## October 14, 2010

### Cohesive ∞-Toposes

#### Posted by Urs Schreiber

The concept of a

axiomatizes properties of a general abstract context of geometric spaces. This has been proposed by Bill Lawvere.

Below is

1. a brief introduction to the definition of cohesive topos

2. an appreciation of Bill Lawvere’s work in this direction.

More is behind the above link.

## The notion of Cohesive Topos

The definition of cohesive topos or category of cohesion aims to axiomatize properties of a topos that make it a gros topos of spaces inside of which geometry may take place.

The idea behind the term is that a geometric space is roughly something consisting of points or pieces that are held together by some cohesion - for instance by topology, by smooth structure, etc.

The canonical global section geometric morphism $\Gamma : E \to Set$ of a cohesive topos over Set may be thought of as sending a space $X$ to its underlying set of points $\Gamma(X)$. Here $\Gamma(X)$ is $X$ with all cohesion forgotten (for instance with the topology or the smooth structure forgotten)

Conversely, the left adjoint and right adjoint of $\Gamma$

$E \stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{Codisc}{\leftarrow}}} Set$

send a set $S$ either to the discrete space $Disc(S)$ with discrete cohesive structure (for instance with discrete topology) or to the codiscrete space $Codisc(S)$ with the codiscrete cohesive structure (for instance with codiscrete=indiscrete topology) .

Moreover, the idea is that cohesion makes points lump together to connected pieces . This is modeled by one more functor $\Pi : E \to S$ left adjoint to $Disc$. In the context of 1-topos theory this sends a cohesive space to its connected components $(\Pi = \pi_0)$. More generally in an $(n,1)$-topos-theory context, $\Pi$ sends a cohesive space $X$ to the $(n-1)$-truncation of its geometric fundamental $\infty$-groupoid $\Pi(X)$.

In total this gives a quadruple of adjoint functors

$(\Pi \dashv Disc \dashv \Gamma \dashv Codisc) : E \stackrel{\stackrel{\overset{\Pi}{\to}}{\overset{Disc}{\leftarrow}}}{\stackrel{\underset{\Gamma}{\to}}{\underset{Codisc}{\leftarrow}}} Set \;$

A cohesive topos is a topos whose terminal geometric morphism admits an extenson to such a quadruple of adjoints, satisfying some further properties.

## A remark on Lawvere’s work

I am very much impressed by what Lawvere is doing here. His work on cohesion is a direct continuation of his thoughts that led to synthetic differential geometry and those expressed in his “Categorical dynamics”-lecture:

He is really at heart a physicist, in the following sense: he is deeply interested in the mathematical model building of reality 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?

Because he is working so close to the fundamental root of everything, 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 the interesting effect that one 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 useful ones. There is a connection between ontology and fundamental physic constructed out of hard math.

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! 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 theorist 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 Jacob Lurie says rightly: Higher 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 $\infty$-toposes . 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 $\infty$-connected $\infty$-topos, and hence also just the assumption that we have a cohesive $\infty$-topos (which implies the former) gives rise to a refinement of the intrinsic cohomology of the $\infty$-topos (which is always there) to intrinsic 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 $\infty$-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 understood without intervention “by hand” as coming from differential cocycles in the corresponding $\infty$-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 $\infty$-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 $\infty$-topos-theory.

(Reproduced from this posting. )

Posted at October 14, 2010 11:31 PM UTC

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## 65 Comments & 0 Trackbacks

### Re: Cohesive ∞-Toposes

So what are some other cohesive toposes? Is Johnstone’s topological topos cohesive? The main thing I’m not sure about there is the existence and behavior of $f_!$.

Posted by: Mike Shulman on October 15, 2010 4:50 AM | Permalink | Reply to this

### examples

So what are some other cohesive toposes?

Let’s try to list and agree on which we think we already know are cohesive toposes over $Set$.

To the extent that my analysis starting at Sites of cohesion is correct – but chances are good that there are still mistakes in there – we’d have:

• every presheaf topos on a site with finite products;

• the topos over the site of contractible topological spaces and open covers – the corresponding 2-topos is the one that contains topological stacks;

• the topos over the site of Cartesian spaces with smooth maps between them and the coverage of good open covers – this is the one that contains diffeological spaces, differentiable stacks, etc., the $\infty$-topos over it is that of $\infty$-Lie groupoids;

• the Cahiers topos , the topos over the site of infinitesimally thickened Cartesian spaces, this is a smooth topos providing a well-adapted model for synthetic differential geometry. It’s $\infty$-version contains $\infty$-Lie groupoids and $\infty$-Lie algebroids.

In his section IV Lawvere says something vague that might be interpreted as saying that he has checked that various of the standard models of synthetic differential geometry are cohesive. That’s certainly his main motivation for the whole concept in te first place, as far as I am aware.

(While probably easy, I am not yet sure how to see it for sites of duals of plain rings with any of the standard topologies. I guess the toposes over these should be cohesive, but the sites are not cohesive in my sense, so my argument based on that does not apply here.)

Posted by: Urs Schreiber on October 15, 2010 9:29 AM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

It might be worth mentioning the notion of a totally connected topos, which (according to the Elephant) is a locally connected topos for which the further left adjoint $\Pi$ preserves all finite limits. It is proven there that if a site is cofiltered and every covering sieve is connected (as a full subcategory of a slice category), then its topos of sheaves is totally connected. I think that the argument given should analogously show that if a site is cosifted and every covering sieve is connected, then the sheaf topos is locally connected and $\Pi$ preserves finite products.

Of course any category with finite products is cosifted (and also “homotopy cosifted”, the relevant ∞-version), and the connectedness of covering sieves looks like the 1-dimensional version of your second condition on contractibility of their nerves. (I don’t understand why you ask that it be a Kan complex rather than just weakly contractible, but I haven’t read carefully through the examples and proof.)

The condition in the Elephant for a sheaf topos to be local is that it have a terminal object which is irreducible, i.e. covered only by the maximal covering sieve. This looks like a consequence of your final condition on points of the Cech nerves, since when $U=\ast$ the space $Hom(\ast,U)$ is nonempty, hence so must $Hom(\ast,C(U))$ be, and hence the covering sieve is maximal. Your condition is much stronger than this, but maybe that’s again the difference in the ∞-world.

I don’t quite understand at a 1-categorical level why these conditions are enough to guarantee the final two parts of the definition, about “continuity” of $f_!$ and monicness of the comparison transformation.

I wonder whether it would be worth distinguishing terminologically between sites whose 1-topoi of sheaves are cohesive from those whose (∞,1)-topoi of sheaves are cohesive? In any case I would find it easier going to read a direct proof in the 1-categorical case first, with the (∞,1)-case later or elsewhere (such as at a page called “cohesive (∞,1)-topos” or “cohesive (∞,1)-site”). (This is not a coded request for you to write something, Urs, more a comment about something I’d be interested in working out—or seeing someone else work out, if they felt like it.)

Posted by: Mike Shulman on October 15, 2010 7:17 PM | Permalink | Reply to this

### cohesive sites

I think that the argument given should analogously show that if a site is cosifted and every covering sieve is connected, then the sheaf topos is locally connected and preserves finite products.

Thanks, very useful hint. I need to think about that.

I don’t understand why you ask that it be a Kan complex rather than just weakly contractible, but I haven’t read carefully through the examples and proof

Right, I do not need to require this for the proofs to go through (because the proofs always hom out of these objects, never into them). But at least the second one – $Hom_C({*},Cech(U))$ – is guaranteed to be a Kan complex. (I was maybe thinking that the first one – $lim_\to Cech(U)$ – is also necessarily Kan. But that does not seem to follow.) I have removed this clause from the definition now.

Your condition is much stronger than this, but maybe that’s again the difference in the $\infty$-world.

I need to think about this. I dreamed up the conditions that I give because they are satisfied in the examples that I am motivated by and because using them it seemed I could prove what I wanted to prove. If one can weaken the assumptions, then all the better.

I don’t quite understand at a 1-categorical level why these conditions are enough to guarantee the final two parts of the definition, about “continuity” of $f_!$ and monicness of the comparison transformation.

And the “continuity” I have in fact not shown yet (as remarked in the text)!

Do you know why we want to demand the continuity axiom in the first place? I am not sure. In Lawvere’s articles it is used in some of the proofs, but I am not sure either why I need the statements thus proven. Instead, in applications of this stuff that I am motivated by, I see – so far – need for all the axioms, but not for this one.

Instead, I see need for what looks like a slight variant of this axiom: it is important for me that $p_!$ (= $\Pi$) preserves homotopy fibers .

I think I have shown that this is the case over a cohesive site, using a result by David Roberts and Danny Stevenson on simplicial principal bundles. The proof is at $\infty$-Lie groupoid – geometric realization.

The relevance of this statement is that it allows to refine principal $\infty$-bundles by lifting them through $\Pi$ and equipping them with “more cohesion”.

