I didn’t read all the way through (I stopped reading midway through the extended airport example), so forgive me if this is answered already; if you say the answer’s in there, I’ll go back and reread. But, in case it’s not, my question is: what would I gain from using category theory for problems like this, instead of graph theory (on which there already exists a vast literature)?
I think, rather than “category theory is about paths in graphs”, it would be more reasonable to say that category theory is about paths in graphs up to equivalence, and in particular about properties of paths which depend on their relations to other paths (more than on their relationship to the vertices)*. If your problem is most usefully conceptualized as a question about paths (finding the shortest path between two vertices, or counting paths, or something in that genre, you should definitely look to the graph theory literature instead)
* I realize this is totally incomprehensible, and doesn’t make the case that there are any interesting problems like this. I’m not trying to argue that category theory is useful, just clarifying that your intuition that it’s not useful for problems that look like these examples is right.
Great question. I don’t think I answer it outright, but the section on natural transformations should at least offer some intuition for the kind of questions which category theory looks at, but which graph theory doesn’t really look at. That doesn’t answer the question of what you’d gain, and frankly, I have yet to see a really compelling answer to that question myself—category theory has an awful lot of definitions, but doesn’t seem to actually do much with them. But I’m still pretty new to this, so I’m holding out hope.
What seems the advantage to me is that category theory lets you account for more stuff within the formalisms.
Case in point, I did my dissertation in graph theory, and what I can tell you from proving hundreds of statements about graphs is that most of the theory of how and why graphs behave certain ways exists outside what can be captured formally by the theory. This is often what frustrates people about graph theory and all of combinatorics: the proofs come generally not by piling lots of previous results but by seeing through to some fundamental truth about what is going on with the collection (“category”) of graphs you are interested in and then using one of a handful of proof tricks (“functors”) to transform one thing into another to get your proof.
Category theory provides a theoretical structure that fits many (all) fields of mathematics. Some non-trivial insights from it are that:
The notion of ‘invertible’ depends on the context (specifically, it needs to let you recover the identity path, which need not be a set-theoretic identity function. I think this is what Eigil is saying above. As an example, consider measure theory where we introduce ‘equal up to sets of measure 0’).
The properties of an object are encoded in the maps we allow to/from the object (as a weak example, a topology induced by a collection of maps. As a strong example: Yoneda’s Lemma).
Some properties or constructions are universal in mathematics (direct products, direct limits, inverse limits, initial objects, final objects), and studying these in a general setting provides a lot of insight in actual applications.
Also personally I’ve had great mileage out of noticing when my wild ideas could not be cast in category-theoretic form, which is a huge red flag and let me identify errors in my reasoning quickly and cleanly.
The problem is that these are all just definitions; they don’t actually say anything about the things they’re defining, and the extent to which real-world systems fit these definitions is not obvious.
Ok, we can define invertibility in a way that depends on the context, but what does that actually buy us? What’s a practical application where we say “oh, this is a special kind of inverse” and thereby prove something we wouldn’t have proved otherwise?
“The properties of an object are encoded in the maps we allow to/from the object” is not a fact about the world, it is a fact about what sort of things category theory talks about. One of my big open questions is whether “maps to/from the object” are actually sufficient to describe all the properties I care about in real-world systems, like a pot of water.
Universal constructions are cute, but I have yet to see a new insight from them—i.e. a fact about a particular system which wasn’t already fairly obvious. Heck, I have yet to even see a general theorem about a universal construction other than “if it exists, then it’s unique up to isomorphism”.
These aren’t rhetorical questions, btw—I’d really appreciate meaty examples, it would make learning this stuff a lot less frustrating.
I’ll give it the good old college try, but I’m no expert on category theory by any means. I’ve always been told that a real-world application of category theory is kind of an in-joke; you’re not allowed to say they don’t exist, but nobody has any. The purpose of category theory is to replace (naive) set theory as a foundation of mathematics. It’s therefore not really surprising that this is far removed from describing properties of a pot of water, for example.
