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1. • CommentRowNumber1.
   • CommentAuthorFosco
   • CommentTimeMar 28th 2014
   • (edited Mar 28th 2014)
   • PermaLink
   Author: Fosco
   Format: MarkdownItexLet $\mathcal{C}$ be a quasicategory; a *factorization system* $\mathbb{F}$ on $\mathcal{C}$ consists of a pair of markings $E, M$ on $\mathcal{C}$ (i.e., two classes of arrows in $\mathcal C$) such that 1. For every morphism $h\colon X\to Z$ in $\mathcal{C}$ we can find a factorization $X\xrightarrow{e} Y\xrightarrow{m} Z$, where $e\in E$ and $m\in M$; 2. $E ={}^{\perp}M$ and $M = E^{\perp}$ (same notion of strong orthogonality adapted to quasicategories: the space of fillings for a lifting problem is contractible). A *weak* factorization system is defined in the same way, only asking that the space of solutions for each lifting problem is nonempty: let's say $E ={}^{\angle}M$ and $M = E^{\angle}$ for the relation of weak orthogonality. This said, everything is right (or at least it seems right to me) with this definition: A *Quillen model structure* on a small-bicomplete quasicategory $\mathcal{C}$ is defined choosing three classes $(W, F, C)$ such that 1. $W$ has the 3-for-2 property (defined in the same way for this context, *mutatis mutandis*) containing all equivalences and closed under retracts; 2. The classes $(C, W\cap F)$ and $( W\cap C, F)$ both form a weak factorization system on $\mathcal{C}$. This seems to reproduce the notion of model category; if I remember well there's also a notion of localization with respect to a given class of edges in an $\infty$-category. Model categories give rise to $\infty$-categories. Now... What am I looking at? I'm not really keen on this kind of speculations, but it seems that I'm re-interpreting a structure in another semantics. What is the meaning of such a *matrioska-esque* procedure? The only other analogy of this situation comes to my mind when I think to Topos Theory, which allows you to define toposes as suitable internal categories in a given topos (but I'm not 100% about this). I feel confused, and I feel I shouldn't. My little knowledge about Logic isn't an help at all.

   Let $\mathcal{C}$ be a quasicategory; a factorization system $\mathbb{F}$ on $\mathcal{C}$ consists of a pair of markings $E, M$ on $\mathcal{C}$ (i.e., two classes of arrows in ) such that

   1. For every morphism $h\colon X\to Z$ in $\mathcal{C}$ we can find a factorization $X\xrightarrow{e} Y\xrightarrow{m} Z$, where $e\in E$ and $m\in M$;
   2. $E ={}^{\perp}M$ and $M = E^{\perp}$ (same notion of strong orthogonality adapted to quasicategories: the space of fillings for a lifting problem is contractible).

   A weak factorization system is defined in the same way, only asking that the space of solutions for each lifting problem is nonempty: let’s say $\mathcal CE ={}^{\angle}M$ and $M = E^{\angle}$ for the relation of weak orthogonality.

   This said, everything is right (or at least it seems right to me) with this definition:

   A Quillen model structure on a small-bicomplete quasicategory $\mathcal{C}$ is defined choosing three classes $(W, F, C)$ such that

   1. $W$ has the 3-for-2 property (defined in the same way for this context, mutatis mutandis) containing all equivalences and closed under retracts;
   2. The classes $(C, W\cap F)$ and $( W\cap C, F)$ both form a weak factorization system on $\mathcal{C}$.

   This seems to reproduce the notion of model category; if I remember well there’s also a notion of localization with respect to a given class of edges in an $\infty$-category. Model categories give rise to $\infty$-categories.

   Now… What am I looking at? I’m not really keen on this kind of speculations, but it seems that I’m re-interpreting a structure in another semantics. What is the meaning of such a matrioska-esque procedure? The only other analogy of this situation comes to my mind when I think to Topos Theory, which allows you to define toposes as suitable internal categories in a given topos (but I’m not 100% about this). I feel confused, and I feel I shouldn’t. My little knowledge about Logic isn’t an help at all.

2. • CommentRowNumber2.
   • CommentAuthoramg
   • CommentTimeMar 29th 2014
   • PermaLink
   Author: amg
   Format: TextHi Fosco, I've actually been working on precisely what you suggest! The fundamental theorem is that when your source is cofibrant and your target is fibrant, then the appropriate notions of "space of left/right homotopy classes of maps" both compute the hom-space in the localization M[W^{-1}]. Can you expand a little on your last paragraph? I'm not sure what you're asking...

