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I was in the differential cohesion group at the HOTT MRC last week and one thing we struggled with was that we only knew one model (sheaves over formally thickened Cartesian space) which is pretty complicated to construct. Also this specific model has extra properties so not all of the axiomatics can really be explained with just one model, for instance Felix told me that every object in this topos is formally smooth.
Plain cohesion, on the other hand, has some very elementary toy models like reflexive graphs and the sierpinski topos that are very nice for getting intuition.
Do we have any analogous toy models for differential cohesion? All the better if they are “thickenings” of sierpinski or reflexive graphs. Ideas are also welcome, I’d be happy to work through some details myself.
I do wonder if there are any tangent categories in the sense of Cockett and Crutwell that are cohesive.
Pick any presheaf site $\mathcal{S}$ for plain cohesion, and then pick a coreflective embedding into another category $\mathcal{S}_{th}$. This induces an adjoint quadruple between the relevant categories of presheaves, and this exhibits differential cohesion.
Don’t you also need the embedding to preserve finite products (or at least be sifted-flat) in order for reduction to preserve finite products? But even this should be very helpful; we should mention it on the page.
By the way, in case Urs or anyone is wondering about the outcome of the MRC project, they made a lot of good progress on writing down type theories for various fragments of differential cohesion (which turns out to be quite hard), and at the end came up with a proposed type theory including all the differential cohesive modalities, but didn’t yet have time to use it for any concrete proofs in differential cohesion.
Don’t you also need the embedding to preserve finite products
Yes, I was just thinking I should add that comment.
By the way, in case Urs or anyone is wondering about the outcome of the MRC project, they made a lot of good progress
That’s great to hear. Please let us know once there is any kind of written account, preliminary as it may be.
Okay, so for instance we have the coreflective and product-preserving embedding $(0\to 2) \hookrightarrow (0\to 1\to 2)$, inducing a thickening of the Sierpinski topos. I think the modalities are:
If we think of $A_2$ as the “real points” and $A_1$ as the “possibly infinitesimal points” while $A_0$ is the “connected components”, then:
Nice, Mike. Let’s start an $n$Lab page for this example.
Really side remark here: the category of simple graphs is a cohesive (topological) Grothendieck quastitopos.
I was wondering about $\mathbf{H}^{(\mathbb{N}, \geq)}$ over here as potentially possessing a bunch of adjoints. Could we get a model for the next tier of super-modalities by some extension to $(0\to 1\to 2 \to 3)$?
Very cool, thanks Urs and Mike!
To expand a little on Mike’s ” infinitesimally thickened sierpinski topos”, it looks like we get a simple notion of “infinitesimally thickened point” which would look like $1 \to D \to 1$ which has a single true point ($A_0 = 1$) but much “purely infinitesimal extension” (the $d\in D$ with $p(*) \neq d$). We can also verify some expected things like any map from an inf thickened point to a coreduced space is constant and we can see what a “formal neighborhood” is by looking at maps from things like $1 \to 2 \to 1$, which would pick out one real point in $A_0$, but would also pick another infinitesimal point in $A_1$ that lies in the same connected component ($A_2$).
Yes, that’s nice. Can we realize $\Im$ as a localization in this model?
I don’t understand localization enough to tell, but it seems to me that there are non-coreduced spaces whose maps out of a thickened point $1 \to D \to 1$ are all constant, say something like $A_2 \to A_1 \to A_1+1$, so I think that means $\Im$ is not localization at all infinitesimally thickened points.
Also, I’d be happy to start this page, would the right terminology be “infinitesimally thickened Sierpinski topos”? It seems slightly off because there’s more than one way to thicken a site.
Also, I noticed that the Sierpinski topos relative to a cohesive topos is itself is a toy model of differential cohesion, as described here, though that model is not as evocative because there are in fact no infinitesimally thickened points because reduction is $\Re(A_{1} \to A_{0}) = 0 \to A_{0}$
How does the jet comonad pan out in this toy model?
I think “infinitesimally thickened Sierpinski topos” is fine, at least for now. The page can remark up-top that there is more than one way to thicken a site, but this example seems a little “canonical” so we might as well call it that.
it seems to me that there are non-coreduced spaces whose maps out of a thickened point $1 \to D \to 1$ are all constant, say something like $A_2 \to A_1 \to A_1+1$, so I think that means $\Im$ is not localization at all infinitesimally thickened points.