For instance for the cohesive $\infty$-topos

$\Gamma : \infty LieGrpd \to \infty Grpd$

we may start with looking at the first fractional Pontryagin class

$\frac{1}{2}p_1 : \mathcal{B} Spin \to K(\mathbb{Z},4)$

in $\infty Grpd$ and then find a smooth refinement of that

$\frac{1}{2}\mathbf{p}_1 : \mathbf{B} Spin \to \mathbf{B}^3 U(1)$

in $\infty LieGrpd$. Under $\Gamma_! = \Pi$ this maps to $\frac{1}{2}p_1$.

Now, the $(\infty,1)$-fiber of $\frac{1}{2}p_1$ is $\mathcal{B}String$ – the classifying space of the topological String group.

Since – by my claim – $\Pi$ preserves $(\infty,1)$-fibers, it follows that the homotopy fiber of $\frac{1}{2}\mathbf{p}_1$ in $\infty LieGrpd$ – which I write $\mathbf{B}String$ and show to be the delooping of the String Lie 2-group –, indeed maps under $\Pi$ to $\mathcal{B}String$.

$\Pi : \left( \array{ \mathbf{B}String &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}Spin &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \mathbf{B}^3 U(1) } \right) \mapsto \left( \array{ \mathcal{B}String &\to& * \\ \downarrow && \downarrow \\ \mathcal{B}Spin &\stackrel{\frac{1}{2}p_1}{\to}& K(\mathbb{Z},4) } \right)$

So this gives an easy re-derivation of the BCSS theorem. But it also generalizes it: generally for every topological classifying map and every refinement of it into a cohesive $\infty$-topos over a cohesive site, the cohesive principal $\infty$-bundle classified by the refined class will have as geometric realization the topological space classified by the original map.

I wonder whether it would be worth distinguishing terminologically between sites whose 1-topoi of sheaves are cohesive from those whose (∞,1)-topoi of sheaves are cohesive?

Sure, as soon as we have a more detailed understanding of which properties on sites induce which kind of cohesion. For the moment I just had one definition and needed a name for it.

In any case I would find it easier going to read a direct proof in the 1-categorical case first, with the (∞,1)-case later or elsewhere

Yes, right. I had the page organized differently first. But the proof in simplicial presheaves rreduces pretty immediately to a 1-categorical proof. The thing is that at severa points we observe that some Quillen adjunction is evident on the global model structure and then invoke the lemma that for it to descent to the local model structure we need only check (since everything is left proper and left Bousfield localization does not change the cofibrations) that the right adjoint preserves locally fibrant objects. In the 1-topos case this is the point where you check that something is a sheaf. So one can obtain this straightforwartdly.

The proof of connectedness of the 1-topos $Sh(Cartsp)$ here I had written after I got the $\infty$-proof, as a toy example of that. In an earlier version of the entry this was placed before the $\infty$-version, then later I decided to put it afterwards. Maybe I should reconsider once more.

In any case, thanks for all the input about cosiftedness, totally connectedness, etc. I’ll try to think about that.

Posted by: Urs Schreiber on October 15, 2010 8:39 PM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

the proof in simplicial presheaves reduces pretty immediately to a 1-categorical proof

That might be true, but I would still find it easier to understand the 1-categorical proof first and then see the generalization. I think the essential ideas are harder to find when there are lots of model structures and fibrant objects floating around. Even the “natural” level of generality for a concept can be an obstacle to initial comprehension of a new idea.

Along these lines, I created cohesive site with a version of the definition and proof for 1-categories (also missing “continuity”). I discovered in doing this that the proof of the monomorphism condition seems to be noticeably different and requires a different hypothesis. One could argue that this is another of those situations in which the ∞-version is better (things don’t get cut off too soon), but if nothing else, seeing the contrast helps me appreciate that.

Posted by: Mike Shulman on October 15, 2010 11:47 PM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

To clarify, is your thought that the step to $\infty$-toposes is required with the shift from continuum mechanics to QFT, as I thought you were suggesting here

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?

or was that step needed already by Lawvere, as you write here

It looks to me that for what he wants to say in section IV the $\infty$-topos-perspective is really mandatory?

Posted by: David Corfield on October 15, 2010 9:15 AM | Permalink | Reply to this

### the passage from cohesive 1-toposes to cohesive ∞-toposes

To clarify, is your thought that the step to $\infty$-toposes is required with the shift from continuum mechanics to QFT […] ?

Not quite, it’s more general than that. Let’s recall:

a cohesive $\infty$-topos – as opposed to just a cohesive 1-topos – comes canonically with its intrinsic notion of path , hence of process. (It was originally Richard Williamson who amplified this):

We have for the functor $\Pi$ left adjoint to $Disc$ that

• on a cohesive 1-topos it sends cohesive spaces to their set of connected components;

• on a cohesive $(2,1)$-topos it sends cohesive spaces to their fundamental groupoid;

• etc.

• on a cohesisive $(\infty,1)$-topos it sends cohesive spaces $X$ to their fundamental $\infty$-groupoid $\Pi(X)$.

This is described in some detail at geometric homotopy groups in an $(\infty,1)$-topos.

We may reflect $\Pi(X)$ back into the cohesive topose, where it becomes the path $\infty$-groupoid $\mathbf{\Pi}(X)$.

This is the object that allows us, roughly speaking, to find dynamics in a cohesive $\infty$-topos: because this is the object that knows about how to traverse paths in an cohesive space.

Notably, the intrinsic “twisted” cohomology of $\mathbf{\Pi}(X)$ in the cohesive $(\infty,1)$-topos is differential cohomology . A cocycle in differential cohomology is what in physics is called a gauge field . These are what encode all forces in physics, such as those exerted by the electromagnetic field or the Yang-Mills field describing the nuclear forces, and also the field of gravity. Or, less fundamentally, a specified force field on a fluid in a problem of fluid-dynamics.

The cohesive 1-topos underlying the cohesive $(\infty,1)$-topos cannot see any of this, because it sees only the 0-truncation $\pi_0(X)$ of $\Pi(X)$, where all paths have been contracted to points.

Of course one can recover $\Pi(X)$ in terms of simplicial objects in a cohesive 1-topos and playing some tricks. This is the traditional way of describing homotopy groups of a 1-topos. But if one looks closely at what these constructions in terms of simplicial objects in the 1-topos do, one sees that they precisely constitute a model for the corresponding $(\infty,1)$-topos. (All this described in some detail behind the above link.) So there is a way to reconstruct the $\infty$-groupoid $\Pi(X)$ by applying $\pi_0$ degreewise on suitable simplicial objects. But this is a hack that loses the nice fact that $\Pi$ is simply the left $(\infty,1)$-adjoint functor to $Disc$. And this is a crucially useful fact.

Posted by: Urs Schreiber on October 15, 2010 9:57 AM | Permalink | Reply to this

### Re: the passage from cohesive 1-toposes to cohesive ∞-toposes

Thanks, that makes sense to me. So one might say that even the physics of the 1860s, Maxwell’s electromagnetism, requires the notion of a cohesive $(\infty, 1)$-topos for its proper mathematical setting. Is that enough, or will you lead us on to cohesive $(\infty, 2)$-toposes?

It’s intriguing to see how the mathematics of a time makes expressible physical laws, and yet later mathematics is needed for a deeper understanding. There is something about this point of view in Michael Friedman’s Dynamics of Reason.

Vladimir Arnold was famous for finding sophisticated mathematics in older physical work, such as a topological proof of the transcendence of Abelian integrals in Newton’s Principia, Huygens and Barrow, Newton and Hooke page 83.

By the way, you mentioned Richard Williamson, who I see is giving a talk Towards homotopy theoretic foundations for mathematics in Oxford on 27 October.

Posted by: David Corfield on October 15, 2010 11:23 AM | Permalink | Reply to this

### fundamental physics from ∞-topos theory

that makes sense to me.

Okay, good.

I went and wrote out a bit more of how fundamental physics springs from $\infty$-topos theory here:

Is that enough, or will you lead us on to cohesive $(\infty,2)$-toposes?

We might need $(\infty,2)$-topos for $\infty$-vector bundles / quasicoherent $\infty$-stacks. In an $(\infty,1)$-topos we have principal $\infty$-bundles naturally. But there are also associated $\infty$-bundles in there. Therefore I am not sure yet if one will strictly need to pass to $(\infty,2)$-toposes. Answering this will require a better understanding of quantization of differential cocycles in an $(\infty,1)$-topos to $n$-dimensional QFTs. I don’t understand this well enough yet to say more.

So one might say that even the physics of the 1860s, Maxwell’s electromagnetism, requires the notion of a cohesive $(\infty,1)$-topos for its proper mathematical setting.

That’s where it naturally takes place in, yes. A derivation that makes this precise is at circle $n$-bundle with connection.

By the way, you mentioned Richard Williamson, who I see is giving a talk Towards homotopy theoretic foundations for mathematics in Oxford on 27 October.

