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1. • CommentRowNumber1.
   • CommentAuthorsanath
   • CommentTimeDec 22nd 2014
   • (edited Dec 22nd 2014)
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   Author: sanath
   Format: MarkdownItexWe can vertically categorify a $1$-category to an $\infty$-category; I was wondering if there was a procedure of categorifying a $1$-category to a $2$-category, a $2$-category to a $3$-category, and so on (or maybe a direct procedure of categorifying a $1$-category to an $n$-category)? (I'm asking this because I want to know what the "$n$-categorification" of the category $Ab$ of abelian groups should be, because it's $\infty$-categorification is the $\infty$-category $Sp$ of spectra.) I hope this question makes sense, because I don't have a very precise idea of what this $n$-categorification should be.

   We can vertically categorify a $1$-category to an $\infty$-category; I was wondering if there was a procedure of categorifying a $1$-category to a $2$-category, a $2$-category to a $3$-category, and so on (or maybe a direct procedure of categorifying a $1$-category to an $n$-category)? (I’m asking this because I want to know what the “$n$-categorification” of the category $Ab$ of abelian groups should be, because it’s $\infty$-categorification is the $\infty$-category $Sp$ of spectra.) I hope this question makes sense, because I don’t have a very precise idea of what this $n$-categorification should be.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 22nd 2014
   • (edited Dec 22nd 2014)
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   Author: Urs
   Format: MarkdownItexThere is never really a "procedure" for categorification. There are always many choices. What is a procedure is decategorification. By categorification one means anything that when followed by decategorification gets one back to where one started. Sometimes one means something even more loose. For instance the decategorification of the $\infty$-category of spectra to a 1-category (by quotienting out all higher morphisms, hence by passing to the homotopy category) is something still much bigger than the category of abelian groups. And nevertheless are spectra usefully regarded as an "$\infty$-categorification" of abelian groups, namely in the sense that they "play the same role in $\infty$-category theory as abelian groups play in 1-category theory", roughly. So if you want to make progress here, you may need to try to focus on something more concrete.

   There is never really a “procedure” for categorification. There are always many choices. What is a procedure is decategorification. By categorification one means anything that when followed by decategorification gets one back to where one started.

   Sometimes one means something even more loose. For instance the decategorification of the $\infty$-category of spectra to a 1-category (by quotienting out all higher morphisms, hence by passing to the homotopy category) is something still much bigger than the category of abelian groups. And nevertheless are spectra usefully regarded as an “$\infty$-categorification” of abelian groups, namely in the sense that they “play the same role in $\infty$-category theory as abelian groups play in 1-category theory”, roughly.

   So if you want to make progress here, you may need to try to focus on something more concrete.

3. • CommentRowNumber3.
   • CommentAuthorsanath
   • CommentTimeDec 22nd 2014
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   Author: sanath
   Format: MarkdownItex@Urs, Ok, I think I understand what you're saying. I guess a bit more concrete question would be, what kind of things play the same role in $n$-category theory as spectra play in $\infty$-category theory and abelian groups play in $1$-category theory. Do you know a way to approach this problem?

   @Urs, Ok, I think I understand what you’re saying. I guess a bit more concrete question would be, what kind of things play the same role in $n$-category theory as spectra play in $\infty$-category theory and abelian groups play in $1$-category theory. Do you know a way to approach this problem?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeDec 22nd 2014
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   Author: Urs
   Format: MarkdownItexLet's be careful and say "$(\infty,1)$-category" and hence "$(n,1)$-category" here, unless you really mean to think about $(n,n)$-categories (recall [[(n,r)-category]]). One possible answer is that the analog of abelian groups in $(n,1)$-category theory are chain complexes of abelian groups concentrated in the lowest $n$ degrees. For more on this see at _[[Dold-Kan correspondence]]_.

   Let’s be careful and say “$(\infty,1)$-category” and hence “$(n,1)$-category” here, unless you really mean to think about $(n,n)$-categories (recall (n,r)-category).

   One possible answer is that the analog of abelian groups in $(n,1)$-category theory are chain complexes of abelian groups concentrated in the lowest $n$ degrees. For more on this see at Dold-Kan correspondence.

5. • CommentRowNumber5.
   • CommentAuthorsanath
   • CommentTimeDec 22nd 2014
   • (edited Dec 24th 2014)
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   Author: sanath
   Format: MarkdownItex@Urs How about in $(n,r)$-category theory? (I think I may be asking a bit too much here!)

   @Urs How about in $(n,r)$-category theory? (I think I may be asking a bit too much here!)

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeDec 23rd 2014
   • (edited Dec 23rd 2014)
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   Author: Urs
   Format: MarkdownItexHave you already read and absorbed the entry on _[[Dold-Kan correspondence]]_?

   Have you already read and absorbed the entry on Dold-Kan correspondence?

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeDec 23rd 2014
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   Author: Mike Shulman
   Format: MarkdownItexAnother possible answer would be spectra with homotopy groups concentrated in degrees $0\le k \lt n$. Although in either case, the lower bound is a bit odd.

