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As a contrasting point of view, the excellent graduate level introduction to algebra, Algebra Chapter 0 by Paolo Aluffi introduces category theory right from th
by danharaj 8y ago
As a contrasting point of view, the excellent graduate level introduction to algebra, Algebra Chapter 0 by Paolo Aluffi introduces category theory right from the start and it's a very good book. It just uses the concepts of a diagram and a universal construction, which have a high power-to-weight ratio.
I agree that the danker category theory is more useful for research than for teaching. In particular when you want to try to import definitions and theorems from one context to another, or you want to decompose a proof into its abstract skeleton and concrete particularities.
As an example of how "A monad is a monoid... etc." can be useful for the abstraction (i.e. library) builder, if you import this definition of a monad into the bicategory of endoprofunctors, you get arrows [1].
[1] http://www.riec.tohoku.ac.jp/~asada/papers/arrStrMnd.pdf http://www.riec.tohoku.ac.jp/~asada/papers/arrStrMnd.pdf
- throwawaymath 8y agoI 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.
- autocorrect23 8y agoWhat is a good textbook in abstract algebra for the beginner?
- umanwizard 8y agoIt depends what "beginner" means -- what's your level of sophistication? Are you a budding mathematician or just an interested programmer?
- deleted 8y ago[deleted]
- autocorrect23 8y agoAssume I know 0 math and want to learn mathematics for mathematics sake. What is the most gentle intro to algebra book?
- Koshkin 8y agoPinter's A Book of Abstract Algebra.
- smadge 8y agoMy 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.
- umanwizard 8y agoIf you know 0 math I would start with basic arithmetic (+, -, *, /) before doing anything else.
- loparal 8y agoTrue 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://truebeautyofmath.com/ https://www.amazon.com/True-Beauty-Math-Foundations/dp/1512065919 https://www.amazon.com/True-Beauty-Math-Foundations/dp/15120... https://www.amazon.com/True-Beauty-Math-Giving-Life/dp/1544830084 https://www.amazon.com/True-Beauty-Math-Giving-Life/dp/15448...
- pdpi 8y agoA good recent release is Bartosz Milewski's Category Theory for Programmers. https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/ https://bartoszmilewski.com/2014/10/28/category-theory-for-p...
- danharaj 8y agoI like Algebra by Mac Lane and Birkhoff, if by beginner you mean undergraduate.
- zozbot123 8y agoSomewhat less of a "textbook", but good at showing applications of category theory - Fong, Spivak: Seven Sketches in Compositionality https://arxiv.org/abs/1803.05316 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!)
- Koshkin 8y agoIn the order of increasing complexity and/or volume: Pinter, Fraleigh, Rotman, Dummit & Foote.
- joppy 8y agoI 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.
- danharaj 8y ago> 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 :)
- zozbot123 8y ago> "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.