Interesting. It seems like maybe that has something to do with the fact that “pieces have points” fails in general for the Sierpinski topos.
Thanks, Max, for starting infinitesimally thickened Sierpinski topos. Is the idea to imitate Sierpinski topos and start with the 1-topos (which you have) and then define the $(\infty, 1)$-topos, and more generally the case over an arbitrary base topos, $\mathbf{H}$?
I don’t really know, I’m sticking to the 1-topos at first, because that’s all I understand.
Ok, but if this is about devising a toy model for HoTT with modalities, such as cohesive homotopy type theory and its differential extensions, surely the target should be $(\infty, 1)$-toposes.
Re my question in #13 about the jet comonad (an answer to which should show us what PDEs look like in this model), presumably base change along $(C_2\to C_1\to C_0) \to \Im(C_2\to C_1\to C_0) = C_2 \to C_0\to C_0$ is
$(A_2 \to A_1 \to A_0) \to (C_2 \to C_0\to C_0) \mapsto (A_2 \to A_1 \times_{C_0} C_1 \to A_0) \to (C_2\to C_1\to C_0).$For presheaf models the broad properties will be the same for 1-toposes and $\infty$-toposes. The existence of the adjoints, the presevation of products etc. all follows from the same kind of general theorems that exist in both cases. If one passes to sheaves instead, then the conditions on the site are in general stronger for the $\infty$-category theoretic construction.
I take it we could also look at sheaves over the topological space on $\{0, 1, 2\}$ with topology $\tau = \{\empty, \{2\}, \{1, 2\}, \{0, 1, 2\}\}$. Does that have a name?
One very interesting question to explore would be if there are other non-toy models for differential cohesion.
Of particular interest should be the topological analogue of the Cahiers topos, if it makes sense.
The site should consist of objects which are products of any topological space $X$ with the germ $G_y(Y)$ of a topological space $Y$ around some point. Hence a morphism $X_1 \times G_{y_1}(Y_1) \to X_2 \times G_{y_2}(Y_2)$ should be a pair consisting of an $X_1$-parameterized function of germes $G_{y_1}(Y_1) \to G_{y_2}(Y_2)$ and the germ of a function $G_{y_1}(Y_1) \to Maps(X_1, X_2)$.
This is meant to be the topological analogue of the site of formal Cartesian spaces. The site of topological spaces embeds coreflectively into this bigger site, with the inclusion preserving products. If one restricts all topological spaces appearing here to be locally contractible, then both sites should also be $\infty$-cohesive.
If this exists, the abstract theory of differential cohesion would imply a non-trivial jet-like calculus for topological spaces. Some discussion I had with Thomas Nikolaus suggested that this does make sense. But I didn’t have the time to follow up on it.
I’ve to infinitesimally thickened Sierpinski topos explicit descriptions of the cohesive modalities, formally etale morphisms and the infinitesimal disk bundle.
I was going to do the jet comonad too but I can’t actually find an explicit description of $\Pi$ in a presheaf topos anywhere on the lab. I’ll work through Maclane-Moerdijk later.
find an explicit description of $\Pi$ in a presheaf topos
Do you mean the left adjoint to the constant presheaf functor? That is the colimit functor, hence the operation that takes a presheaf, regards it as a functor with values in sets, regards that as a diagram in sets, and then takes the colimit over this diagram.
I actually meant dependent product in this case.
Ah, I see.
So there is a general formula for expressing the dependent product as pullback of internal homs in the slice category of a locally Cartesian closed category (here) and there is the statement that the slice of a presheaf topos is presheaves on the slice site (here). Finally with the formula for the internal hom in presheaf toposes (here) this gives a formula for the dependent product in presheaf toposes.
Regarding non-toy models, there should be some examples in algebraic geometry. This is of course not a new thought, I know that it has cropped up in previous discussions, and that Urs has long wanted someone to look at this. I would be willing to give it a go. I have very little time, but if we take it slowly here, we can probably make some progress.
There are I think several possible ways to go in algebraic geometry, but I think it is natural to think that the corresponding ’differential cohomology’ should give something like motivic cohomology/algebraic K-theory. Étale cohomology/étale K-theory might possibly be a slightly easier starting point.
Anyhow, although I had I think a very tiny role in the early days of Urs arriving at these kind of ideas (and in fact my motivation back then was motivic phenomena), I am not especially proficient in the latest forms of differential cohesion. If someone could explain to me the first stumbling block in trying to carry out these ideas in algebraic geometry, I can take a look at that when I get the chance.
