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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 20th 2013
• (edited Oct 21st 2013)

Since people keep asking, here a note on examples of what Lawvere calls “quality types” in the context of the examples of cohesion that we care about.

According to Axiomatic Cohesion,

• a quality type is a category with a full embedding of the base topos such that the embedding has a a left and right adjoint which coincide.

• an extensive or intensive quality on a cohesive topos is a factorization of $\Pi$ or $\Gamma$, respectively, through a quality type.

Here are two examples of quality types that I see:

1. smooth super infinity-groupoids are cohesive over super infinty-groupoids and over plain $\infty$-groupods, where super infinity-groupoids are a “quality type” over $\infty$-groupoids;

2. a tangent (infinity,1)-topos $T \mathbf{H}$ is cohesive over $T \infty Grpd$ and over $\infty Grpd$ where in turn $T \infty Grpd$ is a “quality type” over $\infty Grpd$ by

$(\Pi \dashv \Disc \dashv \Gamma \dashv coDisc) \;\colon\; T \mathbf{H} \stackrel{\overset{T \Pi}{\longrightarrow}}{\stackrel{\overset{T Disc}{\leftarrow}}{\stackrel{\overset{T \Gamma}{\longrightarrow}}{\underset{T coDisc}{\leftarrow}}}} T \infty Grpd \stackrel{\overset{base}{\longrightarrow}}{\stackrel{\overset{0-bundle}{\leftarrow}}{\underset{base}{\longrightarrow}}} \infty Grpd$
• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeOct 21st 2013

Does the extensive/extensive distinction as applied to qualities, have anything to do with that applied to quantities? At Bill Lawvere, it describes a workshop paper ’Extensive and intensive quantities’ as

a proposal for a general axiomatization of homotopy/homology-like “extensive quantities” and cohomology-like “intensive quantities”.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 21st 2013
• (edited Oct 21st 2013)

I am not sure if I see what the extensive/intensive terminology is good for. The articles that cite it which I have seen I didn’t find entirely convincing. I may well be missing something, and anyone who disagrees with me is kindly invited to educate me.

What I do see now are two interesting examples of classes of cohesive $\infty$-toposes which are cohesive over base toposes that themselves are “quality types” over the absolute base topos $\infty Grpd$.

Why it would be useful to say “quality type” here I do not see, to be frank, but I do suppose the way that super-cohesion, tangent cohesion and also synthetic cohesion all involve squares whose horizontal maps are cohesion adjunctions and whose vertical maps are “qualtity type” adjunctions might be relevant for something.

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeOct 21st 2013

They are, naturally enough, Hegelian terms. I can’t say I’m much the wiser, however, reading Sec. 735 of the Science of Logic.

• CommentRowNumber5.
• CommentAuthorZhen Lin
• CommentTimeOct 22nd 2013

I hate to be the voice of dissent, but I doubt Lawvere was thinking about $\infty$-examples. Perhaps the motivating example of cohesion is instead the quadruple adjunction

$\pi_0 \dashv sk_0 \dashv (-)_0 \dashv cosk_0 : Set \to Kan$

which satisfies all the axioms in [Axiomatic cohesion], including “pieces have points” and “pieces of powers are powers of pieces”. Thus, $\operatorname{Ho} Kan$ is a quality type over $Set$ and the “shape” functor $\pi : Kan \to \operatorname{Ho} Kan$ is an extensive quality. On the other hand, the “canonical intensive quality type” consists of the full subcategory of discrete Kan complexes, and the functor that sends a Kan complex to its “canonical intensive quality” is none other than $(-)_0$. It’s quite clear in this example the sense in which extensive qualities capture “form” but not “substance” while intensive qualities capture “substance” but not “form”.

What I have trouble understanding is why Lawvere defines quality type the way that he does. It’s certainly an interesting fact that $Set$ is embedded in $\operatorname{Ho} Kan$ as a full subcategory that has a simultaneous reflector–coreflector, but what does it mean?

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeOct 22nd 2013
• (edited Oct 22nd 2013)

He certainly did not have infinity-examples in mind. And I don’t see what you are dissenting with.

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeOct 22nd 2013

It’s pretty clear at cohesive topos and cohesive (infinity, 1)-topos what’s due to Lawvere, and what’s an extension to the infinity case. Perhaps the second entry could be more explicit in attributing the extension to Urs.

I suppose it might be the case that the concept of ’quality’ seems less convincing as a term when transposed to the infinity case.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeOct 22nd 2013
• (edited Oct 22nd 2013)

Right, and so maybe if it didn’t become clear, let me say it this way:

from just reading Lawvere, it didn’t become clear to me why cohesion is an interesting concept. That became clear to me once I saw that in $\infty$-topos theory cohesion implies a long list of structure of differential geometry and differential cohomology. In fact I first had these, then heard Johnstone speak about cohesion, and then realized that these four adjunctions is also what Lawvere had been talking about.

Similarly, the use of the definition of “quality type” does not become clear to me from Lawvere’s writing. It might be a super-interesting axiom, but right now I don’t yet see how a similar list of nice properties is implied by it. But that’s why I pointed out the above examples: there we do have quality types, and these examples are clearly interesting. So maybe one can find something here derived from the notion of “quality type”. But I am saying: I don’t see it yet-

• CommentRowNumber9.
• CommentAuthorZhen Lin
• CommentTimeOct 22nd 2013

To me at least, the notion of “quality type” is still incomplete. As it stands now, there are some strange examples, even in ordinary category theory: for instance, presheaves on a category with a zero object, e.g. the Cauchy completion of a monoid with zero.

