[The below is -very- confused, but has some insights]

A striking thing about the integral is that it's
-commutative- and -global-.

Consider integrating velocity over time:
- it doesn't matter if you go North first, then West,
  or the reverse:
  you just chop up the pieces of velocity and integrate
  in a big blob.

In reality, on a curved surface, say,
the order does matter.

*** The key technical difficulty is:
    before talking about non-commutativity,
    you need to talk about locality:
    you need to know "where you are".
    Thus -both- the base -and- the target must be topological spaces:
    the base so you can tell in what order to compare things,
    the target actually a -vector bundle- -over- a space
    
    Oh: to have non-commutativity in the base, it must be -ordered-;
    thus you're essentially restricted to:
    Z, R, and perhaps Q, hyperreals, etc.
    
    To discuss "non-commutativity" on a higher dimensional base,
    you'd need some sorta "coherence" statement,
    a la Saunders Mac Lane/Peter May
***

So what's going on?
- the basic integral assumes a fixed -vector space- as target,
  and just explains how to take weighted sums;
  this just requires a -measure space- as domain

- the -physical- interpretation
  (for integrating velocity to get a path)
  does -not- have a fixed vector space as target;
  rather, it has a -trivialized- tangent bundle

!!! oh: the point of a connection is to allow this commensurability
  of different tangent vectors -without- requiring trivialization.
  Why does this matter?
  B/c in a curved space, the tangent bundle is not even -locally-
  trivial, geometrically.

  So what you -actually- need is:
  - a manifold
  - with a vector bundle (not necessarily the tangent one!)
  - with a connection on it

This is much messier to rigorize:
- One simple case is where you have a manifold M
  and a vector field on it, and you want to find integral paths
  (You can also have other fields, like plane fields,
   or several vector fields, and you want to find integral surfaces etc.)
  This shows up when looking at potentials, for instance.

- You have a "space" manifold S of possible states
- You have a "time" manifold T
[hmm...can one do these separately]

BTW, domain cannot be a general measure space,
  as that has no order info/topology

can think of traditional integration as finding integrable
curve/surface "etc." on the abelian M x V

...or as a sort of integral over a point?
------------------------------------
So to take a non-commutative integral:
- domain is R
- the -value- of the integral will be in a Lie group
  (I think this generalizes to homogeneous space)
- the -integrand- (what you're integrating)
  takes values in a Lie algebra
  (which you can interpret as vector fields on the manifold)

You should always be able to integrate a continuous function,
as that's finding an integrable curve for a trivial vector field?
------------------------------------
Upshot:
-noncommutative integrals-
lead you directly to Lie groups and Lie algebras:
"Lie algebras are non-commutative infintesimals"
------------------------------------
These alternate generalization of integrals are -weird-
from the POV of measure theory;
they are very -differential geometric-

to make -vectors- non-commutative,
you need to replace 'em with -vector fields-
(and the Poisson bracket):
follow X then Y is not same as Y then X