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In view of our discussion at adjoint modality I am wondering what the remaining adjoint modalities of differential cohesion would “mean” in the language of unity of opposites.
For the infinitesimal shape modality $\dashv$ infinitesimal flat modality the case seems to be clear, for as the names suggest, this is interpreted directly as the infinitesimal version of the shape modality $\dashv$ flat modality.
But the other one might be interesting, reduction modality $\dashv$ infinitesimal shape modality. Which “unity” does this express?
Notice that this is an interesting case. Since here the infinitesimal shape modality is both left as well as right adjoint, the argument at tangent cohesion applies even for non-stable homotopy types and we find that every at least once deloopable homotopy type $A$ canonically sits in two homotopy pullbacks of the form
$\array{ && \Pi_{dR} \Omega A && \longrightarrow && \flat_{dR}\mathbf{B} A \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{\theta_A}} && \searrow \\ \flat \Pi_{dR} \Omega A && && A && && \Pi \flat_{dR}\mathbf{B} A \\ & \searrow & & \nearrow & & \searrow && \nearrow_{\mathrlap{\Pi \theta_A}} \\ && \flat A && \longrightarrow && \Pi A } \,,$where now $\flat \dashv \Pi$ denote infinitesimal flat modality $\dashv$ infinitesimal shape modality. Here the bottom morphism is the “unity” morphism, in the sense of unity of opposites, which we probably want to say exhibits the “infinitesimal continuum”.
I need to think about what this is telling us for the infinitesimal shape modality.
To clarify what I am after:
the triple
reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality
translates in Science of Logic-terminology to
vanishing of infinitesimals $\dashv$ infinitesimal attraction $\dashv$ infinitesimal repulsion
(the leftmost term appears explicitly, the two terms on the right appear prominently without the “infinitesimal”-qualifier)
So the unity of the pair on the right in Science-of-Logic-language would be “infinitesimal continuum”, which makes good sense.
But what is the unity of the opposites “vanishing of infinitesimals” and “infinitesimal attraction”?
I’m not sure I have a very intuitive picture of coreduction. In this unity we have a space $X$ suspended between its reduction and coreduction? The former is like $X$ but with the infinitesimal fuzz around each point removed. What about the latter? What would happen, say, to an infinitesimally thickened point under $ʃ_{inf}$?
That’s the ’de rham stack’ object, in which all infinitesiml neighour points are made equivalent, more later
Now I have five minutes.
So on an infinitesimally thickened point $\mathbf{p}$ the reduction modality and the infinitesimal path modality both yield the ordinary point $\ast$.
In general, for $X$ a formal manifold, $Red X$ is the underlying ordinary manifold, whereas $\int_{inf}X$ is the de Rham space of the underlying ordinary manifold, which is obtained from $X$ by adding infinitesimal paths, hence equivalences between infinitesimal neighbour points.
The distinction is important, and is probably precisely what i am searching a good English word for. In synthetic differential geoemtry there are, in a way, two kinds of infinitesimals:
the “explicit” infinitesimals; these are what $Red$ removed.
in addition every ordinary space now has “implicit” infinitesimals; this is what $\int_{inf}$ removes.
Maybe we should say that $Red$ is the “moment of tangency”. (?)
One thought: As we discussed, in “Toposes of Laws of Motion” Lawvere proposes the term “atom” (actually a.t.o.m) for objects such as infinitesimally thickened points.
I am not sure if this is something one should follow, but maybe if we are searching for terms that might match, we might want to think about “moment of atomicity” or the like. As in “atoms of the continuum”.
I don’t know, I just brainstorming here. Just because it would be fun to have another nicely named triad here.
Maybe this bit here:
Die Atomistik hat den Begriff der Idealität nicht; sie faßt das Eins nicht als ein solches, das in ihm selbst die beiden Momente des Fürsichseyns und des Für-es-seyns enthält, also als ideelles, sondern nur als einfach, trocken Für-sich-seyendes.
Seems that a moment of Für-es-seyn is meant to be what makes a point more than a “dry” point and “reach out” into the rest of the world a bit. Maybe the reduction modality may usefully be identified with expressing a moment of Für-es-seyn.
