see, if i knew ag, and also read the 487-page phd thesis about the higher category theory underlying this system, i might be able to answer that 😭
Conversation
1
4
What would AG add to this story - does it tell us something about categories? (I know enough category theory to read the 3D diagram - now I know what an idempotent adjunction is - but I know nothing about algebraic geometry.)
1
2
omg haha this is a terrible way to learn what an idempotent adjunction is
no, i guess AG deals with geometric objects that have various kinds of singularities more than, say, differential geometry does, so i might have a better grasp of the significance of singularities
2
3
tldr because diff. geo. is so "floppy" there's basically no hope of being to classify all the ways things can degenerate.
Since alg. varieties are described by a finite amount of data, their singularities are approachable, and you can plausibly turn everything into combinatorics
2
5
My intuition about convex analysis is similar - singularities in convex functions are very well behaved. E.g. in 1d a convex function has to be differentiable except at a countable no. of points, in 2d it can also fail to be differentiable along possibly branching lines, etc.
1
2
i think thats a little different tho bc a convex function is still an infinitary object—it's not like how a variety can be described by a couple of elements of a countable set
1
2
WAIT IM A DUMBASS, R[x] isnt fucking countable LOLLLL
it just feels like it should be, right
3
3
it's countable when R = Z
1
4
okay this is prolly gonna get me canceled but
R is a not-necessarily commutative ring. If you have a commutative ring you have to call it A
5
1
7
Replying to
you’re right and you should say it