   Hi Fosco,
   
   I've actually been working on precisely what you suggest! The fundamental theorem is that when your source is cofibrant and your target is fibrant, then the appropriate notions of "space of left/right homotopy classes of maps" both compute the hom-space in the localization M[W^{-1}].
   
   Can you expand a little on your last paragraph? I'm not sure what you're asking...
3. • CommentRowNumber3.
   • CommentAuthorZhen Lin
   • CommentTimeMar 29th 2014
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexThis seems like an example of [abstraction inversion](https://en.wikipedia.org/wiki/Abstraction_inversion) to me...

   This seems like an example of abstraction inversion to me…

4. • CommentRowNumber4.
   • CommentAuthoramg
   • CommentTimeMar 29th 2014
   • (edited Mar 29th 2014)
   • PermaLink
   Author: amg
   Format: TextHaha. I don't know anything about computer science, but I agree that it might certainly seem gratuitously self-referential. But in fact, I stumbled upon this abstract formulation as a rather elegant solution (elegance entirely thanks to Quillen, of course) to a relatively down-to-earth computational problem. The main project that I'm working on is to generalize Goerss--Hopkins obstruction theory from a particular model category of spectra (satisfying some rather technical but presumably dispensible hypotheses) to an arbitrary presentable \infty-category C. This is an obstruction theory for determining the existence and uniqueness of ring spectra with some specified homology (with respect to some chosen homology theory). Its first main application was to the construction of the cohomology theory of topological modular forms (or TMF, for short), or more precisely to the construction of the sheaf of commutative ring spectra over the moduli stack of elliptic curves of which it's by definition the global sections. However, Goerss--Hopkins obstruction theory is also the key ingredient for giving a "naive" definition of the affine derived scheme associated to a connective commutative ring spectrum, namely as an ordinary affine scheme equipped with a sheaf of commutative ring spectra extending its ordinary structure sheaf. (So in particular, it also underlies the derived-algebro-geometric construction of TMF that's described e.g. in Lurie's "A Survey of Elliptic Cohomology".) Goerss--Hopkins obstruction theory takes place in the E^2-model structure on simplicial spectra. But whereas we know what \infty-category underlies this model category (if you choose a set G of generators for your presentable \infty-category C that's closed under finite coproducts, then this is the \infty-category P_\Sigma(G) of product-preserving presheaves G^{op} --> Spaces), setting up the obstruction theory also relies on all sorts of computations that actually take place in the category of simplicial spectra, and so we also need to have a good understanding of the functor sC --> P_\Sigma(G) (which takes a simplicial object Y to the product-preserving presheaf g ~~> |hom(g,Y)|). After spending quite a while fumbling around trying to prove a single one of the many necessary facts about P_\Sigma(G) by working directly with product-preserving presheaves, it occurred to me that there should be an analogous description to the localization sSpectra --> sSpectra[W_{E^2}^{-1}] in the \infty-categorical context: that is, there should be an E^2-model structure on the \infty-category sC such that the functor sC --> P_\Sigma(G) is precisely the canonical projection to the localization sC[W_{E^2}^{-1}]. As \infty-categories are by their very nature homotopically well-behaved, this notion of "model structure" has less to do with avoiding things like pathological point-set issues, and has more to do with giving a theory of resolutions that's entirely native to the world of \infty-categories. Indeed, the \infty-category P_\Sigma(G) is a/k/a the "nonabelian derived \infty-category" of the base \infty-category C. To head off a totally reasonable complaint, let me be the first to acknowledge that given all the various results we have floating around about how model categories are often Quillen-equivalent to ones with whatever various adjectives, it seems likely that one could also generalize Goerss--Hopkins obstruction theory directly within the world of model categories, and it would work in a large class of examples (probably containing in particular all those model categories that give rise to presentable \infty-categories, namely combinatorial simplicial model categories). My motivation for working instead with \infty-categories was twofold: - As the obstruction groups themselves are defined entirely algebraically, one would expect that the construction of the obstruction theory itself shouldn't depend on the choice of model category of spectra. This assertion is tantamount to saying that the obstruction theory should descend to the unique \infty-category of spectra that underlies all the different model categories of spectra. - The obstruction theory is really very complicated (Goerss & Hopkins wrote over 500 pages in the process of setting it up!). Much of the complexity is truly mathematical, but a decent part of it just has to do with getting point-set stuff taken care of appropriately. The latter sort of complications disappear in the passage from model categories to \infty-categories, and as a bonus, the former sort actually become a little more transparent too. For (way too) much more, feel free to check out these slides from a talk I gave recently about this project: http://math.berkeley.edu/~aaron/writing/thursday-cghost-beamer.pdf

   Haha. I don't know anything about computer science, but I agree that it might certainly seem gratuitously self-referential. But in fact, I stumbled upon this abstract formulation as a rather elegant solution (elegance entirely thanks to Quillen, of course) to a relatively down-to-earth computational problem.
   
