Category Theory

What does Veeky Forums think of Category Theory? I started getting into the subject few days ago and really liked it. Kind of a sweet and gentle. Like those subjects in HS that didn't really matter but professor tried really hard and you felt sorry for it. Same thing here, but without pitty

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bartoszmilewski.com/2016/01/21/tambara-modules
journals.plos.org/plosone/article?id=10.1371/journal.pone.0070585
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Is there more to category theory than babby’s first composition?

Exactly what I said. It looks naive at first. Some things that i didn't bother understanding in algebra, became much simpler now.

Useless. Set Theory is already perfect.

CS folks (well, some of them) seem to like it: bartoszmilewski.com/2016/01/21/tambara-modules

Yes. If you pursue pure math you end up implicitly talking about categories all the time. It's not un-common for people to take a phenomenon which is known in a few specific categories, abstract some of the qualities of these categories into some formal structure, and then produce a proof which works in EVERY category with this formal structure. Then you don't have to repeat the whole proof in another category, you just need to show it has that structure .

On top of this, categories get used extensively as just an extremely nice way to organize information, for example cobordism categories in QFT, constructing classifying spaces of groups, replacing the "fundamental group" of a space with a "fundamental groupoid" that doesn't require you to choose basepoints, etc. I'm mostly only familiar with topological and other pure math applications, but I have a feeling they have applications to computer science as well.

>what's a large category

I mean, not to mention that algebraic topology is "basically just" the study functors out of the homotopy-category of spaces

maclane is shit

literally all of those books except borceux are shit.

Mac Lane is clear, thorough, and better prepares you for working in a general category theoretic setting which may not be concrete.

That first book doesn't even get to functor categories and adjunction until over halfway through the book. Doesn't cover Kan extensions, pullback functors, exact functors, regular categories, coherent categories, tons of other fundamental shit. Fuck, it doesn't even look like it covers full or faithful functors or equivalence of categories.

or like even monads what the hell???

What do you guys think of the Joy of Cats book?

>What is Russel's paradox and how come category theory is able to avoid it with one weird trick?

journals.plos.org/plosone/article?id=10.1371/journal.pone.0070585

>What is Russel's paradox

No, that doesn't exist.

Sets are the perfect mathematical object.

One thing I've always wondered with presentations of category theory that rely on classes is this: How is it possible to define a category where the objects form a class? If a category is defined as a six-tuple (Ob, Mor, id, comp, src, tgt), and Ob is class, then this six tuple is not well-defined because classes cannot be members of anything and tuples are defined as special kinds of classes.

>healthy at every size

>we start abstract and we stay abstract

It's how you may take notes, but not for educational purposes

Really love the poem at the beginning, though.

Second-order logic.

small sets

I think of highly abstract categories as a sort of "proxy" for logical statements rather than "containers" full of things. The objects of a category aren't really that important. It's the properties of morphisms that count.

When you say something general about the category of sets, such as "Set has an initial object", what you're saying is "there is a set X such that for any other set Y, there is exactly one morphism [math] X\rightarrow Y [/math]"

Of course, there's nothing wrong with saying "a box containing every set" because the category of sets is not a set, but it can get messy

Category theory generally doesn't care about foundational issues, but IIRC, category theory can actually be used AS a foundation.

Reflection principles+universes

I think most so/sci/iopaths are ambivalent regarding category theory, and then you of course have the maybe five percent who love it and the other five percent who militantly despise it for whatever reason.

The category theory community definitely has some cancerous members, of whom I think I used to be. Category theorists are like the vegans of the math world: the minute someone mentions something passingly related to categories, the ardent category theory enthusiast has to mention something about that being a special case of algebras for some monad, or that such-and-such construction is really just a Kan extension.

As irritating as this is, they aren't wrong; a vegan definitely says some true things. The environment definitely would improve if we all became vegans, and category theory could be used to formalize most, if not all, of mathematics. However, it is not reasonable or necessary to push one's views on others in this way.

So, I try to politely inform people on categories if they ask, and if they seem interested, I try to expound more and encourage them to study it.

Asking that is like asking if there's anything more to set theory than babby's first membership predicate.