language for infinitary compositions?

language for infinitary compositions?

[Meredith Gregory]

Categorists,

Is there a commonly accepted language for infinitary compositions? Here's
the sort of thing i'm thinking about. There's a purely syntactic
correspondence between a 'braced' notation and an infix notation for
composition. Suppose we have a categorically-friendly notion of composition,
say c. (Meaning the entities c composes can be viewed as morphisms in a
category and c the categorical composition.) Then we can just as easily
write

   - f c g -- c is merely making notationally explicit the interpretation of
   'o' in f o g -- we're coloring the 'o', as it were

or

   - {c| f, g |c} -- we've moved from infix to (not quite) prefix notation
   for the composition.


The braced notation, however, is suggestive of a very powerful notational
mechanism, comprehension notation. We could easily imagine a language
allowing expressions of the form

{c| pattern | predicate |c}

which would denote

pattern{subst_1} c pattern{subst_2} c ...

where subst_i is a substition for 'variables' in the pattern of entities
satisfying the predicate. This would allow reasoning over infinitary
compositions by providing an intensional view of their interior structure.
Surely, such a widget has already been invented. Can someone give me a
reference?

Best wishes,

--greg

-- 
L.G. Meredith
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