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And "these axioms" are indeed the axioms of a monoid operation (for multiplication), plus some sensible notion of "consistency" with the underlying abelian-grou

And "these axioms" are indeed the axioms of a monoid operation (for multiplication), plus some sensible notion of "consistency" with the underlying abelian-group structure. Which is all that the "monoid object" characterization is saying, actually. So, I disagree that the perspective is inherently unintuitive.

throwawaymath 8y ago
I think you may have a different definition of intuitive than the top level comment, or else you two are speaking to different things. Category theory is certainly "elegant" and provides a lot of insight in the connection between various algebraic structures. But I think it would be completely asinine to use category theoretic formalisms in a first course covering algebraic structures like groups, rings and fields. That level of abstraction doesn't make the distinctions between each structure more intuitive; if anything it obfuscates them and requires setting up far more machinery to explain what they even are. It would be highly inappropriate for an undergrad's first exposure. That is my impression of the top level comment's point, and I largely agree with it. If you understand what a group is, I can explain what a ring is to you in a very straightforward way that still shows you the connections between the two things. But setting up category theoretic language would take up a nontrivial portion of the course itself, and at the end you'd have a much less specific understanding of the relevant theorems on rings and groups. You'd be missing all the trees for the forest.
- zozbot123 8y ago
  Look, it's 2019. "Setting up category-theoretic language" is something that should begin in the first lesson of the first course in formal, non-K12 math, starting with the use of ETCS or a comparable "structural" set theory as the 'naïve' foundation of choice. It provides countless advantages not just in abstract algebra, but in a wide variety of other subfields of math as well.
  - throwawaymath 8y ago
    To be clear, how much category theory are you envisioning setting up in a first course on abstract algebra? Are you basically just talking about a modern treatment of morphisms in the abstract sense, or do you also mean discussing things like functors? My perspective on this is basically informed by the following: I like category theory a lot. I've chatted with Baez about this on a few occasions and insofar as I need to use algebra, I like the category theoretic formalisms that come along with it. I think if you had students with a lot of mathematical maturity but no prior exposure to abstract algebra (except maybe a rigorous treatment of linear algebra), you could accomplish what you're proposing. To your point, one of my favorite math textbooks is Aluffi's Chapter 0. You're probably familiar with it, but if not: it builds up abstract algebra in a rigorous and modern fashion alongside category theory. However, there are a few caveats here: 1. Aluffi explicitly states it's a textbook for upper level undergrads and preferably graduates. In my experience, when authors say this they actually mean it's appropriate as a year one graduate course. That's not the time to learn abstract algebra for the first time! 2. Aluffi does a great job of covering things like morphisms and categories, but the size of the book is massive. He doesn't have any nontrivial coverage of deeper category theoretic concepts like functors until much later in the book; it's probably far enough in that you couldn't reasonably cover it in a single semester course. 3. I really don't think most math majors would benefit from it. Those who are applied math majors will get questionable benefit from a slower, more comprehensive approach to algebra than a faster approach that lets them get to applications. Moreover, not all pure math majors have the mathematical maturity to approach category theory before grad school. Proofs in traditional abstract algebra are a lot less abstract than category theory, and it's easier to motivate them even if they aren't ultimately as elegant.
    - zozbot123 8y ago
      The problem with "a faster approach that lets [you] get to applications" is that it will depend a lot on what applications you have in mind. And often it's not even faster in any real sense - there's a whole lot of pointless duplication involved in tailoring things to a lower level of abstraction. It may be better to begin with a more effective explanation of what sorts of "mathematical maturity" we're actually seeking here, so that attaining it is easier in the first place for the average student. For all their supposed "unintuitiveness", category-theory-based explanations do this quite well, in a way that I haven't really seen elsewhere - I mean, this very thread is one where we're discussing software developers learning category theory and benefiting from it, so why couldn't this be also a part of mathematical training?
joppy 8y ago
The consistency with the underlying abelian group could be restated as "multiplication is bilinear, and has a unit", which is the correct perspective to take if you want to "upgrade" from the usual definition to the "monoid object in the category of abelian groups" definition. But understanding bilinearity and its connection to tensor products is the key part of changing between those definitions, and also the most useful step to understand. The final description is pretty uninteresting in contrast with universal mapping properties of a tensor product.

