1. ## Some basic questions about category theory

I currently want to use basic notions from category theory for a graphic design project, but I don't just want to assume that my comprehension is any good. I want to be told if anything I have in mind violates category theory, and if so if it can be fixed "while staying close to the spirit."

The main reference I'm drawing from is
Topoi by Goldblatt where an axiomatic definition of a category is given across pages 24 and 25.

So what I want to say first off is let there be a category
Dust, comprising of just one Dust-object D and four Dust-arrows u, d, l, r (standing for up, down, left, right).

Already at this point I've started to worry that I can't have that many arrows for one object, but then I saw a little later in the book a diagram with more than one arrow to just one object:

So I'm assuming so far so good, no violations to category theory so far – maybe.

Then what I want to do, in a presently undetermined stylized way, is display the "categorical tip of the iceberg" of
Dust, like so (which I think you'll have to click on to enlarge as it's not appearing to me in its original size in the post):

All of my worries rest on whether this violates category theory, along with a few background things I'll want to say (or not want to have to say) when presenting this figure (or some stylized version of it).

In particular, firstly, I haven't noticed the practice of applying two of the same arrows going in opposite directions through the same object instantiations (such as for example how in the upper-left I have u: DD applied twice "going in opposite directions between the same two Ds"). I worry that I can't do this, but I've justified it because it doesn't seem to violate the axioms and because we're talking about domains and codomains and not about domains and ranges (...I'm just saying that's how I justify it at the moment in my relative naivety, not saying that it's necessarily of knowledgeable and lucid thinking).

At least when it comes to this potential violation, I have one potential workaround in mind, and that would be to depict – in 2 dimensions still – that each "D" is on top of another "D," while letting the arrows remain next to each other more or less as they currently appear. But of course I'd want to know if this "stacking" can't be done, either, for whatever reason (i.e., can't be done according to category theory, leaving aside the issue of doing it graphically). Incidentally, if I can't have, at the very least, one and only one "major" object at five and only five "points," these four arrows, and these four arrows composing just like they are here, then that'll truly count as one disaster toward this project.

So the first criterion is for that figure to be able to stand alone and to let us also be able to say that it's like the tip of an iceberg. The next criterion is to let us actually depict the iceberg in its entirety, on some occasions, and any random part of the iceberg, on other occasions, and to do this sometimes by extending arrows from from this initial "tip" (especially when we're showing the whole iceberg, presumably in 3D, which will have to account for all possible compositions, 54 according to my last count) and also sometimes by taking any part whatsoever of the iceberg in isolation and visually representing it in different (but valid) ways.

The final criterion is that we can be somewhat vague in a certain qualified sense about the category's one object and its four arrows.

Suppose we stick with the category name of "Dust" – and we may not – we want to essentially suggest that there's not only one correct way to conceive of it metaphorically, just so long as the conception is rich enough that if the "operations" were to be a little more complicated than up, down, left, right, a lot of bunches of "dust particles" can be operated on "over a long time" and there'd probably still be orders of magnitude more left untouched.

That leaves the nature of the arrows. While presuming the minimum that they behave like functions, such as in a sense that "for every subset of D one and only one operation could've resulted [by the given function]," we want to be able to say something analogous to that the arrows represent – instead of up, down, left, right – dance (with), jump (with), run (with), walk (with), and to be able to let these go largely undefined by us and to let any "user" define them for themselves in such a way that there really are no restrictions on how exactly they can (functionally) define/perform any of those actions for any given subset of D.

So my big hope is that someone sufficiently knowledgeable "can see the sentiment" and help me understand where I’m either dead wrong and hopeless or (preferably) dead wrong with hope.

2. ## Re: Some basic questions about category theory

Here is one problem I see: the way you've drawn the diagram, you appear to be claiming that $uu = 1_D$, when if I understand your analogy correctly, you actually want $ud = du = 1_D$ (you should have opposing pairs $u/d$, and $l/r$).

There is no inherent problem with having several arrows from 1 object to itself, any monoid or group can be regarded as such an obect (the "elements" of the group/monoid become the "arrows", and the composition of arrows is the binary operation between elements). An arrow may mean "do something" and the operation might be "concatenation" so that "do $b$, then $a$, can be regarded as the "composition" $ab$:

$D \stackrel{b}{\to} D\stackrel{a}{\to} D$

3. ## Re: Some basic questions about category theory

Deveno, I realized a bit too late that I forgot to mention that at least in the "tip of the iceberg" part of the category I wasn't too concerned with depicting an identity arrow. I figured that they could coherently just be a part of the iceberg's sub-tip somewhere, part of the background story so to speak. It's my current understanding that, for example, $\displaystyle u \circ d \neq d\circ u$ and I really was attempting to distinguish them by

and .

Your characterization of my arrows-in-sequence as concatenation is precisely right.

And here's an indication of what I mean about iceberg tip and sub-tip, and about being able to show random parts of the category on different occasions while ignoring other parts, etc.:

I'm not saying that I feel that my understanding is necessarily complete for the requirements of the graphic design project (which I've tried to indicate the best I can), only that this is still where my understanding is at.