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Here are some thoughts. Since this is of interest to others here, and since I am busy and distracted with a bunch of things , I thought I’d bring it out in the open here such that others can chime in.
(Speaking of distraction: I am currently on my way to Pittsburgh, where I am visiting Hisham Sati. But instead of being over the Atlantic right this moment, I have only just arrived in London, over twelf hours after getting up this morning, and missed all further connections. Yesterday the cause would have been snow. Today its the trade unions’ strike.)
Our discussion revolves around how to connect in more detail the axioms for “structured geometry” and “étale $\infty$-groupoids” that are encoded
Recall that the observation is that in an ambient “gros” $\infty$-topos $\mathbf{H}_{th}$ equipped with differential cohesion
the Pi_inf-closed morphisms behave as formally étale morphisms, they satisfy most of the closure axioms that have been proposed for open maps and in particular satisfy the axioms for a geometry (for structured (infinity,1)-toposes);
the sub-slices $(\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X}$ on the formally étale morphisms into any given object $X$ are reflective and coreflective, and hence (as they thus inherit the Giraud axioms) are $\infty$-toposes.
We thus say that
for $X \in \mathbf{H}_{th}$, the étale slice $(\mathbf{H}_{th})_{/X}^{fet}$ is the petit oo-topos over $X$;
the structure of a groupoid object on $X$, hence of a 1-epimorphism $X_0 \to X$ such that this is formally étale is the structure of an étale $\infty$-groupoid on $X$.
The latter structure, should have an “external” incarnation as a locally $U$-modeled structured (infinity,1)-topos in a natural way, whose underlying $\infty$-topos is that $(\mathbf{H}_{th})_{/X}^{fet}$, or at least something closely related to it.
By the characterization of structured $\infty$-toposes as $\infty$-toposes over the classifying topos of the given “geometry”, which in the above case is just
$\mathbf{H}_{th} \simeq Sh_{canonical}(\mathbf{H}_{th})$itself, equipped with admissible := formally étale morphisms, all those slices and sub-slices above are canonically $\mathcal{G}$-structured, simply by their etale geometric morphism
$(\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_/X \to \mathbf{H}_{th} \,.$The remaining question is then how to use an étale 1-epimorphism $X_0 \to X$ to exhibit them as being locally representable as structured $\infty$-toposes. Or something like that.
First a few general comments which have not yet much to do with the external interpretation of (internal) étale objects:
First, the theory of structured (∞,1)-toposes gives a possible notion of a morphism of cohesive (∞,1)-toposes - namely a functor inducing a morphism of the canonical geometries living in these toposes (by “Urs’ yoga of cohesive hexagram’s” there are in fact three (or six (or multiples of three if we take into account higher order infinitesimals) candidates deserving the prefix “canonical”).
Another general question is if we find for every admissibility structure a cohesive (∞,1)-topos (or cohesive (∞,1)-category) whose class of étale (or $\Box$-closed for some idempotent monad or command $\Box$) morphisms. The extremes between such admissibility structures can range are: “every morphism is admissible” and “only equivalences are” and I think both situations can be arranged cohesively.
Hi Stephan,
thanks for reacting here! Let’s discuss a bit.
You write:
namely a functor inducing a morphism of the canonical geometries living in these toposes
Let’s see. So you are thinking of having two cohesive $\infty$-toposes $\mathbf{H}_{th}, \tilde \mathbf{H}_{th}$, equipped with differential cohesion $i : \mathbf{H} \hookrightarrow \mathbf{H}_{th}$ and $\tilde i : \tilde \mathbf{H} \hookrightarrow \tilde \mathbf{H}_{th}$. Then you are asking for geometric morphisms
$\phi \colon \mathbf{H}_{th} \to \tilde \mathbf{H}_{th}$that send formally étale morphisms in $\mathbf{H}_{th}$ to formally étale morphisms in $\tilde \mathbf{H}_{th}$.
Right? Is that what you are thinking of?