I think in light of this my previous bullet points weren’t that horrible. If you ask something like “I’m walking down the street, suddenly I see a house on fire. How does category theory help me decide what to do next?” the answer is “it does not, not even the slightest bit”. Instead it helps on a very abstract meta-level: it is there to structure the thoughts you have about fields of mathematics, which in turn will let you gain faster and deeper insight into those fields. Often (for example in algebra) even this will not translate to real-world applications, but you can then use those fields to better absorb something to make predictions about the real world. As an example I’m thinking of Category theory helping you learn Group theory which helps you learn Renormalization theory which gains you real-world insight into complicated many-particle systems, like a gas out of thermodynamic equilibrium. If you are looking for anything more direct than this I flat out don’t have an answer for you.
Finally a bit about the bullet points:
The study of non-invertible linear operators on infinite dimensional vector spaces led to notions of spectrum, and later the Gelfand-Naimark representation and study of Fredholm operators, which are some of the core deep ideas behind mathematical formulations of quantum mechanics (amongst other fields). Also the difference between [a linear map failing to be an isomorphism because it is not bijective] (related to the so-called Point spectrum) and [a linear, bijective map that is still not an isomorphism because its inverse map is not a morphism] (related to the so-called Essential spectrum) is of great importance when using numerical simulations to determine behaviour of differential equations. In particular, the point spectrum depends on the implementation and the essential spectrum does not. As far as I know this is a topic of intense debate in simulations of fluid/air flow and turbulence.
This is a fair point, and I don’t have an answer for you. I do briefly want to remark that (to my knowledge) all of mathematics, both modern and old, falls under the umbrella of category theory somehow. So the claim “I might be looking for properties that this theory cannot capture” has a high burden of proof.
Personally I run into this regularly—statements like “the topology generated by this norm is equivalent to the product topology from these underlying spaces, therefore we now have the following (universal) properties”. On the other hand I don’t have a general theorem about universal constructions for you that says something very exciting. I think universal constructions are more about pre-caching knowledge, so that you may immediately use a range of (in your words fairly obvious) results after you verify some commonly occurring conditions.
I didn’t read all the way through (I stopped reading midway through the extended airport example), so forgive me if this is answered already; if you say the answer’s in there, I’ll go back and reread. But, in case it’s not, my question is: what would I gain from using category theory for problems like this, instead of graph theory (on which there already exists a vast literature)?
I think, rather than “category theory is about paths in graphs”, it would be more reasonable to say that category theory is about paths in graphs up to equivalence, and in particular about properties of paths which depend on their relations to other paths (more than on their relationship to the vertices)*. If your problem is most usefully conceptualized as a question about paths (finding the shortest path between two vertices, or counting paths, or something in that genre, you should definitely look to the graph theory literature instead)
* I realize this is totally incomprehensible, and doesn’t make the case that there are any interesting problems like this. I’m not trying to argue that category theory is useful, just clarifying that your intuition that it’s not useful for problems that look like these examples is right.
Great question. I don’t think I answer it outright, but the section on natural transformations should at least offer some intuition for the kind of questions which category theory looks at, but which graph theory doesn’t really look at. That doesn’t answer the question of what you’d gain, and frankly, I have yet to see a really compelling answer to that question myself—category theory has an awful lot of definitions, but doesn’t seem to actually do much with them. But I’m still pretty new to this, so I’m holding out hope.
What seems the advantage to me is that category theory lets you account for more stuff within the formalisms.
Case in point, I did my dissertation in graph theory, and what I can tell you from proving hundreds of statements about graphs is that most of the theory of how and why graphs behave certain ways exists outside what can be captured formally by the theory. This is often what frustrates people about graph theory and all of combinatorics: the proofs come generally not by piling lots of previous results but by seeing through to some fundamental truth about what is going on with the collection (“category”) of graphs you are interested in and then using one of a handful of proof tricks (“functors”) to transform one thing into another to get your proof.
Category theory provides a theoretical structure that fits many (all) fields of mathematics. Some non-trivial insights from it are that:
The notion of ‘invertible’ depends on the context (specifically, it needs to let you recover the identity path, which need not be a set-theoretic identity function. I think this is what Eigil is saying above. As an example, consider measure theory where we introduce ‘equal up to sets of measure 0’).
