# arrow-structure syntax

**RE01 Rice Brian T. EM2**
BRice@vinson.navy.mil

*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.