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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeMar 2nd 2012
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexJon Beardsley (whom I mentioned in another thread as a graduate student working with Morava) has recently sent me some emails, having seen some stuff I wrote on Boolean algebras and Stone duality on my old blog. With his kind permission, I am posting some extracts, thinking they could be of interest to some people here. > TT: I'm not a topologist by training; I'm more of a category theorist. So I am not immediately familiar with the Bousfield lattice or the Boolean algebra contained in it. Can you tell me a bit more? > JB: Sure! The basic idea is, given a homology theory $E$ (which is the same as a spectrum, since any spectrum $E$ defines a homology theory where $E(X)=\pi_*(E\wedge X)$), we can define the class of $E$-acyclics, denoted $\langle E \rangle$, which is all spectra $X$ such that $E(X)=0$, or equivalently, $E\wedge X$ is contractible. We can then order these classes of acyclics by reverse inclusion, that is, we say $\langle E \rangle \leq \langle F \rangle$ if $\langle F \rangle$ is contained in $\langle E \rangle$. Equivalently, we might say that $\langle E \rangle \leq \langle F \rangle$ if it is the case that $F(X)=0$ implies $E(X)=0$. This work was done by Bousfield in the late 70's who showed a lot of interesting things. For instance, and this is kind of obvious I guess, since the classes of acyclics are localizing subcategories, we can consider localization functors on the homotopy category of spectra. It turns out that for any homology theory, such a functor exists, and its "kernel" is the class of acylics of that homology theory, and its image is what are called the "locals" of that homology theory. Bousfield showed that there are well defined meet and join operations on the collection of all acyclics. It turns out that the join, $\langle E \rangle \vee \langle F \rangle$ is actually the same as wedging (in the topological sense) the spectra $E$ and $F$, i.e. $\langle E \rangle \vee \langle F \rangle = \langle E \vee F \rangle$. The meet is much less well understood and there are some open problems pertaining to it which may rely on some large cardinal axioms or various set theory things, though I don't really know. We also retain the smash product in here though, so that $\langle E \rangle \wedge \langle F \rangle$ can just be defined as $\langle E\wedge F \rangle$ though smash and meet are not the same. People called this collection of acylics the "Bousfield lattice", denoted by $B$, although it wasn't until 89 that this guy Okhawa showed that the collection is a set and not a proper class, hence we really can call it a lattice, and do lattice theory on it. It's also important to note that the partial ordering puts $\langle S \rangle$ (the Bousfield class of the sphere spectrum) at the top of the lattice, and $\langle 0 \rangle$, the Bousfield class of a contractible spectrum, at the bottom. > (cont.) So you may know all of that stuff, but that's the basic rundown. If at any time this is all just too topological, feel free to stop reading and just let me know! I've been staring at this stuff for weeks now and it's all I can really talk about. Fortunately or unfortunately this has resulted in a number of e-mails to mathematicians who may or may not be all that interested. > (cont.) Bousfield also defined a complementation operator $a: B \to B$ which takes a class $\langle E \rangle$ to the join of all $\langle F \rangle$ such that $\langle E \rangle \wedge \langle F \rangle = \langle E\wedge F \rangle = \langle 0 \rangle$. This is well defined and exists. We call $B A$ the sub-poset of all classes of acyclics $\langle X \rangle$ such that $\langle X \rangle \vee a\langle X \rangle = \langle S \rangle$ and $\langle X \rangle \wedge a \langle X \rangle = \langle 0 \rangle$. There's a little more to it I guess, but this gives you a Boolean algebra. One really important point is that in $B A$, the smash product of spectra and the meet are the same, so the meet of two classes has a really nice description. It has some rather easily understood prime/maximal filters, those generated by the so-called Morava K-theories. These are just special spectra that are extremely useful in chromatic homotopy theory. They are special since they are ring spectra (i.e. there is a unit $S \to K(n)$ and a nice map $K(n) \wedge K(n) \to K(n)$) and any module spectrum over $K(n)$ is just a sum of suspensions of $K(n)$, so they are in some sense the fields of brave new algebra (again, sorry if I'm going into way too much detail here). It turns out that they are minimal in the Bousfield lattice, so obviously minimal in $B A$. They generate prime filters. So far, these are the only prime filters I can find, which means they're the only points in $Spec(B A)$, but considering that there's a countable number of these things, that really means we've got a pretty weird topological space (though it's Hausdorff).

   Jon Beardsley (whom I mentioned in another thread as a graduate student working with Morava) has recently sent me some emails, having seen some stuff I wrote on Boolean algebras and Stone duality on my old blog. With his kind permission, I am posting some extracts, thinking they could be of interest to some people here.

   TT: I’m not a topologist by training; I’m more of a category theorist. So I am not immediately familiar with the Bousfield lattice or the Boolean algebra contained in it. Can you tell me a bit more?

   JB: Sure! The basic idea is, given a homology theory $E$ (which is the same as a spectrum, since any spectrum $E$ defines a homology theory where $E(X)=\pi_*(E\wedge X)$), we can define the class of $E$-acyclics, denoted $\langle E \rangle$, which is all spectra $X$ such that $E(X)=0$, or equivalently, $E\wedge X$ is contractible. We can then order these classes of acyclics by reverse inclusion, that is, we say $\langle E \rangle \leq \langle F \rangle$ if $\langle F \rangle$ is contained in $\langle E \rangle$. Equivalently, we might say that $\langle E \rangle \leq \langle F \rangle$ if it is the case that $F(X)=0$ implies $E(X)=0$. This work was done by Bousfield in the late 70’s who showed a lot of interesting things. For instance, and this is kind of obvious I guess, since the classes of acyclics are localizing subcategories, we can consider localization functors on the homotopy category of spectra. It turns out that for any homology theory, such a functor exists, and its “kernel” is the class of acylics of that homology theory, and its image is what are called the “locals” of that homology theory. Bousfield showed that there are well defined meet and join operations on the collection of all acyclics. It turns out that the join, $\langle E \rangle \vee \langle F \rangle$ is actually the same as wedging (in the topological sense) the spectra $E$ and $F$, i.e. $\langle E \rangle \vee \langle F \rangle = \langle E \vee F \rangle$. The meet is much less well understood and there are some open problems pertaining to it which may rely on some large cardinal axioms or various set theory things, though I don’t really know. We also retain the smash product in here though, so that $\langle E \rangle \wedge \langle F \rangle$ can just be defined as $\langle E\wedge F \rangle$ though smash and meet are not the same. People called this collection of acylics the “Bousfield lattice”, denoted by $B$, although it wasn’t until 89 that this guy Okhawa showed that the collection is a set and not a proper class, hence we really can call it a lattice, and do lattice theory on it. It’s also important to note that the partial ordering puts $\langle S \rangle$ (the Bousfield class of the sphere spectrum) at the top of the lattice, and $\langle 0 \rangle$, the Bousfield class of a contractible spectrum, at the bottom.

   (cont.) So you may know all of that stuff, but that’s the basic rundown. If at any time this is all just too topological, feel free to stop reading and just let me know! I’ve been staring at this stuff for weeks now and it’s all I can really talk about. Fortunately or unfortunately this has resulted in a number of e-mails to mathematicians who may or may not be all that interested.

   (cont.) Bousfield also defined a complementation operator $a: B \to B$ which takes a class $\langle E \rangle$ to the join of all $\langle F \rangle$ such that $\langle E \rangle \wedge \langle F \rangle = \langle E\wedge F \rangle = \langle 0 \rangle$. This is well defined and exists. We call $B A$ the sub-poset of all classes of acyclics $\langle X \rangle$ such that $\langle X \rangle \vee a\langle X \rangle = \langle S \rangle$ and $\langle X \rangle \wedge a \langle X \rangle = \langle 0 \rangle$. There’s a little more to it I guess, but this gives you a Boolean algebra. One really important point is that in $B A$, the smash product of spectra and the meet are the same, so the meet of two classes has a really nice description. It has some rather easily understood prime/maximal filters, those generated by the so-called Morava K-theories. These are just special spectra that are extremely useful in chromatic homotopy theory. They are special since they are ring spectra (i.e. there is a unit $S \to K(n)$ and a nice map $K(n) \wedge K(n) \to K(n)$) and any module spectrum over $K(n)$ is just a sum of suspensions of $K(n)$, so they are in some sense the fields of brave new algebra (again, sorry if I’m going into way too much detail here). It turns out that they are minimal in the Bousfield lattice, so obviously minimal in $B A$. They generate prime filters. So far, these are the only prime filters I can find, which means they’re the only points in $Spec(B A)$, but considering that there’s a countable number of these things, that really means we’ve got a pretty weird topological space (though it’s Hausdorff).

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMar 3rd 2012
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   Author: Mike Shulman
   Format: MarkdownItexThanks! Is Jon reading this thread? Can we ask him questions here? I'd like to know whether the statement about wedges being joins is also true for infinite joins? And is the statement about meets being smashes in $B A$ formal, or does it involve some topological input?

