This follows from the discussion held at: MphLee, Generalized Kneser superfunction trick (the iterated limit definition), (January 21, 2021), Tetration Forum.

But said that you shouldn't be overly confident about the variable vs function distinction:

What nonsense is this? It's abstract nonsense!

In category theory, a land where only composition and arrows make the whole ontology, writing \( f \circ g \circ z \) makes perfect sense, not only that, it means exactly what you expect it should.

More than that: I claim that the natural home for general iterated compositions are categories!

In the last part (mostly in the attached pdf+the pdf has a few pages of very gentle introduction to categories) I'll offer moral reasons for that but first let's inspect your "nonsensical" composition.

About evaluation.

In general categories morphisms are just abstract arrows, not functions, and evaluation of them doesn't generally make sense because not every object can be conceived as a bag of something, e.g. points. The philosophy of category theory is exactly this: ignore what's inside, the inner structure of things, and solely observe how your things interact with each other.

There are some very special categories where objects are indeed made of points, e.g. the category of topological spaces, of vector spaces, of abelian groups or the category of bare sets: with this I mean that some categories have among all the objects a special "point object" \( * \).

In these particular categories a morphism from this objectified abstract point to an arbitrary object \( X \) can be thought as (a choice of) a point \( x \) in \( X \)

therefore defining the set points of \( X \) to be the (hom-)set of arrows

For example, for a bare set \( X \) we have \( X^1\simeq X \) where \( 1 \) is a singleton; the set of group homomorphisms from the group of integers to \( G \) is in bijection with the set of group elements of \( G \), i.e. \( {\rm Hom}_{\rm Grp}({\mathbb Z},G)\simeq G \); the linear applications from a field \( \mathbb K \), seen as a vector space, to a \( \mathbb K \)-vector space \( V \) are in bijection with vectors of \( V \), i.e. \( {\rm Hom}_{\rm Vec}({\mathbb K},V)\simeq V \)

and all of this without actually being able to look inside our objects nor knowing what set membership is!

In the case you make, given an abstract arrow \( X\overset{f}{\to}Y \) and a point \( *\overset{x}{\to}X \) we can evaluate \( f \) at \( x \) composing the two and producing a new point \( *\overset{y}{\to}Y \)

Here the TeXed post in pdf+a mini guide on categories.

(2021 02 02) Composition bullet notation and the general role of categories - the softest introduction ever made.pdf (Size: 585.27 KB / Downloads: 412)

Quote:But if I did all this bullet stuff 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.

It is clear to me where you are coming from. Your solution is pleasant, pretty, comfortable and the notation, as usually happens with good notation, hints at new developments, e.g. the differential forms. Even if I don't get \( n \)-forms yet and exterior algebra feels alien to me I feel it's a similarity worth considering: for this and another reason, I like your choice. I feel like you are aiming, not secretly at all, to a general infinitesimal compositional calculus.

But said that you shouldn't be overly confident about the variable vs function distinction:

Quote: \( f \circ g \circ z \)

Wtf is that nonsense? lol

What nonsense is this? It's abstract nonsense!

In category theory, a land where only composition and arrows make the whole ontology, writing \( f \circ g \circ z \) makes perfect sense, not only that, it means exactly what you expect it should.

More than that: I claim that the natural home for general iterated compositions are categories!

In the last part (mostly in the attached pdf+the pdf has a few pages of very gentle introduction to categories) I'll offer moral reasons for that but first let's inspect your "nonsensical" composition.

About evaluation.

In general categories morphisms are just abstract arrows, not functions, and evaluation of them doesn't generally make sense because not every object can be conceived as a bag of something, e.g. points. The philosophy of category theory is exactly this: ignore what's inside, the inner structure of things, and solely observe how your things interact with each other.

There are some very special categories where objects are indeed made of points, e.g. the category of topological spaces, of vector spaces, of abelian groups or the category of bare sets: with this I mean that some categories have among all the objects a special "point object" \( * \).

In these particular categories a morphism from this objectified abstract point to an arbitrary object \( X \) can be thought as (a choice of) a point \( x \) in \( X \)

\( *\overset{x}{\longrightarrow}X \)

therefore defining the set points of \( X \) to be the (hom-)set of arrows

\( {\rm Points}(X):={\rm Hom}(*,X)=\{x\,|\,*\overset{x}{\longrightarrow}X\} \)

For example, for a bare set \( X \) we have \( X^1\simeq X \) where \( 1 \) is a singleton; the set of group homomorphisms from the group of integers to \( G \) is in bijection with the set of group elements of \( G \), i.e. \( {\rm Hom}_{\rm Grp}({\mathbb Z},G)\simeq G \); the linear applications from a field \( \mathbb K \), seen as a vector space, to a \( \mathbb K \)-vector space \( V \) are in bijection with vectors of \( V \), i.e. \( {\rm Hom}_{\rm Vec}({\mathbb K},V)\simeq V \)

and all of this without actually being able to look inside our objects nor knowing what set membership is!

In the case you make, given an abstract arrow \( X\overset{f}{\to}Y \) and a point \( *\overset{x}{\to}X \) we can evaluate \( f \) at \( x \) composing the two and producing a new point \( *\overset{y}{\to}Y \)

Here the TeXed post in pdf+a mini guide on categories.

(2021 02 02) Composition bullet notation and the general role of categories - the softest introduction ever made.pdf (Size: 585.27 KB / Downloads: 412)

MSE MphLee

Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)

S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)