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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 25th 2014

Thomas Holder has been working on Aufhebung. I have edited the formatting a little (added hyperlinks and more Definition-environments, added another subsection header and some more cross-references, cross-linked with duality of opposites).

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeNov 25th 2014

For completeness I have added an brief entry level of a topos and cross-linked a bit.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeNov 25th 2014
• (edited Nov 25th 2014)

Let me look at the most basic case. That $(\flat \dashv \sharp)$ is Aufhebung for $(\emptyset \dashv \ast)$ means that

$\sharp \emptyset \simeq \emptyset$

and that $\sharp$ is minimal with this property. What is the statement regarding conditions under which this is the case?

By adjunction and using that the initial object in a topos is stict, it follows that $\sharp \emptyset \simeq \emptyset$ is equivalent to the statement that the only object with no global points is $\emptyset$ itself, because

$(X \to \sharp \emptyset) \simeq (\flat X \to \emptyset) \simeq (\flat X \simeq \emptyset) \,.$

Is that automatic? (This is probably elementary, please bear with me.) And how to see or check that $\sharp$ is minimal with this property?

• CommentRowNumber4.
• CommentAuthorThomas Holder
• CommentTimeNov 25th 2014

I guess my wording in the entry is a bit careless, as I brought in the general context of categories of being in order to motivate the metaphysical lingo for $\empty\dashv\ast$ and its Aufhebung, though Lawvere proves this only in special cases e.g. in the 1991 ’Hegelian taco’ paper. so unless, a proof for the general case can be cooked up, I am afraid I’ve to row back a bit in the entry.

It maybe worth mentioning that a negative result of a (sufficiently) cohesive topos where this Aufhebungs relation fails to hold would also show the limits of the particular categorical interpretation of Hegel’s logic.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeNov 25th 2014

Let me connect this with something more familiar to me. To say that $(i^*,i_*):E\to F$ is an essential subtopos, with essentiality $i_!$, is almost the same as to say that $(i_!,i^*):F\to E$ is a connected local geometric morphism, except that we don’t ask $i_!$ to be left exact. Right?

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeNov 25th 2014

What does it mean for two levels to be “consecutive”?

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeNov 25th 2014
• (edited Nov 25th 2014)

re #3:

to say that in a more pronounced way, the condition $\sharp \emptyset \simeq \emptyset$ implies that the ambient $\infty$-topos has homotopy dimension $\leq 0$ relative to the sub-$\infty$-topos of $\flat$-modal objects. This is a property enjoyed by all models for which the $\flat$-subtopos is $\infty Grpd$ (by this proposition). So is this “Aufhebung” a strong classicality condition on the $\flat$-modal subtopos?

re #5: I’d say “yes, of course”, but that means I am probably missing some subtlety that you have in mind :-)

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeNov 25th 2014

Are you sure that $\lozenge _i$ and $\Box _i$ are named correctly? If the are intended to suggest the “possibly” and “necessarily” modalities of classical modal logic, then $\lozenge$ should be the monad and $\Box$ the comonad.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeNov 25th 2014
• (edited Nov 25th 2014)

re #6:

this refers to the Idea section here. I just meant two levels where one includes the previous one, I have changed it now to read as follows:

If for two levels $\mathbf{H}_{1} \hookrightarrow \mathbf{H}_2$ the second one includes the modal types of the idempotent comonad of the first one, and if it is minimal with this property, then Lawvere speaks of “Aufhebung” (see there for details) of the unity of opposites exhibited by the first one.

• CommentRowNumber10.
• CommentAuthorThomas Holder
• CommentTimeNov 25th 2014

@#8: the notation was chosen by me probably after having looked in which way the modal operators are adjoint to replace $L\dashv R$ of Lawvere (1989) or the $sk\dashv cosk$ used in KRRZ11. In complies broadly with the intuition to have necessity on the ’right’ side of being/sheaf but feel free to choose something more suitable.

• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeNov 25th 2014

Thomas, can you explain further your reasons for assigning $\lozenge$ and $\Box$ in that order? I don’t understand why necessity would be on the right side.

• CommentRowNumber12.
• CommentAuthorThomas Holder
• CommentTimeNov 25th 2014

My decision came from a short look at p.116 in the ’ generic figures’ book of Reyes et.al. where they define $\lozenge:=i^*\circ i_!$ and $\Box :=i^*\circ i_*$ in context of an inclusion of posets; in which case then $\lozenge\dashv \Box$ whereas the nlab entry defines the composition dually in reverse order. so this should probably be changed, sorry for the confusion!

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeNov 25th 2014
• (edited Nov 25th 2014)

I think Mike’s point is that for the notation here its not the side of the adjunction but whether we have a monad or comonad that decides what the box is. The box should be a comonad as for necessity. That is consistent with what you cite from Reyes, but not with using box for $\sharp$.

(We are lacking an $n$Lab page that says this comprehensively. Something should go at modal operator )

• CommentRowNumber14.
• CommentAuthorThomas Holder
• CommentTimeNov 25th 2014

I agree, my choice was hasty as I looked only for an adjunction between the modal operators without realizing that they were defined dually. Though I must admit that I found the choice esthetically pleasing - it would be good to have some suggestive symbols.

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeNov 25th 2014

I added a remark at modality in a section Notation. Everyone is kindly invited to expand and/or edit.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeNov 25th 2014

Back to the Aufhebung via global points:

so over every infinity-cohesive site it is true that $\sharp\emptyset \simeq \emptyset$, I suppose.

Since here indeed only the initial sheaf has no global points. (E.g. if a sheaf on $CartSp$ has no global points, that means it assigns the empty set to $\mathbb{R}^0$, but that means it must assigns the empty set to each $\mathbb{R}^n$ since there do exist maps $\mathbb{R}^0\to \mathbb{R}^n$).