   Thanks! Is Jon reading this thread? Can we ask him questions here?

   I’d like to know whether the statement about wedges being joins is also true for infinite joins? And is the statement about meets being smashes in $B A$ formal, or does it involve some topological input?

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeMar 3rd 2012
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   Author: Todd_Trimble
   Format: MarkdownItexGreat, thanks! I'm pretty sure Jon has set up or will soon set up an account, and I know he is eager to talk about this material... ask away!

   Great, thanks! I’m pretty sure Jon has set up or will soon set up an account, and I know he is eager to talk about this material… ask away!

4. • CommentRowNumber4.
   • CommentAuthorJon Beardsley
   • CommentTimeMar 3rd 2012
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   Author: Jon Beardsley
   Format: TextHi! I made it! Excuse the minimal use of TeX for now, I'm on my phone at the moment. So yeah, Mike, in B, wedge is meet, entirely. So the join over any arbitrary set of Bousfield classes is just the Bousfield class of the wedge of representatives of those classes (I personally think that's a pretty nice feature of Bousfield classes). As for your other question, I'm not sure I understand. The smash and meet are both defined in B. Before passing to BA, we go through DL, which is a distributive lattice. DL is precisely the class of so-called Bousfield idempotents, i.e. every class whose smash product with itself is just itself again. In B, the smash of classes is always less than or equal to the meet, but in DL, the smash of two classes equals the meet. It is not hard to show this, and if you like I can write it down here, or you can check out Lemma 3.3 and Proposition 3.4 of Hovey and Palmieri's paper. So anyway, BA is in DL and has the same meet, which is smash. So I guess it is formal, but I may not understand the question. Please let me know if you'd like to know more about this, or could perhaps expand on your question. Best, Jon

   Hi! I made it! Excuse the minimal use of TeX for now, I'm on my phone at the moment.
   
   So yeah, Mike, in B, wedge is meet, entirely. So the join over any arbitrary set of Bousfield classes is just the Bousfield class of the wedge of representatives of those classes (I personally think that's a pretty nice feature of Bousfield classes).
   
   As for your other question, I'm not sure I understand. The smash and meet are both defined in B. Before passing to BA, we go through DL, which is a distributive lattice. DL is precisely the class of so-called Bousfield idempotents, i.e. every class whose smash product with itself is just itself again. In B, the smash of classes is always less than or equal to the meet, but in DL, the smash of two classes equals the meet. It is not hard to show this, and if you like I can write it down here, or you can check out Lemma 3.3 and Proposition 3.4 of Hovey and Palmieri's paper. So anyway, BA is in DL and has the same meet, which is smash. So I guess it is formal, but I may not understand the question. Please let me know if you'd like to know more about this, or could perhaps expand on your question.
   
   Best,
   Jon
5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeMar 3rd 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThanks Jon (and welcome)! Yes, you answered my question. It looks like $B$ itself is what category theorists call a [[quantale]]: a complete and cocomplete closed monoidal poset. For a (small) poset, cocompleteness implies completeness, and since the smash product (the monoidal structure) distributes over joins (wedges), cocompleteness also implies closedness. A quantale is a [[frame]] (hence, in particular, a distributive lattice and a [[Heyting algebra]]) if and only if its monoidal structure is idempotent and its unit is the top element. So if you take a quantale whose unit element is already the top (here, the sphere spectrum) and restrict to its idempotent elements, you get a frame: this looks like how you get your $DL$. Finally, the complemented elements in any Heyting algebra form a Boolean algebra, giving $BA$. In particular, it appears that $DL$ also determines a "space" (a [[locale]]). Does that all seem right?

   Thanks Jon (and welcome)! Yes, you answered my question. It looks like $B$ itself is what category theorists call a quantale: a complete and cocomplete closed monoidal poset. For a (small) poset, cocompleteness implies completeness, and since the smash product (the monoidal structure) distributes over joins (wedges), cocompleteness also implies closedness.

   A quantale is a frame (hence, in particular, a distributive lattice and a Heyting algebra) if and only if its monoidal structure is idempotent and its unit is the top element. So if you take a quantale whose unit element is already the top (here, the sphere spectrum) and restrict to its idempotent elements, you get a frame: this looks like how you get your $DL$. Finally, the complemented elements in any Heyting algebra form a Boolean algebra, giving $BA$.

   In particular, it appears that $DL$ also determines a “space” (a locale). Does that all seem right?

6. • CommentRowNumber6.
   • CommentAuthorJon Beardsley
   • CommentTimeMar 3rd 2012
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   Author: Jon Beardsley
   Format: MarkdownItexYep. So here is a point of a little confusion for me though. $DL$ as a frame determines a locale (just the opposite category). However, $DL$ is also a distributive lattice, so generates a coherent frame, $Idl(DL)$, which corresponds to a coherent space. So, it doesn't seem that these should be the same to me, that is, $DL$ is not a coherent frame, but it generates one. Either way we know by Stone's work that the category of distributive lattices is equivalent to the category of coherent spaces, by the process I described. So there's definitely a coherent space we could call $Spec(DL)$, and also a Stone space space $Spec(BA)$. I've been playing with these spaces a little bit recently. So, is every frame a distribute lattice? And if so, since we have a categorical duality between $DistLat$ and $CohSpaces$, and a categorical equivalence between $CohSpaces$ and $CohLoc$, which is dually equivalent to a subcategory of frames, does the category of frames include into itself in this way or something? Again, sorry about the formatting... =\

   Yep. So here is a point of a little confusion for me though. $DL$ as a frame determines a locale (just the opposite category). However, $DL$ is also a distributive lattice, so generates a coherent frame, $Idl(DL)$, which corresponds to a coherent space. So, it doesn’t seem that these should be the same to me, that is, $DL$ is not a coherent frame, but it generates one. Either way we know by Stone’s work that the category of distributive lattices is equivalent to the category of coherent spaces, by the process I described. So there’s definitely a coherent space we could call $Spec(DL)$, and also a Stone space space $Spec(BA)$. I’ve been playing with these spaces a little bit recently.

   So, is every frame a distribute lattice? And if so, since we have a categorical duality between $DistLat$ and $CohSpaces$, and a categorical equivalence between $CohSpaces$ and $CohLoc$, which is dually equivalent to a subcategory of frames, does the category of frames include into itself in this way or something?

   Again, sorry about the formatting… =\

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeMar 3rd 2012
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   Author: Todd_Trimble
   Format: MarkdownItexYes, every frame is by definition a complete lattice where finite meets distribute over arbitrary joins. In particular, a frame is a distributive lattice. Everything hangs of course on what we take the morphisms to be. For frames, the morphisms are functions that preserve finite meets and arbitrary joins. Morphisms of distributive lattice preserve only finite meets and finite joins. Etc. Following what you wrote, we have a string of functors $$Frame \stackrel{forget}{\to} DistLat \stackrel{Idl}{\to} CohSp \hookrightarrow Loc \to Frame$$ where the first and third are covariant functors, and the second and fourth are contravariant equivalences. The notation can get a little confusing, since the points of $Idl(A)$ are actually prime ideals of $A$, not ideals; you should think of the composite as mapping a frame $A$ to the topology on $Idl(A)$, which as a frame is the frame of ideals on $A$. The last three arrows composed give you the left adjoint $DistLat \to Frame$ to the forgetful functor $Frame \to DistLat$ (i.e., if $B$ is a distributive lattice, then the frame of ideals of $B$ is the free frame generated from $B$ as distributive lattice; see the corollary on page 59 of Stone Spaces). So the composite of all four gives you the monad on $Frame$ for the adjunction between frames and distributive lattices.

   Yes, every frame is by definition a complete lattice where finite meets distribute over arbitrary joins. In particular, a frame is a distributive lattice.

   Everything hangs of course on what we take the morphisms to be. For frames, the morphisms are functions that preserve finite meets and arbitrary joins. Morphisms of distributive lattice preserve only finite meets and finite joins. Etc.

   Following what you wrote, we have a string of functors

   $$Frame \stackrel{forget}{\to} DistLat \stackrel{Idl}{\to} CohSp \hookrightarrow Loc \to Frame$$

   where the first and third are covariant functors, and the second and fourth are contravariant equivalences. The notation can get a little confusing, since the points of $Idl(A)$ are actually prime ideals of $A$, not ideals; you should think of the composite as mapping a frame $A$ to the topology on $Idl(A)$, which as a frame is the frame of ideals on $A$. The last three arrows composed give you the left adjoint $DistLat \to Frame$ to the forgetful functor $Frame \to DistLat$ (i.e., if $B$ is a distributive lattice, then the frame of ideals of $B$ is the free frame generated from $B$ as distributive lattice; see the corollary on page 59 of Stone Spaces). So the composite of all four gives you the monad on $Frame$ for the adjunction between frames and distributive lattices.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeMar 3rd 2012
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   Author: Mike Shulman
   Format: MarkdownItex...all of which makes me feel that from an *a priori* category-theoretic viewpoint, the locale determined by the frame $DL$ would be a more natural object to study than the coherent space/locale determined by its underlying distributive lattice, or the Stone space determined by its Boolean algebra of complemented elements. But I expect Bousfield knew what he was doing.

