@ManuelRivera It's not something precise in my head, especially once everything is $\infty$-fied. I seem to want "an $\infty$-category" to qualify as "a piece of algebraic data", but on the other hand I still seem to want to look at an $\infty$-groupoid as some kind of "space". So ultimately I think it boils down to a matter of perspective.

But I can try to explain the example where I feel things clashing a bit better. Goodwillie calculus runs on an analogy where you think of your category $C$ as being a sort of space $X$. Lurie spells it out nicely in the beginning of Ch.6 of Higher Algebra. A point of $X$ is an object of $C$, and so forth.

A functor between categories $C \to D$ is thought of as a map of spaces $X \to Y$. So it's a "covariantly functorial" analogy.

At the same time, if $C$ happens to be a topos, then topos theory invites me to think of $C$ as the category of sheaves on some space $Z$. In this analogy, an object of $C$ does not correspond to a point of $Z$. Rather, it corresponds to an "etale space" over $Z$.

So maybe the "covariant" vs. "contravariant" business isn't the most important thing to say -- the main thing is that I'm not sure how to reconcile these two analogies.

Maybe the thing to accept is that if $C$ is a topos, then the "space" $X$ considered in Goodwillie calculus is simply not the same space as the space $Z$ considered in topos theory.

For example, when $C = Gpd_\infty$, then Goodwillie calculus sees $C$ as a space $X$ which has a point for each $\infty$-groupoid, whereas topos theory sees $C$ as (sheaves on) the one-point space. It's simply a fact that $C$ can be thought of as "being" a space in two incompatible ways.

(One thing to say is that all this really doesn't have much to do with Goodwillie calculus per se, but just the analogy guiding Goodwillie calculus, which I think is much more primitive -- I'd summarize the analogy as "think of a category as the directed fundamental groupoid of a space")

Here's a question: if $X$ is a topological space, and $Et(X)$ is the category of etale spaces over $X$, then what is the relationship between $X$ and $Et(X)$? Do they have the same homotopy type, for instance?

@TimCampion The category of etale spaces over X has a terminal object, so has a contractible classifying space.

I'd be really interested in hearing some deeper answers to your question, though, I don't feel I have built any intuition about this one way or another.

@TimCampion I got the impression that the space we consider in topos theory is merely a topological space, while in Goodwillie calculus it is a smooth manifold. Then the relation between the two reminds me of the Riemann Hilbert correspondence between vector bundles with flat connection and locally constant sheaves. Maybe that's the reason the "0-stage" of both looks similar? this is of course as imprecise as the question, but maybe it helps somehow for the intuition anyways?

@ManuelRivera ah yes, now it is coming back to me. i also remember why i wasn't able to file it away anywhere in my head: what does it "really mean" to remember an $E_\infty$-coalgebra with respect to this stronger notion of weak equivalence? am i remembering correctly that this is defined so that the bar/cobar quillen adjunction is a quillen equivalence?

@ManuelRivera i think this paper of dyckerhoff--jasso--walde gives a homotopical generalization of reflection functors: 1901.06993. in fact, when the arxiv updates in a few hours, there will be a further generalization available at 1910.14602! (this is what my collaborators and i have just finished spending a lotttttt of time on.)

in short, you can reflect composites of morphisms (or more generally morphisms organized by a poset), and when you do so you end up with a sort of abstract form of Verdier duality. more precisely, the category of sheaves on a stratified topological space inherits a stratification, and the equivalence of categories guaranteed by Verdier duality respects the stratifications up to passing through this reflection operation.

as a neat aside, this reflection also gives "half" of lurie's dold--kan correspondence (it equates chain complexes and filtered objects, but not simplicial objects).

what is the role of non-stable semiadditive categories? are there any specifically interesting ones that you might care about, or is this just a natural setting in which to study these notions of height and ambidexterity?

feel free to remain silent on this, but can you imagine actually proving redshift through these means? or maybe here's a better question: presumably in any case that would be hard; assuming so, what is the essential difficulty? is it actually fitting algebraic K-theory into this context?

lastly, what is the significance of allen's counterexample? can you quantify the difference between these two notions of height? perhaps it's too soon to ask, since it sounds like this is all very recent.

@dhy thanks for sharing your perspective, i'd love to understand this better. can you explain what you said in more detail? what is the space (stack?) of representations, and what does the notation -v^TAv mean? (what is v?) in terms of functor-of-points, i'd guess this stack takes a commutative ring R to the groupoid of functors $Q \to Mod_R$?

@dhy what is the hopf algebra structure on this K^0?

and what are the "positive" elements of U of the borel, from this point of view?

@AaronMazel-Gee Yes under a conilpotency condition on the coalgebra side the adjunction is a quillen equivalence between coassociative coalgebras under these cobar-quasi-isomorphisms and associative algebras under quasi-isomorphism.

@AaronMazel-Gee In terms of what you are remembering, well algebraically by definition is the quasi-isomorphism type of the dg associative algebra of the cobar construction of the underlying coassociative structure. The key observation I guess is that the cobar construction of the dg coassociative coalgebraa of singular chains on a path-connected pointed space is naturally quasi-iso to the dg associative algebra of chains on the based loop space.

This last statement was proven by Adams in 1956 for simply connected spaces, but it turns out that essentially the same statement holds for path-connected spaces and oddly enough it was not formulated in the literature until 60 years later.

So to answer your question: if you consider the singular chains coalgebra in this homotopy theory of coalgebras you remember the dg algebra of chains on the based loop space up to quasi-iso (and consequently the fundamental group algebra) together with the coalgebra of chains on the universal cover up to quasi-iso. The latter can also be obtained algebraically from coalgebras under cobar-quasi-iso.

and of course, the conjecture (which we are "close" to proving) is that the E-infinity coalgebra of integral singular chains under cobar-quasi-iso completely determines homotopy types :)