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Nate Cull (.social)Mar 1, 2020

I struggle to get my head even round the beginning parts of category theory. My fundamental problem is that I don't understand why it exists.

Eg, I stumble at this introductory sentence:

https://bartoszmilewski.com/2014/11/04/category-the-essence-of-composition/

<< A category is an embarrassingly simple concept. A category consists of objects and arrows that go between them.>>

And right there I'm screaming: That's a graph. You're literally describing a graph. Why did you invent a whole bunch of new terms to describe what already exists?

Category: The Essence of Composition

I was overwhelmed by the positive response to my previous post, the Preface to Category Theory for Programmers. At the same time, it scared the heck out of me because I realized what high expectati…

Bartosz Milewski's Programming Cafe

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Nate Cull (.social)Mar 1, 2020

Also: things-with-arrows are also literally mathematical relations (and also mathematical functions if the arrows with one label from one thing are guaranteed to only go to one other thing).

So that's my big dumb first question:

We already have plenty of mathematical tools which are 'things-with-arrows'. Why did the category theorists feel that they needed their OWN set of things-with-arrows, and that they couldn't use other people's things-with-arrows?

h/t @charlag for the link

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Starbright Electric Bolt Mar 1, 2020

@natecull @charlag You are confusing abstraction levels. It's like looking at the axioms for a commutative group, saying "a bunch of objects with addition and an identity element? That's literally just the integers.", and wondering why the integers weren't good enough for group theorists and they needed to make up their OWN integers.
An arrow can be a function or relation between sets, but the whole point is not to assume that.

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Nate Cull (.social)Mar 1, 2020

@starbright @charlag

<< An arrow can be a function or relation between sets, but the whole point is not to assume that. >>

Can you explain to me how "something which can be represented in language, or in a computer database, as a network of discrete symbols linked to each other" could possibly NOT be a relation or a graph?

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Nate Cull (.social)Mar 1, 2020

@starbright @charlag

Eg: at what point does the literal graph representing a category stop being a graph and start being that category?

Does a category only exist when a mathematician is thinking about it?

Can a category exist as a representation in a computer, or is part of the idea of categories that they can't ever be represented?

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Starbright Electric Bolt

@natecull @charlag
1. Never; describing a category requires giving more information than is present in a description of its underlying graph.
2. No more or less than any other mathematical object.
3. Categories can be represented. A category is no more or less than two sets and four functions between products of those sets. You can represent functions either as a lookup table or as code that computes that function.