   …all of which makes me feel that from an a priori category-theoretic viewpoint, the locale determined by the frame $DL$ would be a more natural object to study than the coherent space/locale determined by its underlying distributive lattice, or the Stone space determined by its Boolean algebra of complemented elements. But I expect Bousfield knew what he was doing.

9. • CommentRowNumber9.
   • CommentAuthorJon Beardsley
   • CommentTimeMar 4th 2012
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   Author: Jon Beardsley
   Format: MarkdownItexMike, is there a good way of talking about sheaves on locales? I might guess this has something to do with stuff I don't currently understand, i.e. sheaves on topoi or stuff like that. I mean, the reason I'm personally following this trail at the moment is to try to understand some vague comments of Jack Morava's about a structure sheaf on the Bousfield lattice (or subsets thereof) that might lead to an understanding of $Spec(S)$. At the moment I do not entirely understand how to make all this cohere (even as a solid conjecture or anything), but it at least seems useful to have a space to talk about a structure sheaf thereon. That's the main reason I started looking at $DL$ and $BA$ and associated coherent spaces rather than looking at locales straight up. And while Bousfield knew what he was doing, I don't think he looked at associated coherent spaces at all, at least, in these ways that we are trying to. So I'm really open to hearing about other approaches to studying this frame!

   Mike, is there a good way of talking about sheaves on locales? I might guess this has something to do with stuff I don’t currently understand, i.e. sheaves on topoi or stuff like that. I mean, the reason I’m personally following this trail at the moment is to try to understand some vague comments of Jack Morava’s about a structure sheaf on the Bousfield lattice (or subsets thereof) that might lead to an understanding of $Spec(S)$. At the moment I do not entirely understand how to make all this cohere (even as a solid conjecture or anything), but it at least seems useful to have a space to talk about a structure sheaf thereon. That’s the main reason I started looking at $DL$ and $BA$ and associated coherent spaces rather than looking at locales straight up.

   And while Bousfield knew what he was doing, I don’t think he looked at associated coherent spaces at all, at least, in these ways that we are trying to. So I’m really open to hearing about other approaches to studying this frame!

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeMar 4th 2012
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    Author: Mike Shulman
    Format: MarkdownItexThere are all sorts of great ways of talking about sheaves on locales! The most obvious thing to do is to just take the definition of a sheaf (in terms of a functor on the open set category with gluing) and notice that it doesn't refer to points, so it makes perfect sense verbatim for a locale. You can also make sense of the "espace etale" approach, as long as you define "local homeomorphism" correctly for locales (the espace etale for a sheaf on a locale is, of course, itself a locale). A sheaf on a locale is a special case of a sheaf on a topos/site, just as for a sheaf on a topological space, and ultimately that's the "right" way to think about it, but I don't think you need toposes for sheaves on locales any *more* than you do for sheaves on topological spaces.

    There are all sorts of great ways of talking about sheaves on locales! The most obvious thing to do is to just take the definition of a sheaf (in terms of a functor on the open set category with gluing) and notice that it doesn’t refer to points, so it makes perfect sense verbatim for a locale. You can also make sense of the “espace etale” approach, as long as you define “local homeomorphism” correctly for locales (the espace etale for a sheaf on a locale is, of course, itself a locale).

    A sheaf on a locale is a special case of a sheaf on a topos/site, just as for a sheaf on a topological space, and ultimately that’s the “right” way to think about it, but I don’t think you need toposes for sheaves on locales any more than you do for sheaves on topological spaces.

11. • CommentRowNumber11.
    • CommentAuthorJon Beardsley
    • CommentTimeMar 4th 2012
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    Author: Jon Beardsley
    Format: MarkdownItexOh yeah of course. Haha! Interesting. I will have to explore this!

    Oh yeah of course. Haha! Interesting. I will have to explore this!

12. • CommentRowNumber12.
    • CommentAuthorAndrew Stacey
    • CommentTimeMar 5th 2012
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    Author: Andrew Stacey
    Format: MarkdownItex(I'm doubly pleased to see Jon here - for the usual reasons of it being great to have new people, but also because from the error logs it looked as though there were a few technical hitches with the registration process so the fact that Jon is now registered implies that I managed to get those ironed out.)

    (I’m doubly pleased to see Jon here - for the usual reasons of it being great to have new people, but also because from the error logs it looked as though there were a few technical hitches with the registration process so the fact that Jon is now registered implies that I managed to get those ironed out.)

13. • CommentRowNumber13.
    • CommentAuthorJon Beardsley
    • CommentTimeMar 5th 2012
    • PermaLink
    Author: Jon Beardsley
    Format: MarkdownItexYeah thanks Andrew! I think I applied twice or something because the first attempt didn't seem to work. Hope it didn't mess anything up.

    Yeah thanks Andrew! I think I applied twice or something because the first attempt didn’t seem to work. Hope it didn’t mess anything up.

14. • CommentRowNumber14.
    • CommentAuthorJon Beardsley
    • CommentTimeMar 6th 2012
    • (edited Mar 6th 2012)
    • PermaLink
    Author: Jon Beardsley
    Format: MarkdownItexHey Mike. So I don't know if you're still interested in talking about this kind of thing, but if so I'd love to hear your thoughts. I've been looking more at the frame $DL$ is the Bousfield lattice $B$. So, we agree that it's a frame, hence a locale. Now, the goal here, at the moment, is to get some kind of space-type object that I can actually lay hands on, so to speak. What do we mean by the "open" sets of $DL$ as a frame? Do we mean the open sets of the topological space $pt(A)$ which is the collection of locale morphisms $2\to DL$? It seems to me that whether we're looking at $pt(DL)$ or the space associated to $DL$ as a distributive lattice, we're looking at the prime ideals of $DL$, and neither construction seems to be more advantageous.

    Hey Mike. So I don’t know if you’re still interested in talking about this kind of thing, but if so I’d love to hear your thoughts. I’ve been looking more at the frame $DL$ is the Bousfield lattice $B$. So, we agree that it’s a frame, hence a locale. Now, the goal here, at the moment, is to get some kind of space-type object that I can actually lay hands on, so to speak. What do we mean by the “open” sets of $DL$ as a frame? Do we mean the open sets of the topological space $pt(A)$ which is the collection of locale morphisms $2\to DL$? It seems to me that whether we’re looking at $pt(DL)$ or the space associated to $DL$ as a distributive lattice, we’re looking at the prime ideals of $DL$, and neither construction seems to be more advantageous.

15. • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeMar 6th 2012
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    Author: Mike Shulman
    Format: MarkdownItexI'm happy to keep talking about this; I don't really understand what you're trying to achieve, but I can answer questions about locales. (-: The "open sets" of a locale are just the elements of its underlying frame; the topological space $pt(A)$ may not have enough points to support all of those "open sets" as actual sets of points. And the points of a locale are completely prime filters of its underlying frame, which I don't think are the same as prime ideals of that frame *qua* distributive lattice.

    I’m happy to keep talking about this; I don’t really understand what you’re trying to achieve, but I can answer questions about locales. (-: The “open sets” of a locale are just the elements of its underlying frame; the topological space $pt(A)$ may not have enough points to support all of those “open sets” as actual sets of points. And the points of a locale are completely prime filters of its underlying frame, which I don’t think are the same as prime ideals of that frame qua distributive lattice.

16. • CommentRowNumber16.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 6th 2012
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexFrom Mike's #8: > But I expect Bousfield knew what he was doing. I should mention some other things that Jon told me over email the other day which seem sort of curious and intriguing. In speaking of the Boolean algebra (of complemented elements inside the frame $Idem(B)$ of smash-idempotents of the Bousfield lattice), he said that the Morava K-theories are atoms of the Boolean algebra, and that there is a conjecture that the Boolean algebra is in fact atomic (although it is not known, and could very well be false, that the Morava K-theories exhaust the atoms). Actually, I think he said that Morava K-theories are atomic in the Bousfield quantale. This might be semi-obvious to the experts, but I thought it was worth bringing up anyway. It might be nice to have an entry [[Morava K-theory]].

    From Mike’s #8:

    But I expect Bousfield knew what he was doing.

    I should mention some other things that Jon told me over email the other day which seem sort of curious and intriguing. In speaking of the Boolean algebra (of complemented elements inside the frame $Idem(B)$ of smash-idempotents of the Bousfield lattice), he said that the Morava K-theories are atoms of the Boolean algebra, and that there is a conjecture that the Boolean algebra is in fact atomic (although it is not known, and could very well be false, that the Morava K-theories exhaust the atoms).