   The main project that I'm working on is to generalize Goerss--Hopkins obstruction theory from a particular model category of spectra (satisfying some rather technical but presumably dispensible hypotheses) to an arbitrary presentable \infty-category C. This is an obstruction theory for determining the existence and uniqueness of ring spectra with some specified homology (with respect to some chosen homology theory). Its first main application was to the construction of the cohomology theory of topological modular forms (or TMF, for short), or more precisely to the construction of the sheaf of commutative ring spectra over the moduli stack of elliptic curves of which it's by definition the global sections. However, Goerss--Hopkins obstruction theory is also the key ingredient for giving a "naive" definition of the affine derived scheme associated to a connective commutative ring spectrum, namely as an ordinary affine scheme equipped with a sheaf of commutative ring spectra extending its ordinary structure sheaf. (So in particular, it also underlies the derived-algebro-geometric construction of TMF that's described e.g. in Lurie's "A Survey of Elliptic Cohomology".)
   
   Goerss--Hopkins obstruction theory takes place in the E^2-model structure on simplicial spectra. But whereas we know what \infty-category underlies this model category (if you choose a set G of generators for your presentable \infty-category C that's closed under finite coproducts, then this is the \infty-category P_\Sigma(G) of product-preserving presheaves G^{op} --> Spaces), setting up the obstruction theory also relies on all sorts of computations that actually take place in the category of simplicial spectra, and so we also need to have a good understanding of the functor sC --> P_\Sigma(G) (which takes a simplicial object Y to the product-preserving presheaf g ~~> |hom(g,Y)|). After spending quite a while fumbling around trying to prove a single one of the many necessary facts about P_\Sigma(G) by working directly with product-preserving presheaves, it occurred to me that there should be an analogous description to the localization sSpectra --> sSpectra[W_{E^2}^{-1}] in the \infty-categorical context: that is, there should be an E^2-model structure on the \infty-category sC such that the functor sC --> P_\Sigma(G) is precisely the canonical projection to the localization sC[W_{E^2}^{-1}]. As \infty-categories are by their very nature homotopically well-behaved, this notion of "model structure" has less to do with avoiding things like pathological point-set issues, and has more to do with giving a theory of resolutions that's entirely native to the world of \infty-categories. Indeed, the \infty-category P_\Sigma(G) is a/k/a the "nonabelian derived \infty-category" of the base \infty-category C.
   
   To head off a totally reasonable complaint, let me be the first to acknowledge that given all the various results we have floating around about how model categories are often Quillen-equivalent to ones with whatever various adjectives, it seems likely that one could also generalize Goerss--Hopkins obstruction theory directly within the world of model categories, and it would work in a large class of examples (probably containing in particular all those model categories that give rise to presentable \infty-categories, namely combinatorial simplicial model categories). My motivation for working instead with \infty-categories was twofold:
   - As the obstruction groups themselves are defined entirely algebraically, one would expect that the construction of the obstruction theory itself shouldn't depend on the choice of model category of spectra. This assertion is tantamount to saying that the obstruction theory should descend to the unique \infty-category of spectra that underlies all the different model categories of spectra.
   - The obstruction theory is really very complicated (Goerss & Hopkins wrote over 500 pages in the process of setting it up!). Much of the complexity is truly mathematical, but a decent part of it just has to do with getting point-set stuff taken care of appropriately. The latter sort of complications disappear in the passage from model categories to \infty-categories, and as a bonus, the former sort actually become a little more transparent too.
   
   For (way too) much more, feel free to check out these slides from a talk I gave recently about this project: http://math.berkeley.edu/~aaron/writing/thursday-cghost-beamer.pdf
5. • CommentRowNumber5.
   • CommentAuthorFosco
   • CommentTimeMar 30th 2014
   • PermaLink
   Author: Fosco
   Format: MarkdownItexamg, I sent you an email!

   amg, I sent you an email!

