories". My "Categorizing automata is hard enough as we see through Budach, Ehrig, Goguen, Manes, Adamek, etc." is in the realm of "categorical algebra" and monoidal categories. So with graphs understood as being useful also for representing terms with "vertices in trees seen as operator names", like it is done e.g. in tree automata, monoidal categories as underlying categories of term monads is another view and machinery as compared to dealing with the category of graphs as an algebraic category ("categories are algebraic in graphs"). Best, Patrik On 2017-09-28 08:50, Vaughan Pratt wrote:
On 09/26/17 9:28 PM, Patrik Eklund wrote:
The category of graphs may also need revision. Defining a graph as mapping an edge to a pair of vertices hides arities and invites to defining paths. Nevertheless, vertices in trees are seen as operator names.
Even though categories are not algebraic in sets, at least they are algebraic in graphs, which in turn are algebraic in sets.
While I would have little or no quarrel with any revision of "the" category of graphs that preserved this fundamental relation between categories and sets, if the revision you have in mind does not then I would expect at least some of us here would be very interested in why you consider your contemplated revision an improvement.
Vaughan Pratt
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