Perhaps the main idea is supposed to be captured by the “every piece has a unique point” axiom: in good situations, we will be able to decompose every object as a base-indexed coproduct of point-like objects, where by ‘point-like objects’ I mean those that have a unique point; so in some sense a “quality type” is supposed to be a category of sets of “structured points”. But if that’s the case, why aren’t sets equipped with a group action an example of a “quality type” — beyond the boring fact that the left and right adjoints of $\Delta$ don’t coincide? I feel that I am missing a subtle difference.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeOct 22nd 2013
• (edited Oct 22nd 2013)

I think, too, it should be about points.

Let me stick to my examples above, smooth super $\infty$-groupoids and cohesive parameterized spectra. Both of these are $\infty$-toposes over $\infty Grpd$. But regarding them as such discards a whole lot of interesting information that one would want to keep.

And the reason is that “the cohesive points” in both cases are interesting. One would like to have geometric realization and underlying homotopy types also with respect to these cohesive points, not just with respect to the plain base points.

By construction we know in these two examples that they factor their cohesion through “quality types” which remember this structure about points. But of course we could and probably should turn this around and ask for any given cohesive $\infty$-topos whether it sits over an interesting category of cohesive points. That’s what theorem 2 of “Axiomatic cohesion” is getting at.

One thing that makes me wonder, though, is that in my examples above the cohesive points are both “extensive quality type” and “intensive quality type”. The “extensive/intensive” seems to be entirely already encoded in the $\Pi/\flat$.

So right now I’d tend to want to speak about something like “point cohesion” or “cohesive base points” or the like.

If we insisted that it is the nature of the points that gives the objects their “quality” then I could see how “quality type” sounds good. But on the other hand part of the point of cohesion is that there may be important qualities of cohesive objects which are not supported on points.

So I am not sure about “quality types”. But “point cohesion” and that theorem 2 look to me like something worth exploring further.

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeOct 22nd 2013
• (edited Oct 22nd 2013)

Is there something to be said then about how to strip away cohesion successively? So we might go

1. $T(Super \infty Grpd) \to Super \infty Grpd \to \infty Grpd$, or
2. $T(Super \infty Grpd) \to T(\infty Grpd) \to \infty Grpd$.

Starting with $T(Smooth Super \infty Grpd)$ there are yet more paths. And then we’re interested in finding chains of final steps to $\infty Grpd$ amounting to a quality?

I guess paths 1 and 2 make a commutative square. Perhaps there’s a maximal quality type for each cohesion.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeOct 23rd 2013

Yes, I guess this is the kind of story that one should tell.

I will be busy today, but maybe tonight I’ll find the energy to start an entry on quality types…

• CommentRowNumber13.
• CommentAuthorDavid_Corfield
• CommentTimeOct 23rd 2013

Perhaps there’s something like a factorization system at play.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeOct 23rd 2013
• (edited Oct 23rd 2013)

Hm, maybe. But I think in the case you considered the iterative structure is given by the tower of base toposes: $T \mathbf{H}$ is cohesive over $T \mathbf{Base}$ which is point-cohesive over $\mathbf{Base}$; but now if $\mathbf{Base}$ itself is point cohesive over $\infty Grpd$ then one gets such a cube of point cohesions.

I should write this out…

• CommentRowNumber15.
• CommentAuthorDavid_Corfield
• CommentTimeOct 23rd 2013

I meant maybe in general for any cohesive $A$ over $B$, could there be an intermediate largest $C$, such that $C$ is point cohesive over $B$, while there’s no point cohesiveness left for $A$ over $C$? If so, what would be a name for the latter type of morphism, orthogonal to point cohesion?

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeOct 23rd 2013

Right, so this $C$ is what Lawvere calls the universal intensive quality type of the cohesion of $A$ over $B$, I suppose.

What I was saying is that we get more such $C$ if we play this game in turn over a tower of base toposes $B$.

• CommentRowNumber17.
• CommentAuthorDavid_Corfield
• CommentTimeOct 23rd 2013

Oh, so there are two universals. I wonder how the universal intensive and universal extensive qualities relate.

The canonical extensive quality could be called “form” (it seems to neglect substance). By contrast, the canonical intensive quality deﬁned below is called “substance” and seems to neglect form. (This contrast is related to the contrast between “in the large” and “in the small” in traditional analysis. The Poincar´e conjecture expresses the idea that the two canonical qualities could jointly reﬂect isomorphisms.)

So this means there are two extreme ways to factor $A$ cohesive over $B$. Do they lead to incomparable factorization systems?

I wonder what the right environment is for cohesion. If it’s a relative notion, $A$ is cohesive over $B$, do we form the $(\infty, 2)$-category of such pairs. If $A$ is cohesive over $B$, and $C$ over $D$, what would be the morphisms? Some kind of cohesive geometric morphism from $B$ to $D$, with a way to compare the base changed $A$ with $C$, or the other way.

• CommentRowNumber18.
• CommentAuthorDavid_Corfield
• CommentTimeOct 23rd 2013

Presumably the Poincare conjecture appears here because homotopy and cohomology of the 3-sphere are being compared. So maybe intensive/extensive qualities do relate to intensive/extensive quantities:

a proposal for a general axiomatization of homotopy/homology-like “extensive quantities” and cohomology-like “intensive quantities”.

But why “two canonical qualities could jointly reﬂect isomorphisms”? If two things agree as to ’form’ and ’substance’, they’re the same? If $A$ and $B$ are both cohesive over $C$, and their universal intensive and extensive qualities are the same, then they are the same?

• CommentRowNumber19.
• CommentAuthorZhen Lin
• CommentTimeOct 23rd 2013

I don’t think cohomology is an “intensive quality type”: by its nature it reflects “form”, not “substance”. It’s not clear to me that Lawvere’s canonical quality types are actually universal, but it seems reasonable to suppose that “form” and “substance” jointly reflect isomorphism.

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeOct 23rd 2013

Yes, as Zhen Lin says, Lawvere treats the extensive and the intensive case rather asymmetrically.

