now added the details. this proposition
]]>I have added to differential cohesion somewhere statement and proof that the structured -topos of sheaves on any object in differential cohesion is always locally -connected.
But my battery is dying now…
]]>The best text I know is the one by Jacob Lurie that I pointed to,
Notes on Crystals and algebraic D-modules (pdf)
]]>Our entries on D-geometry and crystalline cohomology appear to be kind of stubby. Is there an exposition of them somewhere that I could understand?
]]>In slightly less modern language…
Is this meant ironically? The language of coreduced objects/spaces didn’t exist until a few days ago on this page, did it?
Ah, I didn’t say this well. What I meant was that Grothendieck didn’t formulate crystalline cohomology in terms of D-geometry and didn’t think of this as characterized by an adjoint triple. The “modern language” that I meant is that cited in #40 above — not the ultra-modern language that we are speaking here :-)
]]>Crystalline cohomology is the cohomology of coreduced spaces.
But crystalline cohomology was devised to solve the problem of -adic cohomology of varieties over an algebraically closed field of characteristic . Is this a sign of the return of that old question on p-adic cohesiveness?
]]>In slightly less modern language…
Is this meant ironically? The language of coreduced objects/spaces didn’t exist until a few days ago on this page, did it?
]]>I have started
Both deserve more attention, this is just a quick start. Further discussion of these entries should probably go in its own nForum thread.
]]>So a reduced space is one in which all the infinitesimal directions are “determined canonically by the non-infinitesimal directions”? And a coreduced space is one in which there are no nontrivial infinitesimal directions.
Yes, that’s a good way of saying it.
t’s kind of surprising to me that coreduced spaces even exist, much less that they’re good for anything.
Yeah, but the geometry of coreduced spaces is precisely D-geometry!
In slightly less modern language (of course) this is Grothendieck’s big insight that there is crystalline cohomology. Crystalline cohomology is the cohomology of coreduced spaces.
]]>Okay, this helps. So a reduced space is one in which all the infinitesimal directions are “determined canonically by the non-infinitesimal directions”? And a coreduced space is one in which there are no nontrivial infinitesimal directions.
Put that way, it’s kind of surprising to me that coreduced spaces even exist, much less that they’re good for anything. (-:
]]>Mike,
right, also I didn’t say this well in my previous message. Let me try again.
One way to say it in words is:
a reduced space has no infinitesimal thickening
in a coreduced space all infinitesimal neighbours are already equivalent.
To state these more explicitly, we need to declare with respect to which base geometry we speak of “infinitesimal thickening”. Assume that Cartesian spaces are declared to model non-thickened geometry, and consider infinitesimally thickened points . The former is reduced, the latter not. Neither of them is coreduced. Then
a reduced space is one that is faithfully probed already by only s;
a coreduced space is one whose probes by s are trivial (are probes by the point).
For instance we may start with the “formal manifold” , which is neither reduced nor coreduced. Then
Its reflection to the reduced spaces is (we “strip off” the infinitesimal thickening);
Then the further reflection of that to the coreduced spaces is the de Rham space , a curious non-classical version of the cartesian space which has the same smooth maps from other Cartesian spaces into it as has, but for which all maps out of are constant.
Finally, most classical is the description in terms of the dual function algebras:
a reduced space is the formal dual to an algebra which has no non-zero nilpotent elements;
a nontrivial coreduced space is never the formal dual of an algebra.
Does that help? Maybe we can find together a better way to say all this.
]]>@Urs: okay, I see. But now I am confused about what the difference is between a reduced space and a coreduced one?
]]>functor Π was indexed over the base topos
Can you detail this - and also the contents of your discussion on this. I think you didn’t ensure that it is indexed over the base; at least your references on this do not contain the word “indexed”.
I just thought about how we do escape the internal world over arbitrary bases (for we had ). So the following is still just thinking out loud:…Maybe we can we index Π by something like the “indexed (or parametrized) escape” defined by the hom functor such that is the usual global section functor, then we have of course e.g. . If we restrict to we get for every such (included into or which we consider as indexed over the base (in the way you discussed but which I don’t know yet)) something in and (namely or someting related) and the whole thing might be functorial in the way it should…
]]>Mike,
the ” lift of infinitesimal paths” that you have in mind seems to me to be an echo of the definition of formally smooth scheme. This is about the unit map being an epimorphism.
Or else, since you mention unique lifts, this is an echo of the definition of formally etale maps: a map is formally etale precisely if the unit naturality square
is a pullback. This is a condition on unique lifts of infinitesimal paths. In fact in smooth geometry this translates into
being a pullback, the definition of local diffeomorphisms.
]]>Woops, sorry, of course I had the wrong order, it’s , yes.
Concerning the other point: the only way to have a unique infinitesimal path is to have only the constant path. (On the othe rhand, if something is infinitesimally thickened “in all possible ways”, then it will have lots of infinitesimal paths, hence maps out of .)
The isomorphism
says that the only infinitesimal paths in are the constant paths.
And the interpretation of as forming paths in analogous to that of :
sends every contractible space to the point, , we think of this as forming the fundamental -groupoid of , in that all its points are identified, and in an essentially unique way.
