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1. • CommentRowNumber1.
   • CommentAuthoramg
   • CommentTimeNov 15th 2012
   • (edited Nov 15th 2012)
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   Author: amg
   Format: HtmlHello nForum, I've seen claims made about such-and-such being "the richest possible structure" on the image of a topology-to-algebra functor (e.g. homology), but I haven't ever seen a precise statement of what this actually should mean. So, I <a href=http://math.berkeley.edu/~aaron/writing/universal-algebraic-extensions.pdf>worked out</a> a possible categorical definition, and wrote down a few expected examples. I'd imagine that if this already exists it would be on the nLab, but it's hard to search for such an imprecise concept. So, I'm wondering if (a) this notion already exists, and (b) if not, whether what I've written looks reasonable. Any corrections, extensions, further examples, suggestions for better terminology, etc. are also more than welcome. Of course I'm already aware of one obvious direction to extend: model- and/or \infty-categories! But one step at a time. Also, I feel like one could probably say this whole thing using Kan extensions or something, but I'm not familiar enough with them to see it. Thanks, Aaron Mazel-Gee

   Hello nForum,
   
   I've seen claims made about such-and-such being "the richest possible structure" on the image of a topology-to-algebra functor (e.g. homology), but I haven't ever seen a precise statement of what this actually should mean. So, I worked out a possible categorical definition, and wrote down a few expected examples. I'd imagine that if this already exists it would be on the nLab, but it's hard to search for such an imprecise concept. So, I'm wondering if (a) this notion already exists, and (b) if not, whether what I've written looks reasonable. Any corrections, extensions, further examples, suggestions for better terminology, etc. are also more than welcome. Of course I'm already aware of one obvious direction to extend: model- and/or \infty-categories! But one step at a time. Also, I feel like one could probably say this whole thing using Kan extensions or something, but I'm not familiar enough with them to see it.
   
   Thanks,
   Aaron Mazel-Gee
2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 16th 2012
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   Author: Urs
   Format: MarkdownItexHi Aaron, I have looked at it. Interesting! I am pretty sure that we don't have "this" on the $n$Lab and I can't readily recognize anything really closely related that would be. But possibly somebody else here may have a deeper insight.

   Hi Aaron,

   I have looked at it. Interesting!

   I am pretty sure that we don’t have “this” on the $n$Lab and I can’t readily recognize anything really closely related that would be. But possibly somebody else here may have a deeper insight.

3. • CommentRowNumber3.
   • CommentAuthorKarol Szumiło
   • CommentTimeNov 16th 2012
   • PermaLink
   Author: Karol Szumiło
   Format: MarkdownItexI always assumed that "the richest possible structure" refers to something rather simple. Namely, if $F \colon \mathcal{C} \to \mathcal{D}$ is any functor, then you can consider the monoid $\mathrm{End} F$ of endotransformations of $F$ which acts on $F X$ for every $X \in \mathcal{C}$ just by evaluating transformations at $X$. This monoid action encodes all natural transformations you can put on $F X$. Of course sometimes this monoid itself has more structure, for example if $\mathcal{D}$ is the category of chain complexes then $\mathrm{End} F$ is really a dg-algebra. In general it is a monoid in whatever monoidal category $\mathcal{D}$ is enriched in. You may also want to consider operations of many variables in which case you will get an operad rather than just a monoid (I guess for this $\mathcal{D}$ should be monoidal itself). If you mean something more complicated than that could you give the simplest example that isn't covered by what I suggested? I tried reading examples in your note, but I don't quite follow them.

   I always assumed that “the richest possible structure” refers to something rather simple. Namely, if $F \colon \mathcal{C} \to \mathcal{D}$ is any functor, then you can consider the monoid $\mathrm{End} F$ of endotransformations of $F$ which acts on $F X$ for every $X \in \mathcal{C}$ just by evaluating transformations at $X$. This monoid action encodes all natural transformations you can put on $F X$. Of course sometimes this monoid itself has more structure, for example if $\mathcal{D}$ is the category of chain complexes then $\mathrm{End} F$ is really a dg-algebra. In general it is a monoid in whatever monoidal category $\mathcal{D}$ is enriched in. You may also want to consider operations of many variables in which case you will get an operad rather than just a monoid (I guess for this $\mathcal{D}$ should be monoidal itself).

