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i wanna take the machinery of tambara modules & optics and move it to places that look even less like categories than preordered sets do
let's consider smooth manifolds with boundary instead of categories
let's consider a monoid object there acting on other objects, instead of a monoidal cat acting on other categories
let's consider cobordisms instead of profunctors
what's a tambara module? what's an optic?
let's consider a monoid object there acting on other objects, instead of a monoidal cat acting on other categories
let's consider cobordisms instead of profunctors
what's a tambara module? what's an optic?
i don't know that i have a particular formal generalization in mind that yields this, nor even that I'm aware of one (although I bet the building blocks necessary for one are out there), but i was turning this over just now for a slightly more specific case and i like it a lot:
the case where the monoid doing the acting is M = ([0, ∞), +), so that an object equipped with an M-action is a smooth dynamical system.
then if we have a cobordism W btwn dynamical systems X and Y, I think it makes sense to suggest that a Tambara module structure on W shd be:
then if we have a cobordism W btwn dynamical systems X and Y, I think it makes sense to suggest that a Tambara module structure on W shd be:
an M-action on W that extends the ones on X and Y.
then given points a ∈ X, b ∈ Y and a path p in W from a to b, we get our tambara module action for m ∈ M of m . p a path from m . a to m . b by acting pointwise
having this arise from an overall action of M on W feels like it should be the, like, "dinaturality" or sth
having this arise from an overall action of M on W feels like it should be the, like, "dinaturality" or sth
RIGHT, i was just remembering: you can similarly view a Tambara module structure on a profunctor as equipping its collage category (which feels rather cobordism-y to me) with an appropriate actegory structure :3c
anyway, i guess an optic from (a ∈ X, b ∈ Y) to (s ∈ X, t ∈ Y) should give...
for any tambara module W : X ⇸ Y, it takes paths in W from a to b and gives paths in W from s to t
and i *think* if we still have a coend<->tambara style of theorem in this setup, it'd be like...
for any tambara module W : X ⇸ Y, it takes paths in W from a to b and gives paths in W from s to t
and i *think* if we still have a coend<->tambara style of theorem in this setup, it'd be like...
"every optic is given by acting pointwise on paths with a fixed m ∈ M—i.e., optics are intervals of time over which (a, b) will evolve into (s, t)"
...but my first guess would be that you can't expect to establish a global semi-rigid relationship between systems—like the notion of Tambara module i suggested—purely on the basis of a couple of points bearing a relationship...???
now im wondering about shit like
if u have a circle equipped w/ an M-action that just loops around counterclockwise
and a pair of circles, w/ the same behavior on each in the pair
then can the pair-of-pants cobordism be made a "Tambara module" in the above sense, say
if u have a circle equipped w/ an M-action that just loops around counterclockwise
and a pair of circles, w/ the same behavior on each in the pair
then can the pair-of-pants cobordism be made a "Tambara module" in the above sense, say