Hm, so let’s see, how would that work? First of all I suppose that we’d want to say that $\phi$ preserves differential cohesion if restricted to $\mathbf{H}$ it lands in $\tilde \mathbf{H}$, hence if we have a diagram of geometric morphisms
$\array{ \mathbf{H} &\stackrel{i}{\hookrightarrow}& \mathbf{H}_{th} \\ \downarrow^{\mathrlap{\phi|_{\mathbf{H}}}} && \downarrow^{\mathrlap{\phi}} \\ \mathbf{H} &\stackrel{\tilde i}{\hookrightarrow}& \mathbf{H}_{th} } \,.$in $\infty Topos$.
Hm, but that alone won’t guarantee that formally étale morphisms are taken to formally étale morphisms, will it?. There are some further conditions needed… It seems to me.
Another general question is if we find for every admissibility structure a cohesive (∞,1)-topos (or cohesive (∞,1)-category) whose class of étale (or □-closed for some idempotent monad or command □) morphisms.
This I doubt. After all, the differential cohesion sees formal étaleness. In the smooth case formally étale maps coincide with étale maps in good cases, but in general this won’t be the case.
Also, the axioms on an admissibility structure are rather weak, while the $\mathbf{\Pi}_{inf}$-closed morphisms in a differential cohesive $\infty$-topos satisfy a bunch of further closure properties. That’s not a proof of anything, but makes it seems unlikely that “every” admissibility structure comes from $\mathbf{\Pi}_{inf}$-closure.
Just to have it written here: Lurie defines a morphism of geometries to be a functor such that
it preserves finite limits
carries admissible morphisms to such.
it preserves admissible covers.
If we consider a cohesive ∞-topos $H$ (or some other ∞-category equipped with a sequence of adjoint monads) as a nice machinery to produce geometries. Then for this application I think we should just require of a morphism of such machineries that they induce morphisms of geometries (in Luries sense). As you mention in your second reply, I think that morphisms of $H$’s inducing inducing this is a weaker notion than requiring commutation of the various adjoint functors (which are not even present since we work with $\Box$es), but this is just a guess. Maybe we should also remind that adjunctions and mates live in 2-categories, so far we just considered level 0 and 1. But we (at least not me) have not in mind a particular proposition for which we are forced to define morphisms of $H$’s. What would we do with a “category of categories of generalized spaces”?
In general, and in particular in the specific example you mention, to satisfy condition 2 is implied by condition 1 and commutation of $\Box$ with the functor whereas just to require the restriction condition, I think, does not imply 2.
the axioms on an admissibility structure are rather weak, while the $\mathbf{\Pi}_{inf}$-closed morphisms in a differential cohesive $\infty$-topos satisfy a bunch of further closure properties.
Mike’s post to the ncafé explains the relation how orthogonal (reflective, stable) factorization systems are related to the sub notions of cohesive topos. So the question is how $\Box$-closed classes of morphisms are related to general factorization systems. The fact that in cohesive toposes more closure properties are satisfied is du to the fact that here colimits are universal and $H$ is locally presentable. So if we drop the assumption topos, I think one can nevertheless ask if one can obtain many factorization systems (on a category) by $\Box$-closed classes. I didn’t yet think through if this question makes sense.
But I rather should return to the question of how to externally interpret (internal) étale objects.
By the way: Did you think of the Eilenberg-Moore ∞-categories of the monads defining (infinitesimal) cohesion?
I think that morphisms of H’s inducing inducing this is a weaker notion than requiring commutation of the various adjoint functors
Well, if you want you can define that. But I am not sure why that is a good idea. The axioms of differential cohesion induce a wealth of extra structure. One out of many is the étale-structure. Throwing away everything else and just preserving this may be what one needs in some applications, but I doubt that this is what one wants generically.
Speaking of applications: where are you headed? What are you aiming for?
(which are not even present since we work with $\Box$es)
Well, no, (idempotent) monads are equivalent to the (reflective) adjunctions that induce them. It’s just the the former give a more internal picture. But if we speak of morphisms between the toposes here, we are speaking externally anyway.
morphisms of H’s inducing inducing this The axioms of differential cohesion induce a wealth of extra structure. One out of many is the étale-structure. Throwing away everything else and just preserving this may be what one needs in some applications, but I doubt that this is what one wants generically.