   If you mean something more complicated than that could you give the simplest example that isn’t covered by what I suggested? I tried reading examples in your note, but I don’t quite follow them.

4. • CommentRowNumber4.
   • CommentAuthoramg
   • CommentTimeNov 17th 2012
   • (edited Nov 17th 2012)
   • PermaLink
   Author: amg
   Format: TextHi Karol, Thanks a lot for your comment, it definitely clarifies things a bit. To rephrase, all you're saying is that if we look in the category of D-valued presheaves on C, then our functor is representable and so Yoneda's lemma tells us everything! I'll just briefly say one thing about the Pi-algebra situation. There are some minor notational/conventional issues defining End(F) to give you what it should, namely an action of all the unstable homotopy groups of spheres -- basically just, how do you control who gets to act on what. Anyways, once you nail this down, the structure of a Pi-algebra should amount to nothing more than compatible actions of all the natural transformations F^m => F^n for all m and n. By the failure of excision for homotopy, this is a richer structure than simply asking for some suitably-defined action of all the unstable homotopy groups of spheres. I think this might lead us to the notion of a "colored operad", but I can't say I actually remember what that is. What isn't immediately answered by Yoneda, however, is the question of enriching the algebraic structure on the image of algebras over a monad, which was the original motivation (specifically, the p-adic K-theory functor as applied to the adjunction between Spectra and E_\infty-rings). Or maybe it's answered by Yoneda too, but less obviously; certainly I'll look into this. (If someone can see a way to do this, please say so!) In that case, at best this might serve as a relatively tractable framework in which to address such questions. Maybe this will be conceptually simpler than working in presheaf categories (although mathematically it should be equivalent). I have no idea how one might extract something like the notions of Morava modules and theta-algebras from looking in presheaf categories (although to be fair I don't yet see how to do it in my situation either; currently I'm just interested in a framework, rather than a construction). The definitions are rather involved and nitty-gritty, which is of course my entire point. You can read about them starting on page 52 here: http://www.math.northwestern.edu/~pgoerss/spectra/obstruct.pdf Finally, I'd like to mention that Herman Stel and I just worked out another example, which partially confirms the validity of the setup but also introduces a new wrinkle. I'll post here when I've incorporated it into the document. Herman also informed me of the notion of mated adjunctions (there's an nLab page, of course!), which should definitely factor in here; they should imply that either of the natural transformations in the data of an object of AlgEx determines the other (or equivalently, it will give us a condition that they must satisfy). I'll incorporate that too.

   Hi Karol,
   
   Thanks a lot for your comment, it definitely clarifies things a bit. To rephrase, all you're saying is that if we look in the category of D-valued presheaves on C, then our functor is representable and so Yoneda's lemma tells us everything!
   
   I'll just briefly say one thing about the Pi-algebra situation. There are some minor notational/conventional issues defining End(F) to give you what it should, namely an action of all the unstable homotopy groups of spheres -- basically just, how do you control who gets to act on what. Anyways, once you nail this down, the structure of a Pi-algebra should amount to nothing more than compatible actions of all the natural transformations F^m => F^n for all m and n. By the failure of excision for homotopy, this is a richer structure than simply asking for some suitably-defined action of all the unstable homotopy groups of spheres. I think this might lead us to the notion of a "colored operad", but I can't say I actually remember what that is.
   
   What isn't immediately answered by Yoneda, however, is the question of enriching the algebraic structure on the image of algebras over a monad, which was the original motivation (specifically, the p-adic K-theory functor as applied to the adjunction between Spectra and E_\infty-rings). Or maybe it's answered by Yoneda too, but less obviously; certainly I'll look into this. (If someone can see a way to do this, please say so!) In that case, at best this might serve as a relatively tractable framework in which to address such questions. Maybe this will be conceptually simpler than working in presheaf categories (although mathematically it should be equivalent). I have no idea how one might extract something like the notions of Morava modules and theta-algebras from looking in presheaf categories (although to be fair I don't yet see how to do it in my situation either; currently I'm just interested in a framework, rather than a construction). The definitions are rather involved and nitty-gritty, which is of course my entire point. You can read about them starting on page 52 here: http://www.math.northwestern.edu/~pgoerss/spectra/obstruct.pdf
   
