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Concerning our geologically slow discussion elsewhere, in various other threads, on $\infty$-toposes that contain non-trivial stable homotopy theory.
Here is a trivial thought:
while we grew fond of identifying the $\infty$-category and allegedly $\infty$-topos of parameterized spectra as the tangent (infinity,1)-category to $\infty Grpd$, maybe it’s after all more useful to think of it instead as the full sub-category of the slice $(\infty,2)$-category
$(\infty,1)Cat_{/Spectra}$on the $(\infty,0)$-truncated objects, which we are inclined to write
$\infty Grpd_{/Spectra} \hookrightarrow (\infty,1)Cat_{/Spectra} \,.$But suppose these two obviously plausible facts about $(\infty,2)$-toposes hold true:
slices of $(\infty,2)$-toposes are $(\infty,2)$-toposes;
full subcateories on $(\infty,0)$-truncated objects inside $(\infty,2)$-toposes are $(\infty,1)$-toposes
then it would follow immediately that $\infty Grpd_{/Spectra}$ is an $(\infty,1)$-topos.
Maybe somebody could remind me why this obvious (naive?) strategy for going about it is no good.
Can’t see anything wrong myself.
Is this argument likely to have a smooth or super analogue?
If this argument works, then it is fully general.
It’s not true that a slice of a 2-topos is a 2-topos; you have to restrict to fibrations.
So if we took the fibrational slice over spectra, what would the $(\infty, 0)$-truncated objects be?
Thanks, Mike. What’s a reference for this?
@David, re #5: fibrations won’t do, for our purpose here. For instance we need there to be a tensor unit of parameterized spectra over some $X$, which is the functor constant on the sphere spectrum. This is far from being a fibration.
Has anyone asked Joyal why he believes parameterized spectra form an $\infty$-topos?
I have talked with him on a bus ride to a lobster restaurant in Hallifax, and back. He said he has a general theory of what he calls “loci” that allows him to check this. (I think I remember this correctly, but of course it’s just my memory.) We talked about a lot of things and I didn’t get around to asking for more details on the proof. Unfortunately.
Could he be asked by email?
Sure he could! :-)
He gave a talk about it at IAS. A “locus” is a category $C$ such that the category of indexed families of objects of $C$ form a topos. Since toposes are closed under left exact localizations, so are loci. The category of pointed types is a locus, since families of it are a presheaf category (the category of retractions). Similarly, the category of prespectra is a locus. And the category of spectra is a left exact localization of the category of prespectra. Hence spectra form a locus, so parametrized spectra are a topos.
Eric Finster had another argument for this that has some relationship to Goodwillie calculus, but I don’t remember it.
Heh, didn’t know that you knew the argument.
I follow your paragraph in #12, except for one step. Maybe I am misunderstanding, but how is pointed $\infty$-groupoids paramerized over $\infty$-groupoids an $\infty$-presheaf $\infty$-topos?
A pointed $\infty$-groupoid parametrized over an $\infty$-groupoid is just a map $B\to A$ equipped with a section (which assigns the basepoints of the fibers). So the category of such is the category of diagrams on the walking map-equipped-with-a-section.
Probably worth an entry then for locus. We would need some disambiguation though.
Oh, I see. Thanks.
@David, I have added to your locus the clause that the indexing is specifically over $\infty$-groupoids.
Might something be gained by allowing the indexing to take place over a different base?
So a locus is something like the kind of fiber over $1$ which if pulled back over all of $\infty Grpd$ generates an $\infty$-topos?
Then is there good reason why the tangent category to $\infty Grpd$ at $1$ should be one of these things?
Where does the cohesiveness for parametrized spectra come from? Is it produced by that localization of the topos of parametrized prespectra?
I am not sure how to do the geometric version with this. One evident idea would be to consider presheaves on some site wih values in the $\infty$-topos of parameterized spectra and then localize again at the covers.
I feel I am a bit behind the curve with this topic. Mike, have you thought about this?
I don’t really fully understand it either. Nor am I really happy with the name “locus”. (-:
There should be no reason we can’t vary the base. I think the general notion would be an $E$-indexed locus: an $E$-indexed category $C$ such that the $E$-indexed category $Fam(C)$ is an $E$-indexed topos.
So we need such a concept as locus, however we’d like it be called, because we can’t go directly via left exact localization from, say, parametrized prespectra to parametrized spectra, but need to do this via the unparametrized versions?
I added a link (edit: to locus) to this discussion in a new reference section, and an attribution to Joyal.
Was Joyal just considering 1-toposes? Wasn’t Mike just dropping the $\infty$s?
@David Corfield: yes, the discussion about stable objects only makes sense in $\infty$-toposes.
@David Roberts: sorry, where did you add something?
On locus. I’ll remove DR’s comment to that effect.
Yes, I was slipping into the implicit infinity-category theory convention, sorry. I’m not entirely sure why the locus concept is needed; Joyal might just prefer to talk about it that way.
A development
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