   Another possible answer would be spectra with homotopy groups concentrated in degrees $0\le k \lt n$. Although in either case, the lower bound is a bit odd.

8. • CommentRowNumber8.
   • CommentAuthorRichard Williamson
   • CommentTimeDec 23rd 2014
   • (edited Dec 23rd 2014)
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   Author: Richard Williamson
   Format: MarkdownItexOn the other side of the homotopy hypothesis, the most direct analogue of an abelian group in 2-category theory would be a gadget which is sometimes known as a Picard groupoid: a groupoid equipped with the structure of a symmetric monoidal category, such that objects have inverses with respect to the monoidal structure. One can debate whether the weak or strict notion is more fundamental: I typically (probably against the grain) prefer the strict one. However, there are other possible answers. One which has a more (2,2)-categorical rather than (2,1)-categorical flavour would be: something like the 2-category of abelian categories, possibly with some finiteness condition imposed (Grothendieck abelian categories, for instance).

   On the other side of the homotopy hypothesis, the most direct analogue of an abelian group in 2-category theory would be a gadget which is sometimes known as a Picard groupoid: a groupoid equipped with the structure of a symmetric monoidal category, such that objects have inverses with respect to the monoidal structure. One can debate whether the weak or strict notion is more fundamental: I typically (probably against the grain) prefer the strict one.

   However, there are other possible answers. One which has a more (2,2)-categorical rather than (2,1)-categorical flavour would be: something like the 2-category of abelian categories, possibly with some finiteness condition imposed (Grothendieck abelian categories, for instance).

9. • CommentRowNumber9.
   • CommentAuthorsanath
   • CommentTimeDec 23rd 2014
   • (edited Dec 24th 2014)
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   Author: sanath
   Format: MarkdownItexI was actually thinking about @Mike's method, but was worried about the same thing - the lower bound being odd. @Urs I read the entry on the Dold-Kan correspondence, but I still don't know what the analogue might be in $(n,r)$-category theory. The Dold-Kan correspondence basically states that the Dold-Kan construction $Ch(\mathcal{A})_{0}\to Funct(\mathbf{\Delta}^op,\mathcal{A})$ is fully faithful, and is an equivalence when $\mathcal{A}$ is idempotent complete, right (I forgot to mention that here $\mathcal{A}$ is an additive category)? (Here $Ch(\mathcal{A})_{0}$ consists of those chain complexes $A_\bullet$ where $A_i\simeq 0$ for $i\leq -1$.)

   I was actually thinking about @Mike’s method, but was worried about the same thing - the lower bound being odd. @Urs I read the entry on the Dold-Kan correspondence, but I still don’t know what the analogue might be in $(n,r)$-category theory. The Dold-Kan correspondence basically states that the Dold-Kan construction $Ch(\mathcal{A})_{0}\to Funct(\mathbf{\Delta}^op,\mathcal{A})$ is fully faithful, and is an equivalence when $\mathcal{A}$ is idempotent complete, right (I forgot to mention that here $\mathcal{A}$ is an additive category)? (Here $Ch(\mathcal{A})_{0}$ consists of those chain complexes $A_\bullet$ where $A_i\simeq 0$ for $i\leq -1$.)

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeDec 24th 2014
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    Author: Mike Shulman
    Format: MarkdownItexSpectra are not the same as commutative monoid (is that what you mean by "algebra"?) objects in $\infty Gpd$; the closest connenction is that a *connective* spectrum can be identified with a commutative *group* object in $\infty Gpd$.

    Spectra are not the same as commutative monoid (is that what you mean by “algebra”?) objects in $\infty Gpd$; the closest connenction is that a connective spectrum can be identified with a commutative group object in $\infty Gpd$.

11. • CommentRowNumber11.
    • CommentAuthorsanath
    • CommentTimeDec 24th 2014
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    Author: sanath
    Format: MarkdownItex@Mike Sorry, that was me being stupid! :-) I was thinking, how can we reconcile, in some suitable sense, all these different definitions of the $n$-categorical analogue of abelian groups?

    @Mike Sorry, that was me being stupid! :-) I was thinking, how can we reconcile, in some suitable sense, all these different definitions of the $n$-categorical analogue of abelian groups?

12. • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 24th 2014
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    Author: DavidRoberts
    Format: MarkdownItexSometimes you can't reconcile such things. That's what makes categorification an art, not something canonical.

    Sometimes you can’t reconcile such things. That’s what makes categorification an art, not something canonical.

13. • CommentRowNumber13.
    • CommentAuthorZhen Lin
    • CommentTimeDec 24th 2014
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    Author: Zhen Lin
    Format: MarkdownItexWhat really puzzles me is how to reconcile these facts: * [[Segal's category]] provides a 1-categorical sketch (where the limit cones are the ones coming from the Segal maps _plus_ an extra one to ensure invertibility of elements) whose models in $Set$ are abelian groups and whose homotopy models in $\infty Grpd$ are infinite loop spaces. * The homotopy models for the Lawvere theory for abelian groups are the connective $H \mathbb{Z}$-modules. * The $(\infty, 1)$-category of connective $H \mathbb{Z}$-modules is not a full $(\infty, 1)$-subcategory of infinite loop spaces.