    Actually, I think he said that Morava K-theories are atomic in the Bousfield quantale. This might be semi-obvious to the experts, but I thought it was worth bringing up anyway. It might be nice to have an entry Morava K-theory.

17. • CommentRowNumber17.
    • CommentAuthorJon Beardsley
    • CommentTimeMar 6th 2012
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    Author: Jon Beardsley
    Format: MarkdownItexIn response to Todd's statement above, which I guess is sort of my statement, the other things to look at are the so-called $A(n)$'s, which in some sense measure the failure of the telescope conjecture (which is not known to be true of false yet either, so such $A(n)$'s may not even exist). If they exist, they are atomic as well.

    In response to Todd’s statement above, which I guess is sort of my statement, the other things to look at are the so-called $A(n)$’s, which in some sense measure the failure of the telescope conjecture (which is not known to be true of false yet either, so such $A(n)$’s may not even exist). If they exist, they are atomic as well.

18. • CommentRowNumber18.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 6th 2012
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItex> the other things to look at are the so-called $A(n)$'s, which in some sense measure the failure of the telescope conjecture Can you tell us more about the $A(n)$'s (how are they characterized if they exist; also, is that what people call them , the "A of n's" or "A n's"?). Would the $A(n)$'s plus the Morava $K(n)$'s exhaust the atoms, or could there be other atoms? And finally, what is the telescope conjecture? No doubt answers could be found via google, but if there's anything you'd like to say...

    the other things to look at are the so-called $A(n)$’s, which in some sense measure the failure of the telescope conjecture

    Can you tell us more about the $A(n)$’s (how are they characterized if they exist; also, is that what people call them , the “A of n’s” or “A n’s”?). Would the $A(n)$’s plus the Morava $K(n)$’s exhaust the atoms, or could there be other atoms? And finally, what is the telescope conjecture?

    No doubt answers could be found via google, but if there’s anything you’d like to say…

19. • CommentRowNumber19.
    • CommentAuthorJon Beardsley
    • CommentTimeMar 6th 2012
    • (edited Mar 6th 2012)
    • PermaLink
    Author: Jon Beardsley
    Format: MarkdownItexSorry yeah, I wrote the above in a bit of a rush, knowing that I should probably explain a little more. Let's assume we're working locally to some prime $p$, i.e. we're working in the $p$-localized stable homotopy category, but I will not mention $p$ again... So the telescope conjecture perhaps requires a bit of background, but it's not too tough. I'll remark that any (homotopy class of a) spectrum $X$ in $hoSpectra$ is equivalent to a so-called CW-spectrum, which is a spectrum which is a CW-complex at each level. Since we have these so-called structure maps $\Sigma X_n\to X_{n+1}$ (hrm, subscript not working?) which are homeomorphisms, or we can demand them to be (once we take the homotopy category it won't matter anymore), we see that if the first space has a finite number of cells then so does every other space since when we suspend we're just adding a finite number of cells each time. We can call the homotopy class of this spectrum in $hoSpectra$ a "finite spectrum." A finite spectrum $X$ has "type n" if $K(m)_\ast(X)=0$ for every $m$<$n$ and $K(n)_\ast(X)\neq 0$. We will denote a finite spectrum of type $n$ by $F(n)$. It is known that the Boufield class of $F(n)$ is well defined, in the sense that any finite p-local spectrum of type $n$ will generate the same class (sorry I know I said I wouldn't mention $p$ again, but I felt that it was and important reminder). Now, for any $F(n)$ we have a special "self-map" we call $v_n:\Sigma^{d+i}F(n)\to \Sigma^i F(n)$ which we call a $v_n$-map and literally say "vee-en map." One of the nice things about this map is that $K(m)_\ast(v_n)$ is an isomorphism for $m=n$ and is trivial for $m$>$n$. When $n=0$, $d=0$ and otherwise $d$ is a multiple of $2p^n-2$ (again, we're working p-locally for everything). For our intents this map is unique. So, we have a "telescope" of $v_n$, which is homotopy colimit of the diagram $F(n)\overset{v_n}\to\Sigma^{-d}F(n)\overset{v_n}\to\cdots$, which we might denote by $T(n)$. Again, it is known that the Bousfield class of $T(n)$ is well defined. Now, the telescope conjecture asserts that $\langle T(n)\rangle=\langle K(n)\rangle$. I have heard that Douglas Ravenel has claimed to have both proven and disproven this over the years, and frankly I can't seem to get any consensus on the subject. I think most recently it was "disproven" but somebody said they found a flaw in the disproof. Okay, so, with all that behind us... recall that to every $K(n)$ we have an associated localization functor $L_{K(n)}$. We define $A(n)$ (which as far as I know is just pronounced "ey-en") to be the fiber of the map $T(n)\to L_{K(n)}T(n)$. Note, if it is in fact the case that $\langle K(n)\rangle=\langle T(n)\rangle$ then $L_{K(n)}=L_{T(n)}$ so $L_{K(n)}T(n)=T(n)$. That is the sense in which $A(n)$ measures the failure of the telescope conjecture. Now, actually, looking back on it, I see that it is a conjecture that the $A(n)$'s are even minimal. Very little is known about these $A(n)$'s (of course this may be related to their non-existence, but who knows). However, the conjecture is then the following (which actually rests on the shoulders of two other rather natural conjectures): The atoms of $BA$, the Boolean algebra in the Bousfield lattice, are $\langle K(n)\rangle$ and, for $n\geq 2$, $\langle A(n)\rangle$. Every element of $BA$ can be written as either a finite join of these atoms or as a complement of a finite join of these atoms in a unique way. So, perhaps I have been a little misleading with some of my comments, but that is the precise formulation of the conjectures (all contained in Hovey and Palmieri's "The Structure of the Bousfield Lattice").

    Sorry yeah, I wrote the above in a bit of a rush, knowing that I should probably explain a little more. Let’s assume we’re working locally to some prime $p$, i.e. we’re working in the $p$-localized stable homotopy category, but I will not mention $p$ again…

    So the telescope conjecture perhaps requires a bit of background, but it’s not too tough. I’ll remark that any (homotopy class of a) spectrum $X$ in $hoSpectra$ is equivalent to a so-called CW-spectrum, which is a spectrum which is a CW-complex at each level. Since we have these so-called structure maps $\Sigma X_n\to X_{n+1}$ (hrm, subscript not working?) which are homeomorphisms, or we can demand them to be (once we take the homotopy category it won’t matter anymore), we see that if the first space has a finite number of cells then so does every other space since when we suspend we’re just adding a finite number of cells each time. We can call the homotopy class of this spectrum in $hoSpectra$ a “finite spectrum.” A finite spectrum $X$ has “type n” if $K(m)_\ast(X)=0$ for every $m$<$n$ and $K(n)_\ast(X)\neq 0$. We will denote a finite spectrum of type $n$ by $F(n)$. It is known that the Boufield class of $F(n)$ is well defined, in the sense that any finite p-local spectrum of type $n$ will generate the same class (sorry I know I said I wouldn’t mention $p$ again, but I felt that it was and important reminder).

    Now, for any $F(n)$ we have a special “self-map” we call $v_n:\Sigma^{d+i}F(n)\to \Sigma^i F(n)$ which we call a $v_n$-map and literally say “vee-en map.” One of the nice things about this map is that $K(m)_\ast(v_n)$ is an isomorphism for $m=n$ and is trivial for $m$>$n$. When $n=0$, $d=0$ and otherwise $d$ is a multiple of $2p^n-2$ (again, we’re working p-locally for everything). For our intents this map is unique. So, we have a “telescope” of $v_n$, which is homotopy colimit of the diagram $F(n)\overset{v_n}\to\Sigma^{-d}F(n)\overset{v_n}\to\cdots$, which we might denote by $T(n)$. Again, it is known that the Bousfield class of $T(n)$ is well defined. Now, the telescope conjecture asserts that $\langle T(n)\rangle=\langle K(n)\rangle$. I have heard that Douglas Ravenel has claimed to have both proven and disproven this over the years, and frankly I can’t seem to get any consensus on the subject. I think most recently it was “disproven” but somebody said they found a flaw in the disproof.

    Okay, so, with all that behind us… recall that to every $K(n)$ we have an associated localization functor $L_{K(n)}$. We define $A(n)$ (which as far as I know is just pronounced “ey-en”) to be the fiber of the map $T(n)\to L_{K(n)}T(n)$. Note, if it is in fact the case that $\langle K(n)\rangle=\langle T(n)\rangle$ then $L_{K(n)}=L_{T(n)}$ so $L_{K(n)}T(n)=T(n)$. That is the sense in which $A(n)$ measures the failure of the telescope conjecture.