(Can’t check the English translation right now, need to RUN now, I am late for our seminar…)
So this is
Remark: The Monad of Leibniz.
§ 348
We have previously referred to the Leibnizian idealism. We may add here that this idealism which started from the ideating monad, which is determined as being for itself, advanced only as far as the repulsion just considered, and indeed only to plurality as such, in which each of the ones is only for its own self and is indifferent to the determinate being and being-for-self of the others; or, in general, for the one, there are no others at all. The monad is, by itself, the entire closed universe; it requires none of the others. But this inner manifoldness which it possesses in its ideational activity in no way affects its character as a being-for-self. The Leibnizian idealism takes up the plurality immediately as something given and does not grasp it as a repulsion of the monads. Consequently, it possesses plurality only on the side of its abstract externality. The atomistic philosophy does not possess the Notion of ideality; it does not grasp the one as an ideal being, that is, as containing within itself the two moments of being-forself and being-for-it, but only as a simple, dry, real being-for-self. It does, however, go beyond mere indifferent plurality; the atoms become further determined in regard to one another even though, strictly speaking, this involves an inconsistency; whereas, on the contrary, in that indifferent independence of the monads, plurality remains as a fixed fundamental determination, so that the connection between them falls only in the monad of monads, or in the philosopher who contemplates them.
Let’s try to sort this out.
In §322 we get a clear prescription:
To be ’for self’ and to be ’for one’ are therefore not different meanings of ideality, but are essential, inseparable moments of it.
So we are to find an adjoint modality that expresses
$Ideality \;\colon\; FürSichSein \dashv FürEinsSein$(or possibly the other way around).
The complaint about Leibniz in §348, which we quoted above, makes pretty clear what this is about:
The atomistic philosophy does not possess the Notion of ideality; it does not grasp the one as an ideal being, that is, as containing within itself the two moments of being-forself and being-for-it, but only as a simple, dry, real being-for-self.
Here “atoms” really refers to the decomposition of the continuum into points (atoms of space) because in §337 it says:
The one in this form of determinate being is the stage of the category which made its appearance with the ancients as the atomistic principle, according to which the essence of things is the atom and the void.
But “The one” (The unit) with its repulsion of many I claimed before is well modeled by what $\flat$ produces, the underlying points, the atoms of space.
So in conclusion I get away here with the following: he says that it’s a defect of both the ancients as well as of Leibniz to consider atoms/monads/points which have no way to look outside of themselves into interaction with others, that instead one needs to characterized atoms/monads/points by the above adjoint modality.
Right?
So I conclude that the message is that Eins (“The One”/”The Unit”) is a notion of atom which is similar to what the ancients and Leibniz called atom/monad, only that it improves on that by keeping an additional “moment” which the ancients and Leibniz forgot to retain. Now recall Lawvere’s “Toposes of Laws of Motion” where “atom” is proposed to refer to, essentially, infinitesimally thickened points. Indeed, the “infinitesimal thickening” of the point has something to do with the point “coming out of itself”and interacting with other points.
Summing this up, it seems that the adjoint modality $Red \dashv \int_{inf}$ indeed captures some of this well.
My main problem with concluding this positively now is that it would imply that this adjunction itself would have to be called “ideality”. which seems a term that is out of place.
But the terms/ideas “reduction modality” and “being-for-one” do match well:
here is an infinitesimally thickened point with its infinitesimal antennas reaching out to test what’s going on around
$\array{ -- \bullet -- }$and here is the reduced point, all by itself/for itself
$\array{ \bullet }$Proceeding in this vein, here is another thought: in superalgebra it is traditional to refer to (functions on) those “antennas” above as being the “soul” of (the function algebra on) an infinitesimally thickened point, whereas the (functions on) the point itself is referred to as the “body”.
Maybe this standard mathematics “soul” terminology brings us into useful contact with Hegel’s “ideality” terminology.
Indeed, I can go to a supergeometry conference and advertize with a straight face that $Red \dashv \int_{inf}$ in smooth super-cohesion identifies “souls” in supergeometry, which it does. So why then not “ideality”. Seems to work well.
