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I’ve started sufficiently cohesive topos. Here are a couple of remarks and questions:
The corresponding terminology in def. 2.13 at cohesive topos strikes me as odd: $p_!(\Omega)=1$ is connectedness not contractability.
It isn’t quite clear to me yet at which level of generality to optimally state the definition of ’sufficient cohesion’. It seems that what one wants to get here are the minimal assumptions ensuring that the connectedness of $\Omega$ is equivalent to its contractibility and this presumably requires only preservation of finite products by $p_!$ and not the Nullstellensatz (nor even the existence of $p^!$ !?).
Since the entry so far lives on the (0,1)Lab maybe somebody here has an idea what to say for the ($\infty$,1)-case e.g. assuming connectedness of the (higher) object classifier !?
is connectedness not contractability
Fixed.
(Sometimes it takes less keystrokes to fix a typo than to annonce it. :-)
Can we find a different word than “contractible” for the concept $p_!(X^Y)=1$ used in “sufficient cohesion”? In the context of higher toposes, “contractible” would naturally be the word for $p_!(X)=1$, since there $p_!(X)$ is the fundamental $\infty$-groupoid rather than just the set of connected components. (I think this case is probably what Def 2.13 at cohesive topos had in mind.) I am not sure why Lawvere said “contractible” for $p_!(X^Y)=1$ in the first place; what about something more descriptive like “exponentially connected”?
(Actually, in the higher-topos/HoTT case one wants to qualify this “contractibility” with an adjective like “cohesively contractible” anyway, since there is also the internal notion of “homotopically contractible”, meaning equivalent to $1$, and likewise “homotopically connected”, meaning $\Vert X\Vert_0 = 1$.)
(Note that the syntax for “Remark” is num_remark
, not num_rem
. I’ve fixed it.)
Regarding the level of generality and the $\infty$-case, it’s hard for me to have an opinion without having a motivation for “sufficient cohesion”. What is the purpose of this property? What exactly is it “sufficient” for?
(One remark about the $\infty$-case is that I believe the proof of Corollary 6.5 in Lawvere and Menni works fine there, so that “pieces have points” plus “contractible codiscreteness” implies “sufficient cohesion”. I did a little internal investigation of these axioms in section 10 of BFP in CoHoTT.)
The main motivation for ’contractible’ seems be to that after passage to “homotopy” $Ho\mathcal{E}$ i.e. replacing $Hom(Y,X)$ by $p_!(X^Y)$ contractibles become precisely the terminal objects. So from a 1-categorical perspective the choice appears suggestive and innocuous to me. Within an n-categorical perspective ’exponentially connected’ would sound fine to me.
Instead of ’sufficient cohesion’ which suggests a weak form of cohesion though the concept is actually a strong form I would be inclined to use something like ’having enough connected objects’ or ’having connected power objects’.
I basically stuck to Lawvere’s terminology because the readers of the nLab are expected to come from reading Lawvere’s or Menni’s papers that use the terminology and I think it unwise and impolite to improve on their terminology unless one can also improve on their results. So for the moment I think that one should keep the terminology and probably use the entry’s section on ’terminology’ to make suggestions for alternatives.
Within Lawvere’s story, the concept affords to sort out “degenerate” cohesion where pieces have exactly one point. Intuitively, in a sufficiently cohesive topos ’true’ and ’false’ are not only connected but can be deformed into each other i.e. there is another kind of unity of (logical) opposites here. From a different angle, in such toposes $\Omega$ provides a generalized (nonlinear) interval object which exponentiates connectedly.
[Thanks for catching the bug!]
passage to “homotopy” $Ho\mathcal{E}$ i.e. replacing $Hom(Y,X)$ by $p_!(X^Y)$
This has been stated as a motivation for cohesive 1-toposes since “Axiomatic cohesion”. But it is an open problem how to get the correct homotopy category via this approach. The correct homotopy category is the one given by the above replacement plus the restriction to cofibrant-fibrant objects (in the motivating example of the cohesive 1-topos of simplicial sets at least). Has there been any progress on fixing this?
I think for now the last word on this is the paper by Marmolejo-Menni from a year ago. They work out the details of the passage though they probably leave open the question how this relates to proper homotopy.
There is a certain tension between requiring a category to be a topos, and requiring the result of quotienting hom-sets by homotopy to be the localization at a class of weak equivalences. I wonder what the rough plan is by which this is supposed to be made to work.
Ah, I see! Let me rephrase that in ∞-language. Given a cohesive ∞-topos $p:E\to S$, we can define a “homotopy 1-category” $h E$ with the same objects as $E$ and homs $h E(X,Y) = \| p_! (Y^X) \|_0$, the “connected components of the shape of the homs in $E$”. What Lawvere calls “contractible” means being terminal in $h E$. There is a functor $h E \to \|S\|_0$ defined by $p_!$, and since $p_!$ preserves the terminal object, Lawvere’s “contractibility” implies that $p_! X$ is also contractible (i.e. terminal in $S$). The difference is, as Urs says, that Lawvere’s notion is a “strong contractibility” (equivalent to 1 by an actual homotopy equivalence) whereas contractibility of $p_! X$ is instead about the “weak homotopy type”.
So maybe “contractibility” isn’t so bad, although I’d still prefer to add some adjectives like “strongly cohesively contractible”, at least when the notion is first defined on the page.
I’m not sure why you think “readers of the nLab are expected to come from reading Lawvere’s or Menni’s papers that use the terminology” — I think it at least as likely that a reader of the nLab would encounter the notion on the nLab first and then be pointed to Lawvere and Menni’s papers. And I have no qualms about improving on bad terminology. However, as above, now I see that “contractible” isn’t all that bad, and “sufficiently cohesive” isn’t exactly bad so much as unevocative.
(Plus we should explain somewhere why this is a notion of “contractibility”.)
In general, I’m suspicious of axioms whose only purpose seems to be to rule out models that “aren’t what you had in mind”. If none of your theorems require such an axiom, why bother with it? Your “pathological example” may turn out to be someone else’s “intended example”.
@#10. Lacking reader polls, I extrapolate my own usage of the nLab as well as what I think is compatible with online statements on MO etc.
@#12. ’sort out’ is probably misleading, what Lawvere does is more charting a territory where the inscribed conceptual borders lead to a better understanding of the examples and concepts on both sides. The things ’sorted out’ are just put in another place.
I finally finished the proof of prop.5.7 at sufficiently cohesive topos by replacing the previous non-converging external argument with internal hand waving. Albeit a bit sketchy I am inclined to call it a day resp. a proof but if anybody has a more explicit argument I’d like to see it.
I also added prop.4.5 hinging on the lemma 4.6. Hopefully, I had not noticed this before simply becuse it is false.
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