32
votes

### Reflection principle vs universes

I'm going to go out on a limb and suggest that the book HTT never uses anything stronger than replacement for $\Sigma_{15}$-formulas of set theory. (Here $15$ is a randomly chosen large number, and ...

27
votes

Accepted

### Reflection principle vs universes

Reflecting on Gabe's comment on my original answer, I now think what I wrote is misleading because it conflates two separate (but related) assertions:
The existence of strongly inaccessible cardinals ...

21
votes

### Reflection principle vs universes

I'd like to mention something that I think hasn't been pointed out yet. The original question began with
In set-theoretic language, one fixes some strongly inaccessible cardinal $\kappa$... This ...

20
votes

### A sheaf is a presheaf that preserves small limits

This has nothing to do with $\infty$-categories, but with the fact that we look at the full topos and not an arbitrary site of definition:
Theorem: If $T$ is a (Grothendieck) ($1$-)topos, then a "...

20
votes

Accepted

### Examples of $(\infty,1)$-topoi that are not given as sheaves on a Grothendieck topology

Marc's examples are good ones, but let me add two more (which are closely related to each other):
1) Let $\mathcal{C}$ be an accessible $\infty$-category which admits small filtered colimits, and let ...

18
votes

Accepted

### Is the $\infty$-topos $Sh(X)$ hypercomplete whenever $X$ is a CW complex?

ETA The answer is yes in general. Replace 2 below with a reference to HTT, Prop. 7.1.5.8.
Since this has been open for a while, let me give a partial answer which hopefully is already interesting: I ...

16
votes

### Reflection principle vs universes

OK, I spent much of today trying to figure this out by actually looking in some detail at HTT. It's been quite a ride; I have definitely changed my perspective multiple times in the process. Currently,...

15
votes

Accepted

### Is an ∞-topos of local homotopy dimension $\leq n$ of homotopy dimension $\leq n$?

Let $\mathcal{X}$ denote the $\infty$-topos $\mathcal{S}_{/S^1}$, whose objects are spaces $X$ with a map $X \rightarrow S^1$. Then $\mathcal{X}$ is generated under colimits by the object given by the ...

15
votes

### Reflection principle vs universes

If I understand correctly, you're after a statement of the form :
"If something was proved in HTT using universes, it can be proved without them by restricting to some $V_\kappa$ for $\kappa$ ...

15
votes

### Reflection principle vs universes

Answering this question depends strongly on exactly what you want from Higher Topos Theory, because expressing high logical strength is a different goal from expressing an aptly unified logical ...

14
votes

### A "universally non Hypercomplete" $\infty$-topos via Goodwillie calculus?

As discussed in the comments, I'm writing here the proof of the following fact:
Let $\mathscr{X}_∞$ be the ∞-topos of sheaves on $\mathrm{FinTop}^{op}$ under the atomic topology (the topology where ...

14
votes

### Examples of topoi that are not ordinary spaces

The bicategory of Grothendieck toposes is equivalent to the bicategory
of localic groupoids, with appropriately defined 1-morphisms and 2-morphisms.
"Ordinary spaces" are topological spaces, or, ...

13
votes

Accepted

### Equivalences of categories of sheaves vs categories of $\infty$-Stack

I just found an example, so I thought it would be good to post it here, but if anyone knows other examples, or a more general way to construct some I would be interested to see them as well.
This ...

13
votes

### Reflection principle vs universes

Any theorem $T$ of $\mathsf{ZFC}$ follows from a finite subset of the axioms of $\mathsf{ZFC}$ or, to keep things simple, from $\mathsf{ZFC}$ where the axiom scheme of replacement is restricted to $\...

13
votes

Accepted

### If we replace the spectrally ringed space in the definition of a spectral scheme with an arbitrary infinity-topos, what objects do we get?

Yes, they are more general. This is in fact already the case with ordinary rings. Let's call a classically-ringed $\infty$-topos which is locally the Zariski $\infty$-topos of an affine scheme an $\...

13
votes

Accepted

### Categorical equivalences vs. categories of simplices

No it is not. But there is a replacement for this. There is another model of $\infty$-categories: marked simplicial sets. If $X$ is a simplicial set with a set of marked $1$-simplices $S$, the fibrant ...

12
votes

Accepted

### Monomorphisms, epimorphisms, (co-)images and factorizations in $\infty$-categories

An $n$-monomorphism is a map $A\to B$ for which $Map(X,A)\to Map(X,B)$ has all homotopy fibers $n$-truncated, for all $X$; a space (=$\infty$-groupoid) is $n$-truncated if its homotopy groups all ...

12
votes

Accepted

### Are there continua in $\infty$-topoi?

Every contractible finite CW complex $X$ satisfies these conditions. This follows from results in Section 7.3 of HTT and Appendix A of HA: we have $Shv(X) \otimes Shv(X)=Shv(X\times X)$ since $X$ is ...

12
votes

### Examples of topoi that are not ordinary spaces

This is a translation of Dmitri Pavlov's answer into a more intrinsic, more geometric, and more elementary language. In particular, I will show that the étale topos of a positive-dimensional variety ...

12
votes

Accepted

### Cohesion relative to a pyknotic/condensed base

Let me try to cut through the jargon. One thing that confuses me are two uses of "tangent spaces" here, that I believe are quite unrelated. One is the usual notion of tangent spaces of ...

11
votes

Accepted

### Left Kan extension along Yoneda of pullback-preserving functor preserving pullbacks

If you are wiling to assume that $C$ has a terminal object $1 \in C$, which I assume is the case as you said all finite products, you can do the following:
(As it is not clear if you are interested ...

11
votes

### 2-natural operations on toposes

(For me the category of toposes is the opposite of the category of left exact left adjoint functors and natural transformations, so $Topos^{co}$ in your sense)
The functor $U$ is representable by the ...

10
votes

### Homotopy groups of spheres in a $(\infty, 1)$-topos

If it may be forgiven to resurrect a very old question, it's worth pointing out that these are not the "homotopy groups of spheres" that appear in synthetic homotopy theory / homotopy type theory. ...

10
votes

Accepted

### 2-natural operations on toposes

The "2-natural operations" $U^n \to U$ correspond to functors $\mathbf{FinSet}^n \to \mathbf{Set}$. (edit: As Simon points out, these correspond to the finitary functors $\mathbf{Set}^n \to \mathbf{...

10
votes

### Commutative rings : Topoi = Fields :?

This is a long comment. I would prefer to say that (Grothendieck) topoi are "(some) affine schemes over $\text{Spec } \text{Set}$." Here is my preferred version of the table, sprinkle $\infty$s ...

10
votes

### Reflection principle vs universes

A question that came up in the comments was about the motivation for asking the question. Let me try to address this here.
Foremost, it is about learning! As I mentioned in the original question, I ...

9
votes

### What is an Elementary "Homotopy, Model" Topos?

Since the time when Denis referred in the comments to the relevant nLab page, there has been a new proposal written up by Mike Shulman there:
An elementary $(\infty,1)$-topos is an $(\infty,1)$-...

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 $Fun(\mathsf{FinTop},\mathsf{Top}) \to \mathcal E$ ...

Community wiki

9
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, ...

Community wiki

9
votes

### Reflection principle vs universes

I would consider a conservative extension of ZFC obtained from ZFC by the adition of a constant $\alpha$ and the following axioms:
$\alpha$ is an ordinal ($Ord(\alpha)$).
The sentence $\phi\...

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