   Finally, I'd like to mention that Herman Stel and I just worked out another example, which partially confirms the validity of the setup but also introduces a new wrinkle. I'll post here when I've incorporated it into the document. Herman also informed me of the notion of mated adjunctions (there's an nLab page, of course!), which should definitely factor in here; they should imply that either of the natural transformations in the data of an object of AlgEx determines the other (or equivalently, it will give us a condition that they must satisfy). I'll incorporate that too.
5. • CommentRowNumber5.
   • CommentAuthoramg
   • CommentTimeNov 20th 2012
   • (edited Nov 20th 2012)
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   Author: amg
   Format: TextI've just posted a substantial revision. I think things are quite a bit clearer now; in particular, I've added a baby (entirely algebraic) example which provides some motivation for the definitions, which have been modified somewhat and now fit closer with the intuition anyways. Comments etc. are still appreciated, as always. http://math.berkeley.edu/~aaron/writing/universal-algebraic-extensions.pdf

   I've just posted a substantial revision. I think things are quite a bit clearer now; in particular, I've added a baby (entirely algebraic) example which provides some motivation for the definitions, which have been modified somewhat and now fit closer with the intuition anyways. Comments etc. are still appreciated, as always.
   
   http://math.berkeley.edu/~aaron/writing/universal-algebraic-extensions.pdf
6. • CommentRowNumber6.
   • CommentAuthorAndrew Stacey
   • CommentTimeNov 29th 2012
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   Author: Andrew Stacey
   Format: MarkdownItexI've only taken a brief look at your notes, but it looks a lot like what Sarah Whitehouse and I are thinking about with Tall-Wraith monoids. See [The Hunting of the Hopf Ring](http://loopspace.mathforge.org/discussion/7/the-hunting-of-the-hopf-ring) for details. I was a little bit confused in the section on p-adic K-theory. You wrote $K_*$ but said "$p$-adic K-theory". Do you mean *cohomology* or *homology*? And what is "$MoravaMod$"? With no explanation, I would expect that to mean modules over the coefficients of Morava K-theory (presumably with $n=1$ as you start in ordinary K-theory), but that is mod $p$, not $p$-adic. This section would appear to be referring to Bousfield's work but you don't mention it.

   I’ve only taken a brief look at your notes, but it looks a lot like what Sarah Whitehouse and I are thinking about with Tall-Wraith monoids. See The Hunting of the Hopf Ring for details.

   I was a little bit confused in the section on p-adic K-theory. You wrote $K_*$ but said “$p$-adic K-theory”. Do you mean cohomology or homology? And what is “$MoravaMod$”? With no explanation, I would expect that to mean modules over the coefficients of Morava K-theory (presumably with $n=1$ as you start in ordinary K-theory), but that is mod $p$, not $p$-adic. This section would appear to be referring to Bousfield’s work but you don’t mention it.

7. • CommentRowNumber7.
   • CommentAuthoramg
   • CommentTimeNov 30th 2012
   • (edited Nov 30th 2012)
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   Author: amg
   Format: TextHi Andrew, Thanks for the reference -- this does indeed look like it might possibly be a case of what I'd like to call "universal". I'm not familiar with the ins and outs of Hopf rings, but it seems that perhaps in this setting there should also be colored operations, as there are with Pi-algebras: a based map f : S^m --> \bigvee S^{n_\beta} induces a natural transformation f^* : \prod \pi_{n_\beta}(X) --> \pi_m(X) (of sets!). This is more data than an action of the unstable homotopy groups of spheres, because f may not be a sum of elements of \pi_m(S^{n_\beta}) (as unstable homotopy groups fail to satisfy excision in general). Of course, with Hopf rings we're looking at representing objects (instead of corepresenting objects), so the maps of interest would have signature \prod (E_{n_\beta}) --> E_m. It seems to me that this will again be more structure than having actions of all the hom-groups [E_n,E_m] unless your cohomology theory has Kunneth isomorphisms (at least for products of the representing spaces E_n, of course). In any case, I'll try to see how it fits into my setup at some point. As for your questions: I apologize for the lack of details, this is still very skeletal (in every possible way); I originally posted it here just to see if I was reinventing a wheel of which I simply wasn't aware. I do indeed mean homology, although actually I mean continuous Morava E_n homology: this functor is defined by X |--> \pi_* L_{K(n)} (E_n \smash X), and I'm taking the case n=1. For an arbitrary spectrum X, this carries the structure of what's called a Morava module. At n=1, this roughly means "p-complete Z/2-graded (Z_p^\times)-module". And when X is an E_\infty-ring, this carries the additional structure of a theta-algebra. The precise definitions are in section 2.2 of the Goerss-Hopkins paper I linked for Karol above. (This was the original motivation for my inquiry: there's a free-forget adjunction MoravaMod <=> theta-Alg, and somehow this is supposed to be "universal".) [Sorry, I can't figure out how to put superscripts and subscripts in itex! \underoverset didn't work for me...]