Yes, I just noticed that, too. I haven’t spoken to him in a while. Was eagerly awaiting his thesis to come out. But apparently his interest shifted from the foundations of $\mathbb{A}^1$-homotopy theory to the foundations of foundations.

Posted by: Urs Schreiber on October 15, 2010 12:09 PM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

Although I understand that the point of cohesive toposes is to be a type of big topos (a topos whose objects are “spaces”) I find it intriguing how its basic axiomatization involves many “topological” properties generally studied for little topoi (a topos that itself is regarded as a “space”). In chapter C3 of Sketches of an Elephant, the following hierarchy of properties of topoi (or more generally, of geometric morphisms) can be extracted.

Open                  Compact
Locally connected     Strongly compact (= tidy)
Totally connected     Local
Atomic                Compact Hausdorff (= proper separated)


Each property is a strengthening of those above it, except for the last two rows (neither of which implies the other), and the two columns are “dual” to each other in a certain sense. Then there is the property of “connectedness” which is implied by both total connectedness and locality.

Now the first few axioms of a cohesive topos are that it is locally connected, local (hence connected), and “strongly connected” (a notion in between locally connected and totally connected, which also implies connectedness). In particular, if we were to disregard our instructions and regard a cohesive topos as a generalized space itself, it would be very well-behaved: open, locally connected, strongly compact, connected, etc. Is there some reason why the same properties of interest for little topoi are also important for big ones?

(A few months ago the subject came up on the nForum that when writing in English, there’s really no reason to use the French words “gros” and “petit” in this context when the English ones “big” and “little” do the job perfectly well.)

Posted by: Mike Shulman on October 16, 2010 5:47 AM | Permalink | Reply to this

### thick points

Is there some reason why the same properties of interest for little topoi are also important for big ones?

I can offer my intuition for what’s going on:

I think of a cohesive topos as being the dual of the archetypical fat point on which the geometry is modeled. Maybe using Lawvere’s term I could say: it behaves like the archetypical lump of cohesive points .

The topos $Set = Sh(pt)$ is the dual of the plain point. It encodes geometry modeled on the point $pt$.

The topos $Sh(CartSp)$ behaves abstractly also like a point. It is local and connected. This is even more striking if we look at the $(\infty,1)$-topos $(\infty,1)Sh(CartSp)$: this is even $\infty$-connected! This clearly tells us that we are dealing with a point-like structure.

But of course $Sh(CartSp)$ is “much bigger” than $Sh(pt)$. It behaves like a thick point . We can think of each of the objects $\mathbb{R}^n \simeq B^n$ in $CartSp$, which diffeomorphically are open balls of any dimension, as thick points with smooth structure. Geometry modeled on $CartSp$ is like geometry modeled on $*$ but with the extra information that for each point we know how to find a smooth open neighbourhood of points.

In a way $Sh(CartSp)$ behaves like (the dual of) the general abstract smooth open ball . A bit like $\mathbb{R}^\infty$.

Similarly, consider $Sh(\Lambda^{op})$, where $\Lambda$ is the category of Grassmann algebras. This is the topos of “super-sets”, where super-algebra takes place in. This whole topos behaves like the general abstract super-point .

Of $Sh(\mathbb{L}_{inf})$, the topos of sheaves on infintiesimal loci. This behaves like a general abstract infinitesimally thickened point.

Or the Cahiers topos $Sh(ThCartSp)$ on smooth loci of the form $\mathbb{R}^n$ times an infinitesimal space. This is like the general abstract open ball with infinitesimal thickening .

That’s how I like to think of these toposes, at least.

Of course this point-like structure is also visible in the definition of $\infty$-cohesive site that I considered: the condition that for every covering family $\{U_i \to U\}$ we have that $\lim_\to Cech(U) \simeq *$ says that all objects in the site are contractible, as seen by the Grothendieck topology on the site.

Posted by: Urs Schreiber on October 16, 2010 6:28 PM | Permalink | Reply to this

### Re: thick points

That’s a very beautiful way to think about it, thanks!

Posted by: Mike Shulman on October 16, 2010 8:25 PM | Permalink | Reply to this

### Re: thick points

That’s a very beautiful way to think about it, thanks.

Okay, good. By the way, there is yet one more way in which cohesive $(\infty,1)$-toposes behave like points: they have the same shape as the point in the shape-theoretical sense of shape of an $(\infty,1)$-topos.

I think we can summarize the situation as follows: the main two aspects of the definition of an cohesive $(\infty,1)$-topos say that it is

1. a locally and globally $\infty$-connected $(\infty,1)$-topos;

2. a local $(\infty,1)$-topos.

(I am tending to regard this as the core good definition. Further conditions might be added, but need not.)

Now, geometrically this says two things:

1. local and global $\infty$-connectedness: the shape of a cohesive $(\infty,1)$-topos is that of a point

2. locality: it is a small neighbourhood of the standard point.

I have begun exposing it this way at Cohesive $(\infty,1)$-topos – Interpretation

Posted by: Urs Schreiber on October 17, 2010 1:04 AM | Permalink | Reply to this

### Re: thick points

Would the following also be correct to say? For a topos $E$ regarded as a generalized space, the objects of the category $E$ can also be regarded as generalized spaces, since they can be identified with “local homeomorphisms of topoi” into $E$ (the object $x$ corresponds to the slice topos $E/x$). A local homeomorphism over a “space” $E$ can be thought of as another space “glued together locally out of pieces of $E$”. For most “little topoi” such as $E=Sh(X)$, there is not enough local variety in “pieces of $X$” for this procedure to give you very much freedom in constructing a space: you don’t get things that look very different from $X$. So it’s not usually very useful to work inside such a category $E$ as a “category of generalized spaces.” However, when $E$ is a “big topos,” thought of as a “very thick point,” its local pieces encode a lot of interesting structure, so there is lots of interest in studying spaces glued together from pieces of $E$ as a category of “generalized spaces” in their own right. And of course, even in the latter case one can still consider spaces over $E$ that are not modeled on $E$—those would just be structured spaces (treating $E$ as the classifying topos of a “geometry”).

Posted by: Mike Shulman on October 17, 2010 4:52 AM | Permalink | Reply to this

### Re: thick points

This is certainly how I think about the difference between the big and small Zariski and etale topologies. Although, for a formalism for forming spaces based on local models, I like the Toen-Vaquié approach, as you know.

Posted by: James on October 17, 2010 9:37 AM | Permalink | Reply to this

### Re: thick points

I guess there is a duality at work, which makes the toposes that are big/gros as far as etale geometric morphisms into them are concerned be at the same time very small in that they are not considerably bigger than the point as far as their properties as spaces are concerned.

I like the Toen-Vaquié approach,

Can we see that sites of commutative monoids in a monoidal category as they consider in Au-dessous de $Spec \mathbb{Z}$ are cohesive?

(I have to run now, had no chance yet to think about it myself.)

Posted by: Urs Schreiber on October 17, 2010 12:30 PM | Permalink | Reply to this

### Re: thick points

My apologies, I meant this paper by Toën and Vaquié. See section 2.2.

I don’t actually understand what you mean in your remark above about duality (but then it’s hard for me to read anything on topos theory written in the tradition of the purely category-theoretic school).

Posted by: James on October 17, 2010 10:02 PM | Permalink | Reply to this

### Re: thick points

So does anything pointlike have a corresponding cohesive topos? Is there a $S^{\infty}$ one? (I seem to remember that was contractible). And cohesive toposes for those $k$-flavoured points, $Spec(k)$, for $k$ a field?

Posted by: David Corfield on October 17, 2010 7:12 PM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

I am curious about the inclusion of a “codiscrete functor”. From the pointless point of view, this functor is constant. Is there a situation where one would want to distinguish between codiscreet spaces, or is there some reason that one just needs to know that such an adjoint exists, even if it is trivial?

Posted by: tjs on October 17, 2010 6:27 PM | Permalink | Reply to this

### codisc

I am curious about the inclusion of a “codiscrete functor”. From the pointless point of view, this functor is constant.

Could you maybe expand? I don’t see what you mean. In all examples that we discussed, the functor $Codisc$ is not constant but literally equips a set with its codiscrete topology/smooth structure, etc. (the codiscrete space structure).

Posted by: Urs Schreiber on October 17, 2010 9:27 PM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

I think tjs means that if you believe that locales are more useful and important than topological spaces, then all (nonempty) indiscrete spaces are indistinguishable from each other, since they all have the same underlying locale (namely, the terminal one). So including a codiscrete functor in the definition of cohesive topos means that the objects of a cohesive topos are very much “sets of points equipped with cohesion” – in the sense that the cohesion “all by itself” doesn’t determine the underlying points of the space. I think it’s a good question: it’s not clear to me whether or why that’s a desirable feature. What do we use the indiscrete objects for?

Posted by: Mike Shulman on October 17, 2010 9:46 PM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

Mike,

That is precisely what I meant, but phrased in a much clearer way. Thanks for clarifying.