Of course. It may have been confusing that I wrote this under the title of this thread. I proposed to consider this notion of morphism (of geometries) only in case one concentrates just on étale classes and nothing else.
Speaking of applications
I would be interested in seeing applications which only work if we have all four adjoints at once.
(which are not even present since we work with $\Box$es)
(idempotent) monads are equivalent to the (reflective) adjunctions that induce them.
That’s why working with them simplified some exposition. I thought of taking seriously the game of formulating everything internal if possible. One example where I do not see how it is possible is the statement of the de Rham theorem.
But if we speak of morphisms between the toposes here, we are speaking externally anyway.
Yes, then it does not matter - except perhaps for clarity of the exposition.
Let me see: which specific expression in terms of the adjoints would you need that you are wondering about how to express in terms of the monads?
I would like to say “cohesive infinitesimal (and differential) neighborhood of a cohesive topos” in terms of (co)monads. To have it written here:
$\Box_i\dashv \bigcirc_i$ “$H_th$ is infinitesimal over $H$”
$\Box_c\dashv \bigcirc_c$ “$H$ is cohesive over $\infty Grpd$”
$(d^*\dashv d_*\dashv d^!)$ “$H_th$ is differential over $H$”
such that
$(\Box_t\dashv \bigcirc_t)$ “$H_th$ is cohesive over $\infty Grpd$” and
$(t_!\dashv t^* \dashv t_*)=(c_!i^*\dashv c^* i_* \dashv c_* i^!)$
so in the last condition adjoints of different monads are mixed.
(In correction of what I said above: The statement of the de Rham theorem does not require this factorization.)
I would like to say “cohesive infinitesimal (and differential) neighborhood of a cohesive topos” in terms of (co)monads.
Ah. Very good. That’s what I was getting at when talking about the geometric I Ching.
So we say
a differential cohesive structure is
two adjoint triples of (monadic, idempotent) modalities
$\Pi \dashv \flat \dashv \sharp$
and
$Red \dashv \Pi_{inf} \dashv \flat_{flat}$
such that…
and so we need to say what a set of sufficient conditions on this pair of adjoint triples is such as to guarantee that they come from a pair of adjoint quadruples characterizing differential cohesion.
One of these conditions we already considered:
there is a natural map $\Pi_{inf}(X) \to \Pi(X)$ which factors the units
$\array{ && \Pi_{inf}(X) \\ & \nearrow && \searrow \\ X &&\to&& \Pi(X) }$This is what is currently prop. 3.10.3.
Analogously we have that there is a natural map $\flat A \to \flat_{inf} A$ which factors the counits
$\array{ && \flat_{inf}(A) \\ & \nearrow && \searrow \\ \flat A &&\to&& A }$These two statement reflect the fact that the two adjoint triples are “consecutive”, exhibiting $\mathbf{H}_{th}$ as being essential over the base $\mathbf{H}$ which in turn is cohesive over the absolute base.
What somebody needs to think about is whether these two properties already characterize the relation between the two adjoint triples completeley, or if one needs to remember more.
Ok, I give it a first try:
If $\Box\dashv \diamond$ where $\Box$ is an idempotent monad on $C$ and $\diamond$ is an idempotent comonad on $C$ and $\Box^-$ and $\diamond^-$ are the “restrictions” of the monads to the reflective resp. coreflective subcategories $C^-_R$ resp. $C^-_C$ defined by $\Box$ resp. $\diamond$, and $\Box^+:=\Box + \Box^-$ as the “horizontal composition” . (We think of $C$ as $H_th$, $\Box$ as $\Pi_inf$ and $\Box^+$ as $\Pi$ but only assuming the adjoints I have made explicit). And we write $\bigcirc_l\dashv \bigcirc_r$ as the splitting of a (co)monad into its underlying adjunction. Then apparently $\Box_r=\diamond_l$ implies that $\Box_l\dashv \Box_r=\diamond_l\dashv \diamond_r$ (and we hence have a reflective-coreflective subcategory). But at the moment I don’t see why $\Box\dashv\diamond$ implies that $\Box_r=\diamond_l$. But this seems to be necessary for obtaining the reflective-coreflective adjoint triple.
edit: I also considered the calculus of mates but this is about “parallel” adjunctions and not about “consecutive” ones.
