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Alex Kreitzberg (Apr 06 2025 at 19:28):The space of steady state solutions of linear differential equations are Vector spaces, Hilbert spaces, etc.
Is there some analog in category theory of "specifications or conditions" determining a category, rather than just a vector space?
David Egolf (Apr 06 2025 at 20:42):I recently learned that the extensions of by , where is a group and is an abelian group, organize to form a groupoid.
So this is an example of situation where the solutions to some problem organize to form a category of some kind.
David Egolf (Apr 06 2025 at 20:45):But probably you are mostly interested in examples relating to differential equations!
David Egolf (Apr 06 2025 at 20:46):I suppose one could also pose a question: What are all the functors ? Then the answers to that question organize to form a category.
David Egolf (Apr 06 2025 at 20:48):To try and get a category of solutions to a differential equation, I think you'd want some kind of notion of morphism between solutions of a given differential equation.
(But I'm not sure what would work well for that!)
Alex Kreitzberg (Apr 06 2025 at 20:50):Examples like you have are good, I'm not sure what the appropriate generalizations should be. Algebraic equations specifying functions is the same sort of thing I'm curious about as well.
Just maybe category theory has its own version in categorical language for example
David Egolf (Apr 06 2025 at 20:54):You could imagine a functor where we think of the objects of as coming in different "kinds" depending on the isomorphism class of their image under . We could then ask: which objects are of kind ? (Which objects are such that ?)
We could then take the full subcategory of containing exactly the objects of kind .
David Egolf (Apr 06 2025 at 20:55):(So then the question "What are the objects of of kind ?" would have answers that organize to form a category.)
Mike Shulman (Apr 06 2025 at 21:25):The category of algebras for a monad, or for an endofunctor, or the category of fixedpoints for an endofunctor, seem like reasonable examples to me.
Alex Kreitzberg (Apr 06 2025 at 21:38):Thank you both!
John Baez (Apr 07 2025 at 00:09):If you're interested in category theory and linear differential equations, you might (or on closer examination very possibly might not) want to learn about D-modules. The space of solutions of a system of linear partial differential equations defines a module of a certain noncommutative algebra helpfully called D. Unfortunately most of the work on this is done in the holomorphic context, which is great if you like complex analysis or algebraic geometry, but not relevant otherwise.
In this work, it's not as if a specific differential equation or a specific solution of a specific differential equation is a category. Rather, there's a 'category of D-modules', and some D-modules come from systems of linear differential equations in the manner I just sketched.
Alex Kreitzberg (Apr 08 2025 at 06:14):I'll admit I've struggled to break into algebraic geometry style thinking, but the specific recommendation to look at D-Modules might give me a foot hold.
(Also an aside here, it's somewhat ironic to me that category theory is evidently useful for topology, a very squishy subject, and evidently useful for algebraic geometry; but not evidently useful for the "happy medium" of analysis. It's an odd state of affairs)
John Baez (Apr 09 2025 at 15:55):Algebraic topology largely about discrete structures - groups like or that remain when you 'mod out' by all the squishiness. E.g. there's an interesting study of loops in topological spaces, but when we decree that two loops count as the same if there's a homotopy from one to the other, we're led to the concept of fundamental group.
They say a topologist can't tell the difference between a doughnut and a coffee cup. But an algebraic topologist can't tell the difference between either of those and a circle! And they call this thing because it's the 'walking space with fundamental group '.
John Baez (Apr 09 2025 at 16:01):So, in algebraic geometry the structures involved are naturally somewhat rigid, while in algebraic topology we achieve rigidity by modding out by floppiness.
Mike Shulman (Apr 09 2025 at 19:35):...and in homotopy theory the algebraic geometers work really hard to reintroduce some floppiness so they can mod out by it. (-:
John Baez (Apr 09 2025 at 20:54):Yes, you could say they're trying to make algebraic geometry more like algebraic topology!
Chris Grossack (she/they) (Apr 10 2025 at 06:20):Not exactly what you're looking for, but the [[Fukaya category]] of a symplectic manifold is defined using extremely heavy duty analysis! The objects in this category are lagrangian submanifolds, and arrows between lagrangians are intersection points, which doesn't sound so bad. But composition in this category comes from understanding a moduli space of maps from a disk with 3 punctures into satisfying a differential equation ("J-holomorphicity") and boundary conditions related to your lagrangians and intersection points.
Chris Grossack (she/they) (Apr 10 2025 at 06:30):Oh, and I guess I should mention these fukaya categories are extremely interesting and well studied! They're central in the statement of homological mirror symmetry, which uses ideas from physics to make predictions about how algebraic geometry on one manifold should be related to analysis on another. This is made precise by letting "algebraic geometry" on one manifold mean "derived category of sheaves" and letting "analysis" on the mirror manifold mean "fukaya category".
There's a LOT to say here, but most of it is fairly technical. It's definitely worth looking into, though, if you want analytically defined categories!