Urs,

Sorry for phrasing my question in such an incomprehensible way. Seeing as I have jumped aboard the pointless bandwagon, I don’t like the idea of seeing spaces in a fundamental way as sets of points. Do you know of any situation when you would want to do so? Also, how is the existence of the right adjoint in question used in the theory? Is it significant, or is it just sort of there if you want it?

Posted by: tjs on October 18, 2010 12:03 AM | Permalink | Reply to this

### general definitions of spaces versus concrete classes of examples

tjs writes

Seeing as I have jumped aboard the pointless bandwagon, I don’t like the idea of seeing spaces in a fundamental way as sets of points.

Good, then we are all on the same page: here in the general abstract context that we are talking about, we see spaces as objects of some topos. The generic object in there is far from being like a set of points with extra structure. Even if the topos is cohesive.

But I can see how the intro of the entry possibly gave the wrong imppression: the functor $\Gamma : E \to S$ of a cohesive topos may be thought of as producing the points that underlie a given space – but this does not say that this space is in any way characterized by this set of points. $\Gamma$ just picks the points that happen to be present. There may be very few points present, possibly none.

Do you know of any situation when you would want to do so?

This whole game of generalized spaces is about going back and forth between

• very general abstract definitions of spaces that support a good theory (that admit all the constructions that we want to do with them)

• specific classes of examples of such general spaces that we run into in practice.

Given a very general space in the form of an object in my gros topos, I can ask successive questions about how “tame” it is:

• is it representable?

• is it locally representable?

• does it have an underlying locale?

• does it have an underlying topological space?

• does it have an underlying petit topos?

• etc.

Knowing this is not essential for the general abstract theory, but it is usually very useful to know as much as possible along these lines in concrete applications.

For instance if I know that my space is concrete in the sense that it has an underlying topological space that determines it, then I have, usually, a much easier time computing its geometric homotopy groups: because these will be those of this topological space, and hence I have all the tools of topology at my disposal.

One should remember that people dreamed up concrete sheaves such as diffeological spaces for a purpose (and before other people came and pointed out that they are just special classes of even more general pointless spaces). For instance Chen used them as a tool for computing the cohomology of loop spaces.

Also, how is the existence of the right adjoint in question used in the theory? Is it significant, or is it just sort of there if you want it?

The right adjoint gives a way to say “generalized space that does happen to be concrete” in a completely intrinsic topos-theoretic way that never ever mentions the word “point” explicitly. It serves the purpose of providing the general abstract structure that characterizes what in some concrete examples turns out to be specific class of generalized spaces with underlying point set.

Posted by: Urs Schreiber on October 18, 2010 12:23 PM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

In order to clarify what seemed to be the misunderstanding that a cohesive topos contains only spaces built on an underlying set of points, I added a paragraph to the end of cohesive topos – Idea:

the cohesive topos allows us to ask for the underlying points of a space, but not every space in it is built on a set of points. Rather, the fact that a cohesive topos is local allows us to characterize precisely its sub-quasitopos of those spaces that are.

Posted by: Urs Schreiber on October 18, 2010 12:44 PM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

the functor $\Gamma\colon E\to S$ of a cohesive topos may be thought of as producing the points that underlie a given space — but this does not say that this space is in any way characterized by this set of points. $\Gamma$ just picks the points that happen to be present. There may be very few points present, possibly none.

Actually, as I was thinking about this some more last night, I realized that with Lawvere’s definition, this is not true: objects of a cohesive topos in Lawvere’s sense are required to have a lot of points. For instance, look at the last condition in the definition of a cohesive site: every object of the site $C$ is required to have a global section. In particular, that means that any sheaf $F\colon C^{op}\to Set$ such that $F(U)\neq\emptyset$ for at least one $U\in C$ must also have $\Gamma(F)=F(\ast)\neq\emptyset$.

The purpose of that condition on a cohesive site is to ensure that the transformation $Disc A \to Codisc A$ is monic, or equivalently that the transformation $\Gamma F \to \Pi_0 F$ is epic, which was part of Lawvere’s definition. These are two more ways of saying that the objects of the cohesive topos “have a good deal of points.” The second means that every connected component of a space has at least one point. The first is harder to understand directly, but here’s a nice example I thought of.

Let $cploc$ be a small subcategory of inhabited connected locales, closed under finite products, which contains the point $\ast$ but for which all other locales in $cploc$ have no points. Then $cploc$ is a locally connected site, and also a local site, hence $Sh(cploc)$ is a connected, locally connected, and local topos. But it violates this condition of Lawvere’s; for instance I think $\Pi_0$ will take each representable $U\in cploc$ to $\ast$, since it is connected, whereas $\Gamma$ will take it to the empty set. On the other side, the discrete sheaf on a set $A$ is, as always, the constant sheaf $U\mapsto A$, whereas the codiscrete sheaf is, as always, the sheaf $U\mapsto A^{cploc(\ast,U)}$. But for $U\neq \ast$, we have chosen $cploc(\ast,U)$ to be empty, so for such $U$ we have $Codisc(A)(U) = A^{\emptyset}= \ast$. Hence if $|A|\gt 1$, the transformation $Disc(A)\to Codisc(A)$ is not monic.

Now, maybe we do want to remove that condition of Lawvere’s, as Urs did recently on the nLab page for the $\infty$ version. But now at least I have some idea of what that condition says. (-:

Posted by: Mike Shulman on October 18, 2010 7:49 PM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

every connected component of a space has at least one point.

True. One point. I shouldn’t have said “possibly none”.

Posted by: Urs Schreiber on October 18, 2010 8:10 PM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

Wait a seond: where did the condition come from that objects of a cohesive site need global sections?

Consider the site $\{\emptyset \to *\}$ with trivial topology. Isn’t that a counterexample?

[edit: the details of this simplistic but maybe instructive example are here]

Posted by: Urs Schreiber on October 18, 2010 9:11 PM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

In that comment I specifically said I was talking about cohesive 1-toposes and Lawvere’s axioms. The axioms I wrote down at cohesive site were intended to capture Lawvere’s axioms. And the topos of presheaves on $\emptyset \to \ast$ is not cohesive according to Lawvere’s definition; we have $\Gamma(\emptyset)=\emptyset$ but $\Pi_0(\emptyset) = \ast$. Unless I’m missing something?

Posted by: Mike Shulman on October 18, 2010 9:52 PM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

Unless I’m missing something?

That’s right of course. But let’s just record the punchline: it is only the extra axiom on the comparison map (over local and global connectedness and locality) that rules out spaces with no points.

That seems to be relevant for the discussion we had with tjs.

That simplistic example actually is a kind of instructive: if one looks at it, it does give the most basic notion of “points lumped together by cohesion”. Namely the presheaf category on the interval is the arrow category of $Set$ and $\Gamma$ and $\Pi_0$ interpret an object $S \to I$ as an $I$-indexed family of sets, where the fibers $S_i$ are the cohesive pieces, as seen by $\Pi_0$.

But it does not satisfy Lawvere’s extra axioms, that’s true. I would like to disregard these extra axioms for a bit. Maybe I should invent yet another new term (any suggestions?).

Posted by: Urs Schreiber on October 18, 2010 10:13 PM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

“pre-cohesive”? (-:

Posted by: Mike Shulman on October 19, 2010 12:36 AM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

“pre-cohesive”? (-:

“pointless cohesion”?

Posted by: Urs Schreiber on October 19, 2010 12:54 AM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

If you don’t see a reason for those extra axioms (I don’t, but I haven’t really tried to read much of Lawvere’s paper), then I think it’s okay to use “cohesive” to mean just “locally connected, strongly connected, and local,” with a caveat that Lawvere included some extra axioms.

Posted by: Mike Shulman on October 19, 2010 9:57 PM | Permalink | Reply to this

### extra axioms

If you don’t see a reason for those extra axioms (I don’t, but I haven’t really tried to read much of Lawvere’s paper),

The “continuity” axiom is invoked just once in his article, in the “theorem 1” – but as the author himself notices: it is not really necessary there either.

For the record, that example there is secretly about formulating the homotopy category of simplicial sets in terms of cohesion: the category of simplicial sets is a cohesive topos: $\Pi_0$ sends a simplicial set to its set of connected components. With $Y^X$ the internal hom of simplicial sets we have for Kan complexes $X$ and $Y$ that

$Ho(X,Y) = \Pi_0(Y^X) \,.$

The continuity axiom is supposed to ensure something in this example, but I am not sure I am following. But also the example seems to me to more naturally be about the canonical cohesive $\infty$-topos. So all in all my impression is that the continuity axiom is not something a fundamental definition of cohesive topos needs to contain.