    What really puzzles me is how to reconcile these facts:

    • Segal’s category provides a 1-categorical sketch (where the limit cones are the ones coming from the Segal maps plus an extra one to ensure invertibility of elements) whose models in $Set$ are abelian groups and whose homotopy models in $\infty Grpd$ are infinite loop spaces.
    • The homotopy models for the Lawvere theory for abelian groups are the connective $H \mathbb{Z}$-modules.
    • The $(\infty, 1)$-category of connective $H \mathbb{Z}$-modules is not a full $(\infty, 1)$-subcategory of infinite loop spaces.
14. • CommentRowNumber14.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 25th 2014
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    Author: Dmitri Pavlov
    Format: MarkdownItex@ZhenLin: Doesn't adding an extra limit cone for invertibility of elements force your infinite loop spaces to be group-like? It seems to me that arbitrary infinite loop spaces arise from Segal maps alone.

    @ZhenLin: Doesn’t adding an extra limit cone for invertibility of elements force your infinite loop spaces to be group-like? It seems to me that arbitrary infinite loop spaces arise from Segal maps alone.

15. • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeDec 28th 2014
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    Author: Mike Shulman
    Format: MarkdownItexI don't think Zhen's question is about grouplikeness, but about the difference between (connective) spectra and $H\mathbb{Z}$-modules (the latter being equivalent to chain complexes). But I don't quite understand where it goes from there; what is it that needs reconciling about those three facts?

    I don’t think Zhen’s question is about grouplikeness, but about the difference between (connective) spectra and $H\mathbb{Z}$-modules (the latter being equivalent to chain complexes). But I don’t quite understand where it goes from there; what is it that needs reconciling about those three facts?

16. • CommentRowNumber16.
    • CommentAuthorZhen Lin
    • CommentTimeDec 28th 2014
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    Author: Zhen Lin
    Format: MarkdownItexThe unavoidable conclusion is, of course, that the Lawvere $\infty$-theory generated by the sketch for infinite loop spaces is not equivalent to the Lawvere theory for abelian groups. But how does that come about? What about other Lawvere theories – is the Lawvere theory of monoids equivalent to the Lawvere $\infty$-theory for $A_\infty$-spaces, or not?

    The unavoidable conclusion is, of course, that the Lawvere $\infty$-theory generated by the sketch for infinite loop spaces is not equivalent to the Lawvere theory for abelian groups. But how does that come about? What about other Lawvere theories – is the Lawvere theory of monoids equivalent to the Lawvere $\infty$-theory for $A_\infty$-spaces, or not?

17. • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeDec 29th 2014
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    Author: Mike Shulman
    Format: MarkdownItexAs to how that comes about, it happens in plenty of other cases that categorification does not respect free objects. The inclusion of $n$-categories into $m$-categories for $m\gt n$ is a right adjoint, not a left adjoint, so it doesn't tend to commute with left adjoints. But its left adjoint often does, i.e. if you truncate a free $m$-categorical object down to an $n$-categorical one, you often get the free $n$-categorical object. A simpler example is that the free symmetric monoidal category on a set is not equivalent to the free commutative monoid on that set, but its 0-truncation is.

    As to how that comes about, it happens in plenty of other cases that categorification does not respect free objects. The inclusion of $n$-categories into $m$-categories for $m\gt n$ is a right adjoint, not a left adjoint, so it doesn’t tend to commute with left adjoints. But its left adjoint often does, i.e. if you truncate a free $m$-categorical object down to an $n$-categorical one, you often get the free $n$-categorical object. A simpler example is that the free symmetric monoidal category on a set is not equivalent to the free commutative monoid on that set, but its 0-truncation is.

18. • CommentRowNumber18.
    • CommentAuthorZhen Lin
    • CommentTimeJan 1st 2015
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    Author: Zhen Lin
    Format: MarkdownItexRight, of course. Perhaps a better way of phrasing my question is — are there any heuristics for guessing whether a the Lawvere $\infty$-theory generated by a sketch is already 1-truncated? Of course, if there are "no equations" in the theory (e.g. the theory of objects) then there are no problems, but that's not so useful. For instance, my understanding is that the Lawvere $\infty$-theory generated by $\Delta^{op}$ (as a sketch for monoids) is 1-truncated.

    Right, of course. Perhaps a better way of phrasing my question is — are there any heuristics for guessing whether a the Lawvere $\infty$-theory generated by a sketch is already 1-truncated? Of course, if there are “no equations” in the theory (e.g. the theory of objects) then there are no problems, but that’s not so useful. For instance, my understanding is that the Lawvere $\infty$-theory generated by $\Delta^{op}$ (as a sketch for monoids) is 1-truncated.

19. • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeJan 1st 2015
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    Author: Mike Shulman
    Format: MarkdownItexI can't think of any offhand.

    I can’t think of any offhand.