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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 13th 2010
• (edited Oct 13th 2010)

created cohesive topos.

• wrote an Idea-section that is meant to explain why the concept is very natural, trying to provide some of the chat that one cannot find in the terse (but beautiful in its own way) article by Lawvere

• spelled out the definition in some detail, here, too, trying to fill in things that Lawvere is glossing over, making it all very explicit;

• started an Examples-section:

• copied over the discussion that $\mathrm{Sh}\left(\mathrm{CartSp}\right)$ is a connected topos. checking the remaining axioms for cohesive topos are easy, but i have not typed that yet

• included a little discussion of how diffeological spaces fit in, following our conversation in another thread

• started an analogous section for $\infty \mathrm{Sh}\left(\mathrm{CartSp}\right)$, but just a stub so far

• but added in a section that goes rgrough the various items in Lawvere’s definition and discusses their meaning in a cohesive oo-topos

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeOct 13th 2010
Impressive...
• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 13th 2010
• (edited Oct 14th 2010)

Impressive…

I am very much impressed by what Lawvere is doing here. You can see that his article is a direct continuation of his thoughts that led to SDG and those expressed in his “Categorical dynamics”-lecture:

he is really at heart a physicists, in the following sense: he is deeply interested in the mathematical model building of reality. (See his intro where he talks about the “weatherman’s topos”!) He is searching for those structures in abstract category theory that do reflect the world. He is asking: What is a space in which physics can take place? Concretely: What is the abstract context in which one can talk about continuum dynamics? I gather even though he is an extraordinary mind, he did not push beyond continuum mechanics, otherwise he would also be asking: What is the abstract context in which quantum field theory takes place?

Now, because he is working so close to the fundamental root of everything, of course there is quite a distance from his “very fundamental physics” to the “fundamental physics” that most people will recognize as such. Which conversely leads to this interesting effect that you can see him try to find a dictionary between category theory and ontological concepts in philosophy. What is the precise abstract definition of space, quantity, quality etc? He gives definitions for all this. And I think the right ones (Though I still need to understand his definion of “quality”. I mean I follow the specific definition, but I do not see yet what it means .) Then he continues further towards the more tangible world, defining extensive and intensive quantities, as if writing a book on thermodynamics. And indeed, in his article on cohesion he uses all these analogies with heating and cooling! While one probably has to be careful with how to say these things in the (ignorant) public, but I think it is absolutely admirable how here a pure category theorists is actively working on unravelling the very foundations of reality. I was hoping for many years that more category theorists would see the immense applicability of the theory to theory-modelling in physics. As Lurie says rightly: “Category theory is not theory for its own sake, but for the sake of other theory.” And fundamental theoretical physics is all about scanning the space of theories for those that fit reality (as opposed to the physics that most theoretical physicists do, which is scanning the phenomena of one fixed theory.)

We can see that Lawvere is pushing in this direction, I think. Which is why I wanted to emphasize what his axioms for a cohesive topos are like if we generalize them to cohesive oo-topos . Because then out of the very same set of axioms springs a structure that all by itself gets even closer to being a model for physical reality: this is the point I kept emphasizing here and there: that just the assumption that we have an oo-connected oo-topos, and hence also just the assumption that we have a cohesive oo-topos (which implies the former) gives rise to a refinement of the intrinsic cohomology of the oo-topos (which is always there) to intrisic differential cohomology. In a better world I would walk over to Lawvere and try to tell him that that’s what models real-world physics fundamentally: a bare topos or oo-topos with its notion of cohomology is a context for kinematics (just the configurations, no forces, no dynamics) while differential cohomology encodes the forces and the dynamics. (In case this statement is raising eyebrows with anyone: let me know and we can discuss this in detail, look at examples, etc. This is an important story).