What he calls the universal extensive quality type of a cohesive topos is the construction which we recently discussed in the context of global equivariant homotopy theory:

take a cohesive topos $\mathbf{H}$ to the category whose objects are those of $\mathbf{H}$ and whose spaces of morphisms are the image under $\Pi$ of the internal hom in $\mathbf{H}$.

In cohomological/homotopy-theoretic terms this “universal extensive quality” is the $\infty$-category where cohomology is modified from having coboundaries between cocycles to having “concordances” between cocycles.

• CommentRowNumber21.
• CommentAuthorDavid_Corfield
• CommentTimeOct 23rd 2013

If the tangent $(\infty, 1)$ category of $C$ is to be thought of as having objects which are spaces that are infinitesimal thickenings of those of C, what would the tangent $(\infty, 2)$ category of the $(\infty, 2)$-category $\infty-Toposes$ be like?

For an object in the latter which is cohesive, so modelled on a certain kind of point, shouldn’t we expect its tangent fibre to include cohesive $(\infty, 1)$-toposes modelled on thickened points?

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeOct 23rd 2013
• (edited Oct 23rd 2013)

I have now added to cohesive topos a quick remark on “quality types” in the section Examples – Infinitesimal thickening.

I think we should say “infinitesimal cohesive toposes” for these. I used to use “infinitesimal cohesion” for infinitesimally thickened cohesion but later I switched to calling this differential cohesion, so really “infinitesimal cohesion” should be used for something else.

And it should be used for the infinitesimal fibers of differential cohesion over the underlying cohesion:

it’s like this: from differential cohesion we have extensions

$\mathbf{H}_{inf} \hookrightarrow \mathbf{H}_{th} \longrightarrow \mathbf{H} \,.$

For instance

The stable cohesion thing is different, but similar

• the tangent cohesion $\mathbf{H}_{th} \coloneqq T \mathbf{H}$ sits over $\mathbf{H}$ with fiber $\mathbf{H}_{inf} \coloneqq Stab(\mathbf{H})$.

In all cases $\mathbf{H}_{inf} \hookrightarrow \mathbf{H}_{th}$ is what Lawvere would call “quality type” but which I think we should call “infinitesimal cohesion”

$infinitesimal\;cohesion \hookrightarrow differential\;cohesion \longrightarrow cohesion$
• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeOct 23rd 2013

okay, I have started something at infinitesimal cohesion.

Just a start, not more.

• CommentRowNumber24.
• CommentAuthorDavid_Corfield
• CommentTimeOct 24th 2013

Presumably the upper arrows of the lower horizontal adjoints should be $\Pi$ not $\Gamma$ in the first two examples of infinitesimal cohesion.

I still can’t see the general story. You seem to be saying something like cases of differential cohesion are (a kind of) product of an infinitesimal and an ordinary cohesion. But is the decomposition unique? Presumably not, since one could opt for trivial infinitesimal cohesion.

But maybe one should never just speak of $A$ as differentially cohesive over $B$, since it’s a relative concept in terms of an existing cohesion. So $A$ is differentially cohesive with respect to cohesive $C$ over $B$.

Isn’t it that when $A$ is cohesive over $B$, there are different infinitesimals part of the cohesion. When one of these is used to quotient out to give $C$, we say $A$ is differentially cohesive with respect to cohesive $C$ over $B$. But there’s a maximal infinitesimal part, corresponding to Lawvere’s universal quantity, which when factored out should give a thoroughly non-infinitesimal cohesion.

I feel sure we should be moving up a level. If not my wild speculation at #21, don’t you think there has to be something interesting to say at the $(\infty, 2)$-level?

• CommentRowNumber25.
• CommentAuthorUrs
• CommentTimeOct 24th 2013

Thanks, fixed that typo.

And yes, I don’t claim to understand the general story here. So far all I am saying is: look, these are all examples that involve “quality types”, namely infinitesimal cohesion, and they all play along with differential cohesion in a natural way.

Yes, it would be good to understand more systematically what the big story is here.

• CommentRowNumber26.
• CommentAuthorDavid_Corfield
• CommentTimeOct 24th 2013

I think it’s just something offending my sense of symmetry in those diagrams. It’s only ’diff.cohesion’ relative to the horizontal.

Above I was talking in terms of factorization, but one could easily turn it around and speak of extension. Then the question is, for a particular brand of cohesion, what are the possible infinitesimal extensions. Are they simply ’transported’ from infinitesimal extensions of $\infty Grpd$?

There must be a way of thinking about the modalities in the context of the $(\infty, 2)$-category $\infty Topos$. Then how to characterise in that context what is special about the modalities of infinitesimal extensions, namely, their equivalence?

• CommentRowNumber27.
• CommentAuthorUrs
• CommentTimeOct 24th 2013

Concerning symmetry of the diagram: you should probably better think of the squares as being one horizontal line sitting all over $\infty$-groupoids.

Concerning systematics: I just remembered that I once already claimed that for the case of synthetic cohesion the infinitesimal cohesion is given by pushout of $\infty$-toposes (in the $(\infty,1)$-category of toposes). This is prop 4.5.35 on page 544 here: cohesivedocument131024.pdf

• CommentRowNumber28.
• CommentAuthorDavid_Corfield
• CommentTimeOct 24th 2013

Which shows me a typo:

Definition 4.5.34. InfSmothLoc

and there are three other ’smoth’s in the document.

Then, the sections at the top of p. 537 are in a strange order.

Hmm, isn’t there a whiff of tangency in the air in Proposition 4.5.3.

• CommentRowNumber29.
• CommentAuthorDavid_Corfield
• CommentTimeOct 24th 2013

p. 540 acylic

p. 545

“4.5.3.2 - Lie group cohomology” and “4.5.3.3 1-Lie algebroid cohomology” are duplicated.