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeNov 25th 2014

What I am talking about is now cleaned up here. Let me know if I am hallucinating (it’s late here).

• CommentRowNumber18.
• CommentAuthorMike Shulman
• CommentTimeNov 25th 2014

Ok, I’ve switched $\Box$ and $\lozenge$ in the entry.

• CommentRowNumber19.
• CommentAuthorMike Shulman
• CommentTimeNov 25th 2014

Now I’ve looked at your notation remark at modality; should we use $\bigcirc$ rather than $\lozenge$ at Aufhebung since it is right adjoint to $\Box$? (What’s the origin of $\lozenge$ being left adjoint to $\Box$ and $\bigcirc$ being right adjoint to it? I didn’t think that necessity and possibility were adjoint in either direction; are they?)

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeNov 26th 2014
• (edited Nov 26th 2014)

I did this just by exclusion principle: according to #12 the diamond has been used for the left adjoint, so the circle remains for the right adjoint.

But I am happy with any other convention/tendency.

• CommentRowNumber21.
• CommentAuthorDavid_Corfield
• CommentTimeNov 26th 2014
• (edited Nov 26th 2014)

Cancel what I just wrote having actually read what was written.

If that’s suggesting in modality that $\lozenge$ is less used by modal logicians for possibility, then that’s wrong. As far as I’m concerned it’s used as much as $\bigcirc$:

Also $\lozenge$ is used for a modality, in particular if it is left adjoint to a $\Box$.

On the other hand we could choose to make such a convention of distinguishing left and right adjoint monads. But let’s be explicit.

What about the differential cohesive comonads? Shouldn’t we have notation to distinguish left and right adjoints there too?

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeNov 26th 2014
• (edited Nov 26th 2014)

How do modal logicians choose between $\lozenge$ and $\bigcirc$? What does the choice indicate for them, if anything?

• CommentRowNumber23.
• CommentAuthorDavid_Corfield
• CommentTimeNov 26th 2014

As far as I’m aware it’s just an arbitrary choice of notation, like $\wedge$, $\cdot$, or & is a choice for conjunction.

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeNov 26th 2014

Thanks. I’ll rephrase the statement in the entry then. Just a moment…

• CommentRowNumber25.
• CommentAuthorDavid_Corfield
• CommentTimeNov 26th 2014

If anything I’d say $\lozenge$ is the more commonly used of the two. The SEP modal logic entry chooses it. So, all things equal, I’d have it as the left adjoint monadic modality.

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeNov 26th 2014

Okay, I have tried to edit accordingly here. Please feel invited to further edit if you see further need.

• CommentRowNumber27.
• CommentAuthorDavid_Corfield
• CommentTimeNov 26th 2014
• (edited Nov 26th 2014)

When adjunctions between modalities matter, there is a tendency …

My knowledge of these situations is so small I couldn’t speak of a tendency. I don’t know of any philosophers who have raised the issue of adjunctions. Do people know about computer science, etc.? Even Hermida as mentioned here seems to have the dichotomies lined up.

But what about my final point in #21? Should one use just $\Box$ for left/right adjoint comodalities $Red$ and $\flat_{inf}$? Sometimes people have used L and M for necessity/possibility.

• CommentRowNumber28.
• CommentAuthorUrs
• CommentTimeNov 26th 2014

Okay, I have changed “tendency to use” to “some authors use”. Then I added one more example, namely

• Gonzalo Reyes, A topos-theoretic approach to reference and modality, Notre Dame J. Formal Logic Volume 32, Number 3 (1991), 359-391 (Euclid)

(where the left part of cohesion appears in terms of adjoint modalities on p. 367).

Maybe “some authors” is just “Gonzalo Reyes”, though? Hermida does not seem to use the symbols from modal logic, does he?

Regarding notation for differential cohesion: I may not know what candidates there are from traditional theory. Maybe none? I don’t know. The situation of differential cohesion seems to have been missed, by and large.

• CommentRowNumber29.
• CommentAuthorDavid_Corfield
• CommentTimeNov 26th 2014
• (edited Nov 26th 2014)

Hermida uses $\Box$ and $\lozenge$ in the abstract, but then angled and square parentheses around a relation symbol.

• CommentRowNumber30.
• CommentAuthorUrs
• CommentTimeNov 26th 2014

Ah, right. So under this translation, does he have $\Box$ as the right adjoint?

• CommentRowNumber31.
• CommentAuthorDavid_Corfield
• CommentTimeNov 26th 2014

It seems that $[R]$ is always on the right.

• CommentRowNumber32.
• CommentAuthorUrs
• CommentTimeNov 26th 2014

Thanks. I see it now in remark 3.3. Good, so I’ll add that to the entry modality, too. So then maybe “tendency to use” is not that bad after all. Do we known an author who explicitly does not use that convention?

• CommentRowNumber33.
• CommentAuthorUrs
• CommentTimeNov 26th 2014

Hm, on the other hand, Hermida’s article has $\Box$ be a monad, not a comonad.

• CommentRowNumber34.
• CommentAuthorDavid_Corfield
• CommentTimeNov 26th 2014

Where? I see him say

Monadic interpretation of $\langle - \rangle$,

but that’s expected.

• CommentRowNumber35.
• CommentAuthorUrs
• CommentTimeNov 26th 2014

In that remark 3.3 he says that his $\langle R\rangle$ is the composition of pullback followed by dependent sum, and that his $[R]$ is the composition of pullback followed by dependent product. That makes $[R]$ a monad.