For instance the differential equations in a synthetic smooth topos that Lawvere considered around “Categorical dynamics” can be really understood without intervention “by hand” as coming from differential cocycles in the corresponding oo-topos. This is really what the theory of (derived) D-modules etc. is about. But even though it all flows by itself out of a general abstract source of concepts, unwinding it is a long story. I wish there were more people like Lawvere around, with his perspective on the general abstract basis of everything and at the same time with the overview over modern derived oo-topos theory and the understanding that the richer structure people are seeing in these is Lawvere’s observation that reality springs out of topos theory taken to full blossoming: reality springs out of oo-topos-theory.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeOct 14th 2010

This is beautifully written, Urs. I think you ought to contact Lawvere with your appreciation. He is likely to reply and have further enlightening things to add.

• CommentRowNumber5.
• CommentAuthorDavidRoberts
• CommentTimeOct 14th 2010

I agree with Todd and Zoran. This is very nice stuff.

The four adjoints of a cohesive topos and the structure they give rise to reminds me of the five monoidal structures on the category of Schur functors. (Not that there is any relation, just the rich and constrained structure is amazing to see)

• CommentRowNumber6.
• CommentAuthorEric
• CommentTimeOct 14th 2010

You can see that his article is a direct continuation of his thoughts that led to SDG and those expressed in his “Categorical dynamics”-lecture

• CommentRowNumber7.
• CommentAuthorDavidRoberts
• CommentTimeOct 14th 2010

I had a brief look, but it’s in an Aarhus report, which isn’t likely to be digitised in the usual channels. Maybe someone could get a scan? Also, it might be polite to ask Lawvere if it could be publicly distributed.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeOct 14th 2010

Unless I am misled, one of the conditions in the definition is redundant: the right adjoint ${f}^{!}$ in a local geometric morphism is always fully faithful (see C3.6.1 in the Elephant). As I noted at the other thread, being local is in general (i.e. for $S\ne \mathrm{Set}$) more than just the existence of ${f}^{!}$, full faithfulness of ${f}^{!}$ being one of the equivalent versions of the extra condition, but connectedness of $f$ is another equivalent version which was already assumed earlier in the definition. So I removed that condition from the definition and made it a remark instead.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeOct 14th 2010
• (edited Oct 15th 2010)

Todd,

thanks, I’ll think about that. Maybe I should first polish and streamline the entry on cohesive topos a bit more and put in the things I was thinking about (see below). Then I could ask him for advise about that.

Mike,

thanks, right, you said this before. Thanks for improving this.

Eric,

if memory serves the “Categorical dynamics”-text is not available online, but these lecture notes are which appear to cover a similar ground.

Everyobody,

not sure if I have enough time today, but my plan is to do the following

1. state a precise definition of cohesive oo-topos. The adjunction part is clear, but we need to think about how to interpret the clause that mentions monomorphisms. As we discussed elsewhere, the only really sane definition of mono in the oo-context is “regular mono”, the $\infty$-limit over a cosimplicial diagram. I think with that definition what I say next will give the examples of cohesive oo-toposes that one wants to see.

2. take the various propositions and proofs that I filled into the Examples-section yesterday and streamline them as follows:

one should be able to give a simple definition of a “cohesive site” such that the category of oo-sheaves over it is a cohesive oo-topos. It should go something like this:

A cohesive site is

• a concrete site;

• with products;

• such that the Cech-nerves $C\left(U\right)$ of any covering family is degreewise a coproduct of representable;

(in view of the following condition this is the abstract way of saying that covers are “good covers”)

• such that the simplicial sets ${\pi }_{0}C\left(U\right)$ (degreewise send every representable to the point) is a contractible Kan complex and such that $\Gamma \left(C\left(U\right)\right)$ (degreewise replace every representable by its set of points (using that the site is assumed to be concrete)) is equivalent to $\Gamma \left(U\right)$

(the first is saying that the objects in the site are “contractible” as seen by its topology, the seond is just the oo-version of the conditon on a concrete site, saying that the topology is compatible with the notion of concreteness)

I think that is

1. sufficient for showing that oo-sheaves on this site form an cohesive oo-topos

2. a set of conditions fulfilled by the kinds of sites we are interested in in the practice of modelling gros toposes.

But I will need to write this out now. I already mention it here in case anyone feels like offering help as I go along.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeOct 14th 2010
• (edited Oct 14th 2010)

Okay, I spent some time with the entry:

• created a subsection Sites of cohesion with essentially the above definition and the beginning of a list of examples;

• worked through the propositions and proofs at oo-Topos over a site of cohesion,checking item-by-item that the oo-sheaf topos over a site of cohesion is cohesive.