• CommentRowNumber30.
• CommentAuthorUrs
• CommentTimeOct 24th 2013

Thanks for all this!! Fixed now.

• CommentRowNumber31.
• CommentAuthorDavid_Corfield
• CommentTimeOct 24th 2013

Re the pushout mentioned in #27, do you get similar pushouts replacing the right-hand vertical arrow by

• $Smooth Super \infty Grpd \to Super \infty Grpd$
• $SmoothParameterizedSpec \to ParameterizedSpec$? etc.

Aren’t we just seeing the infinitesimal extensions being transported along a cohesive morphism?

• CommentRowNumber32.
• CommentAuthorUrs
• CommentTimeOct 24th 2013

Yes, right, the same argument applies to supergeometric cohesion. I’ll add that now.

Concerning the “just transported along”: heuristically, yes; formally if we mean by it those pushouts. But beyond that I currently have not good formulation of this. Though there might well be a simple one.

• CommentRowNumber33.
• CommentAuthorDavid_Corfield
• CommentTimeOct 25th 2013
• (edited Oct 25th 2013)

Some rambling while I should be doing something else.

Your comment in #22 seems telling

The stable cohesion thing is different, but similar

Maybe that’s linked to the second of my suggestions in #27 not being a pushout.

Is there something about $T$ and the $J^n$ not being functors on $\infty Topos$, yet approximating $(-)^I$ which is, while $T(\infty Grpd)$ is infinitesimally cohesive, but $(\infty Grpd)^I$ is not? You can’t have both in that case.

But is there a ’super’ functor, that acts, e.g., to send $\infty Grpd$ to $super \infty Grpd$, $smooth \infty Grpd$ to $smooth super \infty Grpd$, etc.? Or maybe one that strip away the ’super’, like a truncation.

• CommentRowNumber34.
• CommentAuthorUrs
• CommentTimeOct 25th 2013

David, you are after something here, but I need to postpone thinking about this for the moment. I’ll get back to you on this later. Sorry.

• CommentRowNumber35.
• CommentAuthorDavid_Corfield
• CommentTimeOct 25th 2013

As they say in Australia, “No worries”. I might start a thread to jot down some ideas, but don’t feel under any pressure to contribute.

By the way, some more typos

is is

p. 433 Proposition 3.10.50. An infinitesimal cohesive 1-groupoid, def. 3.10.49

but this def. defined a formal cohesive 1-groupoid.

p.434 “The bottom row shows the supergeometric refinement of this situation.” Should be top row

p. 442 just by the axioms of cohesion of and stability.

• CommentRowNumber36.
• CommentAuthorUrs
• CommentTimeOct 25th 2013

Thanks for the typoses! Fixed now in my local copy. Will upload latest version a little later tonight. Thanks a whole lot for your all your input!

• CommentRowNumber37.
• CommentAuthorThomas Holder
• CommentTimeAug 20th 2014

In view of the recent Menni papers I thought it good to assemble some information on quality types in a separate entry quality type. Of course, a lot more could and should be added!

• CommentRowNumber38.
• CommentAuthorUrs
• CommentTimeAug 20th 2014
• (edited Aug 20th 2014)

Thanks!

I have taken the liberty of adding the following, please let me know if you’d disagree.

In the very first paragraph after “only a qualitative analysis” I have added:

More geometrically, a cohesive topos which is a “quality type” exhibits infinitesimal cohesion in that the points-to-pieces transform for each object is an isomorphism.

Then in the following sentence “The primary example is the homotopy category.” I have added “that Lawvere had in mind” to make it read

The primary example that Lawvere had in mind is the homotopy category.

This is because in view of the fundamental geometric example of sheaves on infinitesimally thickened points and its role in Lie theory, I find that the homotopy category is a rather exotic example.

This relates to the general fact which I find puzzling, that in Lawvere’s texts there is made little connection between synthetic differential geometry in the sense of the Kock-Lawvere axioms and cohesion. What I find is the archetypical example of what all of cohesion and “quality type” is about is the cofiber sequence of cohesive toposes

$Sh(SmthMfd) \hookrightarrow Sh(FormalSmoothMfd) \to Sh(FormalPoints)$

where the first inclusion exhibits differential cohesion, the topos in the middle is the Cahiers topos modelling snythetic differential geometry, the right map exhibits it as being cohesive not only over the point but over infinitesimal points (which is the key ingredient for the observation at differential cohesion and idelic structure) and finally the rightmost topos is infinitesimally cohesive, hence a “quality type”.

• CommentRowNumber39.
• CommentAuthorThomas Holder
• CommentTimeAug 21st 2014
• (edited Aug 21st 2014)

my apologies for my partial reworking (some of it happened before I came across your comment here - I also thought I would have posted a comment yesterday evening but that doesn’t seem to have uploaded).

In short, my intentions with the entry are mainly pedagogical, meaning I want to stick to Lawvere as close as possible (extensive 1-cats etc.) and make things as 1-pedestrian as possible as the concepts are sufficiently mysterious even then and easier to pick up for applications in cognitive science&philosophy etc. in this form.

the homotopy category is probably not the most important example from a practical point of view but I’d like to stress it because it points to the concept’s origin in reflections on homotopy theory by Lawvere in the 80s. Actually, the real primary example seems to be the simplest topos of graphs in the 89 paper that serves as a quality type for the cohesive reflexive graph topos and puzzled Lawvere as being neither gros nor petit (these contrasts show up as contrast QT vs sufficient cohesion and QT vs. pure variation in 2007). The smooth example hinted at in 2004 then was presumably decisive to elevate QT into a defined concept and let the axioms for gros revolve around it in 2007.

I have rearranged your geometrical view and spelled out the details more in detail. I hope that my intuitions concerning ’infinitesimal cohesion’ as sort of minimal spatiality is acceptable, otherwise feel free to adapt or improve the passage.