Hi Richard,
that’s right, you were the one who made me see the “shape modality” left adjoint, and your name is in the acknowledgements of dcct ever since.
The issue is that the existence of the shape modality need the objects of the site to be locally contractible (as in the argument at infinity-cohesive site), or at least something very similar, and for general varieties this is just not the case. Accordingly in algebraic geometry in general there only exists the “pro-left adjoint” to the constant sheaf/stack functor, which is the reason why étale homotopy groups in algebraic geometry in general are pro-groups.
Hence the question is to find a site of “locally contractible schemes” of sorts, which is still large enough to support an interesting chunk of algebraic geometry. The pro-etale site seemed like a candidate at some point, but I don’t know if it gives cohesion.
And, yes, it does seem that making this work should have an impact on motivic cohomology. The main reason I see is the following: From a bird’s eye perspective, the way for instance Denis-Charles Cisinski describes it in his talks, a key purpose of motivic cohomology is to enforce a decent theory of Chern characters in cohomology. But a decent theory of Chern-characters, that’s precisely what is implied by cohesion, via the differential cohomology hexagon.
And for arithmetic, differential cohesion and idelic structure and Borger’s absolute geometry.
Thanks, David. In fact at the end of differential cohesion and idelic structure is observed that algebraic geometry for non-unital rings is cohesive in an interesting way. (An immediate corollary of Grothendieck-Greenlees-May “local cohomology” theory, as streamlined by Lurie.) Back then I thought that non-unital algebraic geometry is too exotic to be interesting, but later I ran into people who said they were studying it as the right way to go about things more generally. (Thomas Nikolaus had something in this direction.) So maybe this deserves renewed attention.
Ok with the help of Urs’ pointers (thanks, Urs!) I’ve worked through the Jet comonad, I’ve simplified a lot from the raw definition so I might have messed it up, but it looks reasonable to me.
Notation: I’m writing $p : E \to X$ as just p, and the maps as $inf : X_2 \to X_1$, $pc : X_1 \to X_0$ and $x ~ y$ for $pc(x) = pc(y)$. Also I’m going to define them fiberwise because it makes the definition of $J(p)_2$ less redundant, but note that $J(p)_0 \cong E_0$
$J(p)_{0,x_pc} = p_0^{-1}(x_pc)$ $J(p)_{1,x_i} = \sum_{e_{pc} \in p_2^{-1}(pc(x_i))}\prod_{y_i ~ x_i} p_1^{-1}(y_i) \cap pc^{-1}(e_{pc})$ $J(p)_{2,x_r} = \sum_{e_r \in p_2^{-1}(x_r)}\sum_{(-,f_i) \in J(p)_{1,inf(x_r)}} f_i(inf(x_r)) = inf(e_r)$In words, the pieces of $Jet(p)$ are the same as the pieces of $E$. An infinitesimal point over $x_i$ picks a point $e_pc$ above $x_i$’s piece and maps every $y_i ~ x_i$ to something above it in that piece.
Finally a real point over $x_r$ is a real point $e_r$ over it and a map $f_i$ from every $y_i ~ x_r$ to a point above it in $e_r$’s piece that in particular sends $inf(x_r)$ to $inf(e_r)$.
Hi Urs, thanks, that’s very generous of you!
I have a thought a bit about the problem that you mention, but a long time ago. One possibility of course could be to ’embrace the pro-setting’, i.e. just accept the fact that we have a pro-simplicial set and try to expand cohesion to cover that generality (which would be necessary even to treat topological spaces in full generality, i.e. without assuming local contractibility, of course).
But one could also turn the problem around. Are they any results in the literature pertaining to $\mathbb{A}^{1}$-local contractibility in the Nisnevich topos? How general would be the gadgets that we obtain by taking $(\infty,1)$-colimits of the affine spaces $\mathbb{A}^{n}_{k}$ over some field $k$, say? These should be $\mathbb{A}^{1}_{k}$-locally contractible, and naïvely the gadgets seem quite general, but I might be missing something. There is a paper Motivic cell complexes by Dugger and Isaksen which considers homotopy colimits of motivic spheres only, and these are apparently not particularly general, but homotopy colimits of the kind I mentioned seem to me to be much more general.