• CommentRowNumber36.
• CommentAuthorThomas Holder
• CommentTimeNov 26th 2014
• (edited Nov 26th 2014)

Sorry for bringing this up again, as the entry now has $\Box$ on the left, I would suggest wispering to use $\bigcirc$ on the right in the context of Aufhebung as this nicely suggests ’being’ unless this interferes negatively with the convention you just set up. As this is just a tiny detail I would very much like to avoid a discussion on this and propose to drop the subject when you don’t like the idea.

• CommentRowNumber37.
• CommentAuthorUrs
• CommentTimeNov 26th 2014
• (edited Nov 26th 2014)

Yes, that’s exactly what we seem to have agreed on over at modality – Notation.

I’ll change it at Aufhebung, too.

BTW, any comment on #17? We talked about it by email. It now seems to me that it does work for the “standard examples” of cohesion. But let me know if I am missing something.

• CommentRowNumber38.
• CommentAuthorThomas Holder
• CommentTimeNov 26th 2014
• (edited Nov 26th 2014)

Assuming that you use corollary II of Yoneda lemma in the last step of the proof of prop.1 at Aufhebung I think that prop.1 is ok.

As far as I can tell the proof of prop.1 uses only the strictness of $\empty$ and the rest is purely general using just the adjointness and the specific equivalences you assume, so this seems to be valid more generally for extensive cats with an adjunction that relates to $\empty$ in the appropriate way without being necessarily $\flat\dashv\sharp$, no !?

To see through prop.2 I’d need time to acquire some knowledge on infinity-cohesive-sites. This should be easier to see for someone with better grounding in the higher categorical point of view though, I guess.

In any case, it would be nice to have this somewhat more general result for the Aufhebung of $\empty\dashv {\ast}$.

• CommentRowNumber39.
• CommentAuthorUrs
• CommentTimeNov 26th 2014
• (edited Nov 26th 2014)

Thanks for the feedback. For the argument in prop. 2 the homotopy theory is irrelevant (I should have pointed to cohesive site!), you may just as well consider just plain presheaves. It’s meant to be a trivial argument: the site by assumption has a terminal object and there is a morphism from that terminal object to every other object. That implies that if a presheaf (of sets) assigns the empty set to that terminal object, it has to assign the empty set to every other object of the site, too.

• CommentRowNumber40.
• CommentAuthorUrs
• CommentTimeNov 26th 2014

I have edited the formatting of the central definition a bit in order to make the central ideas spring to the eye more vividly. Now it reads as follows:

Let $i,j$ be levels, def. \ref{Level}, of a topos $\mathcal{A}$ we say that the level $i$ is lower than level $j$, written

$\array{ \Box_i &\leq& \Box_j \\ \bot && \bot \\ \bigcirc_i &\leq& \bigcirc_j }$

(or $i\leq j$ for short) when every i-sheaf ($\bigcirc_i$-modal type) is also a j-sheaf and every i-skeleton ($\Box_i$-modal type) is a j-skeleton.

Let $i\leq j$, we say that the level $j$ resolves the opposite of level $i$, written

$\array{ \Box_i &\ll& \Box_j \\ \bot && \bot \\ \bigcirc_i &\ll& \bigcirc_j }$

(or just $i\ll j$ for short) if $\bigcirc _j\Box_i=\Box _i$.

Finally a level $\bar{i}$ is called the Aufhebung of level $i$

$\array{ \Box_i &\ll& \Box_{\bar i} \\ \bot &\searrow& \bot \\ \bigcirc_i &\ll& \bigcirc_{\bar i} }$

iff it is a minimal level which resolves the oppose of level $i$, i.e. iff $i\ll\bar{i}$ and for any $k$ with $i\ll k$ then it holds that $\bar{i}\ll k$.

=–

• CommentRowNumber41.
• CommentAuthorUrs
• CommentTimeNov 26th 2014

It seems to me that this here is proof that over a cohesive site $(\flat \dashv \sharp)$ is Aufhebung of $(\emptyset \dashv \ast)$. But check that I am not being stupid here:

+– {: .num_prop #OverCohesiveSiteBecomingIsAufgehoben}

###### Proposition

Let $\mathcal{S}$ be a cohesive site (or ∞-cohesive site) and $\mathbf{H} = Sh(\mathcal{S})$ its cohesive sheaf topos with values in Set (or $\mathbf{H} = Sh_\infty(S)$ its cohesive (∞,1)-topos ).

Then in $\mathbf{H}$ we have Aufhebung, def. \ref{Aufhebung}, of the duality of opposites of becoming $\emptyset \dashv \ast$.

=–

+– {: .proof}

###### Proof

By prop. \ref{OverCohesiveSiteBecomingIsResolved} we have that $(\flat\dashv \sharp)$ resolves $(\emptyset \dashv \ast)$ and so it remains to see that it is the minimal level with this property. But the subtopos of sharp-modal types is $\simeq$ Set which is clearly a 2-valued Boolean topos. By this proposition these are the atoms in the subtopos lattice hence are minimal as subtoposes and hence also as levels.

=–

• CommentRowNumber42.
• CommentAuthorUrs
• CommentTimeNov 27th 2014
• (edited Nov 27th 2014)

David,

coming back to #34-#35 and also to our discussion in person on the justification of calling a $\Box$-modality the “necessity” modality:

if we turn what Hermida does around, then everything makes sense to me.