As far as I can see, infinitesimals enter the picture merely in the role of generating the discrete base topos via infinitesimal constancy $X=X^T$ as an infinitesimal internal ’foundation’. I am afraid that I don’t have to say anything enlightening about this at the moment. The somewhat strange appearance of Euler infinitesimals at the beginning of the 2008 paper on rig geometry where presumably the right adjointness of core varieties yield QT via their classifying toposes point hopefully to some deeper connection.

As I think of QT as a sort of qualia i.e. they are on the subjective-epistemological side of things, I would expect them to appear as fiber rather than as base in an exact sequence, so would this comply to your intuitions what extends what in the above cofiber sequence ?

• CommentRowNumber40.
• CommentAuthorUrs
• CommentTimeAug 21st 2014
• (edited Aug 21st 2014)

Thanks for the detailed reply and for displaying such detailed knowledge of the history of the development of the axioms.

I am all for the pedagogical exposition, the “$\infty$“-signs at infinitesimal cohesion could just as well be omitted as far as the definition is concerned. In view of pedagogy I find that explaining that $\Pi \simeq \Gamma$ characterizes infinitesimal objects (those with a single point in each piece) is less mysterious than saying that this means that “only qualitative analysis” is possible for these ojects. In fact I don’t understand this remark on qualitative analysis! Also I think right in the vein of avoiding $\infty$-toposes in motivating “quality types” that reference to the homotopy category is less elementary than reference to infinitesimal objects.

For these reasons I still think that if the aim is to explain the actual geometric meaning of “quality types”, then the example $Sh(FormalPoints)$ is the most basic one.

On the other hand of course if the aim is to record Lawvere’s way of arriving at these concepts, then clearly a different route is indicated.

With respect to this though I feel the urge to make an edit on the following, but I’ll refrain before hearing back from you:

Where in the new version of the entry you say that quality types were

intended as ingredients to a synthetic homotopy theory

it’s hard not to notice that with homotopy type theory there has meanwhile arrived an actual synthetic homotopy theory which axiomatizes not just “the homotopy category” – which is well known to be pathological – but its actual homotopy theory. With all due respect to Lawvere, it seems therefore to me that the aim of characterizing homotopy theory via cohesion is dubious, and that instead it seems compelling to interpret cohesion inside synthetic homotopy theory.

• CommentRowNumber41.
• CommentAuthorThomas Holder
• CommentTimeAug 21st 2014
• (edited Aug 21st 2014)
Concerning QT the intuitions I try to insinuate in the idea section is 1. quality (types) point toward qualia or perceptual 'qualities' (casual remarks of Lawvere don't leave doubt that they are intended for this - they are a way to extract information by reflecting a complex object in a simpler object) 2. they propose a general model of such attending to restricted features (inspired by replacing homeomorphism class by homotopy type, and classifying dynamical systems by topological invariants of their flow instead by integration etc.) 3. the collapsed cylinder that diagrammatically underlies the precise definition ultimately derives from the reflection on homotopy theory in the 80s. To call this 'synthetic HOT' is probably misleading and his ideas on homotopy have certainly not caught on among homotopy theorists though Lawvere on several occasions used the cat-list to make his ideas on Sset more widely known in the hope that this might be taken up as research program. From a soft science applications-perspective I find these somewhat vague ideas more suggestive than to roll over the tender sprouts of philosophical insights with the machinery of HOTT, obviously this might look different for a string theorist or working homotopy theorist.

I also think that this broad, somewhat vague perspective stays close to Lawvere's POV who calls it a QT and not a quintessential localization, makes repeatedly the link to quality in colloquial sense and stresses the role of the Hurewicz cat. I have no problem with bringing out more clearly the very important points-pieces role in this although I am not quite sure whether this a felicitous intuition for every QT over extensive cats and for this reason I demoted it after the localization remark.

To sum up, the three intuitions about QT are intended to be suggestive and not pedagogical. The pedagogical intention would be to stick more closely to and make more explicit Lawvere's ideas 1-verbatim. Concerning HOT we probably disagree, although it is somewhat uncertain what the putative synthetic HOT of Lawvere actually amounts to. Nevertheless we should grant Lawvere that his seemingly untimely view turns out to contain deep insights into the concept of space and the role it plays in cognition.
• CommentRowNumber42.
• CommentAuthorUrs
• CommentTimeAug 21st 2014

May I read this as saying that you are against adding a comment and pointer to homotopy type theory where you mention “synthetic homotopy theory”?

To me it’s curious where you identify tender sprouts and where their perils.

• CommentRowNumber43.
• CommentAuthorThomas Holder
• CommentTimeAug 21st 2014
• (edited Aug 21st 2014)
Well, 'sprouts&perils': my idea is that the real potential of QT is for cognitive science. It is a concept where 'qualia', the transcendental role of space in cognition (what it means in general to organize data into a space) and extraction of qualitative information from such space are brought together in a for the moment only vaguely suggestive way. However vague Lawvere threw in the ball in 2007 as a determinate concept.

In order to fulfill such hopes the concept has to be available to researcher in CS or philosophy. In my view a purely technical approach to the concept would be an obstacle to achieve this. Imagine a cognitive scientist reading through Johnstone 1996 - would s/he come up with the idea that this a potential contender for 'qualia' ? I have no doubt that topologists can deal with quintessential localizations without ever having heard of Lawvre's 2007 when they pop up in their practice, I have more doubts, that CS ever reaches n-category theory without passing through more tangible realms before.

If you like to link to HOTT where you see 'synthetic homotopy theory', feel free! Calling Lawvere's ideas 'sHOT' is meant to make more knowledgable people wonder how they might relate to homotopy theory: always connect!
• CommentRowNumber44.
• CommentAuthorUrs
• CommentTimeAug 21st 2014

Okay, thanks. I have added a footnote.