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMar 31st 2014
   • (edited Mar 31st 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAaron, thanks for these pointers to your [notes](http://math.berkeley.edu/~aaron/writing/thursday-cghost-beamer.pdf)! By the way, to produce hyperlinks here, just go [link name](url) (and make sure that below the edit box you have "Markdown" selected). For the moment, I have added a pointer to your notes to our entry _[[Goerss-Hopkins-Miller theorem]]_. But that entry is a stub and generally what your notes are about should get its own entry. For the convenience of bystandars: your discussion of "model $\infty$-categories" starts on slide 60(380 of 528) in the [pdf](http://math.berkeley.edu/~aaron/writing/thursday-cghost-beamer.pdf). Have you thought about calling them instead something like "Quillen localizations of $\infty$-categories" or the like? Because I imagine there will be friction as above ("abstraction inversion") each time you inflict the term "model $\infty$-category" on somebody new. Finally an actual question: I see at the end of your notes the comments on how you imagine applying this to streamline the construction of $tmf$. Also I enjoyed (as I suppose you have seen indications of here) your other notes titled "You could've invented $tmf$!" While I liked those, I wasn''t entirely sure in the end about if there was a claim as follows, and whether I really got it: what is the most abstract characterization of $tmf$ that you can come up with? So we start with the free stable $\infty$-category on the point (spectra), look at the finite $p$-local objects and find that their prime spectra are labled by Morava K-theories. Great. Next we want to glue stuff along the primes at one fixed $K(n)$. Now what exactly would be the statement regarding $tmf$ here? You start saying something like this in your notes, but then you end up stating the usual technical details. If we just stayed at the general abstract nonsense level, can you characterize $tmf$?

   Aaron,

   thanks for these pointers to your notes!

   By the way, to produce hyperlinks here, just go

   [link name](url)
   

   (and make sure that below the edit box you have “Markdown” selected).

   For the moment, I have added a pointer to your notes to our entry Goerss-Hopkins-Miller theorem.

   But that entry is a stub and generally what your notes are about should get its own entry.

   For the convenience of bystandars: your discussion of “model $\infty$-categories” starts on slide 60(380 of 528) in the pdf.

   Have you thought about calling them instead something like “Quillen localizations of $\infty$-categories” or the like? Because I imagine there will be friction as above (“abstraction inversion”) each time you inflict the term “model $\infty$-category” on somebody new.

   Finally an actual question: I see at the end of your notes the comments on how you imagine applying this to streamline the construction of $tmf$. Also I enjoyed (as I suppose you have seen indications of here) your other notes titled “You could’ve invented $tmf$!”

   While I liked those, I wasn”t entirely sure in the end about if there was a claim as follows, and whether I really got it:

   what is the most abstract characterization of $tmf$ that you can come up with?

   So we start with the free stable $\infty$-category on the point (spectra), look at the finite $p$-local objects and find that their prime spectra are labled by Morava K-theories. Great.

   Next we want to glue stuff along the primes at one fixed $K(n)$. Now what exactly would be the statement regarding $tmf$ here? You start saying something like this in your notes, but then you end up stating the usual technical details. If we just stayed at the general abstract nonsense level, can you characterize $tmf$?