Instead, this example makes me think the following: given a cohesive $(\infty,1)$-topos $(\Pi \dashv Disc \dashv \Gamma \dashv Codisc) : \mathbf{H} \to \infty Grpd$ we canonically obtain a new $(\infty,1)$-category $\tilde \mathbf{H}$ with the same objects as $\mathbf{H}$, and with

$\tilde \mathbf{H}(X,Y) := \Pi (Y^X) \,,$

where on the right we have the internal hom $Y^X \in \mathbf{H}$. This works because $\Pi$ preserves products ( this axiom should be kept ) so that we have composition in $\tilde \mathbf{H}$ given by

$\Pi(Y^X) \times \Pi(Z^Y) \stackrel{\simeq}{\to} \Pi(Y^X \times Z^Y) \stackrel{\Pi(comp_{X,Y,Z})}{\to} \Pi(Z^X) \,.$

This canonical $\tilde \mathbf{H}$ encodes some interesting information: for $G \in \mathbf{H}$ any $\infty$-group object and $X \in \mathbf{H}$ any object we have that

$G Bund(X) = \mathbf{H}(X, \mathbf{B}G)$

is the $\infty$-groupoid of $G$-principal $\infty$-bundles on $X$, with equivalences between them as morphisms.

On the other hand, observe that

$\tilde \mathbf{H}(X, \mathbf{B}G)$

is something like the $\infty$-groupoid of concordances of $G$-bundles: objects are $G$-principal $\infty$-bundles on $X$, morphisms are concordances of these, namely $G$-bundles on $X \times I$ restricting over the endpoints of $I$ to given $G$-bundles.

This is an important object to consider. For ordinary groups concordance classes coincide with equivalence classes. In general, the difference is measured by the difference of the hom-spaces of $\mathbf{H}$ and $\tilde \mathbf{H}$.

So this seems to be an important construction in a cohesive $\infty$-topos, which is what Lawvere’s theorem 1 seems to be hinting at. But no additional axiom is needed for it.

Concerning the other axiom, that $\Gamma X \to \Pi_0 X$ is epi: this looks more interesing. In the article it is used in theorem 2 to show that the full subcategory of a cohesive topos on those objects for which $\Gamma X \to \Pi_0 X$ is an iso, is reflective.

These are objects with precisely one point per connected component. Examples are for instance infinitesimal objects in synthetic differential geometry. These are certainly important and interesting objects to consider.

It might be noteworthy that the relevance of the morphism $\Gamma X \to \Pi X$ becomes possibly even more amplified in the $\infty$-topos context: for $G$ an $\infty$-group in a cohesive $\infty$-topos $(\Pi \dashv \Disc \dashv \Gamma \dashv Codisc) : \mathbf{H} \to \infty Grpd$ and $\mathbf{B}G$ its internal delooping,, we have that (under $\infty Grpd \simeq Top$)

• $\Gamma \mathbf{B}G$ is the corresponding Eilenberg-MacLane space $K(G,1)$ for the discrete group underlying $G$;

• $\Pi \mathbf{B}G$ is the corresponding classifying space .

Hence the morphism in question is

$K(G,1) \to \mathcal{B}G \,.$

Posted by: Urs Schreiber on October 19, 2010 11:09 PM | Permalink | Reply to this

### Re: extra axioms

Do you have any feel for why it might be important that those objects are reflective?

Posted by: Mike Shulman on October 21, 2010 3:46 AM | Permalink | Reply to this

### Lie differentiation

Do you have any feel for why it might be important that those objects are reflective?

Thanks for asking. I think the coreflector is Lie differentiation .

Consider the $(\infty,1)$-Cahiers topos $\mathbf{H} = (\infty,1)Sh(ThCartSp)$. This is the $\infty$-topos of “synthetic differential $\infty$-groupoids”.

I claim that every $\infty$-Lie algebra $\mathfrak{g}$ is naturally regarded as an object $\mathbf{B}\mathfrak{g} \in \mathbf{H}$, which is a synthetic $\infty$-groupoid all whose spaces of $k$-morphisms are infinitesimal.

For instance when $\mathfrak{g}$ is the Lie algebra of a Lie group $G$, then $\mathbf{B}\mathfrak{g} \hookrightarrow \mathbf{B}G$ is the inclusion of group elements infinitesimally close to the neutral elements. (this is described at $\infty$-Lie algebroids – Lie algebras).

I claim moreover that on these objects the canonical morphism

$\Gamma \mathbf{B}\mathfrak{g} \to \Pi \mathbf{B}\mathfrak{g}$

is an equivalence. I have typed up the proof of this at $\infty$-Lie algebroids – In a cohesive $(\infty,1)$-topos.

Moreover, I think I have shown at $(\infty,1)$-cohesive site that $\Gamma X \to \Pi X$ is an $(\infty,1)$-epimorphism.

Write $\mathbf{L} \hookrightarrow \mathbf{H}$ for the full sub-$(\infty,1)$-category on the objects $X$ for which $\Gamma X \to \Pi X$ is an equivalence.

Now we can follow Lawvere’s argument in the proof of his theorem 2 on page 5:

since in a cohesive $\infty$-topos $\Gamma$ is a left adjoint, it preserves $(\infty,1)$-colimits. It follows that $\mathbf{L}$ is closed under colimits in $\mathbf{H}$ and the embedding of course preserves them.

If $\mathbf{L}$ is locally presentable (see below), then the adjoint $(\infty,1)$-functor theorem implies that $\mathbf{L}$ is coreflective.

$\mathbf{L} \stackrel{\hookrightarrow}{\underset{Lie}{\leftarrow}} \mathbf{H} \,.$

For $H$ a cohesive $\infty$-group, consider morphisms

$\phi : \mathbf{B}\mathfrak{g} \to \mathbf{B}H \,.$

The adjunct of this is thus

$\mathbf{B}\mathfrak{g} \to Lie \mathbf{B}H \,.$

Since $\phi$ can send the infinitesimal elements in $\mathfrak{g}$ only to infinitesimal elements in $H$, we may put

$\mathbf{B}\mathfrak{h} := Lie \mathbf{B}H \,.$

The unit of the adjunction

$\mathbf{B}\mathfrak{g} \hookrightarrow \mathbf{B}G$

includes an $\infty$-Lie algebra as the collection of infinitesimal $k$-morphisms for all $k$ into the corresponding cohesive $\infty$-group.

Now: is $\mathbf{L}$ locally presentable? This would follow if it were also reflectively embedded. For the 1-categorical case this is what Lawvere does show in the proof of his theorem 2. It looks like this proof should do through verbatim for the $(\infty,1)$-category case. But I need to think about this more. Do you see this?

A writeup of all this is also at Cohesive $(\infty,1)$-topos – Infinitesimal objects.

Posted by: Urs Schreiber on October 21, 2010 1:05 PM | Permalink | Reply to this

### Re: extra axioms

So if I understand right, you’re saying that coreflectivity is interesting, but that reflectivity is maybe only useful as a tool to derive coreflectivity?

is L locally presentable?

In the 1-categorical case, I think you can conclude that from the fact that it is the inverter of the natural transformation $\Gamma \to \Pi$ between two accessible functors. Since the 2-category of accessible categories and accessible functors is closed under (non-strict) 2-limits in Cat, it follows that L is accessible, and so since it is cocomplete (being closed under colimits in the cocomplete H) it must be locally presentable.

Does that argument carry over to the $\infty$-case? Does the $(\infty,2)$-category of locally presentable $(\infty,1)$-categories have 2-limits like inverters? Probably one could prove directly that it has inverters without needing a full $(\infty,2)$-categorical machinery.

The unit of the adjunction… includes an ∞-Lie algebra as the collection of infinitesimal k-morphisms for all k into the corresponding cohesive ∞-group.

I’m not quite following here. The terminal object $\ast$ is in this subcategory, right? So since $\Gamma = Hom(\ast,-)$ it seems like the counit of the coreflection should be inverted by $\Gamma$—in other words, the coreflection doesn’t change the global sections. That means it must be instead adjusting $\Pi$ to match $\Gamma$, i.e. it must be “pulling apart” the cohesion between points in order to put them into separate connected parts. But it sounds like you’re saying that the coreflector should be “throwing away” information about points other than the identity… why is that?

Posted by: Mike Shulman on October 21, 2010 10:33 PM | Permalink | Reply to this

### Re: extra axioms

is $L$ locally presentable?

In the 1-categorical case, I think you can conclude that from the fact that it is the inverter of the natural transformation $\Gamma \to \Pi$ between two accessible functors. Since the 2-category of accessible categories and accessible functors is closed under (non-strict) 2-limits in Cat, it follows that L is accessible, and so since it is cocomplete (being closed under colimits in the cocomplete H) it must be locally presentable.

Thanks for this, Mike! I added this argument in to fill the corresponding gap in the proof of that theorem 2 at cohesive topos – Properties.

I had not been conciously aware of this fact that $AccCat$ is closed under 2-limits. But I see the statement now at accessible category – Properties. Do you have a reference for a proof?

Over at accessible $(\infty,1)$-category – Properties we state closure of $(\infty,1)AccCat$ only under pullbacks (following HTT, of course).

But given the situation for $AccCat$, I suppose it is to be expected that the analogous statement is true for $(\infty,1)AccCat$.