throwawaymath 8y ago

I think you may have a different definition of intuitive than the top level comment, or else you two are speaking to different things. Category theory is certainly "elegant" and provides a lot of insight in the connection between various algebraic structures. But I think it would be completely asinine to use category theoretic formalisms in a first course covering algebraic structures like groups, rings and fields. That level of abstraction doesn't make the distinctions between each structure more intuitive; if anything it obfuscates them and requires setting up far more machinery to explain what they even are. It would be highly inappropriate for an undergrad's first exposure. That is my impression of the top level comment's point, and I largely agree with it. If you understand what a group is, I can explain what a ring is to you in a very straightforward way that still shows you the connections between the two things. But setting up category theoretic language would take up a nontrivial portion of the course itself, and at the end you'd have a much less specific understanding of the relevant theorems on rings and groups. You'd be missing all the trees for the forest.

zozbot123 8y ago
Look, it's 2019. "Setting up category-theoretic language" is something that should begin in the first lesson of the first course in formal, non-K12 math, starting with the use of ETCS or a comparable "structural" set theory as the 'naïve' foundation of choice. It provides countless advantages not just in abstract algebra, but in a wide variety of other subfields of math as well.
- throwawaymath 8y ago
  To be clear, how much category theory are you envisioning setting up in a first course on abstract algebra? Are you basically just talking about a modern treatment of morphisms in the abstract sense, or do you also mean discussing things like functors? My perspective on this is basically informed by the following: I like category theory a lot. I've chatted with Baez about this on a few occasions and insofar as I need to use algebra, I like the category theoretic formalisms that come along with it. I think if you had students with a lot of mathematical maturity but no prior exposure to abstract algebra (except maybe a rigorous treatment of linear algebra), you could accomplish what you're proposing. To your point, one of my favorite math textbooks is Aluffi's Chapter 0. You're probably familiar with it, but if not: it builds up abstract algebra in a rigorous and modern fashion alongside category theory. However, there are a few caveats here: 1. Aluffi explicitly states it's a textbook for upper level undergrads and preferably graduates. In my experience, when authors say this they actually mean it's appropriate as a year one graduate course. That's not the time to learn abstract algebra for the first time! 2. Aluffi does a great job of covering things like morphisms and categories, but the size of the book is massive. He doesn't have any nontrivial coverage of deeper category theoretic concepts like functors until much later in the book; it's probably far enough in that you couldn't reasonably cover it in a single semester course. 3. I really don't think most math majors would benefit from it. Those who are applied math majors will get questionable benefit from a slower, more comprehensive approach to algebra than a faster approach that lets them get to applications. Moreover, not all pure math majors have the mathematical maturity to approach category theory before grad school. Proofs in traditional abstract algebra are a lot less abstract than category theory, and it's easier to motivate them even if they aren't ultimately as elegant.
  - zozbot123 8y ago
    The problem with "a faster approach that lets [you] get to applications" is that it will depend a lot on what applications you have in mind. And often it's not even faster in any real sense - there's a whole lot of pointless duplication involved in tailoring things to a lower level of abstraction. It may be better to begin with a more effective explanation of what sorts of "mathematical maturity" we're actually seeking here, so that attaining it is easier in the first place for the average student. For all their supposed "unintuitiveness", category-theory-based explanations do this quite well, in a way that I haven't really seen elsewhere - I mean, this very thread is one where we're discussing software developers learning category theory and benefiting from it, so why couldn't this be also a part of mathematical training?

joppy 8y ago

The consistency with the underlying abelian group could be restated as "multiplication is bilinear, and has a unit", which is the correct perspective to take if you want to "upgrade" from the usual definition to the "monoid object in the category of abelian groups" definition. But understanding bilinearity and its connection to tensor products is the key part of changing between those definitions, and also the most useful step to understand. The final description is pretty uninteresting in contrast with universal mapping properties of a tensor product.