7. • CommentRowNumber7.
   • CommentAuthoramg
   • CommentTimeApr 1st 2014
   • (edited Apr 2nd 2014)
   • PermaLink
   Author: amg
   Format: MarkdownItexHi Urs! Thanks for the comments/nlab update/question. First of all, I'd be happy to expand the "Goerss--Hopkins--Miller theorem" page; I'll do that when I get a chance sometime soon. I'm not sure this is what you're suggesting, but I think it'd be prudent to wait until I at least arxiv the relevant paper to make an nlab page about "model $\infty$-categories". As for the name, I agree that it can be a little confusing, but I just like it so much better! At the very least, I'll make it very clear even in the abstract of the paper that the point is get a handle on localizations -- just like model structures on 1-categories. As for your question, I share your disappointment that there doesn't seem to be a more intrinsic universal characterization of tmf, or rather that nobody really talks about something that's more clearly universal. The natural guess would be that $M_{fg}$ itself should carry the truly universal sheaf of ring spectra. Paul Goerss talks some about this [here](http://www.math.northwestern.edu/~pgoerss/papers/newton.pdf); right after Problem 1.16, he says that he thinks everyone thinks that this is too much to hope for. My limited and hazy understanding is that this has to do with the fact that given a complex-oriented $E_\infty$-ring spectrum, one can actually recover a $p$-divisible group whose formal component is the given formal group, so despite the fact that chromatic homotopy theory has traditionally concerned itself with $M_{fg}$, this is actually in some sense the wrong place to look if you want interactions with (at least the "totally commutative" -- i.e. $E_\infty$ -- flavor of) derived algebraic geometry. This is why Lurie's representability theorem actually has to do with $M_p(n)$, the moduli of $p$-divisible groups with height $\leq n$ and with 1-dimensional formal component. So presumably the truly universal objects should involve these moduli, but (at least if I recall correctly from a conversation with Paul about exactly this question) such an idea hasn't really been explored because we don't really understand the $M_p(n)$ so well. Instead, we have a better grasp on the moduli of certain global objects (i.e. of elliptic curves and "PEL abelian varieties"). So for instance in the elliptic curve case, the important point is that the formal neighborhood of a supersingular point of $M_{ell}$ is basically just a height-2 Lubin--Tate space, i.e. the local neighborhood in $M_{fg}$ of a point corresponding to a height-2 formal group: to this, the tmf-sheaf assigns the corresponding form of Morava $E_2$. Moreover, since supersingular elliptic curves exist at all primes, we can view this as a reasonable substitute: up to taking fixedpoints by the automorphism groups of these curves, the $E_{2,p}$ all appear as sections of the resulting sheaf (and at any prime $p$, we can recover the product of all of these by applying the Bousfield localization $L_{K(2)}$ to the global sections). This is described in more detail in [these](http://math.berkeley.edu/~aaron/writing/tmf-seminar-talk.pdf) talk notes (especially section 6.2), which give a somewhat more serious treatment of the motivation/construction portion of the "You could've constructed tmf" slides. (As for TAF, I'm much less familiar with the story, but as I recall, there's an analogous theorem about $L_{K(n)}TAF$ being something like a product of $E_{n,p}^{hG}$'s, where $G$ is the automorphism group of some "supersingular" abelian variety. But now, $p$ is fixed from the start, which is a kind of unfortunate aspect of that story in its current form. And then, if I recall correctly, Behrens--Lawson also need to make an arbitrary choice of an imaginary quadratic number field right at the outset, which presents another hurdle in giving any sort of universal characterization.) But then of course, there are lots of other reasons why people care about tmf -- many of them physics-related. I'm sure you know far more about that side of the picture than I do! Incidentally (and this should probably appear elsewhere in the comment), all of this business with $M_{fg}$ having bad properties suggests that perhaps we should start the chromatic story over from the beginning, replacing $\mathbb{C}P^\infty = BS^1$ with the ind-system of $BC_{p^n}$'s (at a fixed prime, or perhaps of $BC_m$'s for all $m \in \mathbb{N}$). That is, we should be keeping track of *$p$-divisible* groups, not *formal* groups. These were used to great and beautiful effect in Nat Stapleton's [thesis](http://math.mit.edu/~nstapleton/papers/tgcm.pdf). And though it's not yet been written up (that I know of, at least), Jacob Lurie has also studied transchromatic homotopy theory via $p$-divisible groups -- you can read Eric Peterson's notes from a talk he gave (beginning on page 14) [here](http://math.berkeley.edu/~ericp/latex/misc/mit-ethy.pdf). Maybe it's also worth mentioning that I've idly pondered from time to time if it's possible to get away from the "dimension-1" constraint on the formal groups that show up in chromatic homotopy theory. Apparently quite a number of people have thought about this, and nobody has really gotten anywhere. This is most likely prohibitively difficult (and the objects probably actually aren't even known to exist), but one "reverse-engineering" sort of way to get a glimpse might be to contemplate the global sections of derived DM stacks of other flavors of abelian varieties (i.e. those that don't necessarily come with some canonical choice of 1-dimensional formal group) and ask about their chromatic properties. It seems hard to believe that higher-dimensional formal groups would have *nothing* to do with chromatic homotopy theory... === P.P.S. Thanks for the tip for fixing the links -- I scanned the "Help" page over and over and still missed it. Some weird sort of dyslexia...

   Hi Urs! Thanks for the comments/nlab update/question.

   First of all, I’d be happy to expand the “Goerss–Hopkins–Miller theorem” page; I’ll do that when I get a chance sometime soon.