Posted by: Urs Schreiber on October 22, 2010 3:11 PM | Permalink | Reply to this

### Re: extra axioms

But given the situation for $AccCat$, I suppose it is to be expected that the analogous statement is true for $(\infty,1)AccCat$.

HTT, prop. 5.4.7.3 asserts that the $(\infty,1)$-category version of $(\infty,1)AccCat$ has all small limits and that these are preserved by the inclusion into the $(\infty,1)$-version of $(\infty,1)Cat$.

Posted by: Urs Schreiber on October 22, 2010 3:35 PM | Permalink | Reply to this

### Re: extra axioms

The proof that $Acc$ has 2-limits is in Makkai-Paré, and I think at least some special cases of it are proven in Adámek-Rosicky.

HTT, prop. 5.4.7.3 asserts that the (∞,1)-category version of (∞,1)AccCat has all small limits and that these are preserved by the inclusion into the (∞,1)-version of (∞,1)Cat.

That’s a step, but it’s not enough for what we need here, since an inverter is not a (∞,1)-categorical limit but rather an (∞,2)-categorical one, since it involves a noninvertible 2-morphism (the one being inverted).

Posted by: Mike Shulman on October 22, 2010 6:42 PM | Permalink | Reply to this

### Re: extra axioms

So if I understand right, you’re saying that coreflectivity is interesting, but that reflectivity is maybe only useful as a tool to derive coreflectivity?

That’s what I was implying, yes.

What if we localize $\mathbf{H}^{op}$ at all morphisms $W = \{(Disc S \to Codisc S)^{op}\}$ to get the reflective subcategory $\mathbf{L}^{op} \hookrightarrow \mathbf{H}^{op}$ full on the $W$-local objects, hence the objects $X \in \mathbf{W}$ such that $\mathbf{H}(X, Disc S \to Codisc S)$ is an equivalence. Then $\mathbf{L} \hookrightarrow \mathbf{H}$ is coreflective.

it must be “pulling apart” the cohesion between points in order to put them into separate connected parts.

Yes, I didn’t say that well: my discussion was all based on the special case where not only $\Gamma X \to \Pi X$ is an equivalence, but moreover goes between two contractible objects.

More generally the coreflector is possibly better described as performing a combination of taking things apart and throwing away cohesive things. I am not sure I’ll get it right this time, but here is an attempt: applied to the delooping of a group $\mathbf{B}G$ it seems to take for every single element of $G$ separately a formal contractible neighbourhood. So it produces a re-thickening of $\Gamma \mathbf{B}G$ by cohesive neighbourhoods around each discrete bit.

Posted by: Urs Schreiber on October 22, 2010 12:14 AM | Permalink | Reply to this

### Re: extra axioms

How do you know that you can localize $H^{op}$ at the large collection of morphisms $W$?

it seems to take for every single element of G separately a formal contractible neighbourhood.

That seems plausible. Now how do Lie algebras come into the picture?

Posted by: Mike Shulman on October 22, 2010 1:34 AM | Permalink | Reply to this

### Re: Lie differentiationn

Mike wrote:

Now how do Lie algebras come into the picture?

The discussion I gave above should still hold, only that $Lie \mathbf{B}G$ is bigger than I said it was.

So for every $L_\infty$-algebra $\mathfrak{g}$ there is an object $\mathbf{B}\mathfrak{g}$ in the $(\infty,1)$-Cahiers topos. It is modeled by the simplicial presheaf

$\int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot \mathfrak{g}_k$

where $\mathfrak{g}_k$ is an infinitesimal object (with a single point) and $\mathbf{\Delta}$ is the fat simplex .

How do we recognize this intrinsically as an $\infty$-Lie algebra? Like this:

1. it is a fat point, $\Gamma \mathbf{B}\mathfrak{g} \simeq * \simeq \Pi \mathbf{B}\mathfrak{g}$;

2. it has the $\mathbb{A}^1$-cohomology of the $L_\infty$-algebra $\mathfrak{g}$ (where $\mathbb{A}^1$ is our line object, which is the real line for the Cahier topos):

$\pi_0\mathbf{H}(\mathbf{B}\mathfrak{g}, \mathbf{B}^n \mathbb{A}^1) \simeq H^n(CE(\mathfrak{g}))$

(this follows with the technology described at function algebras on $\infty$-stacks).

So the question now is what is the universal factorization of morphisms $\mathbf{B}\mathfrak{g} \to \mathbf{B}H$? If we assume we do have the coreflector that we have been talking about, then that will produces it: $\mathbf{B}\mathfrak{g} \to Lie \mathbf{B}H \to \mathbf{B}H$.

And given the above simplicial presheaf model of $\mathbf{B}\mathfrak{g}$ and observing that this is cofibrant, we see in a more precise way what we just said two comments earlier: this $Lie \mathbf{B}H$-thing has to consist of the points in $\mathbf{B}H$ and their infinitesimal neighbourhoods, because that’s the only “cohesive pieces” that the infinitesimal objects $\mathfrak{g}_k$ can hit in each degree.

So first I said we should have $Lie \mathbf{B}H$ is $\mathbf{B}\mathfrak{h}$, but as you remarked rightly, this can’t be. It must be somthing more like “$\mathbf{B}(\mathfrak{h} \times \Gamma(H))$”. Not sure yet.

How do you know that you can localize $\mathbf{H}^{\mathrm{op}}$ at the large collection of morphisms $W$?

Right, I don’t. :-/ I should adhere to my rule and not post after 1 am.

Posted by: Urs Schreiber on October 22, 2010 8:14 AM | Permalink | Reply to this

### Re: Lie differentiationn

is a fat point $\Gamma \mathbf{B}\mathfrak{g} \simeq * \simeq \Pi \mathbf{B}\mathfrak{g}$.

Just for the record:

In

definition 4.6 calls the objects $X$ which preserve fiber propducts and with

$\Gamma X \simeq *$

formula moduli problems (in the $\infty$-topos over the site with trivial topology of formal duals of $E_\infty$-algebras over a field $k$).

(There is no explicit notion of $\Pi$ in his setup. On the other hand, that is still a question we are trying to solve: which sites of duals of $E_\infty$-rings are cohesive?)

In Remark 5.4 the examples of such $X$ obtained by Lie integration

$X = \exp(\mathfrak{g})$

(in the notation of that entry) is considered and this assignment is stated in theorem 5.3 to establish an $\infty$-equivalence between formal moduli problems and dg-Lie algebras ($\simeq$ $L_\infty$-algebras).

Posted by: Urs Schreiber on October 22, 2010 4:41 PM | Permalink | Reply to this

### Vopenca for ∞-categories?

How do you know that you can localize $\mathbf{H}^{op}$ at the large collection of morphisms $W$?

Suppose I would fall back to Vopěnka’s principle.

How would the principle in its incarnation in the context of locally presentable categories lift to the context of locally presentable $(\infty,1)$-categories?

Can one say anything useful about this?

I am aware of its version for combinatorial model categories. It would seem that this can be used to get a statement about $\infty$-colimit-closed full $\infty$-subcategories of locally presentable $\infty$-categories being coreflective. But I am not sure I see it, either way.

Posted by: Urs Schreiber on October 29, 2010 12:56 AM | Permalink | Reply to this

### Re: Vopenka for ∞-categories?

Good question; I don’t know the answer.

Posted by: Mike Shulman on October 29, 2010 7:18 AM | Permalink | Reply to this

### Re: Vopenca for ∞-categories?

It would seem that this can be used to get a statement about ∞-colimit-closed full ∞-subcategories of locally presentable ∞-categories being coreflective.

I found it:

Rosicky, Tholen, Left-determined model categories and universal homotopy theories

This states on page 4

Theorem 2.3 Let $C$ be a left proper combinatorial model category and $Z$ a class of morphisms. Under Vopenka’s principle, the left Bousfield localization $L_Z C$ exists.

With HTT this implies that under Vopenka every locally presentable $(\infty,1)$-category has a reflective localization at every class of morphisms.

Posted by: Urs Schreiber on October 29, 2010 9:35 AM | Permalink | Reply to this

### Re: Vopenca for ∞-categories?

But you wanted to localize $H^{op}$, not $H$, right? Or equivalently to colocalize $H$? I don’t see why $H^{op}$ should be locally presentable.

Posted by: Mike Shulman on October 29, 2010 3:47 PM | Permalink | Reply to this

I don’t see why $\mathbf{H}^{\mathrm{op}}$ should be locally presentable.

Right, the theorem mentioned does not help yet in the given application.

What would help is the $\infty$-analog of another equivalent statement of Vopenka: that every full subcat of a locally presentable one which is closed under colimits is coreflective.

But of course even better than resorting to such statements would be to get a better explicit control of that subcategory.