I am running out of time now. My argument that the oo-morphisms $\mathrm{Disc}X\to \mathrm{Codisc}X$ are $\infty$-regular monos was written a bit hastily. Right now I think it is correct, but I should go over it in more leisure later.

Also I haven’t yet typed the proof that $\Pi$ commutes with powering by Kan complexes, but I think that’s obvious from the other discussion. I’ll type it up later.

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeOct 14th 2010

This is certainly a form of metaphysics, in the best possible sense, working on the constitutive mathematical language in which physical theories must be expressed. Could #3 not go on the Cafe at some point? That might pull in some great mathematicians :).

By the way, without understanding things nearly enough (if only I didn’t have to spend time reading people’s ethics theses), am I hearing hints of what you said above in Sullivan’s Algebra, Topology and Algebraic Topology of 3D Ideal Fluids?

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeOct 14th 2010

Could #3 not go on the Cafe at some point?

I am thinking about that. Need to run now to a colloquium dinner, after that I try to polish the discussion at cohesive topos a bit more, then I try to post something.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeOct 14th 2010

am I hearing hints of what you said above in Sullivan’s

Please give me a page number, I am too busy to read the whole article.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeOct 15th 2010
• (edited Oct 15th 2010)

I have added the remaining propositions and proofs to cohesive topos.

Except for one: I noticed that checking $\Pi \left({X}^{\mathrm{Disc}S}\right)\simeq \Pi \left(X{\right)}^{S}$ is maybe not that easy after all. To start with I might have to think more about how the cartesian closure on $\infty$-stacks is modeled in terms of simplicial presheaves. And once I have that, I need to probably start with a fibrant model for $X$, then form ${X}^{\mathrm{Disc}S}$ in simplicial presheaves and then pass to a cofibrant resolution of that to be able to form the derived $\Pi$. Maybe it’s too late at night now for me, but at the moment I don’t really see anymore why I thought that would be easy to analyse.

• CommentRowNumber15.
• CommentAuthorDavidRoberts
• CommentTimeOct 15th 2010
• such that the Cech-nerves $C\left(U\right)$ of any covering family is degreewise representable;

are you sure about this? Even for Cart this isn’t true, but degreewise a coproduct of representables.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeOct 15th 2010

That’s what I mean. Typo.

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeOct 15th 2010
• (edited Oct 15th 2010)

I fixed this in the above comment. But you are not supposed to be scrutinizing that comment. You are supposed to be looking at the entry cohesive topos instead.

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeOct 15th 2010

I have now posted something over to the blog: here.

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeOct 15th 2010

Next I am trying to get a better feeling for Lawvere’s extensive quality and intensive quality . By def 4 it ought to be true that the terminal geometric morphism itself counts as in intentive quality.

Let me look at that in the $\infty$-context. Then there is the following dictionary between Lawvere’s imagery and the identifications in $\infty$-connected $\infty$-toposes that I have been talking about:

let $A\in \left(\infty ,1\right)\mathrm{Sh}\left(X\right)$

• What Lawvere calls the rarefied substance of $A$ is the map that I write $♭A\to A$ from flat objects to objects (the forgetful functor from flat principal $\infty$-bundles to the underlying $\infty$-bundles).

• What Lawvere calls the condensed substance of $A$ is the map that I write $A\to \Pi \left(A\right)$ – the inclusion of $A$ as the constant paths into its path $\infty$-groupoid.

• What Lawvere calls the cooling map, the composite $♭A\to A\to \Pi A$ is the topologists’s inclusion $K\left(G,1\right)\to BG$.

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeOct 15th 2010
• (edited Oct 15th 2010)

It looks to me that for what he wants to say in section IV the $\infty$-topos-perspective is really mandatory: he is trying to characterize infinitesimal objects now using the properties of a cohesive topos. Unless I am misunderstanding, his discussion there remains inconclusive: we hear “it seems clear that…”, and “weak infinitesimal nature…” and “surprisingly often…” .