Hi Stephan,
some comments:
If $\Box \dashv \Diamond$ where □ is an idempotent monad
It’s “$\Diamond$” wich is usually used for the monad and “$\Box$” is used for the comonad, as in S4 logic.
$\Box^-$ and $\Diamond^-$ are the “restrictions” of the monads to the reflective resp. coreflective subcategories
But since the (co)monads are assumed to be idempotent, these restrictions are identities.
$\Box = \Box + \Box^-$ as the “horizontal composition”
This, too, is the identity, by idempotence.
$\Box^+$ as $\Pi$
So, no, this is not correct. Indeed, in you next line you introduce $\Box_l$, and that would be $\Pi$.
Then apparently $\Box_r = \Diamond_l$ implies that $\Box_l \dashv \Box_r = \Diamond_l \dashv \Diamond_r$.
No. Just look at what this would imply: by combining both statements you’d derived that $\Box_r = \Box_l$.
What is the reasoning behind “apparently” that you were making?
Hi Urs,
there is more than one misunderstanding in notation. For example $\Box_l$ cannot be $\Pi$ in my setup since $\Box_l$ is just a functor and $\Pi$ is a monad. I will send you what I meant in a pdf.
Okay, if you’d rather discuss this privately, then let’s move it there. I just thogutht it might be fun to discuss with more people. But of course since nobody else has chimed in yet, that may just be moot anyway.
Just some quick comments:
a) Let’s stick to the convention to write $\mathbf{\Pi} := Disc \circ \Pi$ for the monad, and $\Pi$ for the reflector that it comes from.
b) You don’t need to show how the single adjoint triple of monads encodes the adjoint quadruple (of course you may if you enjoy to) because this has already been done. What I was referring to above was the question of how to encode that two such adjoint triples come from two consecutive adjoint quadruples as in the axioms of differential cohesion.
Ah, I see an email from you arriving this moment…
What I tried is not that different from:
What somebody needs to think about is whether these two properties already characterize the relation between the two adjoint triples completeley, or if one needs to remember more.
except that I concentrated on $\mathbf{\Pi}\dashv \flat$, $\mathbf{\Pi}_inf\dashv \flat_inf$ since in the conditions
$\array{ && \Pi_{inf}(X) \\ & \nearrow && \searrow \\ X &&\to&& \Pi(X) }$ $\array{ && \flat_{inf}(A) \\ & \nearrow && \searrow \\ \flat A &&\to&& A }$$Red$ and $\sharp$ do not occur explicitly; so forget about them for the moment - if they exist they are uniquely determined by the adjoints we explicate. For the hypothetical reader from “outside” the conditions expressed by the above diagrams are precisely the definitions of the underlying adjunctions expressed by the universal property of the adjunction unit (first diagram) and counit (second diagram). The lack of $Red$ and $\sharp$ corresponds to that we (expect to) have not two underlying quadruples- but two underlying triples of adjoints.
We factor the (co)monads into the underlying adjoint functors which we denote by (the arguably improvable notation):
$\mathbf{\Pi}=\diamond_r\diamond_r^-\diamond_l^-\diamond_l$
$\flat=\Box_l\Box_l^-\Box_r^-\Box_r$
$\mathbf{\Pi}_inf=\diamond_r\diamond_l$
$\flat_inf=\Box_l\Box_r$
We observe that there are in total 8 different adjoints but we want two express that there are precisely 6 different adjoints - composing to two adjoint triples of (co)monads. So our desired condition to be imposed on the two adjoint (co)monads we started with should imply that some of our 8 adjoints are equal (for example $\diamond_r=\Box_l$ and $\diamond^-_r=\Box^-_l$).
So far we did not use that $\mathbf{\Pi}_inf\dashv \flat$.
It’s true that you can concentrate on just two adjoint pairs of monads. If you find necessary and sufficient conditions for them to be “consecutive”, then so are any adjoint triples that they sit in.
Now that you write
$\mathbf{\Pi} = \Diamond_r \Diamond_r^- \Diamond_l^- \Diamond_l$
I for the first time understand what you want to mean by the “$-$“-superscript. But that does not match what you said about these in #12 above.