   I’m not sure this is what you’re suggesting, but I think it’d be prudent to wait until I at least arxiv the relevant paper to make an nlab page about “model $\infty$-categories”. As for the name, I agree that it can be a little confusing, but I just like it so much better! At the very least, I’ll make it very clear even in the abstract of the paper that the point is get a handle on localizations – just like model structures on 1-categories.

   As for your question, I share your disappointment that there doesn’t seem to be a more intrinsic universal characterization of tmf, or rather that nobody really talks about something that’s more clearly universal. The natural guess would be that $M_{fg}$ itself should carry the truly universal sheaf of ring spectra. Paul Goerss talks some about this here; right after Problem 1.16, he says that he thinks everyone thinks that this is too much to hope for.

   My limited and hazy understanding is that this has to do with the fact that given a complex-oriented $E_\infty$-ring spectrum, one can actually recover a $p$-divisible group whose formal component is the given formal group, so despite the fact that chromatic homotopy theory has traditionally concerned itself with $M_{fg}$, this is actually in some sense the wrong place to look if you want interactions with (at least the “totally commutative” – i.e. $E_\infty$ – flavor of) derived algebraic geometry. This is why Lurie’s representability theorem actually has to do with $M_p(n)$, the moduli of $p$-divisible groups with height $\leq n$ and with 1-dimensional formal component.

   So presumably the truly universal objects should involve these moduli, but (at least if I recall correctly from a conversation with Paul about exactly this question) such an idea hasn’t really been explored because we don’t really understand the $M_p(n)$ so well. Instead, we have a better grasp on the moduli of certain global objects (i.e. of elliptic curves and “PEL abelian varieties”). So for instance in the elliptic curve case, the important point is that the formal neighborhood of a supersingular point of $M_{ell}$ is basically just a height-2 Lubin–Tate space, i.e. the local neighborhood in $M_{fg}$ of a point corresponding to a height-2 formal group: to this, the tmf-sheaf assigns the corresponding form of Morava $E_2$. Moreover, since supersingular elliptic curves exist at all primes, we can view this as a reasonable substitute: up to taking fixedpoints by the automorphism groups of these curves, the $E_{2,p}$ all appear as sections of the resulting sheaf (and at any prime $p$, we can recover the product of all of these by applying the Bousfield localization $L_{K(2)}$ to the global sections). This is described in more detail in these talk notes (especially section 6.2), which give a somewhat more serious treatment of the motivation/construction portion of the “You could’ve constructed tmf” slides. (As for TAF, I’m much less familiar with the story, but as I recall, there’s an analogous theorem about $L_{K(n)}TAF$ being something like a product of $E_{n,p}^{hG}$’s, where $G$ is the automorphism group of some “supersingular” abelian variety. But now, $p$ is fixed from the start, which is a kind of unfortunate aspect of that story in its current form. And then, if I recall correctly, Behrens–Lawson also need to make an arbitrary choice of an imaginary quadratic number field right at the outset, which presents another hurdle in giving any sort of universal characterization.)

   But then of course, there are lots of other reasons why people care about tmf – many of them physics-related. I’m sure you know far more about that side of the picture than I do!

   Incidentally (and this should probably appear elsewhere in the comment), all of this business with $M_{fg}$ having bad properties suggests that perhaps we should start the chromatic story over from the beginning, replacing $\mathbb{C}P^\infty = BS^1$ with the ind-system of $BC_{p^n}$’s (at a fixed prime, or perhaps of $BC_m$’s for all $m \in \mathbb{N}$). That is, we should be keeping track of $p$-divisible groups, not formal groups. These were used to great and beautiful effect in Nat Stapleton’s thesis. And though it’s not yet been written up (that I know of, at least), Jacob Lurie has also studied transchromatic homotopy theory via $p$-divisible groups – you can read Eric Peterson’s notes from a talk he gave (beginning on page 14) here.

   Maybe it’s also worth mentioning that I’ve idly pondered from time to time if it’s possible to get away from the “dimension-1” constraint on the formal groups that show up in chromatic homotopy theory. Apparently quite a number of people have thought about this, and nobody has really gotten anywhere. This is most likely prohibitively difficult (and the objects probably actually aren’t even known to exist), but one “reverse-engineering” sort of way to get a glimpse might be to contemplate the global sections of derived DM stacks of other flavors of abelian varieties (i.e. those that don’t necessarily come with some canonical choice of 1-dimensional formal group) and ask about their chromatic properties. It seems hard to believe that higher-dimensional formal groups would have nothing to do with chromatic homotopy theory…

   ===

   P.P.S. Thanks for the tip for fixing the links – I scanned the “Help” page over and over and still missed it. Some weird sort of dyslexia…

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeApr 2nd 2014
   • (edited Apr 2nd 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks! Am on my phone now and about to go offline. Will reply tomorrow, here just briefly on how to get the links: you need square brackets followed by round brackets, not the other way around. You can hit "edit" and fix this. More tomorrow...