   Hi Andrew,
   
   Thanks for the reference -- this does indeed look like it might possibly be a case of what I'd like to call "universal". I'm not familiar with the ins and outs of Hopf rings, but it seems that perhaps in this setting there should also be colored operations, as there are with Pi-algebras: a based map f : S^m --> \bigvee S^{n_\beta} induces a natural transformation f^* : \prod \pi_{n_\beta}(X) --> \pi_m(X) (of sets!). This is more data than an action of the unstable homotopy groups of spheres, because f may not be a sum of elements of \pi_m(S^{n_\beta}) (as unstable homotopy groups fail to satisfy excision in general). Of course, with Hopf rings we're looking at representing objects (instead of corepresenting objects), so the maps of interest would have signature \prod (E_{n_\beta}) --> E_m. It seems to me that this will again be more structure than having actions of all the hom-groups [E_n,E_m] unless your cohomology theory has Kunneth isomorphisms (at least for products of the representing spaces E_n, of course). In any case, I'll try to see how it fits into my setup at some point.
   
   As for your questions: I apologize for the lack of details, this is still very skeletal (in every possible way); I originally posted it here just to see if I was reinventing a wheel of which I simply wasn't aware. I do indeed mean homology, although actually I mean continuous Morava E_n homology: this functor is defined by X |--> \pi_* L_{K(n)} (E_n \smash X), and I'm taking the case n=1. For an arbitrary spectrum X, this carries the structure of what's called a Morava module. At n=1, this roughly means "p-complete Z/2-graded (Z_p^\times)-module". And when X is an E_\infty-ring, this carries the additional structure of a theta-algebra. The precise definitions are in section 2.2 of the Goerss-Hopkins paper I linked for Karol above. (This was the original motivation for my inquiry: there's a free-forget adjunction MoravaMod <=> theta-Alg, and somehow this is supposed to be "universal".)
   
   [Sorry, I can't figure out how to put superscripts and subscripts in itex! \underoverset didn't work for me...]
8. • CommentRowNumber8.
   • CommentAuthorAndrew Stacey
   • CommentTimeNov 30th 2012
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   Author: Andrew Stacey
   Format: MarkdownItexTo use itex you need to use the Markdown+iTeX input formatter. I still don't see anything not covered by Tall-Wraith monoids, but as I said I haven't looked through in any great detail.

   To use itex you need to use the Markdown+iTeX input formatter.

   I still don’t see anything not covered by Tall-Wraith monoids, but as I said I haven’t looked through in any great detail.

9. • CommentRowNumber9.
   • CommentAuthoramg
   • CommentTimeNov 30th 2012
   • (edited Nov 30th 2012)
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   Author: amg
   Format: TextYes, I chose that option when I was trying to submit my post. I'm not sure what happened. I've only skimmed through your intro so far; I'll definitely look into Tall-Wraith monoids more carefully. But in any case, the point of my framework is to give an *abstract* characterization of universality; it may turn out that if your functor happens to be (co)representable by a sequence of objects, then the induced Tall-Wraith (co??)monoid parametrizes the universal algebraic structure. That would be a very nice general statement. (I apologize in advance if my initial impression that T-W monoids arise from having representing objects is off-base.)

   Yes, I chose that option when I was trying to submit my post. I'm not sure what happened.
   
   I've only skimmed through your intro so far; I'll definitely look into Tall-Wraith monoids more carefully. But in any case, the point of my framework is to give an *abstract* characterization of universality; it may turn out that if your functor happens to be (co)representable by a sequence of objects, then the induced Tall-Wraith (co??)monoid parametrizes the universal algebraic structure. That would be a very nice general statement. (I apologize in advance if my initial impression that T-W monoids arise from having representing objects is off-base.)