"Procedures are usually built up from elementary procedures. What these
elementary procedures may be, and how more complex procedures are
constructed from them, is one of the first topics in computer science.
This subject is not hard to understand since there is a precise notion of a
computable function to guide us, and computability relative to a given
collection of initial functions is easy to define.
Procedures operate on members of certain data spaces and produce
members of other data spaces, using in general still other data spaces
as intermediates. A number of operations are known for constructing
new data spaces from simpler ones, but there is as yet no general
theory of representable data spaces comparable to the theory of
Most of the treatments I've seen fall back upon set theory to handle
Gamma, such as saying that items to the left of a turnstile are sets
or that imperative program actions modify a set of known variables.
The closest I've seen to a theory of "representable data spaces"
occurs in the Let Over Lambda book
Are you aware of any references that focus on a "theory of representable
data spaces" that does not involve sets?