The properties of an object are encoded in the maps we allow to/from the object (as a weak example, a topology induced by a collection of maps. As a strong example: Yoneda’s Lemma).
Some properties or constructions are universal in mathematics (direct products, direct limits, inverse limits, initial objects, final objects), and studying these in a general setting provides a lot of insight in actual applications.
Also personally I’ve had great mileage out of noticing when my wild ideas could not be cast in category-theoretic form, which is a huge red flag and let me identify errors in my reasoning quickly and cleanly.
The problem is that these are all just definitions; they don’t actually say anything about the things they’re defining, and the extent to which real-world systems fit these definitions is not obvious.
Ok, we can define invertibility in a way that depends on the context, but what does that actually buy us? What’s a practical application where we say “oh, this is a special kind of inverse” and thereby prove something we wouldn’t have proved otherwise?
“The properties of an object are encoded in the maps we allow to/from the object” is not a fact about the world, it is a fact about what sort of things category theory talks about. One of my big open questions is whether “maps to/from the object” are actually sufficient to describe all the properties I care about in real-world systems, like a pot of water.
Universal constructions are cute, but I have yet to see a new insight from them—i.e. a fact about a particular system which wasn’t already fairly obvious. Heck, I have yet to even see a general theorem about a universal construction other than “if it exists, then it’s unique up to isomorphism”.
These aren’t rhetorical questions, btw—I’d really appreciate meaty examples, it would make learning this stuff a lot less frustrating.
I’ll give it the good old college try, but I’m no expert on category theory by any means. I’ve always been told that a real-world application of category theory is kind of an in-joke; you’re not allowed to say they don’t exist, but nobody has any. The purpose of category theory is to replace (naive) set theory as a foundation of mathematics. It’s therefore not really surprising that this is far removed from describing properties of a pot of water, for example.
I think in light of this my previous bullet points weren’t that horrible. If you ask something like “I’m walking down the street, suddenly I see a house on fire. How does category theory help me decide what to do next?” the answer is “it does not, not even the slightest bit”. Instead it helps on a very abstract meta-level: it is there to structure the thoughts you have about fields of mathematics, which in turn will let you gain faster and deeper insight into those fields. Often (for example in algebra) even this will not translate to real-world applications, but you can then use those fields to better absorb something to make predictions about the real world. As an example I’m thinking of Category theory helping you learn Group theory which helps you learn Renormalization theory which gains you real-world insight into complicated many-particle systems, like a gas out of thermodynamic equilibrium. If you are looking for anything more direct than this I flat out don’t have an answer for you.
Finally a bit about the bullet points:
The study of non-invertible linear operators on infinite dimensional vector spaces led to notions of spectrum, and later the Gelfand-Naimark representation and study of Fredholm operators, which are some of the core deep ideas behind mathematical formulations of quantum mechanics (amongst other fields). Also the difference between [a linear map failing to be an isomorphism because it is not bijective] (related to the so-called Point spectrum) and [a linear, bijective map that is still not an isomorphism because its inverse map is not a morphism] (related to the so-called Essential spectrum) is of great importance when using numerical simulations to determine behaviour of differential equations. In particular, the point spectrum depends on the implementation and the essential spectrum does not. As far as I know this is a topic of intense debate in simulations of fluid/air flow and turbulence.
This is a fair point, and I don’t have an answer for you. I do briefly want to remark that (to my knowledge) all of mathematics, both modern and old, falls under the umbrella of category theory somehow. So the claim “I might be looking for properties that this theory cannot capture” has a high burden of proof.
Personally I run into this regularly—statements like “the topology generated by this norm is equivalent to the product topology from these underlying spaces, therefore we now have the following (universal) properties”. On the other hand I don’t have a general theorem about universal constructions for you that says something very exciting. I think universal constructions are more about pre-caching knowledge, so that you may immediately use a range of (in your words fairly obvious) results after you verify some commonly occurring conditions.