Regarding “qualia”: as I mentioned before, I happen to be somebody who does not understand why quality types are regarded as

a concept where ’qualia’, the transcendental role of space in cognition (what it means in general to organize data into a space) and extraction of qualitative information from such space are brought together

Could you explain this?

• CommentRowNumber45.
• CommentAuthorThomas Holder
• CommentTimeAug 22nd 2014
I wish I could do this! alas, the sprouts are tender indeed. But let me expand a bit why QT might be connected to 'qualia' or 'qualities' in a cognitive sense:

1. Lawvere wrote a message to conceptualmathematics in about 2008 that qualities like 'red' are intended to be modeled by ex/intensive qualities in the technical sense. I doubt that this claim is entirely thought through. Nevertheless due to his epistemological realism Lawvere has thought a lot about such things and e.g. has a sort of categorical theory of induction (most explicitly sketched in the objective logic 1994 paper, and hinted at in last year's Lawvere-fest abstract); so such a suggestion should probably not be dismissed easily. also from a more down to earth view QT occured as qualities in the analysis of cohesive graph toposes.

2. An example of 'qualitative analysis' in the non-technical sense in cogntion was given by Jean Petitot, who derived the categorization of the phonic continuum into the discrete vowel space with methods from catastrophe theory, suggesting the jet space example that Lawvere mentions. In general, mathematical models of cognition often take the form of neural networks which more or less are a form of field theory and differential geometry. So the idea that methods for QFT provide synthetic methods for cognition is not entirely farfetched. The highly abstract form DG takes in cohesion I refer to as 'transcendental', meaning 'concerning the conditions of possibility of perception' , a Kantian term.

(to be continued, I have to run!)
• CommentRowNumber46.
• CommentAuthorThomas Holder
• CommentTimeAug 23rd 2014
I have expanded the footnote on sHOT therefor slightly altering your text. I hope you still find your views adequately represented.
• CommentRowNumber47.
• CommentAuthorUrs
• CommentTimeAug 23rd 2014
• (edited Aug 23rd 2014)

Hi Thomas,

would it be fair to say in the entry,then, that it remains unclear why an ambidextrous adjoint cyclinder faithfully axiomatizes “qualia”?

By the way, I once followed the implicit suggestion of Lawvere’s of reading Hegel’s Science of Logic and trying to formalize each “opposite of dualities” mentioned there by an adjoint modality, starting with identifying “becoming : nothing $\dashv$ being” with $\emptyset \dashv \ast$, as first suggested in Some Thoughts on the Future of Category Theory.

Now Hegel talks about quality, of course. Reading through what he writes and trying to match available adjoint cyclinders as faithfully as possible with Hegel’s poetry, I did arrive at an identification of “quality” with certain cohesive structure. This is the content of the entry Science of Logic (scroll down a good bit to see the relevant diagrams).

Regarding that footnote: sure, looks good.

• CommentRowNumber48.
• CommentAuthorThomas Holder
• CommentTimeAug 23rd 2014
• (edited Aug 23rd 2014)

Well, to throw in ’qualia’ was probably not a very lucky choice of me, as they tend to wake up bats and other favorite beasts of philosophers i.e. they are related to the question ’what it feels to have certain perceptions’. I used it to point to the psychological dimension and potential of this cohesion story, so I prefer to constrain my remarks to ’quality’ with a subjective dimension in the following.

Before doing this let me point out there is a mathematical model for qualia by Balduzzi-Tononi (pdf) which was morphed into an approach using presheaves in a Bayesian framework by David Balduzzi (here) where at one point he uses a tangent space metaphor for prior update which somewhat suggests laws of (mental) motions.

Most broadly QT suggest qualitative analysis of space, when you consider QT as degenerate or simpler spaces which replace the original space by valuing ex/intensive quality functors there - vaguely a projection to lower ( =infinitesimal) dimensions. The Poincaré remark pointing in the direction that several partial views permit to reconstitute the original space.

The subjective dimension enters then primarily by the necessity of being an earthling that cannot grasp the original space. In the kinship paper Lawvere uses at some point the term of ’rational neglect’ for the homotopical contraction I think. Sure a reasonable thing to do for a being with finite ressources.

More optimistically, one would try to link the infinitesimal space to the minimal soil of concious awareness (sort of Weber-Fechner pyschophysics like) - this is suggested e.g. by Petitot’s phonetics to phonology collapse provided it can be accomodated in cohesion+QT: although the full signal is processed by the auditory tract what enters concious awareness pertains to discrete phonological features.

To conclude with some Hegelian musings: Hegel’s WdL heavily depends on Kant’s KdrV which contains a dialectics as well as intriguing passages on the intensive (?) quality of conciousness which probably goes back to at least the substance dualism in Descartes, in short Hegel actually draws on a rich tradition here.

When I recall correctly Hegel links ’quality’ to determinate being or ’bestimmtes dasein’ for which you could interpret the determinateness as the pointedness and the connectedness as the being, probably the collapse of the cylinder as negation of becoming - a having come into determinate (self-identical) being. Well, I guess a cohesive reading of Hegel would be a fruitful undertaking for Hegel experts and should pay close attention to the text of WdL even at the cost that it does not fit the categorical bill.

• CommentRowNumber49.
• CommentAuthortrent
• CommentTimeAug 23rd 2014

Sorry for the off topic question, but, Thomas - which do you think is of more interest for mathematicians*: the mind/brain or financial markets?

