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3
votes
Reflection principle vs universes
In response to the edit which nails things down to a formal system involving cardinals $\kappa_{-1} < \kappa_0 < \kappa_{1/2} < \kappa_1$:
I'm going to go out on a perhaps more ill-advised limb and pr …
3
votes
Generalizations of tangent $\infty$-topos
Most directly, this fact generalizes to the whole Goodwillie tower, which are certainly "constructed in a similar way".
I don't think it generalizes to all algebraic theories.
However, I think one c …
9
votes
A "universally non Hypercomplete" $\infty$-topos via Goodwillie calculus?
To summarize the discussion in the comments:
$Fun(\mathsf{FinTop},\mathsf{Top})$ classifies objects
(Proof: since a left adjoint functor $Fun(\mathsf{FinTop},\mathsf{Top}) \to \mathcal E$ correspon …
4
votes
Is every conservative, left exact left adjoint comonadic, $\infty$-categorically?
The answer is no. That is, a conservative, exact left adjoint need not be comonadic, $\infty$-categorically.
For a counterexample, recall Thm 6.7 of Bousfield's Localization of spectra with respect to …
3
votes
What's an example of an $\infty$-topos not equivalent to sheaves on a Grothendieck site?
Here is a conditional answer to the question. Consider the following
Hypothesis: If $\mathcal Y$ is a sheaf $\infty$-topos, then for any set of objects $Y_0 \subset \mathcal Y$, there exists a small, …
6
votes
Accepted
Does the concept of a $\infty$-category have a natural definition in the $\infty$-world?
Homotopy type theory (HoTT) gives a natural internal language for studying $\infty$-groupoids. Riehl and Shulman give an extension of HoTT which gives an analogous internal language for studying $\inf …
5
votes
Commutative rings : Topoi = Fields :?
I've played a little bit with this question in the past, and I don't have anything firm, but here are some thoughts:
It seems to me that the characteristic properties of fields have a lot to do with …
10
votes
What's an example of an $\infty$-topos not equivalent to sheaves on a Grothendieck site?
Not an answer -- the question is very much open! But I think it's worth compiling together some of the observations made in the comments (this answer is community wiki; feel free to add, correct, chan …
7
votes
Comonadicity of spaces over spectra?
Let me record this as an answer: Blomquist and Harper show that $\Sigma^\infty$ is comonadic on simply-connected spaces.
It occurs to me that if your ultimate goal is to understand unstable homotopy …
3
votes
When is a stable $\infty$-category the stabilization of an $\infty$-topos?
Here is another sort of constraint. I'll write $Sp(\mathcal C)$ instead of $Stab(\mathcal C)$.
Claim: If $\mathcal A \simeq Sp(\mathcal X)$ for a nontrivial [1] $\infty$-topos $\mathcal X$, then for a …