Caveat: I’m a bit of an algebraic abstractologist with a not-that-deep understanding of category theory. Thus I might represent the worst of both worlds, someone defending the abstract for the abstract without concrete arguments.
My first reaction when seeing this post is that it was a great idea to give an intuitive explanation of category theory. My second reaction, when reading the introduction and the “Path in Graphs” section, was to feel like every useful part of category theory had been thrown away. After reading the whole post, I feel there are great parts (notably the extended concrete example for functors), and others with which I disagree. I will try to clarify my disagreement.
First, category theory is not about graphs. Categories are not graphs. There not even graphs with a tiny bit of structure on top. Categories are abstractions of mathematicals structures, which can be represented as graph. One can even argue that applications of category theory are simply about structuring the underlying objects and relations in such a way that what we want to prove is shown by a basic diagram. An intuition for this is that you can do category theory without using the graphs.
There are very good reason to use sets and the category of sets as primarily examples. The most obvious one is that category theory aims to provide a fondation of mathemathics in place of set theory. Thus starting with sets, which any student of modern mathematics know, is being nice to the mathematical reader. Also, pretty much any book on category theory uses the category of sets as a foil, to show that categorical notions should not be given an intuition based only on set theory and functions on sets, as this intuition breaks down in other cases. For example, the fact that an isomorphism in category theory is not just a bijection as in the category of sets—for the category of posets, isomorphisms must be order-preserving while bijections are not necessarily.
Lastly, about the applications you seem to search for category theory. I feel like you want to use category theory as a tool to solve a problem, like you apply an inequality to derive a new theorem. But as far as I know, all uses of category theory come after the fact. It is a formal and structured perspective you can take on mathematical objects and how they relate. It might give you an understanding of the underlying structure behind your approach or results, but it rarely, if ever, gives you the means to accomplish an effective goal. Even the most concrete examples like monads in Haskell come from people already committed to find a solution within category theory, not because category theory was the best tool to solve the problem at hand.
All that being said, I still think trying to boil down the main concepts of category theory is a great idea. I’m only worried that your approach throws away too much of what makes the value of category theory.
My second reaction, when reading the introduction and the “Path in Graphs” section, was to feel like every useful part of category theory had been thrown away.
I was expecting/hoping someone would say this. I’ll take the opportunity to clarify my goals here.
I expect “applied category theory” is going to be a field in its own right in the not-so-distant future. When I say e.g. “broader adoption of category theory is limited in large part by bad definitions”, that’s what I’m talking about. I expect practical applications of category theory will mostly not resemble the set-centered usage of today, and I think that getting rid of the set-centered viewpoint is one of the main bottlenecks to moving forward.
(A good analogy here might be group-theory-as-practiced-by-mathematicians vs group-theory-as-practiced-by-physicists—a.k.a. representation theory.)
I generally agree that the usage of category theory today benefits from not thinking of things as graphs, from using set as a primary example, etc. But today, all uses of category theory come after the fact. I want that to change, I’ve seen enough to think that will change, and that’s why I’m experimenting with a presentation which throws out most of the existing value.
I don’t know if I should feel good or bad for taking the bait… let’s say it make the debate progress.
From what you are saying, I think I get what you are aiming for. But I am also not sure that this issue you see is really there. Because every textbook on category theory that I read pointed in its first chapter or section that you should not use your intuition for sets in the abstract setting of category theory. Even the historical origins of category theory comes from algebraic topology, where the influence of sets and set theory is very dimmed.
That being said, maybe there is some “Cantor bias” in the modern practice of mathematics, even when the point is to replace set theory. That’s actually one aspect of the Physics, Topology, Logic and Computation: A Rosetta Stone paper mentioned in my other answer that I like a lot: the authors give different categories, and argue that the category best capturing applications in Physics and Computer Science is not the category of sets, but another behaving differently.
I think I have two questions following your comment:
First, could you expand your example of group theory in math vs group theory in physics? I know a bit a mathy group theory, and I know it is applied to various parts of physics, but I’m curious of clear cut differences between the uses.
Second, does the already existing field of Applied Category Theory (intro paper and intro book) fits your bill for applications?
I do like that Rosetta Stone paper you linked, thanks for that. And I also recently finished going through a set of applied category theory lectures based on that book you linked. That’s exactly the sort of thing which informs my intuitions about where the field is headed, although it’s also exactly the sort of thing which informs my intuition that some key foundational pieces are still missing. Problem is, these “applications” are mostly of the form “look we can formalize X in the language of category theory”… followed by not actually doing much with it. At this point, it’s not yet clear what things will be done with it, which in turn means that it’s not yet clear we’re using the right formulations. (And even just looking at applied category theory as it exists today, the definitions are definitely too unwieldy, and will drive away anyone not determined to use category theory for some reason.)
I’m the wrong person to write about the differences in how mathematicians and physicists approach group theory, but I’ll give a few general impressions. Mathematicians in group theory tend to think of groups abstractly, often only up to isomorphism. Physicists tend to think of groups as matrix groups; the representation of group elements as matrices is central. Physicists have famously little patience for the very abstract formulation of group theory often used in math; thus the appeal of more concrete matrix groups. Mathematicians often use group theory just as a language for various things, without even using any particular result—e.g. many things are defined as quotient groups. Again, physicists have no patience for this. Physicists’ use of group theory tends to involve more concrete objectives—e.g. evaluating integrals over Lie groups. Finally, physicists almost always ascribe some physical symmetry to a group; it’s not just symbols.
So your path-based approach to category theory would be analogous to the matrix-based approach of group theory in physics? That is, removing the abstraction that made us stumble into theses concepts in the first place, and keeping only what is of use for our applications?