Unless I am really missing something, one is certainly getting all schemes over $k$ (with some finiteness condition that I cannot remember the name of) by glueing the $\mathbb{A}^{n}_{k}$’s using ordinary colimits when using the Zariski topology, without any $\mathbb{A}^{1}$-localisation. So the question is how much of this picture is preserved when one works with a) homotopy colimits instead b) the Nisnevich topology instead c) we throw in $\mathbb{A}^{1}$-localisation. My apologies if there is some obvious reason why we do not get much that is interesting, I have not properly thought it through; it is just a first idea that came to me.
Re #30, so now we should have a version of:
The Eilenberg-Moore category of coalgebras over the Jet comonad has the interpretation of the category of partial differential equations
These coalgebras are just the images of base change along $X \to \Im(X)$, as Urs points out.
One possibility of course could be to ’embrace the pro-setting’,
Yes, one can do that, but then one is no longer in an $\infty$-topos.
Same is true for the usual $\mathbb{A}^1$-local homotopy category: while it is of course a “shape-like” loclization, by construction, it is not an $\infty$-topos.
A cohesive version of algebraic geometry, in the usual $\infty$-topos theoretic sense should have some relation with $\mathbb{A}^1$-homotopy theory, but must be a little different.
There are maybe two different goals to be distinguished: The first is to put cohesive $\infty$-toposes to any usage in a context of algebraic geometry, and possibly interpret the result in terms of the general idea of motivic geometry, and the second is to explore relations specifically to $\mathbb{A}^1$-homotopy theory. I feel the former might be more interesting.
Unless I am really missing something, one is certainly getting all schemes over $k$ (with some finiteness condition that I cannot remember the name of) by glueing the $\mathbb{A}^{n}_{k}$’s using ordinary colimits when using the Zariski topology,
Could you dig out the precise statement that you are thinking of here?
Hm, I didn’t say that well: That the base is not a topos need not be that bad. But what matters is that the two functors from the base are fully faithful, so that the three (co-)monads that are induced are indeed idempotent.
Yes, one can do that, but then one is no longer in an $\infty$-topos.
Yes. My feeling was that one might be able to get it to work nevertheless, but this could be naïve. One possibility might to work with something like a pro-analogue of a topos, but also one might be able to carefully pass on a case-by-case basis from the usual topos-theoretic arguments to ones that can work with when we have a pro-homotopy type instead.
Same is true for the usual $\mathbb{A}^{1}$-local homotopy category: while it is of course a “shape-like” localization, by construction, it is not an $\infty$-topos.
Yes, my feeling about this was that it might not be a really fundamental difficulty. If we just imagine for now that it is a topos, I think we could probably find some way around the obstruction if everything else works.
On the other hand, I do feel that the $\mathbb{A}^{1}$-localisation is very formal, and quite ‘forced’; in some ways, it is quite remarkable that one can say anything at all. There might be a more sensitive approach. I think this agrees with the following remark which you made.
There are maybe two different goals to be distinguished: The first is to put cohesive $\infty$-toposes to any usage in a context of algebraic geometry, and possibly interpret the result in terms of the general idea of motivic geometry, and the second is to explore relations specifically to $\mathbb{A}^{1}-homotopy theory. I feel the former might be more interesting.
Yes, I think that is a reasonable point of view. My reason for including the $\mathbb{A}^{1}$-localisation was that I think one then has a reasonably large collection of locally contractible spaces, and so has a chance of producing the left adjoint without passing to any kind of pro-setting.
Could you dig out the precise statement that you are thinking of here?
My thoughts were simply as follows.
1) Every scheme is a colimit of affines in the big Zariski topos over $k$ (every presheaf is a colimit of representables).
2) Every finitely generated $k$-algebra is a colimit in the category of $k$-algebras of the $k[x_{1}, \ldots, x_{n}]$’s for varying $n$.
I forgot to dualise 2), so the correct statement (more or less tautological) would be something like: every scheme over $k$ which is locally a finitely generated $k$-algebra can be obtained using limits and colimits of the affine spaces $\mathbb{A}^{n} = Spec(k[x_{1}, \ldots, x_{n}]$.
Thus my question in #31 should be modified to consider $(\infty,1)$-colimits and limits of the affine spaces $\mathbb{A}^{n}$.
One might look at fibred toposes rather than pro-toposes. SGA4 has a bunch of material on them. A simple change of perspective…
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