But I think the situation is clear: in the $\infty$-topos the map ${p}_{!}$ is $\Pi$ – geometric realization. The objects in the kernel of $\Pi$ are therefore cohesive structures whose underlying topologial space is the point. These are the infinitesimal objects. For instance take the infinity-Lie algebroids as discussed there, under $\Pi$ they map to the realization of their space of objects.

So the “canonical intensive quality” $L$ that Lawvere speaks about, the full sub-$\infty$-topos on those objects $A$ for which $♭A\to \Pi \left(A\right)$ (the “cooling map”) is an equivalence contains precisely those cohesive $\infty$-groupoids which are “discrete up to an infinitesimal thickening”. For instance disjoint unitions of ${L}_{\infty }$-algebras etc.

Notice that over at function algebras on infinity-stacks we are making this precise by finding full subcategories of objects in the kernel of $\Pi$ and seeing that these are indeed $\infty$-Lie algebroids.

• CommentRowNumber21.
• CommentAuthorMike Shulman
• CommentTimeOct 15th 2010

the only really sane definition of mono in the oo-context is “regular mono”

Why is that? What’s wrong with monomorphism in an (infinity,1)-category? (Regular monos might be the right thing to take here—I don’t have any intuition for that, since in a 1-topos all monics are regular. But it seems to me that non-regular monos have some role to play in (∞,1)-topoi.

• CommentRowNumber22.
• CommentAuthorMike Shulman
• CommentTimeOct 16th 2010

As I mentioned at the cafe, I created cohesive site to talk about the 1-categorical version. I still like the idea of moving the definition of “(∞,1)-cohesive sites” and proof that they give cohesive (∞,1)-topoi to a page like (infinity,1)-cohesive site – then the pages cohesive topos and cohesive (infinity,1)-topos can be more about the behavior of the topoi themselves.

• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeOct 16th 2010

okay, I started splitting things off: created

and tried to move the material appropriately. There is more to be done, but I have to quit now.

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeOct 19th 2010
• (edited Oct 19th 2010)

following the latest Café-discussion and so that the nLab definitions are not secretly inconsistent with Lawvere’s terminology but clear on where they differ, if they differ, I have

• added two more paragraphs to the end of the Idea-section of cohesive topos, explaining the idea of the additional two axioms;

• edited the definition such that all three groups of axioms have their own name:

1. cohesive topos

2. cohesive pieces have points

3. pieces of products are products of pieces

• CommentRowNumber25.
• CommentAuthorMike Shulman
• CommentTimeOct 19th 2010

I changed “pieces of products are products of pieces” to “pieces of powers are powers of pieces” since that axiom is actually only about powers of a single space, not products of arbitrary families. (Have you figured out yet what property of a site ensures that property of a topos?)

I also reworded your reference to “the generic object of a cohesive topos” which confused me for a bit since it sounded like a reference to the generic model in a classifying topos.

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeOct 19th 2010

Right, thanks.

No, I haven’t thought about that axiom any more. I need some motivation for this axiom before I have energy to think about it.

• CommentRowNumber27.
• CommentAuthorUrs
• CommentTimeOct 19th 2010
• (edited Oct 19th 2010)

In Lawvere’s article, the “continuity”-axiom is invoked only once, in “Theorem 1” on page 4.

But right below that he remarks that with a little bit of homotopy theory, one could drop the continuity axiom after all!

The theorem 1 itself seems to be getting at something interesting: I suppose the “$YX$” is a typo for ${Y}^{X}$ and we are to keep in mind the case where $X$ and $Y$ are topological spaces, so that ${p}_{!}{Y}^{X}$ is homotopy-classes of continuous maps $X\to Y$. So I guess we are to think of this as an indicating relations of cohesive toposes to localization theory. But as far as I follow what’s meant to be going on, I’d think that cohesive $\infty$-toposes would do the trick in a better way. But maybe I am missing something.

• CommentRowNumber28.
• CommentAuthorUrs
• CommentTimeOct 19th 2010

I started an Examples-section simplicial sets. But really need to call it quits now.

