I think this is a good point. I find the language of category theory an invaluable tool in my own understanding of abstract algebra, and universal properties are a large part of that. I think that a lot of undergraduate mathematicians could benefit from seeing the universal properties of a product/coproduct, and reflecting on what this means in categories that they're comfortable working in. Universal properties are usually quite useful.

My specific gripe is with expecting that a statement like "a Lie group is a group object in the category of manifolds" is going to lead to any new theorems in the theory of Lie groups. It's just not, and you are not going to become any better at Lie groups by understanding that statement. It's good to understand at some point, because it unifies the definition of group / Lie group / algebraic group, etc, but it will really not give you anything new to work with.

> It's good to understand at some point, because it unifies the definition of group / Lie group / algebraic group, etc, but it will really not give you anything new to work with.

I think that's true at an elementary level, but on the other hand when one doesn't know what concepts they're looking for in a new area, category theory can be a machete cutting a swathe through the jungle. I think Grothendieck was right :)

> "a Lie group is a group object in the category of manifolds"

The point is of course to motivate the definition of a Lie group. Whether such strong motivation counts as "a better understanding of Lie groups" is largely a matter of semantics that there's some room to disagree over. Same goes for the categorical definition of a ring that's been mentioned at the start of this thread, or indeed the definition that OP talks about.

I love Aluffi's Chapter 0 and mentioned it in another comment. But I don't think it's appropriate as a first pass on abstract algebra, even for pure math majors. Maybe for an Honors Intro to Abstract Algebra course...otherwise, I think Aluffi is pretty clear in that while he thinks category theory should be covered early on, it's really a text for graduates and not undergrads. Abstract algebra should be covered before you get to grad school.

What is a good textbook in abstract algebra for the beginner?

Somewhat less of a "textbook", but good at showing applications of category theory - Fong, Spivak: Seven Sketches in Compositionality https://arxiv.org/abs/1803.05316

(Notably, this includes application to a "graphical" treatment of fairly-elementary linear algebra, which a different HN user is mentioning even as I write this, as a part of math where CT cannot possibly be useful and will only ever confuse students!)

It depends what "beginner" means -- what's your level of sophistication? Are you a budding mathematician or just an interested programmer?

Assume I know 0 math and want to learn mathematics for mathematics sake. What is the most gentle intro to algebra book?

True Beauty of Math Volumes I and II. These build up math from the ground up starting with sets and assume you have zero knowledge of math. What's cool is that you don't use any theorems, structures, or notation that you haven't proven or introduced previously in one of the books. I've found them to be very valuable in both understanding the topics that they cover and giving an intuitive understanding of math as process (which the all the classes I took in college tended to gloss over).

The first volume is roughly about set theory and the second volume is roughly about abstract algebra.

https://truebeautyofmath.com/

https://www.amazon.com/True-Beauty-Math-Foundations/dp/15120...

https://www.amazon.com/True-Beauty-Math-Giving-Life/dp/15448...

Bought the first book. Thanks for the recommendation!

If you know 0 math I would start with basic arithmetic (+, -, *, /) before doing anything else.

My undergraduate Abstract Algebra course used Gallian’s “Contemporary Abstract Algebra.” Unfortunately if you are truly looking for an introduction assuming no mathematics, you will need to learn high school algebra and a bit about sets and proofs first.

Pinter's A Book of Abstract Algebra.

A good recent release is Bartosz Milewski's Category Theory for Programmers.

https://bartoszmilewski.com/2014/10/28/category-theory-for-p...

In the order of increasing complexity and/or volume: Pinter, Fraleigh, Rotman, Dummit & Foote.

I like Algebra by Mac Lane and Birkhoff, if by beginner you mean undergraduate.