A lot to catch up on here. For the moment, Leibniz in The Monadology presents a world of windowless monads, which as such have no influence on one another. On the other hand, they do reflect what is occurring in other monads, and vary in this power of reflection. This corresponds to the scale from matter, through plants and animals, to humans and supernatural creatures. There is a Monad of monads which has perfect knowledge of other monads. Pre-established harmony ensures that monads reflections cohere.
So we’re seeing Hegel object to the lack of interaction between the windowless monads.
Re #11, you mean smooth super cohesion as infinitesimal over $Smooth \infty Grpd$? So the $\mathbf{Red}$ gets rid of any super-halo around points, and the $\int_{inf}$ does what?
So we’re seeing Hegel object to the lack of interaction between the windowless monads.
Yes, he says Leibniz’s monads are just “für sich” , but not “für eins”, whereas his “Eins” is both and thus has the required Ideality.
The orginal German here is maybe more evocative than the English translations that I have seen.
“für sich sein”, which standard sources translate as “being-for-self”, really means to be alone and undisturbed. One says: “Ich gehe jetzt in mein Büro, ich muss mal für mich sein um mich zu konzentrieren.” (I’ll retreat to my office to be alone and undisturbed.)
“für eins sein” which Hegel uses, is not proper German and probably wasn’t even at his time, but it is clearly meant to rhyme with “für sich sein”, and the similar phrase that does exist is “für einander sein”, which means: to be available for others.
Re #11, you mean smooth super cohesion as infinitesimal over Smooth∞Grpd?
Yes. In particular I am thinking of the full square of levels of cohesion (and the same kind of square exists for synthetic differential cohesion).
In this full picture indeed $\flat$ produces the underlying fat points, or as I now should maybe say, “die ideellen Einsen”, the points with their infinitesimal “halos”, their für-eins-sein.
And $Red$ gets rid of that “halo”, hence of the für-eins-sein, it retains only the reduced points, alone and undisturbed.
And $\int_{inf}$ makes infinitesimally close points equivalent. You may usefully think of $\int_{inf} X$ as the sub-thing in the fundamental path infinity-groupoid of $X$ which consists only of the infinitesimal paths.
By the way, did you agree with what you might call a full cube of levels of cohesion, as here, by making present or absent each of those three kinds of dimension?
Yes, I did and do! Forgot to reply back then. Did just now.
In this full picture indeed $\flat$ produces the underlying fat points, or as I now should maybe say, “die ideellen Einsen”, the points with their infinitesimal “halos”, their für-eins-sein. And $Red$ gets rid of that “halo”, hence of the für-eins-sein, it retains only the reduced points, alone and undisturbed. And $\int_{inf}$ makes infinitesimally close points equivalent.
All this talk of monads is making me interpret this in the sense of nonstandard analysis. Given a (say) uniform space $X$, $Red(X)$ is the subspace of $X$ consisting of the standard points; while $\int_{inf}X$ is the quotient space of $X$ consisting of the monads (which are also called halos, according to Wikipedia). We have a map $Red(X) \to \int_{inf}X$, mapping each standard point to its monad; this is injective iff $X$ is $T_1$ and surjective iff $X$ is compact. I can't tell if this fits with what you're talking about or not.
Ah, thanks for saying this. It would be nice to see these modalities realized in nonstandard anlysis.
I didn’t know that use of “monad” and “halo” in non-standard analysis. That fits nicely into our recent Hegelian exegesis. I’d want to add some comments on this later.
But you may have to help me a bit, or generally we might want to write out more details of the relevant ingredients on nLab pages.
Notably, can we define $Red$ and $\int_{inf}$ in detail as (co-)mondas on a category of maybe uniform spaces? Maybe it’s obvious to you. But then maybe you’d enjoy briefly writing it out on the nLab somewhere?
I wonder if the difference between nilpotent and invertible infinitesimals has something to do with the not altogether straightforward relationship between category theory and model theory, that we once discussed. I mean, you never make any use of that very model-theoretic transfer principle with nilpotent infinitesimals, do you? Perhaps this relates to the difference between geometric and logical morphisms in toposes.
It's not obvious to me that $Red$ and $\int_{inf}$ (on uniform spaces as I defined them) are monads or comonads, but probably because I'm not comfortable enough with nonstandard analysis. I'm only pretty sure that they're functors!
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