(I’m asking because - while physics is the field** with the richest connections to mathematics - I think it is important for mathematicians (if they are exploring fields outside of mathematics) to explore multiple fields so as to keep their intuition general and not tied to any one particular strata of reality. People like Petitot & Andree Ehresmann bring alot of interesting mathematics into cognitive science …. but there are also lots of physicists & mathematicians on wall st who bring interesting math into financial markets. If one is already studying physics, finance has the advantage that (going from physics to chemistry to biology to neuro science etc..) finance is even more removed from physics than cognitive science is (and if you get even more removed from physics than finance and economics … the terrain is too subjective/fuzzy to be brought into rigorous interaction with math). Cognitive Science allows one to attempt to understand the nature of consciousness, whereas finance allows one to attempt to understand the nature of risk and probability.

I have no idea whether this book is any good, but just because we’re talking about taking methods from QFT and applying them to cognition: here is a book which applies QFT to financial markets.)

*by for mathematicians, I mean for people who care about the “physical mathematics” approach to a discipline outside of math just as much as they care about the “mathematical physics” approach **outside of the formal sciences

• CommentRowNumber50.
• CommentAuthorTodd_Trimble
• CommentTimeAug 24th 2014

Trent, you aren’t by same chance the same as the Trent who asked a question at MathOverflow here?

• CommentRowNumber51.
• CommentAuthortrent
• CommentTimeAug 24th 2014

I am the same Trent, good eye.

• CommentRowNumber52.
• CommentAuthorTodd_Trimble
• CommentTimeAug 24th 2014

And here as well? ;-)

• CommentRowNumber53.
• CommentAuthortrent
• CommentTimeAug 24th 2014

Same guy.

• CommentRowNumber54.
• CommentAuthorThomas Holder
• CommentTimeAug 24th 2014
• (edited Aug 24th 2014)

@49. Well, finance vs. CS, a delicate matter: a guy like the late Francis Crick who always looked for the big intellectual adventure took to biology after WW II and then to conciousness at the end of his career. In the first case his Pascalian wager was a huge sucess and in the second case less so, and in any case he probably could have earned more on Wallstreet than the Nobel prize money.

I have no problems with mathematical finance where a lot interesting math lies around though. I would probably view it in his current form as part of applied statistical physics as the blueprint of equilibrium models there since Walras are actually taken from thermodynamics. So the choice seems to be a matter of personal choice, in general, it seems to me a much safer bet to think on risk at Wallstreet, than otherwise. So I guess it all comes down to the question whether you are interested in risk from a theoretical side or from a practical side. By the way speaking of risk, there is a paper on categorical risk in the recent TAC here though I haven’t looked at it yet.

A last remark on QFT: the thing I particular find intriguing with Lawvere’s approach to cohesion is that permits to reason synthetically on such methods and models i.e. it very much cuts across from QFT to petty graph toposes. E.g. that neural networks are field theory in disguise and as such hopefully amenable to cohesive analysis is probably less important per se than that cohesive SDG permits to throw out the particular models used and to focus on the really important features of the models. This seems to me to be a prerequisite to tackle the real big question which is not how the brain works but why it works the way it does (the problem of transcendental deduction in Kantian terms).

• CommentRowNumber55.
• CommentAuthorUrs
• CommentTimeAug 24th 2014
• (edited Aug 24th 2014)

Hi Thomas,

one aspect that you are maybe alluding to and which I know better how to relate to is the idea of factoring cohesion.

By this I mean given a cohesive topos $\mathbf{H}$, then it may happen that we have another cohesive topos $\mathbf{H}_{inf}$ and a geometric morphism $\mathbf{H} \longrightarrow \mathbf{H}_{inf}$ which exhibits $\mathbf{H}$ as cohesive over $\mathbf{H}_{inf}$.

Since both $\mathbf{H}$ and $\mathbf{H}_{inf}$ are assumed to be cohesiver over the given base topos $\mathbf{B}$, this means that the absolute cohesion of $\mathbf{H}$ “factors” through that of $\mathbf{B}$

$\array{ \mathbf{H} &&\longrightarrow&& \mathbf{H}_{inf} \\ & \searrow && \swarrow \\ && \mathbf{B} } \,.$

In such a situation it makes good sense to say that as we pass to the right along

$\mathbf{H} \longrightarrow \mathbf{H}_{inf} \longrightarrow \mathbf{B}$

that we “lose quantitative information” and “retain only more qualitative information”. This is simply so by the standard interpretation of the cohesive modalities, as we pass to the right with the left adjoint for instance we remember of spaces only their connected components (or their homotopy types in the homotopy context) and so in the middle stage we have something in between the full “quantitative” information and the bare “qualitative” information of $\pi_0$ (or $\Pi_\infty$).

This would be a sense in which I would understand why “quality types” refer to “quality”.

Of course in the above $\mathbf{H}_{inf}$ need not be a quality type for this interpretation to work, but a quality type would be a special case of this formalization of “qualitative information”.

Might it be possible to show that $\mathbf{H}_{inf}$ being a quality type makes it somehow universal among non-degenerate such factorizations?? In other words, could we show that factoring cohesion through a quality type is a “smallest non-trivial factorization possible”? That would be useful to know.

• CommentRowNumber56.
• CommentAuthorThomas Holder
• CommentTimeAug 24th 2014
• (edited Aug 24th 2014)

An interesting suggestion. I need to think to about it. As Lawvere has pointed out on conceptualmathematics QT over S organize themselves into a 2-category, so I would be somewhat surprised when it would be possible to characterize a single QT as universal, though it might hopefully be possible to characterize the whole 2-cat somehow.

In any way, one would have to consider the lattice of essential localizations and look through the Johnstone papers which characterizes QT over S. Another possibility is to use the orthogonality to sufficient cohesion or ’pure variation’. I intend to go through the literature on QT soon also in order to report the useful results for the entry.

