A development

- Marc Hoyois
*Topoi of parametrized objects*, (arXiv:1611.02267)

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.

]]>On locus. I’ll remove DR’s comment to that effect.

]]>@David Corfield: yes, the discussion about stable objects only makes sense in $\infty$-toposes.

@David Roberts: sorry, where did you add something?

]]>Was Joyal just considering 1-toposes? Wasn’t Mike just dropping the $\infty$s?

]]>I added a link (edit: to locus) to this discussion in a new reference section, and an attribution to Joyal.

]]>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 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.

]]>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?

]]>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?

]]>@David, I have added to your *locus* the clause that the indexing is specifically over $\infty$-groupoids.

Oh, I see. Thanks.

]]>Probably worth an entry then for locus. We would need some disambiguation though.

]]>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.

]]>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?

]]>I thought Goodwillie was somewhere around. That’s about taking the tangent at the one-point space, according to Finster. Hmm, can one ’transport’ the tangent space around along morphisms to $1$ in the underlying $(\infty, 1)$-category?

]]>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.

]]>Sure he could! :-)

]]>Could he be asked by email?

]]>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.

]]>Has anyone asked Joyal why he believes parameterized spectra form an $\infty$-topos?

]]>@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.

]]>Thanks, Mike. What’s a reference for this?

]]>So if we took the fibrational slice over spectra, what would the $(\infty, 0)$-truncated objects be?

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