Namely: if we

• read “necessarily” as the name specifically for the “for all”-operation turned into a comonad;

• read “possibly” as the name specifically for the “there exists”-operation turned into a monad;

then first of all $\Box$ is a comonad as desired and moreover then its interpretation as formalizing “necessity” is indeed justified:

for let $X$ be a “context” which you may want to think of as the “type of all possible worlds”, and if $P$ is a proposition about terms of type $X$, then

• the statement that “for $x \colon X$ it is necessarily true that $P(x)$” is just another way to say in Enlish that “for all $x \colon X$ it is true that $P(X)$”;

• the statement that “for $x \colon X$ it is possibly true that $P(x)$” is just another way to say in Enlish that “there exists $x \colon X$ such that it is true that $P(X)$”;

So, more formally, given the adjoint triple of dependent sum $\dashv$ context extension $\dashv$ dependent product

$\mathbf{H}_{/X} \stackrel{\stackrel{\sum_X}{\longrightarrow}}{\stackrel{\stackrel{X^\ast}{\longleftarrow}}{\underset{\prod_X}{\longrightarrow}}} \mathbf{H}$

then it makes justified sense to call the induced (co-)-monads

$(\lozenge_X \dashv \Box_X) \coloneqq ( X^\ast \underset{X}{\sum} \dashv X^\ast \underset{X}{\prod}) \colon \mathbf{H}_{/X} \to \mathbf{H}_{/X}$

“possibility” and “necessity”, respectively, because

• by the established interpretation of $\sum_X$ as “there exists $x \colon X$” we have that $\lozenge_X P$ holds for a given $x\in X$ precisely if it holds for at least one $x \in X$, hence if it is possible in the context $X$ for it to hold at all;

• by the established interpretation of $\prod_X$ as “for all $x \colon X$” we have that $\Box_X P$ holds for a given $x\in X$ precisely if it holds for all $x \in X$ hence if it is inevitable, hence necessary, in the context $X$ that if holds.

• CommentRowNumber43.
• CommentAuthorThomas Holder
• CommentTimeNov 27th 2014

Sorry for the interruption, but I had to fix the definition of Aufhebung as the minimality is defined relative to the usual order of the levels in the literature. For this I reused $\leq$ as the order of levels and tried to use predecessor-equality for the resolution relation but the local teX doesn’t know my dialect.

• CommentRowNumber44.
• CommentAuthorUrs
• CommentTimeNov 27th 2014

Let me see if I understand – you changed the notation in the first clause in the definition, but should you not also change it in the last line then? Maybe I am missing something.

By the way, regarding your email: the reason that I oriented these diagrams as I did as opposed to with levels going upwards is just because I wouldn’t know how to typeset the order relation going vertically.

• CommentRowNumber45.
• CommentAuthorDavid_Corfield
• CommentTimeNov 27th 2014

Urs, re #40, I was trying to do something like that back here.

I had the feeling Neel was onto something too a few comments later.

• CommentRowNumber46.
• CommentAuthorThomas Holder
• CommentTimeNov 28th 2014
• (edited Nov 28th 2014)

ad 44#: to me the definition appears to be as intended unless I forgot to switch notation at some place. The problem was previously it used $\ll$ for the minimality where it should have used the essential subtopos order which I then thought best to denote with the least marked $\leq$. I think I’ll try later to use $\sqsubset$ and $\lhd$ instead of $\prec$ and $\ll$ as they have rotated versions $\sqcup$ and $\bigtriangledown$.

• CommentRowNumber47.
• CommentAuthorUrs
• CommentTimeNov 28th 2014

Thomas, if it is as intended, then my question would be why this is intended. Why use in the last line of the definition not the same order relation as introduced in the first line? What’s the rationale for this?

• CommentRowNumber48.
• CommentAuthorThomas Holder
• CommentTimeNov 28th 2014
• (edited Nov 28th 2014)

I start to see your point here, the way I stated the definition is cooked up out of Lawvere 89 who defines this for graphic toposes and states everything in terms of the pretty-well behaved ideals in the underlying category, so I kept his terminology but looked for the definitions in terms of subtoposes to KRRZ11 which I’ve read in the way that $\prec$ and $\leq$ are two different things, but actually life becomes much easier if they are the same. I was already looking for a proposition showing their compatibility in order to prove that quintessential localizations are their proper Aufhebung. I guess you are right.

• CommentRowNumber49.
• CommentAuthorUrs
• CommentTimeNov 28th 2014
• (edited Nov 28th 2014)

It seems we’d have the freedom to make a definition either way, after all this is to formalize something that Hegel said, and depending on how we feel about that and depending on which mathematics we would like to see developed, we may feel that different definitions are appropriate. I am just thinking that in any case there should be some kind of justification. The order relation via subcollections of modal types seems well motivated, but then switching to a different relation along the way seems to call for a reason.

I’ll be happy with whatever definition leads to something interesting. In any case I am presently short of examples of subtoposes which would be in relation in one of the senses under consideration, but not in the other. Maybe we should try to get hold of some examples for such a phenomenon to get a feeling for which kind of subtlety the definition should try to take care of.

• CommentRowNumber50.
• CommentAuthorUrs
• CommentTimeDec 9th 2014
• (edited Dec 9th 2014)

This is long overdue: I have started at differential cohesion – relation to infinitesimal cohesion to add some first notes on how differential cohesion

$\array{ \Re &\dashv& ʃ_{inf} &\dashv& \flat_{inf} \\ && \vee && \vee \\ && ʃ &\dashv& \flat &\dashv& \sharp }$

induces “relative” shape and flat $ʃ^{rel} \dashv \flat^{rel}$ – such that when this extends to a level

$\array{ \flat^{rel} &\dashv& \sharp^{rel} \\ \vee && \vee \\ \flat &\dashv& \sharp \\ \vee && \vee \\ \emptyset &\dashv& \ast }$

then this level exhibits infinitesimal cohesion.

This is the case in particular for the model of formal smooth ∞-groupoids and all its variants (formal complex-analytic $\infty$-groupoids, etc.).

But I think in all these cases $(\flat^{rel} \dashv \sharp^{rel})$ does not provide Aufhebung for $(\flat \dashv \sharp)$.