A caveat at this quantity-quality contrast. I brought up it in the entry because i thought it suggestive to think in these terms about topological invariants for dynamical systems, but the real contrast is indeterminateness-quality. It seems better to think of them as minimally cohesive spaces. Probably also hopeful to look at the graph example in TAC 2007-Como 2008 with the respective ex/intensive qualities to understand how they pick up information from the cohesive graphs in the domain.

In the 1991 Como paper Lawvere suggests on pp.9-10 that the SDG-infinitesimal spaces are of dimension $\epsilon$ which I guess means they are supposed to be just above (below?) the ground level 0 . So perhaps QT are more generally a sort of atoms of the essential subtopos lattice but Johnstone would have surely remarked on this if true. I need more time to think this over.

• CommentRowNumber57.
• CommentAuthorDavidRoberts
• CommentTimeAug 25th 2014

Off-topic, but the blog conceptualmathematics was moved to private some time ago, unfortunately. If this was run by Lawvere, I would hope he would make it open again, not least because I vaguely recall it hosted some hard-to-find papers of his.

• CommentRowNumber58.
• CommentAuthorUrs
• CommentTimeAug 25th 2014

Thomas, could you recall the orthogonality of QT with “sufficient cohesion” for me? Thanks.

• CommentRowNumber59.
• CommentAuthorThomas Holder
• CommentTimeAug 25th 2014
• (edited Aug 26th 2014)

This is e.g. proposition 3 of TAC 2007 p.47:

If $\mathcal{E}$ over $\mathcal{S}$ is both sufficiently cohesive and a QT, then $\mathcal{S}$ is inconsistent.

In the 1989 graph example $\mathcal{S}^{\{0,1\}}$ this occurs as the nonconnectedness of $\Omega$ and in the 1991 literally infinitesimal spaces he comments on this as lack of enough connected objects.

I recall also a remark on a factorization result for subtoposes of Paré, Rosebrugh(!?), Wood from 1989 in an Australian Journal that he points regularly to and calls relevant for infinitesimal generation of a topos in the ’Como 2008 lectures’.

• CommentRowNumber60.
• CommentAuthorUrs
• CommentTimeAug 25th 2014
• (edited Aug 25th 2014)

Oh, I thought you were referring to an arthogonal factorization system. That might have been the answer to what I was after: the factorization of any cohesive geometric morphism into something followed by a QT.

• CommentRowNumber61.
• CommentAuthorDavid_Corfield
• CommentTimeAug 25th 2014

the factorization of any cohesive geometric morphism into something following by a QT

Does the order matter? We spoke once of a ’cube’ of cohesion. It seemed like you could take out the maximal infinitesimal part first or the orthogonal part.

• CommentRowNumber62.
• CommentAuthorUrs
• CommentTimeAug 25th 2014

Right, that cube arises from there being two different kinds of infinitesimal directions. I suppose if one were to formalize a factorization then that would be accounted for by the base topos itself factoring over yet smaller base toposes.

But I don’t have a systematic way (yet) to produce this factorization from just one cohesive topos. What I know how to do and did discuss is that

a) starting with an inclusion $\mathbf{H}_{reduced}\hookrightarrow \mathbf{H}$ that exhibits differential cohesion to produce its cofiber $\mathbf{H} \longrightarrow \mathbf{H}_{inf}$ where $\mathbf{H}_{inf}$ has infinitesimal cohesion.

b) or conversely, starting with $\mathbf{H} \longrightarrow \mathbf{H}_{inf}$ produce $\mathbf{H}_{reduced}$.

What Thomas’ remarks eventually made me think of is that maybe given just $\mathbf{H}$, we might be able to find some $\mathbf{H}_{inf}$ as a “smallest non-pointlike” cohesive base topos over which $\mathbf{H}$ still sits (maybe in that if you could find any other non-trivial such base topos, it would also have to sit over $\mathbf{H}_{inf}$).

This is vaguely plausible from the geometric interpretation of “infinitesimal cohesion”/”quality types”. But I don’t see yet how to make it precise.

• CommentRowNumber63.
• CommentAuthortrent
• CommentTimeAug 26th 2014
• (edited Aug 26th 2014)

I have no problems with mathematical finance where a lot interesting math lies around though.

The most interesting portion of mathematical finance, I think, is where the interesting math that is lying around is not yet known to pure mathematicians. Elie Ayache (a paradoxically practical and theoretical person … an options trader with an enginering background / philosopher / ceo & founder of a financial software company) claims that - in his book The Blank Swan (not to be confused with Taleb’s The Black Swan) he has managed to “free the mathematics of contingency from the mediation of probability”: in philosophical form in an article like this The Turning (Wilmott Magazine) and in financial form in an article like this Actuarial Value vs Financial Price (Wilmott Magazine). (But, I should start a new thread rather than hijack this one if anyone is interested in generalized probability theory. The reason I brought this up is b/c we’re talking about mathematizing hegel in this thread and in other threads on the nforum, and perhaps Ayache is the hegel of the 21st century (in the sense that he is the only 21st century philosopher I am aware of who has written something that looks like it might (like Hegel’s Science of Logic) prefigure in conceptual form future mathematics). (There are actually 2 20th century books which have been called sequels to Hegel’s Science of Logic - Deleuze’s Cinema Books, where he elaborates a semiotics more advanced than Charles Sanders Peirce’s - but that’s another story. I just mention those books b/c if it is the case that they also prefigure math, it’s interesting that Hegel was able to do so with language, Deleuze (possibly) with cinema, and Ayache (possibly) with the actions of traders. Getting way ahead of things - what type of mathematics prefiguring medium could possibly come next?)

edit/addition: anyone who finds both philosophical and financial language alien, but still wants to see what a mathematics of contingency beyond probability can be can read Borges’ (very) short story Pierre Menard, Author of the Quixote. The terrain that distinguishes Menard’s Quixote from Cervantes’ Quixote is the terrain that contingency captures and that probability is blind to.