    Now, actually, looking back on it, I see that it is a conjecture that the $A(n)$’s are even minimal. Very little is known about these $A(n)$’s (of course this may be related to their non-existence, but who knows). However, the conjecture is then the following (which actually rests on the shoulders of two other rather natural conjectures):

    The atoms of $BA$, the Boolean algebra in the Bousfield lattice, are $\langle K(n)\rangle$ and, for $n\geq 2$, $\langle A(n)\rangle$. Every element of $BA$ can be written as either a finite join of these atoms or as a complement of a finite join of these atoms in a unique way.

    So, perhaps I have been a little misleading with some of my comments, but that is the precise formulation of the conjectures (all contained in Hovey and Palmieri’s “The Structure of the Bousfield Lattice”).

20. • CommentRowNumber20.
    • CommentAuthorJon Beardsley
    • CommentTimeMar 7th 2012
    • (edited Mar 7th 2012)
    • PermaLink
    Author: Jon Beardsley
    Format: MarkdownItexOh and Mike, thanks for you comment above. So lets say I was just playing around and did something like define a presheaf $F:DL\to Cat$ by saying $F(\langle E\rangle)=L_{E}hoSpectra$, i.e. on a Bousfield class it is just the category of (homotopy classes of) $E$ local spectra. A lot of the work for this is already done in some sense, since we know that the restriction maps are fine (here we assume that we've got a morphism $\langle E\rangle\to\langle F\rangle$ whenever $\langle E\rangle\leq\langle F\rangle$) since in general, if $\langle E\rangle\leq\langle F\rangle$ then $L_E L_F=L_E$. Does anything funny happen with it being a presheaf of categories? Would you recommend looking into any higher category for this thing, or can I just plug along and try to either show the equalizer diagram or the basic properties (like gluing and so forth)?

    Oh and Mike, thanks for you comment above. So lets say I was just playing around and did something like define a presheaf $F:DL\to Cat$ by saying $F(\langle E\rangle)=L_{E}hoSpectra$, i.e. on a Bousfield class it is just the category of (homotopy classes of) $E$ local spectra. A lot of the work for this is already done in some sense, since we know that the restriction maps are fine (here we assume that we’ve got a morphism $\langle E\rangle\to\langle F\rangle$ whenever $\langle E\rangle\leq\langle F\rangle$) since in general, if $\langle E\rangle\leq\langle F\rangle$ then $L_E L_F=L_E$. Does anything funny happen with it being a presheaf of categories? Would you recommend looking into any higher category for this thing, or can I just plug along and try to either show the equalizer diagram or the basic properties (like gluing and so forth)?

21. • CommentRowNumber21.
    • CommentAuthorMike Shulman
    • CommentTimeMar 7th 2012
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItex> we have these so-called structure maps $\Sigma X_n\to X_{n+1}$ which are homeomorphisms, or we can demand them to be Really? It seems to me that if those structure maps are homeomorphisms, then your spectrum must be a suspension spectrum, and not every spectrum is even equivalent to a suspension spectrum. In particular, suspension spectra are connective. Am I confused? > We define $A(n)$ to be the fiber of the map $T(n)\to L_{K(n)}T(n)$. So, given that you just defined $A(n)$, it seems to me that it is clear that $A(n)$ exists. The question appears to be rather whether $A(n)$ is contractible or not. This is of course just a point of language, but do people really phrase the question as "whether $A(n)$ exists"?

    we have these so-called structure maps $\Sigma X_n\to X_{n+1}$ which are homeomorphisms, or we can demand them to be

    Really? It seems to me that if those structure maps are homeomorphisms, then your spectrum must be a suspension spectrum, and not every spectrum is even equivalent to a suspension spectrum. In particular, suspension spectra are connective. Am I confused?

    We define $A(n)$ to be the fiber of the map $T(n)\to L_{K(n)}T(n)$.

    So, given that you just defined $A(n)$, it seems to me that it is clear that $A(n)$ exists. The question appears to be rather whether $A(n)$ is contractible or not. This is of course just a point of language, but do people really phrase the question as “whether $A(n)$ exists”?

22. • CommentRowNumber22.
    • CommentAuthorMike Shulman
    • CommentTimeMar 7th 2012
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexRe 20, I would indeed recommend looking into higher category theory of some sort. In general, the first question for a presheaf of categories is whether it is a strict presheaf or only a "pseudo" presheaf. Many presheaves of categories that appear in nature are only pseudo presheaves, necessitating at least some 2-category theory, but sometimes it is possible to make choices carefully to make them into a strict presheaf, and that may be the case here. However, once you start asking questions about gluing and sheafiness, then I think it becomes almost totally wrong to ask a presheaf of categories to satisfy the naive equalizer version of the sheaf condition: on the one hand it will rarely be true, and on the other hand it isn't usually what you want anyway; you want instead a [[stack]] condition that replaces the equalizer with a "descent object". Furthermore, I kind of doubt that a presheaf of *homotopy* categories would be well-behaved enough to be a stack, since passing to homotopy categories involves some brutal quotienting of all higher homotopies, and stack conditions depend on having well-behaved higher homotopies. So the question I would be inclined to ask is to consider the $(\infty,1)$-presheaf that sends $\langle E\rangle$ to the $(\infty,1)$-category obtained from that of spectra by localization at $E$, and question whether that is an $(\infty,1)$-stack. Unfortunately, this is kind of a complicated thing, but my guess is that it would be the "right" thing to look at.

    Re 20, I would indeed recommend looking into higher category theory of some sort.

    In general, the first question for a presheaf of categories is whether it is a strict presheaf or only a “pseudo” presheaf. Many presheaves of categories that appear in nature are only pseudo presheaves, necessitating at least some 2-category theory, but sometimes it is possible to make choices carefully to make them into a strict presheaf, and that may be the case here.

    However, once you start asking questions about gluing and sheafiness, then I think it becomes almost totally wrong to ask a presheaf of categories to satisfy the naive equalizer version of the sheaf condition: on the one hand it will rarely be true, and on the other hand it isn’t usually what you want anyway; you want instead a stack condition that replaces the equalizer with a “descent object”.

    Furthermore, I kind of doubt that a presheaf of homotopy categories would be well-behaved enough to be a stack, since passing to homotopy categories involves some brutal quotienting of all higher homotopies, and stack conditions depend on having well-behaved higher homotopies. So the question I would be inclined to ask is to consider the $(\infty,1)$-presheaf that sends $\langle E\rangle$ to the $(\infty,1)$-category obtained from that of spectra by localization at $E$, and question whether that is an $(\infty,1)$-stack. Unfortunately, this is kind of a complicated thing, but my guess is that it would be the “right” thing to look at.

23. • CommentRowNumber23.
    • CommentAuthorJon Beardsley
    • CommentTimeMar 7th 2012
    • (edited Mar 7th 2012)
    • PermaLink
    Author: Jon Beardsley
    Format: TextMike, with regards to the suspension spectrum situation, what you say makes good sense and there's a good chance I'm mistaken. I guess I was thinking that that's the way they do it in EKMM but I have never really learned all the details - that is, the "pre-spectra" or whatever they start out with have homeomorphisms as structure maps that they then apply a lot of hardware to (operads and so forth) to get a good "poInt-set" level category for spectra. But after checking I notice that these are homeomorphisms to loops on function spectra. I guess the point is we just need some topological notion of finiteness. I don't know the best way to define this. I will try to figure this out. Perhaps it works for now for me to just demand finite "stable cells" when we're thinking in terms of CW spectra? I do believe that all of our spectra are equivalent to CW-spectra at least, so we can talk about stable cells, but I am a significant distance away from being an "expert" on the subject by any metric. As regards my use of "exists," you are absolutely right. I should have said "is contractible". I guess I just meant that they don't exist as something interesting in some sense. I do indeed tend to speak rapidly and imprecisely, a bad habit given my chosen profession. I highly doubt anyone phrases it that way! :-) And lastly, I am rather excited at the thought of having to use/learn a little higher category theory. I haven't had too many chances yet!

    Mike, with regards to the suspension spectrum situation, what you say makes good sense and there's a good chance I'm mistaken. I guess I was thinking that that's the way they do it in EKMM but I have never really learned all the details - that is, the "pre-spectra" or whatever they start out with have homeomorphisms as structure maps that they then apply a lot of hardware to (operads and so forth) to get a good "poInt-set" level category for spectra. But after checking I notice that these are homeomorphisms to loops on function spectra. I guess the point is we just need some topological notion of finiteness. I don't know the best way to define this. I will try to figure this out. Perhaps it works for now for me to just demand finite "stable cells" when we're thinking in terms of CW spectra? I do believe that all of our spectra are equivalent to CW-spectra at least, so we can talk about stable cells, but I am a significant distance away from being an "expert" on the subject by any metric.
    