So let me get back to the construction that Lawvere indicates. When he says “the reflection can be constructed as the pushout of the adjunction map along the basic epimorphism” possibly he means to say that the reflector $X \to L X$ is given by the pushout

$\array{ Disc \Gamma X &\to& X \\ \downarrow && \downarrow \\ Disc \Pi X &\to& L X } \,.$

For when we hit this diagram with $\Gamma$ and use that it preserves colimits and that $Disc$ is full and faithful it becomes the pushout

$\array{ \Gamma X &\stackrel{\simeq}{\to}& \Gamma X \\ \downarrow && \downarrow \\ \Pi X &\to& \Gamma L X }$

and if we hit it with $\Pi$ and use that this also preserves pushouts and use the assumption that $\Gamma X \to \Pi X$ is an epi we get

$\array{ \Gamma X &\to& \Pi X \\ \downarrow && \downarrow \\ \Pi X &\to& \Pi L X & \simeq \Pi X } \,.$

So on $L X$ we have that $\Pi$ and $\Gamma$ agree.

I am fond of a statement that is vaguely similar: for $* \to A$ a pointed object, we can form the pullback that I call $\mathbf{\flat}_{dR} A$:

$\array{ \mathbf{\flat}_{dR} A &\to& Disc \Gamma A \\ \downarrow && \downarrow \\ {*} &\to& A }$

I know that this is the coefficient for “$A$-valued differential forms”. There is a canonical map $\Omega A \to \mathbf{\flat}_{dR} A$ that is the Maurer-Cartan form : for $A = \mathbf{B}G$ for $G$ an ordinary Lie group this is the ordinary MC form regarded as a morphism of sheaves $G \to \mathbf{\flat}_{dR} \mathbf{B}G = \Omega^1_{flat}(-,\mathfrak{g})$.

So this is also a kind of Lie differentiation. Notably we have

$\Gamma ( \mathbf{\flat}_{dR}A ) \simeq *$

(since $\Gamma$ preserves pullbacks and $\Gamma Disc \simeq Id$).

Posted by: Urs Schreiber on November 1, 2010 9:47 PM | Permalink | Reply to this

I am beginning to grow wary of the idea that the inverter of $\Gamma \to \Pi$ is the right thing to consider in the first place.

For a while I have been suggesting (to myself, mainly) that Lie integration/differentiation in a locally $\infty$-connected $\infty$-topos $\mathbf{H}$ is the adjunction

$*/\mathbf{H} \stackrel{\overset{\mathbf{\Pi}_{dR}}{\leftarrow}}{\underset{\mathbf{\flat}_{dR}}{\to}} \mathbf{H} \,,$

where the right adjoint $\mathbf{\flat}_{dR}$ forms the pullbacks

$A \mapsto \left( \array{ \mathbf{\flat}_{dR} &\to& Disc \Gamma A \\ \downarrow && \downarrow \\ * &\to& A } \right)$

and the left adjoint $\mathbf{\Pi}_{dR}$ the pushouts

$X \mapsto \left( \array{ X &\to& * \\ \downarrow && \downarrow \\ Disc \Pi X &\to& \mathbf{\Pi}_{dR}(X) } \right) \,.$

I have decent control over $\mathbf{\flat}_{dR}A$ in that I can compute this explicitly in large classes of examples. This shows that this is the coefficient object for flat $A$-valued differential forms.

Accordingly then by adjunction a morphism $\mathbf{\Pi}_{dR} \to A$ encodes a flat $A$-valued differential form on $X$. (Reading the above diagrams, one sees that it encodes actually a flat $A$-principal $\infty$-bundle on $X$ subject to the constraint that the underlying $\infty$-bundle is trivial: such a flat connection on a trivial bundle is a flat form!)

Of course most of $A$ is irrelevant for values of differential forms. For instance if $A = \mathbf{B}U(1)$ the differential forms with values in it see only the linear $\mathbf{B}\mathbb{R}$-approximation. This is formalized by noticing that by adjunction every morphism $\mathbf{\Pi}_{dR} \to A$ factors universally through the counit

$\mathbf{\Pi}_{dR} \mathbf{\flat}_{dR} A \to A \,.$

This object on the left therefore plays the role of an infinitesimal approximation to $A$. Let me write it $Lie A$.

My problem with making progress with this has been (and still is) that I have trouble computing this left hand. I can compute $\mathbf{\flat}_{dR} A$. But then for the remaining homotopy pushout I need to form a cofibrant replacement of that and so forth, and this I have not yet been able to express in a useful form.

But I have been considering this in the locally $\infty$-connected case so far. What changes when we assume the context is a cohesive $\infty$-topos?

One thing is this:

we have

$\Pi Lie A \simeq {*}$

and

$\Gamma Lie A \simeq {*}$.

The first generally since

$\Pi \mathbf{\Pi}_{dR} X \simeq {*}$.

The second similarly with $\Gamma \mathbf{\flat}_{dR} A \simeq {*}$ and using that $\Gamma$ also preserves pushouts in a cohesive $\infty$-topos.

So in a cohesive $\infty$-topos the object $Lie A$ is indeed a fat point, as seen by both $\Gamma$ and $\Pi$.

Posted by: Urs Schreiber on November 2, 2010 3:00 AM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

I think $\Pi_0$ will take each representable $U \in cploc$ to $*$, since it is connected,

I’d think this follows as soon as constant presheaves are sheaves, hence follows for your locally connected site.

Because then $\Pi_0$ is given by the colimit, and the colimit over a representable functor is always the point.

Posted by: Urs Schreiber on October 18, 2010 8:25 PM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

Or more generally, if you believe that locales are more useful and important than topological spaces, you might ask: what are non-sober spaces ever good for?

(Can someone tell me whether the Zariski topology on an affine variety, say, has to be sober? What about the spectrum of a commutative ring?)

Posted by: Tom Leinster on October 18, 2010 1:01 AM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

Both are sober. The thing to check is that every closed irreducible set is the closure of a unique point. In the case of the Zariski topology of an affine variety, each closed irreducible corresponds to a unique prime ideal of the ring that gives rise to the affine variety.

Posted by: Todd Trimble on October 18, 2010 1:30 AM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

Well, that’s true if by “affine variety” you mean some kind of affine scheme. I’m sure that’s what working algebraic geometers all do nowadays, but IIRC the baby intros to algebraic geometry that I read first included only maximal ideals when talking about spaces called “varieties.”

In fact, I believe that the Zariski topology on $Spec R$ (the usual affine scheme whose points are prime ideals of $R$) is the soberification of the Zariski topology on $Spm R$ (the space of maximal ideals). In other words, the topology on $Spm R$ is not sober, but if you add in new points for all the irreducible closed sets, you get $Spec R$. So believing in locales is another good reason for the scheme-theoretic viewpoint.

Posted by: Mike Shulman on October 18, 2010 6:23 AM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

I thought about saying something about “variety” vs. “scheme”, but figured Tom meant “scheme” in his question. I seem to hear “variety” a lot these days even though I think what is meant is the scheme.

Posted by: Todd Trimble on October 18, 2010 11:40 AM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

Actually, I really did mean variety, but that’s OK: with your reply and Mike’s, I’ve got the answers I asked for. Thanks.

It would be nice if what Mike said were true, that one Zariski topology is the soberification of the other.

Posted by: Tom Leinster on October 18, 2010 11:49 AM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

I think what Mike said is right. The points of the soberification (what a word!) are in one-to-one correspondence with frame maps from the topology to the two-element frame, and these may be checked to be in one-to-one correspondence with closed irreducible subsets of the space. For the Zariski topology of a affine variety, “closed” corresponds to ideal (each closed set in the variety consists of all maximal ideals which contain some given ideal) and “irreducible” corresponds to the condition of the ideal being prime. So the points of the soberification are in one-to-one correspondence with the prime ideals, which is what one is considering for the points of the affine scheme.

Posted by: Todd Trimble on October 18, 2010 2:36 PM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

included only maximal ideals when talking about spaces called “varieties.”

And that’s what Wikipedia does too (maximal ideals of $k[x_1,\dots,x_n]$ being identifiable with points of $k^n$). Unless I’ve misremembered the bit of algebraic geometry I know, which is entirely possible.

Posted by: Mike Shulman on October 18, 2010 6:32 AM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

Thanks a lot. So, the question remains.

Posted by: Tom Leinster on October 18, 2010 1:55 AM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

Possibly you might want to talk to Tim Porter about this, who I think is preparing some notes on finite topological spaces (and their utility for homotopy theory) for the nLab. If I recall correctly, the category of finite topological spaces (which most of the time are non-sober) is equivalent to the category of finite preorders; basically, for each finite space, say $x \leq y$ if the closure of $x$ is contained in the closure of $y$. This in turn is equivalent to the opposite of finite distributive lattices.

Well, this may not be such a great answer, and it’s generally true that non-sober spaces are far-removed from the traditional concerns of geometry. But spaces which seem pathological from the point of view of geometry do make their appearance in logical studies, for example in topological domain theory. I guess someone like Alex Simpson would be good to talk to about this.