So what’s your conclusion now? Can you state something like a proposition and a proof?
If it is clear when precisely one adjoint triple of functors comes from one adjoint pair of monads and how, I think it remains to show that the factorization of the unit resp. counit implies, that the reflective resp. coreflective subcategories lie in each other as expected. That e.g. the reflective case is consecutive one can see as follows: Let $\epsilon^+$ be the unit of an idempotent monad on $C$ factoring as
$\epsilon^+=(\phi \epsilon^i)=:(id_C\stackrel{\epsilon^i}{\to}rl\stackrel{\phi}{\to}RL)$where $rl$ denotes an idempotent monad on $C$. $rl$ defines a reflective subcategory $C^-\stackrel{\overset{l}{\leftarrow}}{\underset{r}{\hookrightarrow}}C$. $R$ necessarily factors through $r$ into a full and faithful functor $r^-$; i.e. $R=r r^-$. In other words $r^-$ defines a full subcategory $C^{--}\stackrel{r^-}{\hookrightarrow}C^-$. Since $R$ and $r$ have left adjoints, also $r^-$ has a left adjoint $l^{-}$ (namely the factorization of $L$ through $l$) and $C^{--}\stackrel{\overset{l^-}{\leftarrow}}{\underset{r^-}{\hookrightarrow}}C^-$ is reflective and $\phi$ conjugates $r^- l^-$ with the idempotent monad $rl$ (through which $RL$ was factored) to $RL=r r^- l^- l$.
I think more relations between the adjoint pairs of monads we do not need to fix to have them consecutive, since we know by the above argument how the monads factor and we know that they are adjoint. When these adjoint monads happen to come from two adjoint triples of functors I am not yet sure…
Hi Stephan,
you write above:
since we know by the above argument how the monads factor
But by which argument? Didn’t you start your last message with making this an assumption when writing:
Let $\epsilon^+$ be the unit of an idempotent monad on C factoring as …
? If I am missing your argument, it might help me if you write it out more formally.
Let me recall what the question is that I asked in #11:
Given two adjoint triples of (monadic, idempotent) modalities
$\Pi \dashv \flat \dashv \sharp$
and
$Red \dashv \Pi_{inf} \dashv \flat_{flat}$
(which, you may assume, we know come from two adjoint quadruples)
such that there is a natural map $\Pi_{inf}(X) \to \Pi(X)$ which factors the units
$\array{ && \Pi_{inf}(X) \\ & \nearrow && \searrow \\ X &&\to&& \Pi(X) }$then: does it follow that the underlying quadruples of adjoints are consecutive in the sense of differential cohesion? If not, what further condition on the modalities would need to be stated?
We don’t need to continue this if you’d rather stop. But if we continue, then this is the question that I am asking you to think about.
Hi Urs,
We don’t need to continue this if you’d rather stop. But if we continue, then this is the question that I am asking you to think about.
That I didn’t reply the whole day is just due to the fact that in my daily routine pm and am have changed.
But by which argument? Didn’t you start your last message with making this an assumption when writing:
That the unit of the monad factors is your assumption 2 above. I didn’t assume that $L$ factors through $l$. I assumed the following:
Given two idempotent monads whose units factor as
$\array{ && \Pi_{inf}(X) \\ & \nearrow^{\epsilon^i} && \searrow^\phi \\ X &&\stackrel{\epsilon^+}{\to}&& \Pi(X) }$Since $\Pi$ and $\Pi_inf$ are idempotent they factor each into a full and faithful functor having a left adjoint, say $\Pi=RL$ and $\Pi_inf=rl$.
$\array{ && rl(X) \\ & \nearrow^{\epsilon^i} && \searrow^\phi \\ X &&\stackrel{\epsilon^+}{\to}&& RL(X) }$Moreover $\Pi$ and $\Pi_inf$ define reflective subcategories, call them $C^{--}$ and $C^-$. Now the adjoint of the unit $\epsilon^+$ is the identity transformation on $L$. Since $\epsilon^+$ factors through $\epsilon^i$, by the universal property of adjoint functors also $id_L$ factors: If we denote the adjoint of a morphism $g$ by $\hat g$ we have $(\phi \epsilon^i)^\hat=\eta^+ L(\phi)L(\epsilon^i)$ is a factorization of $id_L$. Now here an argument is missing which implies that then $L$ factors through $l$ (I have to think about this). If this is true than we can set off at the point from my previous post.
