Talk:Category Theory/Categories

About notation for composition
Juxtaposition for function composition may be confusing

I am not Category theory expert, but the juxtaposition notation $$g f$$ instead of composition operation $$g\circ f$$ is confusing because $$(g\circ f)x = g(f x)$$ is clear to me, but what does $$gfx$$ mean? $$(gf)x$$ or $$g(fx)$$. In the usual notation, function application $$f g h x$$ means $$((f g)h)x$$

That is ok, because composition is associative $$(f\circ g)\circ h f\circ (g\circ h)$$, in category theory.

But what about higher order functions, as used in languages like Haskell, where category theory is applied. There the class like, is defined as all datatypes having defined the function where the law holds. How should this law be written using juxtaposition $$gf$$ instead of $$g\circ f$$. Without a symbol for composition operator the above law can not be clearly written: $$fmap\ f\ fmap\ g=fmap\ (fg)$$

Other symbol for composition

Some times, mainly when comparing with commutative diagrams, the relation composition symbol is a better choice. $$(f\stackrel{\circ}{{}_9}g)x=(g\circ f)x=g(fx)$$ where the $$\stackrel{\circ}{{}_9}$$ should look like a hollow semicolon, and read $$f$$ before $$g$$

With relational composition symbol it is more clear de correspondence with commuting diagrams as $$f:A\to B,g:B\to C$$ and $$f\stackrel{\circ}{{}_9}g:A\to C$$ is clearly depicted in diagrams $$A\stackrel{f}{\longrightarrow}B\stackrel{g}{\longrightarrow}C=A\ \stackrel{f\,\stackrel{\circ}{{}_9}\,g}{\longrightarrow}C$$

Some authors reverse arrows in signatures to achieve the same effect: $$g:C\leftarrow B, f:B\leftarrow A, g\circ f:C\leftarrow A$$ But is not clear as the $$f {\stackrel{\circ}{_9}}g$$ notation as the commutative diagrams are not written in this direction: $$C\stackrel{\ g}{\longleftarrow} B\stackrel{\ f}{\longleftarrow} A =C\stackrel{\ g\circ f}{\longleftarrow} A$$

Adopting the $$f\stackrel{\circ}{_9}g$$ notation

I think that the adoption of the $$f\stackrel{\circ}{_9}g$$ notation, instead of juxtaposition $$gf$$ would make this wikibook clearer, but as I mentioned earlier I am not expert in category theory and my opinion is constrained to the content of the whole wikibook on this subject in order to have a consistent notation. I do not still know it, but the authors of the present text should take it into account, as this opinion may be common to other readers with just a little experience with category theory in programming like me.