• CommentRowNumber29.
• CommentAuthorUrs
• CommentTimeOct 21st 2010

I realize that I do not fully follow Lawvere’s proof of his theorem 2 when it comes to this bit

it follows that this subcategory is closed under arbitrary subobjects and arbitrary products and is hence by completeness epi-reflective;

I understand closure under subobjects. Not sure why it is closed under non-finite products. And not sure how to get the reflector from this.

Can anyone help me?

• CommentRowNumber30.
• CommentAuthorMike Shulman
• CommentTimeOct 21st 2010

Can you translate the proof out of the language of “intensive quantities” and “quality types” into a statement about categories and functors?

• CommentRowNumber31.
• CommentAuthorUrs
• CommentTimeOct 21st 2010

Okay, I have now written down the statement and Lawvere’s proof in ordnary terms at cohesive topos – Properties.

I inserted markers

  (....details....)


at all points where one should make the reasoning more explicit. Some of them I can fill myself. But some of them I am not sure about.

• CommentRowNumber32.
• CommentAuthorUrs
• CommentTimeOct 21st 2010

This here should be a way to see that $L↪E$ is a reflective subcategory:

• by definition it is the full subcategory on objects $X$ such that $\Gamma X\to \Pi X$ is an iso

• that’s equivalent to saying that for all $Y$ we have that $\mathrm{Hom}\left(\Gamma X\to \Pi X,Y\right)$is an iso.

• that should be equivalent to saying that for all $Y$ we have that $\mathrm{Hom}\left(X,\mathrm{Disc}Y\to \mathrm{Codisc}Y\right)$ is an iso

• that would mean that ${L}^{\mathrm{op}}$ is the category of $W$-local objects for $W=\left\{\mathrm{Disc}Y\to \mathrm{Codisc}Y\right\}$.

• That implies that ${L}^{\mathrm{op}}$ is reflective hence that $L$ is coreflective.

• CommentRowNumber33.
• CommentAuthorUrs
• CommentTimeOct 22nd 2010

Oh, I only now see Mike’s latest reply over at the Cafe.

• CommentRowNumber34.
• CommentAuthorUrs
• CommentTimeJan 3rd 2011

there should be a discussion of the following natural :

Question: for $ℰ$ a cohesive topos, what is the condition (if any) on an object $X\in ℰ$ so that the slice $ℰ/X$ is itself is cohesive?

It is easy to see that $X$ being connected helps (${\Pi }_{0}X\simeq *$): that makes $ℰ/X$ be connected.

Intuitively one expects that $\Gamma \left(X\right)\simeq *$ should imply that $ℰ/X$ is local, but I am not yet sure (that ought to be easy to see, though).

Then what else? Intuitively I would expect that somehow $X$ needs to be “maximally non-concrete”. But not sure.

• CommentRowNumber35.
• CommentAuthorUrs
• CommentTimeJan 4th 2011
• (edited Jan 4th 2011)

Then what else?

I think $X$ being a tiny object is sufficient for $ℰ/X$ to be local (easy proof now at local topos). So if $X$ is both connected as well as tiny (or contractible and tiny in the $\infty$-case), then $ℰ/X$ satisfies all the conditions of a cohesive topos except that possibly ${\Pi }_{0}$ may fail to preserve binary products (and nothing said yet about the extra axioms “pieces have points”).

I am trying to see if slicing cohesive $\infty$-toposes over suitable objects can be used as an alternative route to derived geometry: if we start with a 1-localic cohesive $\infty$-topos such as $\infty \mathrm{Sh}\left(\mathrm{CartSp}\right)$, then the slice over an $n$-truncated object will be $n$-localic. We know that for the corresponding derived geometry ($\infty$-sheaves over the $\infty$-site of simplicial smooth rings^op) it should be $\infty$-localic. So possibly there is a slice over an un-truncated object that achieves this.

Possibly one could consider obects $X$ to slice over such as the $\infty$-stack of all simplicial ${C}^{\infty }$-rings (i.e. $U↦{N}_{\mathrm{hc}}\left({C}^{\infty }\left(U\right)/\left({C}^{\infty }\mathrm{Ring}{\right)}^{{\Delta }^{\mathrm{op}}\right)}$) or the like. I don’t know, this is just to indicate what kind of idea I am after here.