For a conclusive proposition and a proof (also in regard to your question) I have to think further about it. If you have a guess you may say it.
But that does not match what you said about these in #12 above.
I am not sure if the misunderstanding because of my notation isn’t still virulent because it is still the same as in #12. The words “restriction” and “composition” were misleading since they are not restrictions or compositions; I used these words since the diagram defining consecutivity remind a bit of horizontal pasting diagrams.
That I didn’t reply the whole day
I didn’t mean to suggest that you are not reacting quickly enough. Not at all. I just wanted to say that we don’t need to continue this discussion in public if you’d rather not.
That the unit of the monad factors is your assumption 2 above.
This is not an assumption. This is either, depending on what we discuss, a consequence of two consecutive adjoint quadruples, or else an axiom. The question that I am asking is: is this axiom strong enough to recover the consecutiveness of the two adjoint quadruples?
Moreover $\Pi$ and $\Pi_{inf}$ define reflective subcategories, call them C −− and C −.
I vote for contuing to call them $\infty Grpd$ and $\mathbf{H}$. Not to proliferate notation and in order to ease communication.
For a conclusive proposition and a proof (also in regard to your question) I have to think further about it.
Do that! And let’s come back to this either if you have the solution or if you give up or if you have further questions first.
Coming to this thread very late, this looks like something I should be interested in, but I can’t understand what you are talking about because I haven’t yet understood the ’infinitesimal cohesion’ business. Can you state the main question you are trying to answer here in plain categorical language?
The question was: how do we say that an $\infty$-topos $\mathbf{H}_{th}$ is equipped with differential cohesion entirely in the internal language of $\mathbf{H}_{th}$?
Externally, the structure of differential cohesion on $\mathbf{H}_{th}$ is
a cohesive structure $(\Pi_{th} \dashv Disc_{th} \dashv \Gamma_{th} \dashv coDisc_{th}) : \mathbf{H}_{th} \to \infty Grpd$ on $\mathbf{H}_{th}$;
a fully faithful inclusion $(i_! \dashv i^* \dashv i_* \dashv i^!) : \mathbf{H} \hookrightarrow \mathbf{H}_{th}$ of another $\infty$-topos $\mathbf{H}$ which is itself equipped with cohesive structure over $\infty Grpd$.
Internally, the structure of differential cohesion on $\mathbf{H}_{th}$ is reflected by the existence of two adjoint triples of idempotent monadic (co)modalities
$\array{ comonad &\dashv& monad &\dashv& comonad &\dashv& monad \\ && \mathbf{\Pi} &\dashv& \flat &\dashv& \sharp \\ \mathbf{Red} &\dashv & \mathbf{\Pi}_{inf} &\dashv& \flat_{inf} }$satisfying some compatibility condition.
What I suggested Stephan might think about is which compatibility conditions on these two adjoint triples are necessary and sufficient to ensure that this internal data is equivalent to the external one.
(I guess all that one really needs to add is a condition that makes the composite top left adjoint preserve products.)
What compatibility between $(i_! \dashv i^* \dashv i_* \dashv i^!)$ and the cohesive structures of $\mathbf{H}_{th}$ and $\mathbf{H}$ is required?
Externally there is no further condition. Because by the essential uniqueness of the geometric morphism from any Grothendieck $\infty$-topos to $\infty Grpd$, the cohesive structure of $\mathbf{H}$ is the composite of $i$ with that of $\mathbf{H}_{th}$:
$(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) \simeq ( \Pi_{th}\circ i_! \dashv i^* \circ Disc_{th} \dashv \Gamma_{th}\circ i_* \dashv i^! \circ coDisc_{th})$and so there is nothing else to say.