Posted by: Todd Trimble on October 18, 2010 2:52 PM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

One thing I like about the Toën-Vaquié approach is that it doesn’t work with underlying point sets. (This is useful in algebraic geometry, where you have many underlying point sets, one for every ring.) They just take a topos, a subcategory of “local models”, and certain allowable gluing maps, and then they consider the closure of the subcategory under the allowable gluing. If point set topology is pointalism, then they do collage.

I like their approach a lot, but it’s definitely not in final form. I’d love to see it reworked and polished by a competent category theorist.

Posted by: James on October 18, 2010 3:43 AM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

That sounds like it has something in common with part of a PhD thesis I just read, Nicolas Michel’s Categorical Foundations of $K$-Theory. (It hasn’t yet completed the formalities of being accepted, so it’s probably not publicly available.) He takes an axiomatic approach to “locally trivial” objects, a major example being schemes. The two main categorical tools are fibrations and Grothendieck topologies.

Posted by: Tom Leinster on October 18, 2010 3:59 AM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

Here’s one place I know of where indiscrete spaces appear. In any quasitopos, one can define an intrinsic notion of “coarse object” and prove that the coarse objects are reflective and form a topos. In topological-looking quasitopoi, the coarse objects are the “indiscrete” ones. This gives you a way to recover “non-cohesive” objects canonically from the category of “cohesive” ones, if I may borrow the suggestive word—the definition of coarse objects doesn’t depend on any “underlying-set” functor or geometric morphism to be given a priori.

On the other hand, for a cohesive topos we already have the “non-cohesive” objects sitting inside the cohesive ones as the discrete objects, which seems intuitively (to me) to be a more natural embedding. So why do we also need the indiscrete ones?

Posted by: Mike Shulman on October 18, 2010 6:27 AM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

Here’s a bit of easy playing around that helped me understand what “codiscrete objects” look like a bit better. Consider a site $ctop$ consisting of (inhabited) connected topological spaces, with its usual open cover topology. (I’m going to stay in the 1-categorical world because I understand it better. If we were in the ∞-world, we’d look at contractible spaces instead.) We can’t take all of them for size reasons, so take some small subcategory closed under products. This is a 1-cohesive site over $Set$; a “space” in the cohesive topos $Sh(ctop)$ is a sheaf on $ctop$, i.e. a functor $ctop^{op}\to Set$ that glues along open covers. Evidently any topological space $X$ (not necessarily connected or contractible) induces such a sheaf, so $Top$ maps into $Sh(ctop)$, and it’s probably a full embedding on a big part of $Top$ (depending on what spaces we put into $ctop$).

Now the discrete space on a set $X$ is the constant sheaf $U\mapsto X$. (The constant presheaf is a sheaf by the axioms of a cohesive site.) In other words, when we probe this discrete space $Disc X$ by mapping out of connected spaces, any such map must be constant, and there are exactly $X$ such constant maps. But this is exactly the same sheaf induced by the discrete topological space $X_d \in Top$ in the above way, so here the discrete objects are exactly what you would expect.

Next, the codiscrete space on a set $X$ is the sheaf $U \mapsto X^{|U|}$, where $|U| = \Gamma(U)$ is the underlying set of points of $U$. (That it has to be this follows pretty directly from the adjunctions; that this presheaf is actually a sheaf follows from the definition of 1-cohesive site.) In other words, when we probe a codiscrete space $Codisc X$ by mapping out of connected spaces, the maps are just the set-maps from $U$ into the set $X$. But this is exactly the same sheaf induced by the codiscrete topological space $X_c \in Top$ as above, so again we get what we expect. (Since nonempty codiscrete spaces are connected and even contractible, many of these sheaves will actually be representable.)

But here’s the kicker: we could just as easily replace $ctop$ by a similar category $cloc$ of (inhabited) connected locales, with a similar functor $Loc \to Sh(cloc)$. The discrete and codiscrete spaces on a set will be defined in the same way as above, and the discrete ones will again be represented by discrete locales, but the codiscrete ones won’t be represented by anything in the world of locales. It’s as if passing to the sheaf topos has “adjoined indiscrete objects” to our category of locales, or “caused all our objects to become equipped with underlying sets.”

Anyway, one final point is that the functor $Top \to Sh(ctop)$ preserves limits (of course) but not many colimits other than open covers. That’s one reason Johnstone introduced the “topological topos” I mentioned above: it admits a full embedding from a good-sized subcategory of $Top$ (the sequential spaces) that preserves all limits and a lot of colimits too. But I’m inclined to think now that this topos is probably not cohesive, because its “cohesion” is defined in terms of sequences which are not a “connected” notion. (The site Johnstone gives is not a locally connected site, but I’m not sure how to prove that it couldn’t also be defined by some better-behaved site.) (I also wonder now what the ∞-topos of ∞-sheaves on that same site would be like.)

Posted by: Mike Shulman on October 18, 2010 7:17 AM | Permalink | Reply to this

### Re: Cohesive ∞-Toposes

I am now reminded of the “topological systems” in Steve Vickers’ book Topology via Logic, which “interpolate” between topological spaces and locales. That is, they have both a set of points and a frame of opens, but we require neither that an open is determined by the points it contains (as in a topological space) or that a point is determined by the opens containing it (as in a locale). I feel like the objects of a cohesive topos behave quite similarly to this: they have an underlying set (or ∞-groupoid, if we’re in the ∞-world) of points, as well as some “cohesion,” but neither points nor “cohesion” necessarily determines the other.

From that point of view, it seems reasonable to ask whether we can characterize some subcategories of a cohesive topos which act like the topological spaces and the locales sitting inside the “topological systems.” For an analogue of topological spaces, I think we do have a good candidate. The endofunctor $Codisc \circ \Gamma$ of a cohesive topos is a left-exact idempotent monad, which (on a topos) can be identified not just with a “subtopos” (in this case, the subtopos is $Set$ or whatever the base topos is) but also with a Lawvere-Tierney topology whose sheaves are the images of objects of $Set$ (in this case, the codiscrete spaces). But we can also talk about separated objects for a Lawvere-Tierney topology which form a reflective quasitopos. It seems like these should be the spaces where the “cohesion is determined by the points”. In particular, when the site of our cohesive topos is a “concrete site,” these separated objects can be identified with the concrete sheaves.

So maybe here we have one answer to the question of what codiscrete spaces are good for: as a tool to identify the separated spaces, i.e. the concrete sheaves. But like all good answers, it immediately suggests another question: is there a notion of “co-concrete sheaf” which identifies the spaces (analogous to locales) in which the points are determined by the cohesion?

(It also doesn’t really answer tjs’ original question, since if you really believe locales are better than spaces (or topological systems), then there doesn’t seem to be much reason for you to care about identifying the concrete sheaves either. But it’s interesting.)

Another important question this suggests is: is there a good notion of “separated object” for a left-exact reflector on an (∞,1)-topos? The 1-topos-theoretic notion of “separated object” involves monomorphisms, whose ∞-analogue is a little slippery (at least, to me)….

Posted by: Mike Shulman on October 18, 2010 7:32 AM | Permalink | Reply to this

### concrete sheaves

So maybe here we have one answer to the question of what codiscrete spaces are good for: as a tool to identify the separated spaces, i.e. the concrete sheaves.

Yes, that was the motivation that led to the discussion which eventually led to this blog entry here:

I had been happily looking at locally and globally $\infty$-connected $\infty$-toposes for a while, when in discussion with Dave Carchedi it became manifest that for bringing concrete $\infty$-sheaves into the picture in a general abstract way, we ought to be also making the locality of the $\infty$-topos manifest.

It also doesn’t really answer tjs’ original question, since if you really believe locales are better than spaces (or topological systems), then there doesn’t seem to be much reason for you to care about identifying the concrete sheaves either. But it’s interesting.

Two remarks:

1. What precisely a concrete sheaf is depends on which topos is the base topos. Maybe there is a base topos other than $Set$/ $\infty Grpd$ better suited for tjs’ needs.

2. We should generally care about organizing all objects of our gros topos into hierarchies of ranging from wild to tame , simply because that is part of understanding the whole topos and because it means having more tools available in special cases in practice.

One bit of the hierarchy is the one Jacob Lurie discusses in Structured Spaces , as summarized here:

• representables (affines)

• $\hookrightarrow$ locally representables (schemes)

• $\hookrightarrow$ with underlying $\infty$-topos and local structure (“structured $\infty$-topos”)

• $\hookrightarrow$ with underlying $\infty$-topos

• $\hookrightarrow$ everything

I think concrete $\infty$-sheaves should be one more data point sitting in between schemes and the structured $\infty$-toposes: they are those for which the underlying petit $\infty$-topos is that on a genuine topological space.

It can’t hurt to pay special attention to these, even if not every example one runs into will be of this kind.

In this context remember also that Andrew Stacey was amplifying all the nice extra properties that concrete sheaves and notably Isbell-self-dual concrete sheaves have. Can’t hurt to make all this extra information explicit, in cases where it happens to be available.

Posted by: Urs Schreiber on October 18, 2010 8:59 AM | Permalink | Reply to this

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