   Thanks! Am on my phone now and about to go offline. Will reply tomorrow, here just briefly on how to get the links: you need square brackets followed by round brackets, not the other way around. You can hit “edit” and fix this.

   More tomorrow…

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeApr 2nd 2014
   • (edited Apr 2nd 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexBusy day today, I need to get back to you later. But one quick remark, in a stolen second: since you mentioned it: yes, I am well aware of the role and motivation of $tmf$ in physics, and that's precisely why I am wondering how "canonical" or "god given" it is mathematically. Because the derivation in physics is via a drastic truncation: the Witten genus, being the superstring partition function is only a tiny aspect of the full string theory. That's roughly why Witten ended his seminal article with the following wise words, worth repeating > A properly developed theory of elliptic cohomology is likely to shed some light on what string theory really means. ([Witten 87, very last sentence](http://ncatlab.org/nlab/show/elliptic%20cohomology#Witten87)). You see, one would like to turn this around: one is looking for the proper mathematical formulation of what physicists have a bunch of hints for. One realizes that one such hint says that an aspect of the whole story is the String-orientation of $tmf$. Now one would like to say: ah, and given that $tmf$ is the follwoing canonical and conceptually clear construction, that must mean that the full physics picture is really suchandsuch. Anyway, whether or not that works out, that's a natural thought. And that's why I am wondering to which degree the existence of $tmf$ might follow from just some basic axioms. (I should pre-empt one evident comment: yes, I am of course well aware of the Stolz-Teichner proposal. That is in some sense opposite to what I am after here. Where they propose to explain tmf as the space of certain 2d field theories, I would like to see 2d field theories explained by canonical structures in stable homotopy theory.)

   Busy day today, I need to get back to you later. But one quick remark, in a stolen second:

   since you mentioned it: yes, I am well aware of the role and motivation of $tmf$ in physics, and that’s precisely why I am wondering how “canonical” or “god given” it is mathematically.

   Because the derivation in physics is via a drastic truncation: the Witten genus, being the superstring partition function is only a tiny aspect of the full string theory. That’s roughly why Witten ended his seminal article with the following wise words, worth repeating

   A properly developed theory of elliptic cohomology is likely to shed some light on what string theory really means. (Witten 87, very last sentence).

   You see, one would like to turn this around: one is looking for the proper mathematical formulation of what physicists have a bunch of hints for. One realizes that one such hint says that an aspect of the whole story is the String-orientation of $tmf$. Now one would like to say: ah, and given that $tmf$ is the follwoing canonical and conceptually clear construction, that must mean that the full physics picture is really suchandsuch.

   Anyway, whether or not that works out, that’s a natural thought. And that’s why I am wondering to which degree the existence of $tmf$ might follow from just some basic axioms.

   (I should pre-empt one evident comment: yes, I am of course well aware of the Stolz-Teichner proposal. That is in some sense opposite to what I am after here. Where they propose to explain tmf as the space of certain 2d field theories, I would like to see 2d field theories explained by canonical structures in stable homotopy theory.)

10. • CommentRowNumber10.
    • CommentAuthoramg
    • CommentTimeJan 17th 2015
    • PermaLink
    Author: amg
    Format: MarkdownItexI have a general question about the nLab, but related to this thread (in light of [this](http://arxiv.org/abs/1412.8411) recent paper): Should I create/update nLab pages based on my own work? I'm worried that this might be seen as overly self-promoting or obnoxious, especially depending on the general interest and/or on the importance of the work. Moreover, it seems like it might be better for somebody else to digest the paper and decide for themselves what's worth including in an nLab entry and what's not. On the other hand, that actually requires a fair bit of work of someone else, so maybe I should view this as an opportunity to summarize the paper in a more relaxed and informal format than simply e.g. in its abstract. Relatedly (but unrelatedly to this thread), I saw that Adeel updated the [page](http://ncatlab.org/nlab/show/Quillen+adjunction) on Quillen adjunctions to reflect my [other](http://arxiv.org/abs/1501.03146) recent paper. Given that this subsumes the analogous result on simplicial model categories (there listed as Proposition 3), would it be inappropriate to remove that one? Alternatively, I could explain why it's recovered as a special case -- this requires the additional observation that for a simplicial model category, the two notions of "underlying $\infty$-category" (either the full simplicial subcategory on the bifibrants or the hammock localization of the whole thing) agree. Actually I haven't checked that these give equivalent adjunctions under the equivalence between these two notions, but I think the idea that they might give distinct adjunctions is too fantastical to entertain.