This is because: for $X$ being $\flat$-modal hence being a discrete object, then maps $U \to \sharp^{rel} X$ out of any object $U$ in the site, which are equivalently maps $\flat^{rel}U \to X$, are maps out of the disjoint union of all formal disks in $U$ into $X$. These are again representable (we are over the site of formal smooth manifolds) and so these maps are equivalent to $X(\flat^{rel}U)$. But $X$ is discrete and hence constant as a sheaf on the Cahiers-site, and so these are equivalent to $X(\flat U)$ which in turns is equivalent to maps $\flat U \to X$ and hence to maps $U \to \sharp X$. So by Yoneda we conclude that $\sharp^{rel} X \simeq \sharp X$ in this case, but this is in general not equivalent to $X$.

• CommentRowNumber51.
• CommentAuthorThomas Holder
• CommentTimeDec 9th 2014
• (edited Dec 9th 2014)

So, do you mean that $\flat\dashv\sharp$ lacks Aufhebung ? If this is the case, couldn’t $\flat^{rel}\dashv\sharp^{rel}$ be regarded as a sort of homotopy approximation to the Aufhebung i.e. is construction of $\flat^{rel}\dashv\sharp^{rel}$ from $\flat\dashv\sharp$ sufficiently canonical !?

• CommentRowNumber52.
• CommentAuthorUrs
• CommentTimeDec 9th 2014
• (edited Dec 9th 2014)

I haven’t really thought about under which conditions $(\flat \dashv \sharp)$ has Aufhebung, all I meant to say here is that there is naturally this level $(\flat^{rel} \dashv \sharp^{rel})$ sitting above it, but that in the standard model this level, at least, is not even a resolution of $(\flat \dashv \sharp)$. There might be others that are, though.

And regarding this being canonical: the claim is that if differential cohesion is given, then $(\int^{rel} \dashv \flat^{rel})$ is canonically given. Think of it this way: differential cohesion is not really a level above cohesion, because of the “carrying” of the adjoints to the left. But it canonically induces $\flat^{rel}$ and so as a soon as that happens to have an adjoint $\sharp^{rel}$, then thereby it induces something that is a level over cohesion.

So maybe it’s good to think of this as some kind of “carrying back to the right”-operation, if you wish.

• CommentRowNumber53.
• CommentAuthorThomas Holder
• CommentTimeDec 10th 2014
• (edited Dec 10th 2014)

Unfortunately, I am not sufficiently familiar with your example to fully understand the details but I asked because I have the understanding that $\flat^{rel}\dashv\sharp^{rel}$ is actually a quality type hence its own Aufhebung !? So my idea is that in some sense it comes reasonably closest to provide Aufhebung for a level which otherwise lacks Aufhebung.

In order for this to make sense, one would probably like to demand that $\flat^{rel}\dashv\sharp^{rel}$ is the smallest quality type that subsumes $\flat\dashv\sharp$.

• CommentRowNumber54.
• CommentAuthorUrs
• CommentTimeDec 10th 2014
• (edited Dec 10th 2014)

Yes, right, in the given model the $(\flat^{rel} \dashv \sharp^{rel})$-level exhibits what “we” here had decided to call “infinitesimal cohesion”, which is essentially another word for what Lawvere had called a “quality type”.

And yes, I’d agree that it would make much sense to regard $(\flat^{rel} \dashv \sharp^{rel})$ as being the “next” level after $(\flat \dashv \sharp)$. After all, the sequence of inclusions of levels

$\array{ \flat^{rel} &\dashv& \sharp^{rel} \\ \vee && \vee \\ \flat &\dashv & \sharp \\ \vee && \vee \\ \emptyset && \ast }$

reads in words “a) the single point, b) collections of points, c) collections of points with infinitesimal thickening”.

And it seems clear in the model (though I’d have to think about how to prove it) that $\flat^{rel} \dashv \sharp^{rel}$ (when given by first-order infinitesimals) should be the smallest nontrivial level above flat $\flat \dashv \sharp$.

• CommentRowNumber55.
• CommentAuthorUrs
• CommentTimeDec 10th 2014
• (edited Dec 10th 2014)

Ah, I should be saying this more properly (and this maybe highlights a subtlety in language that we may have not properly taken account of somewhere else in the discussion):

in the topos over the site of formal smooth manifolds, the sub-topos of $\flat^{rel}$-modal types is “infinitesimally cohesive” in that restricted to it the map $\flat \to \int$ is an equivalence.

• CommentRowNumber56.
• CommentAuthorUrs
• CommentTimeJan 5th 2015

coming back to #50:

on the other hand, of course $\flat_{inf}$ does provide Aufhebung of cohesion in the sense that $\flat_{inf} \int \simeq \int$.

Of course this follows trivially here, since we are one step to the left and both of $\int$ and $\flat$ correspond to the same subcategory.

• CommentRowNumber57.
• CommentAuthorUrs
• CommentTimeJan 6th 2015
• (edited Jan 6th 2015)

coming back to #16:

I used to think and say that in the axioms of cohesion the extra exactness condtions on the shape modality seem to break a little the ultra-elegant nicety of the rest of the axioms. There is an adjoint triple of (co-)monads, fine… and in addtition the leftmost preserves the terminal object – what kind of axiomatics is that?!

But Aufhebung now shows the pattern: that extra condition on the shape modality

$ʃ {}_\ast \simeq {}_\ast$

is just a dual to the “Aufhebung of becoming”

$\sharp \emptyset \simeq \emptyset\,.$

Maybe a co-Aufhebung, or something.

• CommentRowNumber58.
• CommentAuthorUrs
• CommentTimeJan 7th 2015
• (edited Jan 7th 2015)

That makes me want to experiment with re-thinking about a possibly neater way of defining differentially cohesive toposes.