But internally we cannot appeal to this essential uniqueness. So internally we need to add some condition that says that the full subcategory $\mathbf{H}$ of $\mathbf{H}_{th}$ defined by either of $(\mathbf{Red} \dashv \mathbf{\Pi}_{inf} \dashv \flat_{inf})$ is itself a cohesive $\infty$-topos.
Okay, now I understand the question, thanks!
So, left-exactness of $\mathbf{\Pi}_{inf}$ implies that that subcategory is a topos, and we of course have the composite adjoint string exhibiting it as a topos over the same base with both additional adjoints. So now I understand #24 too (where finite products include the terminal object, of course, to make the new topos connected). Is there a reason we can’t just stipulate directly that the relevant functor preserves finite products?
Is there a reason we can’t just stipulate directly that the relevant functor preserves finite products?
We can do that. And I’d think that’s what one should do.
I just felt that I didn’t (and still don’t) have the leisure (too many other things to concentrate on) to think through all this so as to convince myself of which theorems precisely to state confidently. So I was trying to motivate Stephan to do it! :-)
Very good! (-:
I also did other things, but now I will look into this.
What do you mean in #26 by $coDisc=i^! co Disc$? (Don’t say the second is $co Disc_th$.)
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.
This seems to be a simpler condition
Is the relevant reflective subcategory easily characterizable inside $\mathbf{H}_{th}$?
Don’t say the second is $co Disc_th$
Why not? It seems clear to me that that’s what he meant.
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 $i^!$ 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 $\infty Grpd$.
I have fixed now the missing subscript.
But this also follows
Yes, if we already assume that we have declared $\mathbf{H}$ to be cohesive, then – and this is what I said in #26 – it must be true that, in particular, $coDisc_{\mathbf{H}} \simeq i^! \circ coDisc_{\mathbf{H}_{th}}$ and hence – even so there is no explicit condition on $i^!$ – that $i^! \circ coDisc_{\mathbf{H}_{th}}$ is fully faithful.
Conversely, if we decide to have explicit axioms only for $\Gamma_{th} : \mathbf{H}_{th} \to \infty Grpd$ to be cohesive and for $i$ to exist, then we need to add axioms which ensure that $\Gamma_{th} \circ i : \mathbf{H} \to \infty Grpd$ is also cohesive.
Hmm, from the point of view of $\mathbf{H}_{th}$ (but categorically, not type-theoretically yet) we have
Externally, this gives us the two adjoint strings, with $Disc_th$ and $Codisc_th$ and $i_!$ and $i_*$ being fully faithful. Then we can ask the discrete objects to be an exponential ideal, which ensures that $\Pi_th$ (or $ʃ_th$ or $$_th$) 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 $i ^*\circ Disc_th$ 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 $H_th$, 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?
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.
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?
Why is that?
We are talking about the full subcategory defined by $\mathbf{\Pi}_{inf} = i^* i_*$, right?
As that notation is supposed to suggest, this modality plays a role like that of the path $\infty$-groupoid functor $\mathbf{\Pi}$, only that it forms infinitesimal paths. Since $\mathbf{\Pi}$ defines discrete objects, $\mathbf{\Pi}_{inf}$ would define an infinitesimal analog of this.
In fact $i_*$ sends a space $X$ to its “de Rham space” $\mathbf{\Pi}_{inf}(X) = X_{dR}$, which is the space obtained by identifying infinitesimal neighbourhood points, since
$Hom(U \times D, i_* X) \simeq Hom(i^* (U \times D), X) \simeq Hom(U , X)$for $U$ a reduced space and $D$ an infinitesimally thickened point.
Do you mean $i_* i^*$? I thought $Red = i_!i^*$, $\Pi_inf = i_* i^*$, and $\flat_inf = i_* i^!$?
I just don’t understand why $\Pi_inf$ forms “infinitesimal paths”. It seems to me that the isomorphism $Hom(i_! U \times D, i_* X) \simeq Hom(i_! U , X)$ says that $i_*X$ is $X$ uniquely thickened infinitesimally in all possible ways: for any thickening $i_!U\times D$ of $U$, any morphism $U\to X$ can be uniquely extended to a morphism $i_!U \times D\to i_*X$. In fact, I don’t see how an operation which starts with $i^*$ can say anything at all about infinitesimal paths: doesn’t $i^*$ just forget all the infinitesimal thickenings?
