Timeline for Is the logic of the ($\infty$-?)topos of simplicial sets "contradictory up to homotopy"?
Current License: CC BY-SA 4.0
26 events
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| Oct 14 at 6:02 | comment | added | მამუკა ჯიბლაძე | What I could not add since I still do not understand it: what is the status of those maps $A\to B$ for which the induced map from the fibre to $A$ is nullhomotopic? | |
| Oct 14 at 5:59 | answer | added | მამუკა ჯიბლაძე | timeline score: 2 | |
| Oct 14 at 5:50 | vote | accept | მამუკა ჯიბლაძე | ||
| Oct 14 at 5:50 | comment | added | მამუკა ჯიბლაძე | @PeterLeFanuLumsdaine Having thought about what you wrote for a while, I came up with something additional, please have a look at my answer, does it make sense? | |
| Oct 12 at 16:31 | comment | added | Peter LeFanu Lumsdaine | @მამუკაჯიბლაძე: You’re right — my previous comment was mistaken. The resolution instead is that retract-inclusions are not generally ∞-monos; for instance, the basepoint inclusion $1 \to S^1$ is not an ∞-mono (its homotopy fibre is $\mathbb{Z}$). And it’s just ∞-monos (i.e. –1-connected maps) which will be classified by $1 \to 2$ (i.e. represented homotopically-uniquely as ∞-pullbacks thereof). | |
| Oct 12 at 16:23 | comment | added | მამუკა ჯიბლაძე | @PeterLeFanuLumsdaine This I don't understand. For $X$ connected there are only two maps to $2$, but it can have lots of retracts distinct in all possible senses, no? | |
| Oct 12 at 16:13 | comment | added | Peter LeFanu Lumsdaine | @მამუკაჯიბლაძე: any retract-inclusion can be represented as a homotopy-pullback of 1 —> 2 — not generally a strict pullback, but for the ∞-categorical universal property, ∞-pullbacks (i.e. homotopy pullbacks) are what’s required. | |
| Oct 12 at 16:02 | answer | added | Peter LeFanu Lumsdaine | timeline score: 13 | |
| Oct 12 at 14:29 | comment | added | მამუკა ჯიბლაძე | @aws Also want to say one thing. I believe retracts are subobjects in any possible sense, homotopy or otherwise. And you cannot classify retracts with $2$, right? While with $\Omega$ you can. | |
| Oct 12 at 14:28 | comment | added | მამუკა ჯიბლაძე | @ZhenLin Acknowledged. Maybe the way I formulated the question is misleading. I am rather interested in whether properties of $\Omega$ as an $\infty$-groupoid in a 1-topos where homotopies are present can say anything about the internal logic of this 1-topos. | |
| Oct 12 at 13:07 | comment | added | Zhen Lin | These properties are not really visible in the ∞-topos. They rely on strict equality, which is not homotopy-invariant. | |
| Oct 12 at 12:54 | comment | added | მამუკა ჯიბლაძე | @ZhenLin Well, it might be not very interesting but still somehow interesting. It is a contractible Kan complex, and every object embeds into its power, for example. Also it is a simplicial Heyting algebra. In the "ordinary world", I believe, no nontrivial connected Heyting algebras exist. | |
| Oct 12 at 12:43 | comment | added | Zhen Lin | Yes. But it is contractible, as you noted, hence not very interesting… | |
| Oct 12 at 12:30 | comment | added | მამუკა ჯიბლაძე | @ZhenLin I believe in any case an internal $\infty$-groupoid in a 1-topos is a legitimate thing to consider, no? | |
| Oct 12 at 12:28 | comment | added | მამუკა ჯიბლაძე | @aws Most probably what you say constitutes an optimal answer. Could you provide references for booleannes of the $\infty$-topos of simplicial sets, and what does homotopy subobject mean? | |
| Oct 12 at 12:27 | comment | added | მამუკა ჯიბლაძე | @AndrejBauer Not that this improves anything, but I've added the soft-question tag | |
| Oct 12 at 12:26 | history | edited | მამუკა ჯიბლაძე |
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| Oct 12 at 11:19 | comment | added | aws | This is the same point as Zhen Lin, but just to emphasis it, this is not in any way problematic and it is very misleading to say simplicial sets is "contradictory up to homotopy" because the subobject classifier for the 1-topos has very little to do with the subobject classifier of the $\infty$-topos. When we're working with a model category, I would refer to the latter as "homotopy subobject classifier," but I'm not sure how standard that is. Simplicial sets is boolean as an $\infty$-topos, so the homotopy subobject classifier is 2, which is not connected. | |
| Oct 12 at 11:02 | comment | added | Andrej Bauer | I find this question unclear, as it asks us to answer something that is placed in derision quotes (in the title), while the first paragraph asks for homotopy-theoretic significance (very close to asking for an opinion). What would an answer possibly constitute? | |
| Oct 12 at 10:49 | comment | added | Zhen Lin | As an ∞-topos the subobject classifier is a different thing. | |
| Oct 12 at 10:27 | comment | added | მამუკა ჯიბლაძე | As for isomorphism of true and false, here I mean that in the $\infty$-groupoid there are morphisms in both directions between these objects, there are homotopies between their composites and identities, these homotopies are homotopic to identities of identities, etc. | |
| Oct 12 at 10:25 | comment | added | მამუკა ჯიბლაძე | @AndrejBauer By "contractible" I mean that there is a homotopy between the identity of $\Omega$ and $\Omega\to1\to\Omega$, for some/any $1\to\Omega$. | |
| Oct 12 at 10:20 | comment | added | Andrej Bauer | Also, am I correctly reading the statement that $\mathbf{true}$ and $\mathbf{false}$ being isomorphic as just saying that there is an automorphism $m : \Omega \to \Omega$ such that $\mathbf{false} = m \circ \mathbf{true}$? (For instance, in a Boolean topos we could take $m$ to be negation.) | |
| Oct 12 at 10:18 | comment | added | Andrej Bauer | Could you please be more explicit about the notion of "contractible". In type theory "contractible" means "equivalent to the unit type", but that can't be what you're talking about. | |
| Oct 12 at 10:11 | history | edited | მამუკა ჯიბლაძე | CC BY-SA 4.0 |
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| Oct 12 at 9:57 | history | asked | მამუკა ჯიბლაძე | CC BY-SA 4.0 |