Why Category Theory?

[timelets]

Why should we use Category Theory where a "normal" math would do?


My main motivation, in Kant's language, is Apodeictical: CT is fundamentally visual, therefore we can clearly demonstrate mathematical concepts to people who have trouble grasping "normal" math language. We know from psychological studies that the right visual representation improves problem understanding and solving from 10 to 100 times. Unfortunately, our standard modes of explanation and testing for intelligence are skewed toward algebraic and linguistic expressions, mostly because they are easier for multiple choice problems. As the result, we waste a powerful communication channel and we put at a disadvantage people who have high visual intelligence.

By contrast with programming, where the emphasis is on performance rather than explanation, in natural and artificial sciences the need for communicating concepts is essential to the success of a theory and its models. Furthermore, in this field of inquiry concept construction is often more important than actual computation. Therefore, adoption of CT for developing ideas should be promoted in fields where analytical results have to be aligned with the need to communicate them to lay people who may lack in math background, but have plenty of smarts and practical experience.

Comments

[ecreet]

Then why should we use category theory where set theory (with the same visual representation) would do? I.e. when your categories don't have morphisms between different objects, or have at most one morphism (thus being effectively posets)?

[timelets]

Because 1) we do have more than one morphism between objects; 2) we need functors to show relationships b/w categories, esp. adjoints.

[ecreet]

Can you give an example where you use categories to explain common notions (like here: https://timelets.dreamwidth.org/832278.html) and the categories have morphisms beyond identity morphisms?

[timelets]

That was more than a year ago! :)

As of today, I need a category to describe a technology-based business model that includes at least technology/process, consumer desires, choices, added value and trade-offs. Besides, I want to apply Cooke's work on Process Theory and he uses monoidal categories.

Could do all of it with sets? Probably.

[ecreet]

So, no example.

If you are not defining the morphisms, you are already doing all of it with sets. A category with only identity morphisms IS a set, and a functor between such categories is a map of sets. (There are no nontrivial natural transformations though, since a natural transformation is a bunch of morphisms in the target category). You can do a lot with sets! You can visually depict things with diagrams. You can impose conditions such as commutativity of those diagrams. You can take limits and colimits of diagrams, including coproducts and products. You can use those diagrams to express things like rules of algebra (aka "normal" math language). This is exactly what you describe in your post.

[timelets]

What do you mean by "defining the morphisms"? I just want to make sure I understand the issue.

[ecreet]

A (small) category consists of a set of objects, and for any two objects, a set of morphisms (maps, arrows) between them. If we have a category whose objects are, say, consumer desires, then what are the arrows between two different desires? Can you describe an associative composition of those arrows?

[timelets]

Got it. Here's a simple example. Does it make sense?

d2 = f ◦ d1
d1 = f^-1 ◦ d2

[timelets]

Actually, on my walk I realized that the diagram I posted was wrong. f is not an isomorphism. At least...

[timelets]

Disregard the earlier stuff; it was all old and mistaken. Here's a category I'm actually contemplating now. Does it make sense?

[ecreet]

Are all of the morphisms (except identities) in the category pictured here, or are you considering a category generated by those with relations listed on the left?

[ecreet]

Actually, disregard the question: it can't be all morphisms, since it misses any morphisms between A and R (and we have ue and tr). So, a category given by generators and relations. Great, this is definitely a category!

What I am questioning is not your ability to consider a category, but the usefulness of these models applied to non-mathematical notions. A category has a binary operation (composition) that needs to be associative; this restricts the options for "real life" interpretations of morphisms in a category severely.

[timelets]

First, I really appreciate your comments and questions. They really help me focus on both consistency and usefulness of the model. Frankly. putting together a category is still a bit of challenge; mapping it to a real life situation is even a greater challenge. In short, you are asking all the right questions and I should've been asking them myself more often. And, overall, I agree with your point that most of it can be done in sets. For now.

Second, with regard to usefulness, just this summer we applied/tested this approach, albeit informally, to help several startups to evaluate and rework their business models. You'll be surprised how many questions and insights these little formulas te=1A and ur =IR generate in a conversation between technologists and investors. Of course it's a small sample, but I'm encouraged with the early results.

Third, I have a feeling that composition of arrows can be translated into real life using the concept of Point of View. For example, the process has to commute from points of view of J, A, H and R. Still working on it, though.

[ecreet]

1) Glad to be of help.

2) I'll take your word for it: me, I am a greenhouse flower, never having seen an investor in my life :).

[timelets]

Sometimes it's useful to think like an investor because you do invest your efforts into something that may or may not pay off, even though your capital at risk and returns are not necessarily financial.