arrow-structure syntax

RE01 Rice Brian T. EM2
Sat, 5 Dec 1998 17:03:02 +0800

> A question came to my mind: what is the link between the arrow system
> and Category Theory ? I guess the arrow stuff won't only be a syntactic
> system.....
> (please, don't answer "None !" :):)
well, i have a few books which are just collections of papers on arrow
theory, and one of the papers walks through the relationship between the two
like this:  categories can be viewed as arrow-graphs with special
restrictions, although the inversion, identity, and combinator operators are
all applicable.  it basically amounts to the exclusion of multigraphs.  in
fact, the author goes on to create a special category theory description
equivalent to 'full' (first-order) arrow theory and then makes a vague
conjecture about the equivalent category theory for n-dimensional arrow
theory (don't ask, since i'm just now sorting out the idea myself).

there's also one interesting part of category theory: the implicit 'worlds'
constructed within each category, which, according to the papers i have,
amount to _informal_ sets of axioms.  i've been thinking through an idea of
using a "category of categories" to build the conceptual framework of Tunes,
but this seems rather difficult from where we are.

> you bet the "arrow type" is a tough notion !... I'm not sure enumeration 
> will suffice....we should provide some recursive means of constructing
> arrow types... and we have to work out all the fundamentals (domain
> equations or any other stuff... to seriously build the whole system
> theoretically, to prove its consistency...and so on :)
yeah, i'm trying to sort out some ideas about that right now.  maybe i'll
have some results in a few days.