• CommentRowNumber36.
• CommentAuthorUrs
• CommentTimeJan 8th 2011

added to the Definition-section of cohesive topos the remark that the dual of “pieces have points” is equivalently “discrete objects are concrete” (!)

• CommentRowNumber37.
• CommentAuthorUrs
• CommentTimeJan 10th 2011
• (edited Jan 10th 2011)

For emphasis, I have expanded the above remark to the statement that precisely if a cohesive topos satisfies that extra axiom “pieces have points” then also its sub-quasitopos of concrete objects is cohesive.

• CommentRowNumber38.
• CommentAuthorUrs
• CommentTimeFeb 17th 2011

have added material to the Properties-section at cohesive topos.

I spell out the proof that ${p}_{*}X\to {p}_{!}X$ is epi for all $X$ precisely if ${p}^{*}S\to {p}^{!}S$ is mono for all $S$ a bit more explicitly than Peter Johnstone does. This is a curious kind of exercise. It feels like doing th most trivial of computation, but without the right formalism. I guess it’s all a tautology with a suitable strng diagram calculus, but that this lives in 4d or something, where visualization breaks down.

• Already a few days back I had added one further subsection Relations between the axioms, summarizing Johnstone’s observations.

• CommentRowNumber39.
• CommentAuthorDavidRoberts
• CommentTimeFeb 17th 2011

I’m thinking that a lot of these properties/structures would be great in settings more general than topoi. For example, in pretopoi or quasitopoi (I know this is mentioned in the article), or even lextensive cartesian closed categories or similar. Certainly we can ask for the four adjoints, for ’pieces have points’ and ’pieces of powers are powers of pieces’ in much more general setting.

Similarly, for categories of simplicial objects in one of the above ’generalised cohesive categories’, there should be a generalisation of a cohesive (oo,1)-topos.

• CommentRowNumber40.
• CommentAuthorUrs
• CommentTimeFeb 17th 2011

Yes, lots of things still to be done. In Axiomatic cohesion more general categories than toposes are envisioned. The restriction of the concept to toposes in the $n$Lab entry is due to me. I find the point of view that the ambient context is a topos with possibly some non-topos subcategories of interest to be useful. But that should not stop anyone from going in other directions.

• CommentRowNumber41.
• CommentAuthorUrs
• CommentTimeNov 4th 2011

I have started a new section at cohesive topos, titled Internal modal logic. This is induced from discussion with Mike on the HoTT blog here, but still a bit experimental.

• CommentRowNumber42.
• CommentAuthorDavid_Corfield
• CommentTimeNov 5th 2011

Do you have any sense that this will make any contact with what a philosopher understands by modal logic? There was discussion of a kind of modality in a category here.

• CommentRowNumber43.
• CommentAuthorUrs
• CommentTimeAug 27th 2012
• (edited Aug 27th 2012)

In the course of brushing-up the references at allegory I came to Bob Walters’ blog and from there to his Como Category Theory Archive and there I found a link to a pdf with notes on that Como 2008 lecture by Lawvere on cohesive toposes, of which so far I only new a video of the first 10 minutes existed.

So I added links to these pdf notes to cohesive topos and to Cohesive Toposes – Combinatorial and Infinitesimal Cases .

(However, looking over these notes there is nothing much in there which is not also in Axiomatic Cohesion. While bornological spaces make an appearance, they are not put into a cohesive topos.)

• CommentRowNumber44.
• CommentAuthorDavid_Corfield
• CommentTimeAug 27th 2012

Why does Lawvere see bornological spaces as so important? He often speaks of them?

I see here I mentioned that

Lawvere speaks in ’Volterra’s Functionals and Covariant Cohesion’ of bornology involving a notion of covariant cohesiveness.

• CommentRowNumber45.
• CommentAuthorUrs
• CommentTimeAug 27th 2012
• (edited Aug 27th 2012)

I don’t know. In general I find that the collection of examples or non-examples that he gives for cohesion in his writings is not the one I would find most immediate for the presentation of the topic.