Generalized Kneser superfunction trick (the iterated limit definition)

[JmsNxn]

\(
\int_b^c f(s,z)\,ds\circ \int_a^b f(s,z) ds\circ z = \int_a^c f(s,z)\,ds\circ z\\
\)

So, if I get it, this kind of interpretations problems arise often in category theory (when u have to compose lot of weird stuff).
Let's test my understanding: If I get this right the following should make sense for you as well

\(
\int_b^c f(s,-)\,ds\circ \int_a^b f(s,-) ds = \int_a^c f(s,-)\,ds\\
\)

That is the same practical reason, let's ignore the historical one, of the use of \( ds \) to specify the variable you are integrating over.

Yes, it's exactly the same thing!

Especially when I write,

\(
\Omega_{j=1}^\infty \phi_j(s,z)\\
\)

Does this mean,

\(
\lim_{n\to\infty}\phi_1(s,\phi_2(s,...\phi_n(s,z)))\\
\)

or does it mean,

\(
\lim_{n\to\infty} \phi_1(\phi_2(...\phi_n(s,z)...,z),z)\\
\)

So we add a bullet to bind the variable. Then, when I write,

\(
\Omega_{j=n}^{m} \phi_j(s,z)\bullet \Omega_{j=m+1}^k\phi_j(s,z)\bullet z = \Omega_{j=n}^k \phi_j(s,z)\bullet z\\
\)

Tell me this doesn't look better than writing:

\(
\Omega_{j=n}^{m} \phi_j(s,z)\bullet z \circ \Omega_{j=m+1}^k\phi_j(s,z)\bullet z
\)

But if I did all this bullet suff with \( \circ \)--that's not really how \( \circ \) is usually used, so I'd be overriding the meaning of an existent symbol within this context. Better to use a new symbol and be fresh. This is especially beneficial when we talk about \( ds\bullet z \) which is almost like a differential form. Writing \( ds\circ z \) would be going a step too far I think.

Edit:

I think a good idea is to think of it this way.

\(
f(g(z)) = f \circ g = f\bullet g \bullet z\\
\)

When we use a bullet, we should declare what we're binding it to. We don't need to do that with \circ. It would be wrong to override this and write,

\(
f \circ g \circ z\\
\)

Wtf is that nonsense? lol

Plus, now we can write,

\(
\Omega f \bullet g \bullet z\\
\)

and it's **almost** like a differential form