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)

]]>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.

]]>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?

]]>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.

]]>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...

]]>But check.

Checked. Seems to be ok.

by the way, the coverage defined on should be the coverge induced by the coverage of via the functor ; I guess this is a general construction, but at induced toplogy there's only a hint of this. can any expert expand that? ]]>

i) assume is an arbitrary affine scheme; can we build a natural dgca out of the cotangent complex of ?

ii) how is this dgca related to (where we thicken by taking its tangent (oo,1)-category)? ]]>

Here is what I mean in the 1-categorical case

Morphisms of sites -- Examples -- Injections into tangent categories.

But check.

]]>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...

]]>there's a question which has been puzzling me all the afternoon, and maybe now I'm finally able to write it in a not too obscure way. fix a thickening of . then for every object of we have a notion of and morphisms stemming out of this are differential forms on . but now, classically, I'd like to interpret differential forms as sections of suitable powers of the cotangent bundle (or better, complex..) of . and this seems indeed to be the case, by the very definition of : it is the adjoint of and so a functor stemming from with values in is the same thing as a functor stemming from with values in , which I'd like to think as a section of "something built from and the cotangent complex of . vice versa, having a functorial notion of cotangent complex of (equivalently, of its sections) would give us a natural definition of differential forms on , and so would define a apparently without any notion of thickening.

this is still very confused, but I'm thinking of the following principle: a thickening of is the same thing as endowing each object of of a cotangent complex in a functorial way, i.e., it is giving the cotangent complex functor. even better: "thickening is taking the (oo,1)-tangent category".

so we have now to answer a few technical questions, yours: the tangent (oo,1)-category of an (oo,1)-topos is an (oo,1)-topos? projection the base is essentially geometric? and mine: do the notions of differential forms coming from the cotangent complex functor and the one coming from coincide? ]]>

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.

]]>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?

]]>(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.

]]>(sorry, I said something stupid here, which I decided to remove...) more later...

]]>edit: here there were a few considerations on integration as a Kan extension, but they were nonsense as in an (oo,1)-topos there is no distinction between extension and Kan extension. maybe they apply to integration in an (oo,2)-topos, but I am not going to think to this right now ]]>

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.

]]>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?

what we had thought with Gabriele was that the obstruction theor on X had to be the datum of some algebroid (e.g., for X a smooth compact differential manifold, one would have taken the tangent algebroid, recovering the classical fundamental class; however one could have taken the 0 algebroid, and would have got 0-dimensinal fundamental calsses). the algebroid role was to say which were the "directions" in which one was allowed to move.

now I see this is precisely the role of the thickening . and also in this case the thickening can be arbitrary, but in some cases, e.g., a smooth topos, there is a canonical one.

I'll think more to this and will be back. ]]>

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.

]]>since I feel the relative oo-path groupoid has interest on its own, I would distiguish it from the thickening stuff, to which I would devote a distinct subsection; i.e., I would have "Relative homotopy oo-groupoid" and "Thickenings and and infinitesimal oo-groupoid"

typo in the definition of : infinitsimal ]]>