Something like this:

A differential cohesive topos is (…of course…) a topos $\mathbf{H}$ equipped with two idempotent monads $\sharp,\Re : \mathbf{H} \to\mathbf{H}$ such that there are adjoints $\int \dashv \flat \dashv \sharp$ and $\Re \dashv ʃ_{inf} \dashv \flat_{inf}$ (…but now:) and such that

1. (clear:) $ʃ {}_{\ast}\simeq {}_{\ast}$ and $\sharp \emptyset \simeq \emptyset$

2. (maybe:) $\flat_{inf} \Pi \simeq \Pi$ and $\Re \flat \simeq \flat$.

I need to go through what I have to see what the minimum needed here is. I certainly need $\flat_{inf} \flat \simeq \flat$ for the relative infinitesimal cohesion to come out right. Also $\Re \ast \simeq \ast$, which would follow from the above.

In any case, I feel now one should think of these axioms as describing a picture of the following form (the Proceß)

$\array{ &\stackrel{}{}&& id &\stackrel{}{\dashv}& id \\ &\stackrel{}{}&& \vee && \vee \\ && & \Re &\dashv & ʃ_{inf} & \\ &&& \bot && \bot \\ &&& ʃ_{inf} &\dashv& \flat_{inf} \\ &&& \vee && \vee \\ &&& ʃ &\dashv& \flat & \\ &&& \bot && \bot \\ &&& \flat &\dashv& \sharp & \\ &&& \vee && \vee \\ &&& \emptyset &\dashv& \ast & \\ }$

and the question is what an elegant minimal condition is to encode the $\vee$-s.

• CommentRowNumber59.
• CommentAuthorUrs
• CommentTimeApr 22nd 2015
• (edited Apr 22nd 2015)

I wrote out a more detailed proof of the statement here that the bosonic modality $\rightsquigarrow$ preserves local diffeomorphisms.

It seems the proof needs not just Aufhebung in that

$\rightsquigarrow \Im \simeq \Im$

but needs also that this is compatible with the $\Im$-unit in that $\rightsquigarrow$ sends the $\Im$-unit of an object $\stackrel{\rightsquigarrow}{X}$ to itself, up to equivalent

$\rightsquigarrow( \stackrel{\rightsquigarrow}{X} \stackrel{\eta_{\stackrel{\rightsquigarrow}{X}}}{\longrightarrow} \Im \stackrel{\rightsquigarrow}{X} ) \;\;\; \simeq \;\;\; ( \stackrel{\rightsquigarrow}{X} \stackrel{\eta_{\stackrel{\rightsquigarrow}{X}}}{\longrightarrow} \Im \stackrel{\rightsquigarrow}{X} )$

This is true in the model of super formal smooth $\infty$-stacks, so I am just adding this condition now to the axioms. But it makes me wonder if one should add this generally to the concept of Aufhebung, or, better, if I am missing something and this condition actually follows from the weaker one.

• CommentRowNumber60.
• CommentAuthorUrs
• CommentTimeJul 18th 2015
• (edited Jul 18th 2015)

added to the discussion here of Aufhebung $\sharp \empty \simeq \empty$ over cohesive sites pointer to lemma 4.1 in

• William Lawvere, Matías Menni, Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness, Theory and Applications of Categories, Vol. 30, 2015, No. 26, pp 909-932. (TAC)

which obverseves this more generally when pieces-have-points.

• CommentRowNumber61.
• CommentAuthorUrs
• CommentTimeNov 12th 2015
• (edited Nov 12th 2015)

At MPI Bonn this Aufhebungs-announcement is flying around (full pdf by In Situ Art Society).

I have take the liberty to add it to the entry Aufhebung.

• CommentRowNumber62.
• CommentAuthorDavid_Corfield
• CommentTimeNov 12th 2015

The quote from Hegel is 113 from here:

Cancelling, superseding, brings out and lays bare its true twofold meaning which we found contained in the negative: to supersede (aufheben) is at once to negate and to preserve.

• CommentRowNumber63.
• CommentAuthorMatt Earnshaw
• CommentTimeJul 4th 2016

the functors L and R must actually correspond to inclusions of disjoint subcategories

I take it this is not necessary in general, and is only true here since the composites $T \circ L, T \circ R$ are equal to the identity. I think that in general, when the composites are merely isomorphic to the identity, the subcategories have intersection given by the equalizer of the subcategory inclusions. Is this reasoning sound?

• CommentRowNumber64.
• CommentAuthorThomas Holder
• CommentTimeJul 5th 2016

From a quick glance at the section you link, to the disjointness property is meant only for the particular example. In the general case, the two subcategories are usually far from disjoint, in fact, one can think of the process of Aufhebung as a gradual level-to-level augmentation of the objects in the intersection, that contains the ’true thoughts’ where content (left inclusion) coincides with notion (right inclusion), starting from $0\cap 1=\emptyset$ up to $id_\mathcal{E}\cap id_\mathcal{E}=\mathcal{E}$.

1. Disclaimer: This is my first post here. I'm a philosopher by training, but I'm trying to teach myself enough category theory in order to eventually participate in the project in a meaningful way.

I wanted to offer a reason to reconsider the following quote from the entry:

"However critical these idealist systems had been to the claims of traditional metaphysics and epistemology they all left the traditional logic untouched and in this respect fell behind Leibniz. It is at this point where Hegel starts: he sets out to extend the critical examination of the foundations of knowledge to logic itself."