    As regards my use of "exists," you are absolutely right. I should have said "is contractible". I guess I just meant that they don't exist as something interesting in some sense. I do indeed tend to speak rapidly and imprecisely, a bad habit given my chosen profession. I highly doubt anyone phrases it that way! :-)
    
    And lastly, I am rather excited at the thought of having to use/learn a little higher category theory. I haven't had too many chances yet!
24. • CommentRowNumber24.
    • CommentAuthorMike Shulman
    • CommentTimeMar 7th 2012
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexThere are a lot of different models for spectra. In EKMM, the "spectra" they start with have homeomorphisms $X_n \cong \Omega X_{n+1}$ (this kind of spectrum goes back to Lewis-May); then they add some more stuff and put a model structure on it. For "diagram spectra" (symmetric spectra, orthogonal spectra, etc.) one allows arbitrary maps $X_n \to \Omega X_{n+1}$ (or equivalently $\Sigma X_n \to X_{n+1}$) in the objects of the underlying category, but then in the model structure the fibrant objects have the property that the maps $X_n \to \Omega X_{n+1}$ are weak equivalences. But all of these models have a notion of "cell" and "cell spectrum", and a corresponding notion of "finite cell spectrum" obtained by attaching finitely many cells, which is I think what you want. See for instance the general categorical notion of [[cell complex]].

    There are a lot of different models for spectra. In EKMM, the “spectra” they start with have homeomorphisms $X_n \cong \Omega X_{n+1}$ (this kind of spectrum goes back to Lewis-May); then they add some more stuff and put a model structure on it. For “diagram spectra” (symmetric spectra, orthogonal spectra, etc.) one allows arbitrary maps $X_n \to \Omega X_{n+1}$ (or equivalently $\Sigma X_n \to X_{n+1}$) in the objects of the underlying category, but then in the model structure the fibrant objects have the property that the maps $X_n \to \Omega X_{n+1}$ are weak equivalences. But all of these models have a notion of “cell” and “cell spectrum”, and a corresponding notion of “finite cell spectrum” obtained by attaching finitely many cells, which is I think what you want. See for instance the general categorical notion of cell complex.

25. • CommentRowNumber25.
    • CommentAuthorJon Beardsley
    • CommentTimeMar 8th 2012
    • (edited Mar 8th 2012)
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    Author: Jon Beardsley
    Format: MarkdownItexThanks for all the comments Mike. I'm sort of trying to approach this $(\infty,1)$-stack notion that you described above. It is clear to me that, from my current vantage point, even properly stating the question will be difficult. Do you recommend any specific references? I have begun sort of picking at HTT and some stuff by Toen and Vezzosi, both of which have been recommended to me. Is there anything else that you think might simplify this approach? Thanks again!

    Thanks for all the comments Mike. I’m sort of trying to approach this $(\infty,1)$-stack notion that you described above. It is clear to me that, from my current vantage point, even properly stating the question will be difficult. Do you recommend any specific references? I have begun sort of picking at HTT and some stuff by Toen and Vezzosi, both of which have been recommended to me. Is there anything else that you think might simplify this approach?

    Thanks again!

26. • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeMar 8th 2012
    • (edited Mar 8th 2012)
    • PermaLink
    Author: Urs
    Format: MarkdownItex> I'm sort of trying to approach this (∞,1)-stack notion that you described above. I haven't been following closely, but if I read correctly what you are talking about, you indeed want genuine [[(∞,2)-sheaves]] here, meaning genuine $(\infty,1)$-category-values $\infty$-functors. There is little to nothing about this currently in the literature. But of course the right answer can be built from the $\infty$-stacks = $(\infty,1)$-sheaf case that does have an exhaustive discussion in the literature. There should be at least the following three ways to approach it, which should all be equivalent,if done right. Let $C$ be some [[infinity-site]]. Write $\mathcal{C} := Sh_{(\infty,1)}(C)$ for the [[(infinity,1)-sheaf (infinity,1)-category]] over $C$, hence for the $\infty$-groupoid valued $\infty$-functors that satisfy [[descent]]. Then, for $C$ nice enough, the $(\infty,1)$-category valued functors on $C$ that satisfy descent should be equivalently * [[internal (infinity,1)-categories]] in $\mathcal{C}$ ([[complete Segal space]] objects in $\mathcal{C}$); * $\infty$-limit preserving functors $\mathcal{C}^{op} \to (\infty,1)Cat$; * the localization of $Funct_{\infty}(C^{op}, (\infty,1)Cat)$ at the covering sieves.

    I’m sort of trying to approach this (∞,1)-stack notion that you described above.

    I haven’t been following closely, but if I read correctly what you are talking about, you indeed want genuine (∞,2)-sheaves here, meaning genuine $(\infty,1)$-category-values $\infty$-functors.

    There is little to nothing about this currently in the literature. But of course the right answer can be built from the $\infty$-stacks = $(\infty,1)$-sheaf case that does have an exhaustive discussion in the literature.

    There should be at least the following three ways to approach it, which should all be equivalent,if done right.

    Let $C$ be some infinity-site. Write $\mathcal{C} := Sh_{(\infty,1)}(C)$ for the (infinity,1)-sheaf (infinity,1)-category over $C$, hence for the $\infty$-groupoid valued $\infty$-functors that satisfy descent.

    Then, for $C$ nice enough, the $(\infty,1)$-category valued functors on $C$ that satisfy descent should be equivalently

    • internal (infinity,1)-categories in $\mathcal{C}$ (complete Segal space objects in $\mathcal{C}$);

    • $\infty$-limit preserving functors $\mathcal{C}^{op} \to (\infty,1)Cat$;

    • the localization of $Funct_{\infty}(C^{op}, (\infty,1)Cat)$ at the covering sieves.

27. • CommentRowNumber27.
    • CommentAuthorJon Beardsley
    • CommentTimeMar 8th 2012
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    Author: Jon Beardsley
    Format: MarkdownItexThanks Urs. For your second bullet point below, is that typo, or are we indeed looking at functors from curly C op to $(\infty,1)Cat$?

    Thanks Urs. For your second bullet point below, is that typo, or are we indeed looking at functors from curly C op to $(\infty,1)Cat$?

28. • CommentRowNumber28.
    • CommentAuthorJon Beardsley
    • CommentTimeMar 8th 2012
    • PermaLink
    Author: Jon Beardsley
    Format: MarkdownItexAdditionally... we know that a frame is canonically a site... can we just then call it trivially an $\infty$-site? I mean, like an $(\infty,0)$-site or something? I want a big $\infty$-stack over a really um, non-$\infty$ category. Or in other words, I know I can make a space into an $\infty$-category, but can I extend that to locales/frames?

    Additionally… we know that a frame is canonically a site… can we just then call it trivially an $\infty$-site? I mean, like an $(\infty,0)$-site or something? I want a big $\infty$-stack over a really um, non-$\infty$ category. Or in other words, I know I can make a space into an $\infty$-category, but can I extend that to locales/frames?

29. • CommentRowNumber29.
    • CommentAuthorMike Shulman
    • CommentTimeMar 8th 2012
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexRe 28: Yes, every (0,1)-category (i.e. poset, such as a frame) is in particular a 1-category, and thus in particular a (2,1)-category, and thus ... in particular an $(\infty,1)$-category. Thus, every [[posite]] (such as a frame with its canonical topology) is in particular an $(\infty,1)$-site. Re 26-27: Yes, I think we do need $(\infty,2)$-sheaves to do it right. However, one could look at the underlying $(\infty,1)$-sheaf of that $(\infty,2)$-sheaf (which amounts to ignoring everything except the 0-object in Urs' first bullet point). Depending on what one's goal is, descent for that might be good enough.

    Re 28: Yes, every (0,1)-category (i.e. poset, such as a frame) is in particular a 1-category, and thus in particular a (2,1)-category, and thus … in particular an $(\infty,1)$-category. Thus, every posite (such as a frame with its canonical topology) is in particular an $(\infty,1)$-site.

    Re 26-27: Yes, I think we do need $(\infty,2)$-sheaves to do it right. However, one could look at the underlying $(\infty,1)$-sheaf of that $(\infty,2)$-sheaf (which amounts to ignoring everything except the 0-object in Urs’ first bullet point). Depending on what one’s goal is, descent for that might be good enough.

30. • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeMar 8th 2012
    • (edited Mar 8th 2012)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI had posted a reply before Mike's above, but in posting my connection broke down. It overlaps now with Mike's but just for the record, here is what I had written a few minutes back: > For your second bullet point below, is that typo It's not a typo, but I should have made this more explicitly: Compare to ordinary topos theory: given a [[Grothendieck topos]] $\mathcal{C}$, it is equivalent to sheaves _on itself_ when we regard $\mathcal{C}$ itself as a site equipped with the [[canonical topology]]. A functor $\mathcal{C}^{op} \to SET$ is a sheaf for this topology precisely if it preserves limits (sends colimits in $\mathcal{C}$ to limits in $SET$ (capitals for [[universe enlargement]], but never mind that)). So $\mathcal{C} \simeq Sh_{can}(\mathcal{C})$. The same remains true for $\infty$-toposes and $\infty$-groupoid valued sheaves. Moreover, every object $X$ in an $\infty$-topos $\mathcal{C}$ can be understood as an "internal groupoid", (see [[groupoid object in an (infinity,1)-topos]]). Then under this equivalence we identify $X \in \mathcal{C}$ with the functor $$ \mathcal{C}(-, X) : \mathcal{C}^{op} \to \infty GRPD $$ that it represents. Now the idea is to use this equivalence to implicitly say what a _category object_ in $\mathcal{C}$ is, namely now an $\infty$-limit preserving $\infty$-functor $\mathcal{C}^{op} \to (\infty,1) CAT$ which we implicitly think of as being represented by an "actual" category object $X \in \mathcal{C}$, which in turn we identify with an [[(infinity,2)-sheaf]] on the site $C$ of $\mathcal{C}$. This is fairly well understood in the 1-categorical context. See for instance section 3 of the article "Strong stacks" that is referenced at [[2-sheaf]]. > we know that a frame is canonically a site... can we just then call it trivially an ∞-site? Yes. > I mean, like an (∞,0)-site or something? No: a _[[(0,1)-site]]_ ! :-)

    I had posted a reply before Mike’s above, but in posting my connection broke down. It overlaps now with Mike’s but just for the record, here is what I had written a few minutes back:

    For your second bullet point below, is that typo

    It’s not a typo, but I should have made this more explicitly:

    Compare to ordinary topos theory: given a Grothendieck topos $\mathcal{C}$, it is equivalent to sheaves on itself when we regard $\mathcal{C}$ itself as a site equipped with the canonical topology. A functor $\mathcal{C}^{op} \to SET$ is a sheaf for this topology precisely if it preserves limits (sends colimits in $\mathcal{C}$ to limits in $SET$ (capitals for universe enlargement, but never mind that)).

    So $\mathcal{C} \simeq Sh_{can}(\mathcal{C})$.

    The same remains true for $\infty$-toposes and $\infty$-groupoid valued sheaves. Moreover, every object $X$ in an $\infty$-topos $\mathcal{C}$ can be understood as an “internal groupoid”, (see groupoid object in an (infinity,1)-topos). Then under this equivalence we identify $X \in \mathcal{C}$ with the functor

    $$\mathcal{C}(-, X) : \mathcal{C}^{op} \to \infty GRPD$$

    that it represents.

    Now the idea is to use this equivalence to implicitly say what a category object in $\mathcal{C}$ is, namely now an $\infty$-limit preserving $\infty$-functor $\mathcal{C}^{op} \to (\infty,1) CAT$ which we implicitly think of as being represented by an “actual” category object $X \in \mathcal{C}$, which in turn we identify with an (infinity,2)-sheaf on the site $C$ of $\mathcal{C}$.

    This is fairly well understood in the 1-categorical context. See for instance section 3 of the article “Strong stacks” that is referenced at 2-sheaf.

    we know that a frame is canonically a site… can we just then call it trivially an ∞-site?

    Yes.

    I mean, like an (∞,0)-site or something?

    No: a (0,1)-site ! :-)

31. • CommentRowNumber31.
    • CommentAuthorUrs
    • CommentTimeMar 8th 2012
    • (edited Mar 8th 2012)
    • PermaLink
    Author: Urs
    Format: MarkdownItex> However, one could look at the underlying (∞,1)-sheaf of that (∞,2)-sheaf In fact, probably under mild conditions an $(\infty,2)$-presheaf is an $(\infty,2)$-sheaf precisely if its [[core]]-$(\infty,1)$-presheaf is an $(\infty,1)$-sheaf. For $2$-sheaves Joyal-Tierney state this as a fact in section 3 of their "Strong stacks"-article.

    However, one could look at the underlying (∞,1)-sheaf of that (∞,2)-sheaf

    In fact, probably under mild conditions an $(\infty,2)$-presheaf is an $(\infty,2)$-sheaf precisely if its core-$(\infty,1)$-presheaf is an $(\infty,1)$-sheaf.

    For $2$-sheaves Joyal-Tierney state this as a fact in section 3 of their “Strong stacks”-article.

32. • CommentRowNumber32.
    • CommentAuthorJon Beardsley
    • CommentTimeMar 8th 2012
    • PermaLink
    Author: Jon Beardsley
    Format: MarkdownItexIt does indeed appear that you all are the right people to ask about $\infty$. :-))

    It does indeed appear that you all are the right people to ask about $\infty$. :-))

33. • CommentRowNumber33.
    • CommentAuthorMike Shulman
    • CommentTimeMar 8th 2012
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItex> For 2-sheaves Joyal-Tierney state this as a fact in section 3 of their "Strong stacks"-article. But they are working only with internal categories in the topos of sheaves, not with arbitrary 2-presheaves. For a general 2-presheaf, saying that its core is a stack only gives you gluing of objects and of isomorphisms, not gluing of noninvertible morphisms.

    For 2-sheaves Joyal-Tierney state this as a fact in section 3 of their “Strong stacks”-article.

    But they are working only with internal categories in the topos of sheaves, not with arbitrary 2-presheaves. For a general 2-presheaf, saying that its core is a stack only gives you gluing of objects and of isomorphisms, not gluing of noninvertible morphisms.

34. • CommentRowNumber34.
    • CommentAuthorJon Beardsley
    • CommentTimeMar 9th 2012
    • (edited Mar 9th 2012)
    • PermaLink
    Author: Jon Beardsley
    Format: MarkdownItexOkay... so, amid all of this, let me respond with a really basic sort of... synopsis to see if I understand. We want to look at a sheaf of categories, in fact, a sheaf of $(\infty,1)$-categories. That is,we have our [[posite]], which happens to be a locale. For our specific purposes, its elements happen to be localizing subcategories of the stable homotopy category (now of course I'm thinking about doing this for a general [[tensor]] [[triangulated category]] and also my apologies, I just realized you could double-bracket stuff in the forum) with the coverings given by inclusion (since the ordering on the frame is originally given by reverse inclusion). Now the way I understand the next step is by analogy with the following: A stack (at least according to some people) is a groupoid valued functor. So, for instance, we can take a Hopf-algebroid and recognize this as a stack over the category of rings (or some algebras over something or whatever) by homming out of it. This is an "internal" groupoid object I think (trying to pick up the terminology here as I go). So, we take the $(\infty,1)$-category of $(\infty,1)$-sheaves on our locale DL (which as far as I can tell are essentially sheaves of spaces or simplicial sets) which gives us an $(\infty,1)$-topos which Urs denotes by $\mathcal{C}$ I believe. By analogy with the above situation, we're now looking for _internal_ $(\infty,1)$-categories in $\mathcal{C}$. _If_ I can show that my special sheaf is such a thing then I can, in good faith, call it an $(\infty,2)$-sheaf?

    Okay… so, amid all of this, let me respond with a really basic sort of… synopsis to see if I understand.

    We want to look at a sheaf of categories, in fact, a sheaf of $(\infty,1)$-categories. That is,we have our posite, which happens to be a locale. For our specific purposes, its elements happen to be localizing subcategories of the stable homotopy category (now of course I’m thinking about doing this for a general tensor triangulated category and also my apologies, I just realized you could double-bracket stuff in the forum) with the coverings given by inclusion (since the ordering on the frame is originally given by reverse inclusion).

    Now the way I understand the next step is by analogy with the following: A stack (at least according to some people) is a groupoid valued functor. So, for instance, we can take a Hopf-algebroid and recognize this as a stack over the category of rings (or some algebras over something or whatever) by homming out of it. This is an “internal” groupoid object I think (trying to pick up the terminology here as I go).

    So, we take the $(\infty,1)$-category of $(\infty,1)$-sheaves on our locale DL (which as far as I can tell are essentially sheaves of spaces or simplicial sets) which gives us an $(\infty,1)$-topos which Urs denotes by $\mathcal{C}$ I believe. By analogy with the above situation, we’re now looking for internal $(\infty,1)$-categories in $\mathcal{C}$. If I can show that my special sheaf is such a thing then I can, in good faith, call it an $(\infty,2)$-sheaf?

35. • CommentRowNumber35.
    • CommentAuthorMike Shulman
    • CommentTimeMar 9th 2012
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItex@Jon, I think that's about right! There is a niceness in the $\infty$-case that you don't have in your analogy, of course: an internal groupoid object (such as a Hopf algebroid) will not in general be a stack. But an internal category in an $(\infty,1)$-topos *is* automatically an $(\infty,2)$-sheaf. (Actually, this isn't due to the $\infty$-ness per se, only to the fact that we're starting from a category of sheaves in which all the invertible morphisms are already present. The same thing would be true for internal 1-categories in a (2,1)-topos.)

    @Jon, I think that’s about right!