Woops, sorry, of course I had the wrong order, it’s $\mathbf{\Pi}_{inf} = i_* i^*$, 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 $D$.)
The isomorphism
$Hom(D, i_* X) \simeq Hom(*, X)$says that the only infinitesimal paths in $i_* X$ are the constant paths.
And the interpretation of $\mathbf{\Pi}_{inf}$ as forming paths in analogous to that of $\mathbf{\Pi}$:
$\mathbf{\Pi}$ sends every contractible space to the point, $\mathbf{\Pi}(\mathbb{R}^n) \simeq *$, we think of this as forming the fundamental $\infty$-groupoid of $\mathbb{R}^n$, in that all its points are identified, and in an essentially unique way.
Now similarly $\mathbf{\Pi}_{inf}$ 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 $\mathbf{\Pi}_{inf}(X)$ is written $X_{dR}$ instead, for instance on p. 7 of
But for instance as in
one finds explicitly cohomology over $X_{dR}$ expressed by “crystalline cohomology” which makes sheaves eqivariant with respect to infinitesimal neighbourhood relation (“infinitesimal paths”).
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 $X \to \mathbf{\Pi}_{inf}(X) = X_{dR}$ being an epimorphism.
Or else, since you mention unique lifts, this is an echo of the definition of formally etale maps: a map $f : X \to Y$ is formally etale precisely if the unit naturality square
$\array{ X &\to& \mathbf{\Pi}_{inf}(X) \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\mathbf{\Pi}_{inf}(f)}} \\ Y &\to& \mathbf{\Pi}_{inf}(Y) }$is a pullback. This is a condition on unique lifts of infinitesimal paths. In fact in smooth geometry this translates into
$\array{ T X &\to& X \\ \downarrow^{\mathrlap{d f}} && \downarrow^{\mathrlap{f}} \\ T Y &\to& Y }$being a pullback, the definition of local diffeomorphisms.
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 $S$ (for $\infty Grpd$ we had $\Gamma [X,Y]=hom(X,Y)$). 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 $\Gamma_Z X:=hom(Z,X)$ such that $\Gamma_* =\Gamma$ is the usual global section functor, then we have of course e.g. $\Gamma_{i^* Z}=\Gamma_Z\circ i_*$. If we restrict to $Z\in \infty Grpd$ we get for every such $Z$ (included into $H$ or $H_th$ which we consider as indexed over the base (in the way you discussed but which I don’t know yet)) something in $H$ and $H_th$ (namely $\Gamma_Z$ or someting related) and the whole thing might be functorial in the way it should…
@Urs: okay, I see. But now I am confused about what the difference is between a reduced space and a coreduced one?
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 $\mathbb{R}^n$ are declared to model non-thickened geometry, and consider infinitesimally thickened points $D$. 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 $\mathbb{R}^n$s;
a coreduced space is one whose probes by $D$s are trivial (are probes by the point).
For instance we may start with the “formal manifold” $\mathbb{R}^n \times D$, which is neither reduced nor coreduced. Then
Its reflection to the reduced spaces is $\mathbb{R}^n$ (we “strip off” the infinitesimal thickening);
Then the further reflection of that to the coreduced spaces is the de Rham space $\mathbf{\Pi}_{inf}(\mathbb{R}^n)$, a curious non-classical version of the cartesian space which has the same smooth maps from other Cartesian spaces into it as $\mathbb{R}^n$ has, but for which all maps out of $D$ 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.
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. (-:
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.
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.
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?
Crystalline cohomology is the cohomology of coreduced spaces.
But crystalline cohomology was devised to solve the problem of $p$-adic cohomology of varieties over an algebraically closed field of characteristic $p$. 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?
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 :-)
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?
The best text I know is the one by Jacob Lurie that I pointed to,
Notes on Crystals and algebraic D-modules (pdf)
I have added to differential cohesion somewhere statement and proof that the structured $\infty$-topos $Sh_{\mathbf{H}}(X)$ of sheaves on any object in differential cohesion is always locally $\infty$-connected.
But my battery is dying now…
now added the details. this proposition
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