I can't speak for Fichte, but there's some good evidence that this is not the case for Kant. Dorothea Achourioti and Michiel van Lambalgen have an excellent paper titled 'A Formalization of Kant's Transcendental Logic' (http://philpapers.org/rec/ACHAFO) that makes the case that Kant's logic is actually geometric logic. The paper is aimed at philosophers, and so goes out of its way to not use category theory (it uses inverse limits of a system of inverse sets), but I have confirmed with van Lambalgen that this was a very deliberate de-categorification for philosophical consumption. I actually think that the semantics they provide is probably way too simple, but the exegetical case for interpreting Kant's logic as geometric is very good in my view, and the connection to Grothendieck topoi provides various interesting connections between Kant's transcendental psychology and mathematics/compsci (e.g., the idea that object synthesis is about tracking local invariants in varying heterogeneous data (intuition) attached to/organised by some base topology (forms thereof)).

If you want a potted example, Kant's distinction between negative judgments and infinite judgements makes perfect sense from the perspective of Steve Vickers's ideas about geometric type theory. For Kant, the crucial difference between the two (corresponding to propositional vs. predicate negation) is that the latter has existential import (relation to an object) and the former does not. Vickers's idea is that the infinitary disjunctions of geometric logic can be re-interpreted using typed existential quantification, and this is essentially what provides the existential import/objective validity of the infinite judgment from the Kantian perspective. Using a standard and overly simplistic example, judging that an extended object is non-blue excludes a determinate range of possibilities for that object, because the type of extended objects includes a colour attribute with a strictly delimited but potentially infinite range of possible variations. There's more that could be said here, but I think this indicates that Kant is more interesting for the nLab project than the above quote suggests.

I'll close with one more observation about the history of logic. I'm no expert, but the history of logic between Leibniz and Boole is really not very well articulated, and it seems that there are a lot of lines of influence that really aren't properly understood. It's all too easy to represent it as a sort of logical dark age where nothing took place. I honestly don't know how well Hegel would have understood Kant's perspective on logic, given that much of what we now understand comes from the lectures he gave on the topic, which wouldn't have been widely available at the time. However, it seems that Hegel and Schelling were significantly influenced by Ploucquet, from whom they seem to have contracted the idea of the reversibility of subject and predicate. Ploucquet is one of the significant figures recognised in the tradition post-Leibniz, but his work doesn't seem to be widely studied (it's all in latin, with no English translation from what I can gather) either on its own terms or as an influence on Hegel's logical views. It's worth avoiding 'logical dark age' rhetoric, whether it is between Aristotle and Frege or Leibniz and Hegel, as there are hidden lines of research and influence that were still in the process of uncovering and reconstructing.
• CommentRowNumber66.
• CommentAuthorDavid_Corfield
• CommentTimeJul 6th 2016

Welcome Peter, from another philosopher.

I’m very interested in your “potted example”. The reading group I belong to will soon be reading Paul Redding on the lost subtleties of negation possible in term logic in his ’Analytic Philosophy and the Return of Hegelian Thought’.

Have you written on the Kant-Vickers connection?

2. Hi David, thanks for the welcome. I still feel somewhat out of my depth here, even if I've been browsing nLab for at least a year at this point. I considered submitting something for your HoTT and philosophy workshop, but nervousness and the need to move to South Africa on short notice mitigated against it.

I haven't written anything up on the Kant-Vickers connection yet. It's part of a cache of insights I've been slowly tripping over since discovering the A&L paper on transcendental logic and getting serious about understanding sheaves and Grothendieck topoi. Strangely, I got into all this by trying to figure out what was so unsatisfactory about Brandom's formal incompatibility semantics, which has some Hegelian philosophical inspiration, but ends up being horribly gerrymandered into classical rubbish. One of the inconsistencies between Brandom's philosophical inspirations and his formalism (as pointed out to me by Ken Westphal) is precisely that it ignores Kant's distinction between negative and infinite judgments, collapsing everything back into (classical) propositional negation, and thereby being completely unable to account for the sorts of concrete incompatibilities between predicates he starts from (e.g., between blue and the various predicates - green, red, yellow, etc. - that fall under non-blue). There are more complaints I could make on this front, as learning what's wrong in Brandom's project has been quite enlightening, but I'll stop there.

The overarching project that these ideas belong to is the development of what I'm calling 'computational Kantianism', reading Kant's transcendental psychology as essentially already the project of AGI, using contemporary work in logic/maths/compsci to make sense of Kant and using Kant to provide some overarching structure connecting this same work. I gave a talk in Dublin recently that went over some of the overarching methodological ideas of this approach, but it's not written up. I'm due to give a seminar at a Summer school in NYC directly on computational Kantianism later this month, but I'm still working from notes, trying to condense things down into something tractable. I think one can draw a useful line between Kant's insistence on the primacy of judgment as a starting point for transcendental psychology and Harper's idea of computational trinitarianism, and that this (along with certain stories one can tell about subterranean connections between Kant and constructivism in the history of mathematics: i.e., Brouwer-Heyting, topos theory, Curry-Howard, HoTT) opens up the possibilities for reflecting back and forth between Kant and contemporary work I'm proposing. The really novel thing I think can be imported back from Kant is his conception of the relationship between mathematical and empirical judgment/cognition, which I think can be roughly understood through the duality between intuitionistic and co-intuitionistic logic (judgment) and computational data and co-data (cognition). I think this line of thinking inevitably leads you from Kant to Hegel, as the relationship between imagination and understanding needs to be supplemented with that between understanding and reason, but it's nice to start with Kant's emphasis on our (computational) finitude and build up to Hegel from there. It also has the advantage of suggesting how to extend computational trinitarianism beyond HoTT, insofar as one can project something like a co-intuitionistic dual of HoTT for empirical cognition. There's a few more things I could say about this, but I'll stop myself before I get further out of my depth!

Is this reading group online? If so, I'd very be interested in tagging along. One of the things reading the work being done here has taught me is how much was lost in the transition from the Aristotelian to the Fregean logical paradigm.
• CommentRowNumber68.
• CommentAuthorDavid_Corfield
• CommentTimeJul 6th 2016

Well you’d have been very welcome to the workshop anyway even if just to attend.