    There is a niceness in the $\infty$-case that you don’t have in your analogy, of course: an internal groupoid object (such as a Hopf algebroid) will not in general be a stack. But an internal category in an $(\infty,1)$-topos is automatically an $(\infty,2)$-sheaf. (Actually, this isn’t due to the $\infty$-ness per se, only to the fact that we’re starting from a category of sheaves in which all the invertible morphisms are already present. The same thing would be true for internal 1-categories in a (2,1)-topos.)

36. • CommentRowNumber36.
    • CommentAuthorJon Beardsley
    • CommentTimeMar 12th 2012
    • PermaLink
    Author: Jon Beardsley
    Format: MarkdownItexSo, are there categories in which internal groupoid objects are always stacks? In which cases is this not true? I mean, I guess a stack has to be an internal groupoid object that satisfies descent in some way, depending on the category?

    So, are there categories in which internal groupoid objects are always stacks? In which cases is this not true? I mean, I guess a stack has to be an internal groupoid object that satisfies descent in some way, depending on the category?

37. • CommentRowNumber37.
    • CommentAuthorUrs
    • CommentTimeMar 12th 2012
    • (edited Mar 12th 2012)
    • PermaLink
    Author: Urs
    Format: MarkdownItex> In which cases is this not true? The simplest kind of example is this: let $G$ be a sheaf of groups over some site. This is naturally a group object in the corresponding sheaf topos. Accordingly, there is an internal groupoid in this 1-topos whose object of objects is the terminal object, and whose object of morphisms is $G$. While this is an internal groupoid, regarded as a groupoid-valued presheaf on the given site it will be a stack only if all $G$-torsors over the given site are trivial. However, one should keep in mind that this groupoid object nevertheless uniquely determines a stack, namely its stackification. In practice it is often most useful to regard that non-stack sheaf of groupoids as a stand-in for the stack that stackifies it.

    In which cases is this not true?

    The simplest kind of example is this: let $G$ be a sheaf of groups over some site. This is naturally a group object in the corresponding sheaf topos. Accordingly, there is an internal groupoid in this 1-topos whose object of objects is the terminal object, and whose object of morphisms is $G$. While this is an internal groupoid, regarded as a groupoid-valued presheaf on the given site it will be a stack only if all $G$-torsors over the given site are trivial.

    However, one should keep in mind that this groupoid object nevertheless uniquely determines a stack, namely its stackification. In practice it is often most useful to regard that non-stack sheaf of groupoids as a stand-in for the stack that stackifies it.

38. • CommentRowNumber38.
    • CommentAuthorJon Beardsley
    • CommentTimeMay 31st 2012
    • PermaLink
    Author: Jon Beardsley
    Format: MarkdownItexHello again! Not sure if anyone is still interested in what we were talking about here, but I've been thinking about it a lot, and would like to see what people think. So here's the general set-up (and this idea seems to apply pretty easily to the stable homotopy situation): We've got a stable homotopy category, so, a closed tensor-triangulated category, and let's assume whatever we want, i.e. it has compact generators, it's well-generated, it's combinatorial, it's simplicial, whatever. We can take the Bousfield lattice of this category. Then, we can look at the distributive lattice which is a sub-poset of the Bousfield lattice, and also, it turns out, is a locale. Now, since we have a locale whose "open sets" correspond to localizing (and hence also colocalizing) subcategories, we can talk about AT LEAST sheaves on it. One sort of obvious sheaf (stack...) is the one that to each open set associates the colocalizing subcategory associated to that open set already. You have restriction functors. Now, it turns out that in stable homotopy, by these so-called chromatic fracture squares, there are very specific situations in which we can glue a finite number of sections together. However, that appears to be all you can really do. So, I'm interested in just straight up taking the sheafification (in the sense of the new value of the functor on an "open set" is the holim of that big simplicial diagram). I guess my question is the following: does this destroy the whole situation? That is, does this new sheaf even take values in that same colocalizing subcategory? This question really comes down to, given a lattice of colocalizing subcategories (or we could take localizing) when can we glue covering categories together to get the category they cover? Does anyone have any intuition about this, or have experience with it? Haha, not sure if anyone will see this. This thread is pretty old. But I'd love to talk to someone about it. =-) -Jon

    Hello again!

    Not sure if anyone is still interested in what we were talking about here, but I’ve been thinking about it a lot, and would like to see what people think.

    So here’s the general set-up (and this idea seems to apply pretty easily to the stable homotopy situation):

    We’ve got a stable homotopy category, so, a closed tensor-triangulated category, and let’s assume whatever we want, i.e. it has compact generators, it’s well-generated, it’s combinatorial, it’s simplicial, whatever. We can take the Bousfield lattice of this category. Then, we can look at the distributive lattice which is a sub-poset of the Bousfield lattice, and also, it turns out, is a locale. Now, since we have a locale whose “open sets” correspond to localizing (and hence also colocalizing) subcategories, we can talk about AT LEAST sheaves on it. One sort of obvious sheaf (stack…) is the one that to each open set associates the colocalizing subcategory associated to that open set already. You have restriction functors. Now, it turns out that in stable homotopy, by these so-called chromatic fracture squares, there are very specific situations in which we can glue a finite number of sections together. However, that appears to be all you can really do. So, I’m interested in just straight up taking the sheafification (in the sense of the new value of the functor on an “open set” is the holim of that big simplicial diagram). I guess my question is the following: does this destroy the whole situation? That is, does this new sheaf even take values in that same colocalizing subcategory? This question really comes down to, given a lattice of colocalizing subcategories (or we could take localizing) when can we glue covering categories together to get the category they cover? Does anyone have any intuition about this, or have experience with it?

    Haha, not sure if anyone will see this. This thread is pretty old. But I’d love to talk to someone about it. =-)

    -Jon

39. • CommentRowNumber39.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 1st 2012
    • (edited Jun 1st 2012)
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexZoran might be the person to ask about this, it looks like non-commutative algebraic geometry a la Rosenberg, where there is a (non-commutative generalisation of) a co-pretopology given by families of localisation functors. Regarding some of your earlier questions about higher category theory (I didn't follow the thread until now), some of that stuff looks like it can be approached using (triangulated) derivators. Mike has written a bit about this at the n-category cafe. Regarding your questions about internal groupoids which are not stacks, consider say, a finitely complete category $S$ with the canonical topology (actually take it to be a [[superextensive site]] as well, makes things easy). Then the (2,1)-category of internal groupoids can be localised at a particular class of arrows--the fully faithful and essentially surjective, in an internal sense--to get the (2,1)-category of _geometric_ stacks on $S$. These are stacks $X$ which admit an epimorphism of stacks from a representable stack, namely one equivalent to a stack of the form $\underline{a} :=S(-,a) \to Set \to Gpd$ for some object $a$ of $S$. Also, we require $\underline{a}\to X$ to be a cover in some sense, and $a$ is then called an _atlas_ for the stack. Not all stacks are of this form, in that they are 'too big' to be covered by an atlas, or that certain 'hom-sheaves' associated to the stack are not representable by objects of $S$. One model for the 2-category of geometric stacks is the 2-category of internal groupoids, [[anafunctors]] and transformations. Using this 2-category geometric stacks _are_ representable by internal groupoids. Now of course there are stacks of categories, too, such as the stack of coherent sheaves on a scheme with all morphisms of sheaves, not just isomorphisms. People tend to say these are not geometric stacks, or that stacks of categories in general are not geometric, but there is no reason not to consider them to be so (in fact I am working on this at the moment).

    Zoran might be the person to ask about this, it looks like non-commutative algebraic geometry a la Rosenberg, where there is a (non-commutative generalisation of) a co-pretopology given by families of localisation functors.

    Regarding some of your earlier questions about higher category theory (I didn’t follow the thread until now), some of that stuff looks like it can be approached using (triangulated) derivators. Mike has written a bit about this at the n-category cafe.

    Regarding your questions about internal groupoids which are not stacks, consider say, a finitely complete category $S$ with the canonical topology (actually take it to be a superextensive site as well, makes things easy). Then the (2,1)-category of internal groupoids can be localised at a particular class of arrows–the fully faithful and essentially surjective, in an internal sense–to get the (2,1)-category of geometric stacks on $S$. These are stacks $X$ which admit an epimorphism of stacks from a representable stack, namely one equivalent to a stack of the form $\underline{a} :=S(-,a) \to Set \to Gpd$ for some object $a$ of $S$. Also, we require $\underline{a}\to X$ to be a cover in some sense, and $a$ is then called an atlas for the stack. Not all stacks are of this form, in that they are ’too big’ to be covered by an atlas, or that certain ’hom-sheaves’ associated to the stack are not representable by objects of $S$.

    One model for the 2-category of geometric stacks is the 2-category of internal groupoids, anafunctors and transformations. Using this 2-category geometric stacks are representable by internal groupoids.

    Now of course there are stacks of categories, too, such as the stack of coherent sheaves on a scheme with all morphisms of sheaves, not just isomorphisms. People tend to say these are not geometric stacks, or that stacks of categories in general are not geometric, but there is no reason not to consider them to be so (in fact I am working on this at the moment).