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Thanks for the comments!!
concerning the first one: I added a link to nice topological space and removed the -subscript throughout. It is not of any importance there anyway.
Now I have also edited the entries Top and nice topological , in an attempt to polish them a bit more.
concerning your second comment:
okay, so I have edited that introducory paragraph a little bit,following your suggestions, to make it slightly more accessible.
Thanks, these are very good points.
I'll work these in as soon as I have a second.
Concering the notation of versus : I have thought about which one to use and found that using the specific names helps greatly to see what's actually going on. You are of course right that this notation does not amplify how general-abstract the construction is, but i found it helps to amplify the meaning of these general abstract constructions here.
Okay, I see. I edited the entry a bit, put in the right at the beginning. Also rephrased the sentence where local contractibility is introduced. I'll see if I rename the -occurences later, have to run now...
Thanks, Domenico, very helpful comments.
I only had a handful of minutes to spare and quickly implemented what you suggested about : I renamed that to , which is indeed of course a much smoother perspective. The thing it sits over I named and then .
Already have to run again. But thanks for the comments. Will look at this more in a little while.
Thanks Domenico, that sounds interesting. I wasn't aware that you are in direct contact with Gabriele Vezossi. That's nice.
I am also still thinking about how to say "infintiesimal thickening" correctly. I feel like I should be able to simply put all the available ingredients together. Here is a thought:
start with some , an (oo,1)-category "of spaces" of sorts, so that is the category of "function algebras" on these spaces.
To find its infinitesimal thickening, we should form its tangent (infinity,1)-category .
Because we know from the classical case (as reviewed at module) that if , then is the category of all modules over all rings, but with each module over a ring regarded as the square-0-extension ring . But this is of course in precisely an infinitesimal thickening of the space corresponding to .
Do you see what I mean?
So I am beginning to think that the tangent (oo,1)-category of an (oo,1)-topos is what gives its infinitesimal thickening. And I suppose it is no coincidence that the terms do match this way.
One technical point that i am unsure about: for an (oo,1)-topos,
is itself an (oo,1)-topos?
is the canonical morphism a (essential) geometric morphism?
I have added some more details to tangent (infinity,1)-category.
If one looks at the cotangent-complex adjunction
that does begin to look a bit like the infinitesimal topos-thickening
that we would like to see, using and noticing that under opposing categories left adjoint become right adjoints, and vice versa.
But I am not sure if this can be made to work. Maybe that's not white what the relation is. But something like this might make sense.
(sorry, I said something stupid here, which I decided to remove...) more later...
(second attempt, my previous one was slightly flawed...)
Domenico,
very good point about the retract.
Reminds me of the observation at my discussion of the path oo-groupoid on oo-toposes of oo-shesves on sites of "geometrically contractible objects". That makes ooGrpd be a retraxct of these beasts in a special way, which implies that these (oo,1)-toposes have the "shape" - in the sense of shape of an (infinity,1)-topos -- as the point. (Which I suppose makes good sense for these gros oo-toposes of "all spaces".)
This makes me think it would make sense if an infinitesimal thickening of such an oo-topos should still have the "shape" of the point, in particular any infinitesimal thickening should probably have the same "shape" as what it is a thickening of.
I wrote a remark on this at shape of an (infinity,1)-topos in the new section Shape of an essentialretract.
Here is an observation:
if we start with some oo-topos of sheaves on Rings^op and then pass to its tangent oo-category, then we know that the new objects are something like pairs consisting of the old objects and a module, turned into a nilpotent ideal.
So this new tangent category might look not unlike oo-stacks on the category of pairs (rings, nilpotent ideal) which are the kinds of sheaves considered in Zoran's work with Durov, p. 22 here.
Do you see what I mean?
Maybe we want to be looking at this:
Let be a site and its tangent category. Then we want to put a site structure on such that the projection is a morphism of sites. Then this induces a geometric morphism of oo-stack oo-toposes.
Now, by exactly the same kind of argument that I use to show that oo-toposes are locally contractible, we should be able to see that is locally contractible relative to .
This should follow when the site structure is such that sheaves on that are constantly extended to are still sheaves there, which is just what we expect of a topology on "infintiesimally thickened spaces". Then this should imply that we
.
Do you see what I mean? Now that I wrote this it looks to me that this should be an essentially trivial generalization of the non-relative situation discussed so far.
the tangent (oo,1)-category of an (oo,1)-topos is an (oo,1)-topos? projection the base is essentially geometric?
For the time being I think it may be useful to circumvent this problem by being very slightly non-intrinsic and talking about sites:
I believe for C any site we should be able to canonically equip with the structure of a site such that the 0-section is a morphism of sites.
For instance for the example of in which case should be thought of as Mod we just take a cover in Mod to be anything which is a cover in Cring after forgetting the square-0--extensions. The "0-section" here is the functor that sends .
My best current understanding of the resulting picture I have now indicated in the section relative theory over a base at structres in an (oo,1)-topos
(which does follow a few of your earlier suggestions)
Concerning the other point that you raise, let me think a bit...
Here is what I mean in the 1-categorical case
Morphisms of sites -- Examples -- Injections into tangent categories.
But check.
But check.
Checked. Seems to be ok.
Thanks. I wanted to write tangent category (entirely 1-categorical case) to make it fully precise. Now I realize that I need to think harder about how to correctly talk about morphisms going between objects over a different base. Still thinking...
Oh, wait, it is correct, I managed to confuse myself.
I am talking about the definition of morphisms at tangent category (explicit 1-categorical version). I got worried that what I wrote as the general definition of morphisms in does not match the morphisms in Mod when CRing. But now I think it does.
but to be honest I found it a bit hard to read the proof as it is now
Sure, maybe this is the kind of thing best done on paper and then omitted.
a morphism of abelian group objects over .
Yes, but first I was worried that the pullback abelian group object as I defined it does not actually match the notion of pullback of modules. But I was just being dense, it all works of course.
Dunno, should I better remove the proof I typed again? Is it more confusing then enlightning?
Okay, thanks. I removed my proof and retained yours.
All right, so coming back to the infinitesimals now.
I still need to write out a fully formal proof that the infinitesimal path oo-groupoid functor is equivalent to . By analogy with the finite case, for this it is sufficient to show that on representables it has the desired action, namely that it sends some to the presheaf which takes probes of X only after reducing away all infinitesimal directions of the probe object.
I started typing something along these lines in the section Properties at infinitesimal interval object, but this needs to be expanded.
For the moment though I need to take care of something else.
I polished a bit here and there and then expanded the discussion and the proofs in the section on the infinitesimal path oo-groupoid (on my personal web)
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