@AaronMazel-Gee i think there's something to be said about basically building any such thing out of this kind of functor

I'm pretty sure @DenisNardin understands this a lot better than I do.

@JonBeardsley you mean, like, descent? i suppose this is all local on BGL_1(R), after all

a/k/a the canonical presentation of a presheaf as a colimit of representables, i guess

ah i can't remember right now. i'm feeling very unsettled about all of this. i think i just found a relatively significant flaw in some of the stuff i've been (trying) to write down

@AaronMazel-Gee right, something like... the "representable" presheaves here all look like $\Omega X$ or something.

Let's see, actually I think this is pertinent to the issue I'm having...

a representable presheaf on a space Y looks like the space of paths based at the representing point, i.e. it's contractible

ermmmm, but it takes every point to $\Omega Y$ I thought?

what does, hom_Y(-,y)?

yeah

well, a clearer answer is that hom_Y(z,y) = {paths in Y from z to y}

so yes, but noncanonically

well of course.

right.

but I'm also fine assuming that our space $Y$ is connected, maybe even has a unique 0-simplex

and then this is the generator of the cat of presheaves on Y

(under colimitS)

that is, the functor $y\mapsto \Omega Y$

which corresponds to the path loop fibration

in Top_{/Y}

or $\ast\to Y$

does this mean that $y\mapsto \Sigma^\infty_+\Omega Y$ is the generator of $Fun(Y,Spectra)$?

By something along the lines of $Stab(Fun(Y,Top))\simeq Fun(Y,Spectra)$

and presumably stabilization preserves colimits?

given a point $y \in Y$, the presheaf determined by $X \to Y$ assigns $y \mapsto \lim(\{y\} \rightarrow Y \leftarrow X) =: X_y$, the fiber. any presheaf is a colimit of representables, indexed by the corresponding total space, i.e. pairs $(y,\alpha \in X_y)$, i.e. just $X$ itself. the diagram of interest takes such a point to the corresponding representable $h_y$, which is just the functor $X \to Y$. so in the end, the presheaf determined by $X \to Y$ is just

the colimit of the composite $X \to Y \xrightarrow{\mathrm{Yo}} \mathrm{PShv}(Y)$

so in that sense, if you have a colimit-preserving functor, such as apparently the thom spectrum functor (?), then its behavior is entirely determined by what happens on the constant maps

the thom spectrum functor is colimit preserving

i think that's what i meant to say

sooooo, are we in agreement then? i think?

i used this fact to prove that thom spectra are comodules

yup! i was just trying to say what i was trying to say a little more precisely

(and that comodules are closed under colimits)

i'm just effing terrible at saying these things precisely.

i think it's good practice to say things as precisely as possible

("good practice" as in "it makes you better at it" -- i'm not trying to be prescriptive here)