I've been learning about all these crazy structures and I'm trying to understand not just each one, but how it ties and connects to all the other ones I know.

Inspired by the level shifting at the end of chapter one, I [posed a question](https://forum.azimuthproject.org/discussion/comment/19563/#Comment_19563) on whether it's possible to define a type of preorder where the objects are arbitrary algebraic structures and \\( \leq \\) is inclusion.

So if a structure B is structure A + some extra properties, then \\( A \leq B \\) ("A is less defined than B").

I drew a part of the Hasse diagram of such a preorder:

![](https://image.ibb.co/cptDx8/Screenshot_20180622_085940.png)

It mainly consists of things talked about in the first two and a half chapters (things I know so far). First of all, does this look correct?

I'm asking because this closely corresponds to my internal view of the topic - when I'm learning what these abstract structures are - I'm internally arranging them in a preorder! It's only after so many years I realized that this _thing_ I'm internally arranging in my head corresponds to a preorder. I guess this is one of the wonders of CT - it allows me to more clearly communicate what I have in my head.

Now, some of this structure is captured linguistically, by adding various prefixes to a word. But I feel that language doesn't properly capture all the intricacies, as is for the example of the unital commutative quantale. [Just knowing the word](https://www.fs.blog/2015/01/richard-feynman-knowing-something/) quantale doesn't tell me how it's connected to a preorder.

But seeing it as a part of a larger diagram immediately tells me many things! At a first glance, I can tell that it's a symmetric monoidal closed preorder with _some extra structure_. I can even fill in the blanks in some cases with minimal effort just by realizing some part of a square is missing. I think some famous mathematician said (Tao, perhaps?) that most of his papers were just completing the hypercube in this sort of way (although I can't find the reference).

---

So my question is, is this sort of diagrammatic reasoning useful? Does this scale as I add more objects?

I assume it would be like Fredrick commented to my question; perhaps it'd become unmanageable, where each structure could be factorized by _all_ the axioms that apply to it.

Googling indeed yields [many](https://commons.wikimedia.org/wiki/File:Algebraic_structures.png) [results](https://en.wikipedia.org/wiki/Map_of_lattices#/media/File:Lattice_v4.png) that are very cluttered!

But then again, all these pictures present a sort of a flat, unnested hierarchy.

Could _smart nesting_ somehow alleviate the problem? What if we tried to _define_ our structures in such that the resulting diagram becomes nested and self-referential? And to try to answer that question, aren't we trying to do _exactly that_ with category theory?

As far as I understand, with all the self-referential objects in CT (Category of categories, preorder of preorder structures...) one of the things we're trying to do is to find the 'best' way to define things such that there are no leaky abstractions and that theorems naturally follow.

To illustrate what I mean: I realized I have some redundancy in my previous graph and this is the improved version:

![](https://image.ibb.co/bz7Z4o/Screenshot_20180622_090212.png)

This notation follows closely Seven sketches, where the inside of the boxes represents objects of certain categories and dotted arrows represent functors, or in this case, monoidal maps. But those categories themselves can be arranged in a preorder!

So this shows us that we have several preorders (one of which contains the object preorder itself) between which there is a preorder structure! We can say that \\( \mathcal{F} \leq \mathcal{M} \\) (the entire F is less defined than entire M)

Now, I'm aware that this diagram can be further improved, but I think I've managed to get the point across.

I'd love to be able to see more of this diagram; to have a 'world map' of sorts where I can locate myself and observe both the nearby landscape as well as the distant mountain ranges.

Is this sort of approach feasible?

Inspired by the level shifting at the end of chapter one, I [posed a question](https://forum.azimuthproject.org/discussion/comment/19563/#Comment_19563) on whether it's possible to define a type of preorder where the objects are arbitrary algebraic structures and \\( \leq \\) is inclusion.

So if a structure B is structure A + some extra properties, then \\( A \leq B \\) ("A is less defined than B").

I drew a part of the Hasse diagram of such a preorder:

![](https://image.ibb.co/cptDx8/Screenshot_20180622_085940.png)

It mainly consists of things talked about in the first two and a half chapters (things I know so far). First of all, does this look correct?

I'm asking because this closely corresponds to my internal view of the topic - when I'm learning what these abstract structures are - I'm internally arranging them in a preorder! It's only after so many years I realized that this _thing_ I'm internally arranging in my head corresponds to a preorder. I guess this is one of the wonders of CT - it allows me to more clearly communicate what I have in my head.

Now, some of this structure is captured linguistically, by adding various prefixes to a word. But I feel that language doesn't properly capture all the intricacies, as is for the example of the unital commutative quantale. [Just knowing the word](https://www.fs.blog/2015/01/richard-feynman-knowing-something/) quantale doesn't tell me how it's connected to a preorder.

But seeing it as a part of a larger diagram immediately tells me many things! At a first glance, I can tell that it's a symmetric monoidal closed preorder with _some extra structure_. I can even fill in the blanks in some cases with minimal effort just by realizing some part of a square is missing. I think some famous mathematician said (Tao, perhaps?) that most of his papers were just completing the hypercube in this sort of way (although I can't find the reference).

---

So my question is, is this sort of diagrammatic reasoning useful? Does this scale as I add more objects?

I assume it would be like Fredrick commented to my question; perhaps it'd become unmanageable, where each structure could be factorized by _all_ the axioms that apply to it.

Googling indeed yields [many](https://commons.wikimedia.org/wiki/File:Algebraic_structures.png) [results](https://en.wikipedia.org/wiki/Map_of_lattices#/media/File:Lattice_v4.png) that are very cluttered!

But then again, all these pictures present a sort of a flat, unnested hierarchy.

Could _smart nesting_ somehow alleviate the problem? What if we tried to _define_ our structures in such that the resulting diagram becomes nested and self-referential? And to try to answer that question, aren't we trying to do _exactly that_ with category theory?

As far as I understand, with all the self-referential objects in CT (Category of categories, preorder of preorder structures...) one of the things we're trying to do is to find the 'best' way to define things such that there are no leaky abstractions and that theorems naturally follow.

To illustrate what I mean: I realized I have some redundancy in my previous graph and this is the improved version:

![](https://image.ibb.co/bz7Z4o/Screenshot_20180622_090212.png)

This notation follows closely Seven sketches, where the inside of the boxes represents objects of certain categories and dotted arrows represent functors, or in this case, monoidal maps. But those categories themselves can be arranged in a preorder!

So this shows us that we have several preorders (one of which contains the object preorder itself) between which there is a preorder structure! We can say that \\( \mathcal{F} \leq \mathcal{M} \\) (the entire F is less defined than entire M)

Now, I'm aware that this diagram can be further improved, but I think I've managed to get the point across.

I'd love to be able to see more of this diagram; to have a 'world map' of sorts where I can locate myself and observe both the nearby landscape as well as the distant mountain ranges.

Is this sort of approach feasible?