Now similarly sends every infinitesimally thickened point to the point. We may hence think of it as identifying all infinitesimally close points by essentially unique paths.
In algebraic geometry often is written instead, for instance on p. 7 of
But for instance as in
one finds explicitly cohomology over expressed by “crystalline cohomology” which makes sheaves eqivariant with respect to infinitesimal neighbourhood relation (“infinitesimal paths”).
]]>Do you mean ? I thought , , and ?
I just don’t understand why forms “infinitesimal paths”. It seems to me that the isomorphism says that is uniquely thickened infinitesimally in all possible ways: for any thickening of , any morphism can be uniquely extended to a morphism . In fact, I don’t see how an operation which starts with can say anything at all about infinitesimal paths: doesn’t just forget all the infinitesimal thickenings?
]]>Why is that?
We are talking about the full subcategory defined by , right?
As that notation is supposed to suggest, this modality plays a role like that of the path -groupoid functor , only that it forms infinitesimal paths. Since defines discrete objects, would define an infinitesimal analog of this.
In fact sends a space to its “de Rham space” , which is the space obtained by identifying infinitesimal neighbourhood points, since
for a reduced space and an infinitesimally thickened point.
]]>by the structure which we see in the context of cohesion, it would make sense to call the “coreduced objects” “infinitesimally discrete”
Why is that? I’ve been thinking of the coreduced objects as being “infinitesimally thickened in all possible ways”; why is that “discrete”?
We have a long and honored tradition that I ignore this point and you keep pointing it out to me.
Hmm, I have completely lost my memory of this, then. Can you point me to our previous discussions? Does it help if the modality Π underlies a stable factorization system?
]]>Thanks, Mike, that sounds good, yes.
And yes, it’s good to look for words like “reduced objects” and “coreduced objects”. Maybe these are indeed the good words for these inclusions. I haven’t really thought about that.
The good thing about the term “reduced” is that it is standard in the sense of “reduced scheme”. The bad thing is that, on the other hand, already for schemes I don’t find it a very well-chosen term, but probably it’s a bad idea to fight this.
But then, by the structure which we see in the context of cohesion, it would make sense to call the “coreduced objects” “infinitesimally discrete” objects. If we did this then the reduced objects would be (not “infinitesimally codiscrete” but) something like “infinitesimally exceptionally discrete”, maybe. Ah, I don’t know.
how did we ensure that Π is indexed in the internal version of ordinary cohesion?
We have a long and honored tradition that I ignore this point and you keep pointing it out to me. :-) I guess the way to go is to make Frobenius reciprocity an axiom.
]]>Hmm, from the point of view of (but categorically, not type-theoretically yet) we have
Externally, this gives us the two adjoint strings, with and and and being fully faithful. Then we can ask the discrete objects to be an exponential ideal, which ensures that (or or $) preserves finite products.
If we had a way to ensure that the induced functor Π was indexed over the base topos, then the induced geometric morphism from H to the base topos would be locally connected, so that it would be connected (and hence Disc and Codisc would be fully faithful) as soon as Π preserves the terminal object. But I don’t see any way around asserting at least that explicitly. Actually, I don’t remember: how did we ensure that Π is indexed in the internal version of ordinary cohesion?
Now for Π preserving binary products, the corresponding subcategory that must be an exponential ideal is the image of in H, which is to say the subcategory of reduced objects that are the coreflections of discrete ones, or equivalently coreduced objects that are the reflections of discrete ones. And the coreduced objects are an exponential ideal in , so in particular inherit their exponentials from it. Thus, I guess the right condition is that the coreduced reflections of discrete objects are an exponential ideal. Does that seem right?
]]>I have fixed now the missing subscript.
But this also follows
Yes, if we already assume that we have declared to be cohesive, then – and this is what I said in #26 – it must be true that, in particular, and hence – even so there is no explicit condition on – that is fully faithful.
Conversely, if we decide to have explicit axioms only for to be cohesive and for to exist, then we need to add axioms which ensure that is also cohesive.
]]>Is the relevant reflective subcategory easily characterizable inside Hth?
The relevant subcategory are the discrete objects and in your post you describe how to do this - however the codiscrete case would be easier.Do you have a better idea now?
Why not? It seems clear to me that that’s what he meant.
I just wondered if this is full and faithful (since is not). But this also follows, as I see now, from the essential uniqueness of the terminal geometric morphism. At least if the base is .
]]>This seems to be a simpler condition
Is the relevant reflective subcategory easily characterizable inside ?
Don’t say the second is
Why not? It seems clear to me that that’s what he meant.
]]>There is a theorem of the Elephant (stated at reflective subcategory) saying that if the reflector of a reflective subcategory (of a cartesian closed category) preserves finite product iff the reflective subcategory is an exponential ideal in the category into which it embeds. This seems to be a simpler condition (at least simpler to state) than to require that in some factorization diagram some composite functor preserves finite products.
I didn’t find any particular use for the factorization of the unit of the adjunctions in question. Also in principle, it is not clear to me how to infer from a factorization of a natural transformation between functors any factorization property of the involved functors.
]]>