Interesting you mention Brandom. I jotted down a note which sounds like it may be in the same direction as your criticism. (By the way, I overlapped here with Ken Westphal for a couple of years.)

When you have something to read on what you describe in the 3rd paragraph, I’d be very interested.

Unfortunately, the reading group is just a bunch of us in a room thrashing things out.

3. I'll pass on to you anything that gets written up, or perhaps recordings of the talk if they appear first.

This is probably a silly question, but would the right etiquette be to edit your Brandom note if I wanted to respond to your thoughts on the topic?
• CommentRowNumber70.
• CommentAuthorThomas Holder
• CommentTimeJul 6th 2016
• (edited Jul 6th 2016)

@#65: Peter, concerning the passage you criticize though I would admit the sin of ’rhetoric’ I would deny the charge of ’dark age rhetoric’. The intention there is to provoke the reader with the idea that the ’logical lightweight’ Hegel outdoes Kant when it comes to having a critical attitude to traditional logic (note the implied suggestion to view Hegel as an expansion of the Kantian project to logic) and more generally that the postKantian philosophers of the 1790 were quick to dismiss practically all preceding ’dogmatic’ metaphysics but often took the traditional laws of formal reasoning for granted. I am probably willing now to exempt at least some of the postKantians from this charge since some of them felt indeed that the critical philosophy demanded a revision of traditional logic e.g. Salomon Maimon published a ’Neue Theorie des Denkens’ in 1794, Jacob Siegismund Beck, a mathematician from Kant’s inner circle published a ’Lehrbuch der Logik’ in 1820 introducing transcendental concepts into traditional logic, and Fichte in 1808 lectured on ’transcendental logic’ producing a large posthumously published text. The point is that Kant did not feel this need, the Jaesche-Logik contains the famous quote that general and pure logic is dull and short and basically a closed chapter since antiquity (or something like this), a quote that made it into the 1928 textbook of Hilbert and Ackermann who obviously did not think that chapter quite as closed neither did Leibniz before them.

This does not mean that the Jaesche-Logik is unimportant for the philosophy of logic nor that Kant’s transcendental logic cannot not fruitfully confronted with geometric logic, though calling the later ’Kant’s logic’ runs into the problem that Kant admitted traditional logic as a valid form of reasoning regardless of the objective content of the concepts employed i.e. to the extent that Kant ’had’ a logic traditional logic is a better candidate for it, in my view.

That Kant’s reasoning is inherently constructive is due to his attempt to model philosophy on the reasoning with constructions in Euclid’s geometry and the later is also an albeit remote source of geometric logic. Anyway, I am the last person to belittle Kant who is in fact one of the brightest stars on my philosophical firmament. In the later passages of the nLab article the continuity between Kant and the postkantian systems and Hegel is stressed. I generally find it useful to view thinkers like Kant, Fichte, Schelling and Hegel to be involved in a common project of transcendental philosophy which in my view is highly relevant to contemporary philosophy or cognitive science and deserves to be formalized by methods of modern mathematics.

Concerning Ploucquet, there is a German-Latin edition of his Logic by Michael Franz available as well as an article by Redding exploring the connection between Hegel and Ploucquet called THE ROLE OF LOGIC “COMMONLY SO CALLED” IN HEGEL’S SCIENCE OF LOGIC presumably available from his homepage as a preprint. In the context of cognitive underpinning for sheaf theory the link to the Petitot paper at Aufhebung might be interesting as well.

In any case, feel free to edit or expand Aufhebung when you cannot stomach certain passages. Additional insights or views are always appreciated and generally encouraged by the nLab!

• CommentRowNumber71.
• CommentAuthorDavid_Corfield
• CommentTimeJul 7th 2016

@#69 Peter, regarding editing pages, the general rule is that anything on nLab can be edited, with announcement here if substantial. For others’ private webs, I just correct typos.

As for my own, where that Brandom note is, I just collect together some sketchy thoughts there. I’d be happy to read your thoughts there, if you could designate them as yours.

• CommentRowNumber72.
• CommentAuthorDavidRoberts
• CommentTimeJul 7th 2016

@Peter,

there is a syntax for query boxes, we don’t use it much these days, with discussion being held here instead, but you could still use it. That would help separate your questions/comments from what David C wrote.

4. Thanks guys. I'll try to figure out the neatest way of commenting that separates out my comments from David's thoughts.

Thomas, thanks for the reference to the Redding paper, it looks great!

As for Kant/Hegel, I appreciate that what you're doing is provocation. It is of course incredibly important to get people to consider Hegel as an actual logician, and indeed, thereby to broaden their understanding of what logic is. I simply think it is worth pushing the envelope further back and giving Kant the same treatment, and indeed, thereby allowing us to draw more interesting *logical* lines from Kant to Hegel. Here are a couple further reasons to reconsider Kant's status as a logician, responding to your points:

1. One of van Lambalgen's students, Riccardo Pinosio, wrote an excellent MSc thesis reconstructing Kant's philosophy of mathematics based on the idea that transcendental logic is geometric logic (https://www.illc.uva.nl/Research/Publications/Reports/MoL-2012-13.text.pdf). It's really good, and it does an excellent job of dismissing Friedman's claim that Kant's constructivism was motivated by the inability of his logic to formulate ∀∃ judgements. There's more work to be done along these lines, as the role of temporality in Kant's account of construction is still not completely explicated, but I think it's a great start.

• CommentRowNumber74.
• CommentAuthorThomas Holder
• CommentTimeSep 30th 2019

Added a reference to the recent article by Marmolejo-Menni on “level $\epsilon$”.

• CommentRowNumber75.
• CommentAuthorThomas Holder
• CommentTimeDec 8th 2019