I would like to see that. I’m not sure that your own proposition is the right one, but the idea is exciting.
Caveat: I’m a bit of an algebraic abstractologist with a not-that-deep understanding of category theory. Thus I might represent the worst of both worlds, someone defending the abstract for the abstract without concrete arguments.
My first reaction when seeing this post is that it was a great idea to give an intuitive explanation of category theory. My second reaction, when reading the introduction and the “Path in Graphs” section, was to feel like every useful part of category theory had been thrown away. After reading the whole post, I feel there are great parts (notably the extended concrete example for functors), and others with which I disagree. I will try to clarify my disagreement.
First, category theory is not about graphs. Categories are not graphs. There not even graphs with a tiny bit of structure on top. Categories are abstractions of mathematicals structures, which can be represented as graph. One can even argue that applications of category theory are simply about structuring the underlying objects and relations in such a way that what we want to prove is shown by a basic diagram. An intuition for this is that you can do category theory without using the graphs.
There are very good reason to use sets and the category of sets as primarily examples. The most obvious one is that category theory aims to provide a fondation of mathemathics in place of set theory. Thus starting with sets, which any student of modern mathematics know, is being nice to the mathematical reader. Also, pretty much any book on category theory uses the category of sets as a foil, to show that categorical notions should not be given an intuition based only on set theory and functions on sets, as this intuition breaks down in other cases. For example, the fact that an isomorphism in category theory is not just a bijection as in the category of sets—for the category of posets, isomorphisms must be order-preserving while bijections are not necessarily.
Lastly, about the applications you seem to search for category theory. I feel like you want to use category theory as a tool to solve a problem, like you apply an inequality to derive a new theorem. But as far as I know, all uses of category theory come after the fact. It is a formal and structured perspective you can take on mathematical objects and how they relate. It might give you an understanding of the underlying structure behind your approach or results, but it rarely, if ever, gives you the means to accomplish an effective goal. Even the most concrete examples like monads in Haskell come from people already committed to find a solution within category theory, not because category theory was the best tool to solve the problem at hand.
All that being said, I still think trying to boil down the main concepts of category theory is a great idea. I’m only worried that your approach throws away too much of what makes the value of category theory.
I was expecting/hoping someone would say this. I’ll take the opportunity to clarify my goals here.
I expect “applied category theory” is going to be a field in its own right in the not-so-distant future. When I say e.g. “broader adoption of category theory is limited in large part by bad definitions”, that’s what I’m talking about. I expect practical applications of category theory will mostly not resemble the set-centered usage of today, and I think that getting rid of the set-centered viewpoint is one of the main bottlenecks to moving forward.
(A good analogy here might be group-theory-as-practiced-by-mathematicians vs group-theory-as-practiced-by-physicists—a.k.a. representation theory.)
I generally agree that the usage of category theory today benefits from not thinking of things as graphs, from using set as a primary example, etc. But today, all uses of category theory come after the fact. I want that to change, I’ve seen enough to think that will change, and that’s why I’m experimenting with a presentation which throws out most of the existing value.
I don’t know if I should feel good or bad for taking the bait… let’s say it make the debate progress.
From what you are saying, I think I get what you are aiming for. But I am also not sure that this issue you see is really there. Because every textbook on category theory that I read pointed in its first chapter or section that you should not use your intuition for sets in the abstract setting of category theory. Even the historical origins of category theory comes from algebraic topology, where the influence of sets and set theory is very dimmed.
That being said, maybe there is some “Cantor bias” in the modern practice of mathematics, even when the point is to replace set theory. That’s actually one aspect of the Physics, Topology, Logic and Computation: A Rosetta Stone paper mentioned in my other answer that I like a lot: the authors give different categories, and argue that the category best capturing applications in Physics and Computer Science is not the category of sets, but another behaving differently.
I think I have two questions following your comment:
First, could you expand your example of group theory in math vs group theory in physics? I know a bit a mathy group theory, and I know it is applied to various parts of physics, but I’m curious of clear cut differences between the uses.
Second, does the already existing field of Applied Category Theory (intro paper and intro book) fits your bill for applications?
I do like that Rosetta Stone paper you linked, thanks for that. And I also recently finished going through a set of applied category theory lectures based on that book you linked. That’s exactly the sort of thing which informs my intuitions about where the field is headed, although it’s also exactly the sort of thing which informs my intuition that some key foundational pieces are still missing. Problem is, these “applications” are mostly of the form “look we can formalize X in the language of category theory”… followed by not actually doing much with it. At this point, it’s not yet clear what things will be done with it, which in turn means that it’s not yet clear we’re using the right formulations. (And even just looking at applied category theory as it exists today, the definitions are definitely too unwieldy, and will drive away anyone not determined to use category theory for some reason.)
I’m the wrong person to write about the differences in how mathematicians and physicists approach group theory, but I’ll give a few general impressions. Mathematicians in group theory tend to think of groups abstractly, often only up to isomorphism. Physicists tend to think of groups as matrix groups; the representation of group elements as matrices is central. Physicists have famously little patience for the very abstract formulation of group theory often used in math; thus the appeal of more concrete matrix groups. Mathematicians often use group theory just as a language for various things, without even using any particular result—e.g. many things are defined as quotient groups. Again, physicists have no patience for this. Physicists’ use of group theory tends to involve more concrete objectives—e.g. evaluating integrals over Lie groups. Finally, physicists almost always ascribe some physical symmetry to a group; it’s not just symbols.
So your path-based approach to category theory would be analogous to the matrix-based approach of group theory in physics? That is, removing the abstraction that made us stumble into theses concepts in the first place, and keeping only what is of use for our applications?
I would like to see that. I’m not sure that your own proposition is the right one, but the idea is exciting.
Yup, that’s basically the idea.