    I have a general question about the nLab, but related to this thread (in light of this recent paper): Should I create/update nLab pages based on my own work? I’m worried that this might be seen as overly self-promoting or obnoxious, especially depending on the general interest and/or on the importance of the work. Moreover, it seems like it might be better for somebody else to digest the paper and decide for themselves what’s worth including in an nLab entry and what’s not. On the other hand, that actually requires a fair bit of work of someone else, so maybe I should view this as an opportunity to summarize the paper in a more relaxed and informal format than simply e.g. in its abstract.

    Relatedly (but unrelatedly to this thread), I saw that Adeel updated the page on Quillen adjunctions to reflect my other recent paper. Given that this subsumes the analogous result on simplicial model categories (there listed as Proposition 3), would it be inappropriate to remove that one? Alternatively, I could explain why it’s recovered as a special case – this requires the additional observation that for a simplicial model category, the two notions of “underlying $\infty$-category” (either the full simplicial subcategory on the bifibrants or the hammock localization of the whole thing) agree. Actually I haven’t checked that these give equivalent adjunctions under the equivalence between these two notions, but I think the idea that they might give distinct adjunctions is too fantastical to entertain.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJan 17th 2015
    • (edited Jan 17th 2015)
    • PermaLink
    Author: Urs
    Format: MarkdownItex> Should I create/update nLab pages based on my own work? Please do. Contrary to Wikipidia, we don't impose the "neutral point of view", we just impose the [[nPOV]] (and not really that, it's mostly a fun pun). The point of the $n$Lab is for expert people who do research to make notes that they find useful, and the most useful notes anyone typically produces is on subjects that he or she is expert on. You are an expert, so the $n$Lab and those using it will benefit from you adding the stuff you are doing research on. The only thing you are strongly asked to do is to announce whatever you add briefly here on the $n$Forum, so that others who might care get a chance to be alerted of what's happening, so that they may give feedback or even join into the editing.

    Should I create/update nLab pages based on my own work?

    Please do. Contrary to Wikipidia, we don’t impose the “neutral point of view”, we just impose the nPOV (and not really that, it’s mostly a fun pun). The point of the $n$Lab is for expert people who do research to make notes that they find useful, and the most useful notes anyone typically produces is on subjects that he or she is expert on. You are an expert, so the $n$Lab and those using it will benefit from you adding the stuff you are doing research on.

    The only thing you are strongly asked to do is to announce whatever you add briefly here on the $n$Forum, so that others who might care get a chance to be alerted of what’s happening, so that they may give feedback or even join into the editing.

12. • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 17th 2015
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexThe tone of #10 and other writings suggests that you (amg), or anyone else for that matter, have nothing to worry about in terms of your coming off as self-promoting, etc. There have been a few people in the past who have really pushed it in this regard, but generally anyone who uses common sense should feel free to quote their own work wherever relevant.

    The tone of #10 and other writings suggests that you (amg), or anyone else for that matter, have nothing to worry about in terms of your coming off as self-promoting, etc. There have been a few people in the past who have really pushed it in this regard, but generally anyone who uses common sense should feel free to quote their own work wherever relevant.

13. • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeJan 18th 2015
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexI third what Urs and Todd said. As for your second paragraph, my inclination would be to add the explanation of why it's recovered as a special case.

    I third what Urs and Todd said. As for your second paragraph, my inclination would be to add the explanation of why it’s recovered as a special case.

14. • CommentRowNumber14.
    • CommentAuthoramg
    • CommentTimeJan 19th 2015
    • PermaLink
    Author: amg
    Format: MarkdownItexUrs, Todd, and Mike -- Thanks for your clarification and encouragement. I'll try to do this at some point when I'm up for it.

    Urs, Todd, and Mike – Thanks for your clarification and encouragement. I’ll try to do this at some point when I’m up for it.