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Stream: learning: questions

Topic: shapes that compose

David Egolf (Sep 28 2022 at 22:17):

I was thinking about combining generalized elements to make a generalized element of the same shape. That is, say we have two morphisms $m,n: B \to G$ in some category. I think of these as $B$-shaped generalized elements of $G$. Sometimes, if two generalized elements are compatible, we can combine them to make another of the same shape - so we get another morphism $p: B \to G$ that is in some way the composite "$n \circ m$" $B$-shaped generalized element. For example, if $B$ is the category with two objects and a single non-identity morphism from the first object to the second, then a morphism $m: B \to G$ in $Cat$ corresponds to a morphism of $G$. Then for two "adjacent" generalized $B$-shaped elements of $G$, we can compose them using the composition of morphisms in $G$. However, for this to work, I think the shape of $B$ needs to satisfy a "two copies of this shape combine to give this shape" property.

Some shapes, when combined with themselves, make another of that shape. For example, two composable morphisms - which we draw as directed edges - when placed end to end form a third directed edge by composition. As another example, we can take two commutative squares that share an edge and "glue them along the common edge" to obtain another commutative square by taking the outer edges. We can think of the left edge of a commutative square as the "source" of the square and the right edge as the "target", and then commutative squares with compatible source and target can be composed. Maybe there's a way to do a similar thing with triangles too, although I'm worrying about assigning a "source" and "target" to each triangle, in order to be able to talk about composing them. I don't see how to glue together pentagons to yield another pentagon, though.

Which shapes (or diagrams) satisfy the condition that attaching two examples of that shape along smaller "edge" shapes yields another example of that shape?

Amar Hadzihasanovic (Sep 29 2022 at 09:45):

I don't think the question as you are asking it is well-posed because it has a trivial answer. Take any $n, m \geq 1$ and consider a diagram with $n$ source arrows and $m$ target arrows. Fix any $k \leq \min (n, m)$. Then you can

• compose two diagrams of this shape along the rightmost $k$ target arrows of the first and the leftmost $k$ source arrows of the second, which yields a composite diagram with $2n - k$ source arrows and $2m - k$ target arrows; then
• compose the leftmost $n - k + 1$ arrows of the source of the first diagram, and the rightmost $m - k + 1$ arrows of the target of the second diagram.

The result is another diagram with $n$ source arrows and $m$ target arrows.

Amar Hadzihasanovic (Sep 29 2022 at 09:48):

(You can dualise this by turning “leftmost” into “rightmost” and vice versa, and it gives you another option.)

David Egolf (Sep 29 2022 at 14:51):

Thanks for your response, Amar! I appreciate you giving my rather vague question a little thought.

I'm trying to follow your example. I don't understand what you mean by "compose the leftmost $n-k+1$ arrows of the source of the first diagram, and the rightmost $m-k+1$ arrows of the target of the second diagram". You mention two collections of arrows here, but they can't be composed pair-wise, as they have a different number of elements. Are you talking about a scenario where the source arrows of the first diagram can be composed with themselves?

David Egolf (Sep 29 2022 at 15:05):

In case it helps clarify, here's an example of where shape isn't preserved under composition.

Here's an example of where the shape isn't preserved under composition:
shape isn't preserved

In this example, we have two diagrams, each having a designated source and target. We can stick the two together if the target of one is compatible with the source of another. (They are compatible when the $t_1$ object is the same as the $s_2$ object). However, the resulting shape is different than what we started with - there are now 5 arrows in the resulting diagram instead of 3, for example).

I'm afraid my discussion above is still very vague. I guess what I'm interested in is associative ways to combine (compatible) diagrams that can each be viewed as functors from a "shape" category $S$, to yield another diagram that can also be viewed as a functor from $S$. I am most interested in ways of doing this that reduce to composition of arrows when $S$ is the category with two objects and a single non-identity morphism between them.

Nathanael Arkor (Sep 29 2022 at 16:42):

This isn't precisely an answer to your original question, but your motivation sounds very similar to that of operadic categories. I quite like these notes of Markl on the subject.

David Egolf (Sep 29 2022 at 17:08):

Nathanael Arkor said:

This isn't precisely an answer to your original question, but your motivation sounds very similar to that of operadic categories. I quite like these notes of Markl on the subject.

Thanks for the link to those notes! I was hoping someone would come along who could point out related topics :smile: .

Jorge Soto-Andrade (Sep 29 2022 at 17:58):

Hi David, Interesting question. I do not think it is an ill posed question I gave some thought to this question some years ago. Comments: your square example boils down to the well-known category whose objects are the arrows of a previous category.

Jorge Soto-Andrade (Sep 29 2022 at 18:00):

Sorry, I did not mean to send my message yet. Second comment: you have something more interesting if you take the notion of shape seriously and you recall the geometric approach to dimension, which is quite fruitful regarding *fractal * dimension.,

Jorge Soto-Andrade (Sep 29 2022 at 18:08):

Now you need to compose by trios, for instance, in the case of (equilateral) triangles, or three legged anchors, and so on. I guess that this is related to operadic categories... Thank Nathanael for the nice reference. On the other hand, it seems to me that Amar example is related to spiders or metabolic categories (biochemical reaction = spider whose sources are the reactants and whose targets are the products; they compose in a "biochemically obvious" way Food for thought ...