Composition, bullet notation and the general role of categories

[MphLee]

EDIT: Oh and I look forward to reading the PDF!

Oh but it is at the end of my first post!

Hey, Mphlee. I absolutely agree with you, but I think you misread me. I write,

I'm sorry if I did misread you. Please tell me exactly what I'm missing.


I think I understood that you wanted to make a distinct notation; as you say I agree that typographically it is better; I agree that writing \( f\circ g\circ z \) someone could be confused believing it is an abuse of notation and, at the end I do believe that this is a very smart choice indeed: firstly for the similarity to forms, because it remembers the \( dz \) making it analogue to integration and secondly, this is the reason I omitted in the post, because it is perfect when you deal with multivariable functions.

When I say

for this and another reason, I like your choice.

the other reason is that the bullet seems to me to fit perfectly when you manipulate expressions with multiple variables that you don't want to hide, i.e. the standard in math, e.g.

\( f(z_0,...,z_j,...,z_n)\bullet g(z_0,...,z_j,...,z_n)\bullet z_j=f(z_0,...,g(z_0,...,z_j,...,z_n),...,z_n) \)

I admit that for me there are some gray zones in it's usage, but I'll ask you you when I'll need to use your notation.

I'm very sorry but reading backwards I realize that I was not very clear and I did not make explicit some of my thoughts. I did this post as a side effort: I'm tackling the multi-valued case for the superfunction trick but I believed making this post would be helpful to make my stance more sensible.

To clear myself a little bit: what I don't believe is that \( f\circ g\circ z \) is actually a real abuse of notation. It seems an abuse but I say it is perfectly formal and legit. What I believe is that we are not leaving the world of composition. That's not a reason at all to abandon the bullet but it is important to notice imho.

ps: the \leftarrow notation it's horrible, even categorically...