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- CommentRowNumber1.
- CommentAuthordomenico_fiorenza
- CommentTimeMay 3rd 2010
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Author: domenico_fiorenza
Format: MarkdownItexI just noticed we miss an entry [[n-vector space]]. I'd like to start it, but I only have a very vague idea, recursively implementing the notion of [[Baez-Crans 2-vector space]]. something like: an $n$-vector space is an $(n-1)$-category of modules over a $(n-1)$-Vect enriched symmetric monoidal $(n-1)$-category. how far is this from the correct notion?

I just noticed we miss an entry n-vector space. I’d like to start it, but I only have a very vague idea, recursively implementing the notion of Baez-Crans 2-vector space. something like: an $n$ -vector space is an $(n-1)$ -category of modules over a $(n-1)$ -Vect enriched symmetric monoidal $(n-1)$ -category.

how far is this from the correct notion?
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeMay 3rd 2010
- (edited May 3rd 2010)
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Author: Urs
Format: MarkdownItex> I just noticed we miss an entry n-vector space. We do have an entry [[2-vector space]], though. There are various different notions of n-vector spaces. One notion is: an n-vector space is a chain complex of vector spaces in degrees 0 to n. For n=2 this is Baez-Crans. This is useful for lots of things, but tends to be too restrictive in other contexts. Another is, recursively: an (n-1)-algebra object (or its (n-1)-category of modules) in the n-category of (n-1)-bimodules. For higher n this is mentioned towards the end of [[Topological Quantum Field Theories from Compact Lie Groups]]. For n=2 this subsumes various definitions that are in the literature, such as notably Kapranov-Voevodsky.

I just noticed we miss an entry n-vector space.

We do have an entry 2-vector space, though.

There are various different notions of n-vector spaces.

One notion is: an n-vector space is a chain complex of vector spaces in degrees 0 to n. For n=2 this is Baez-Crans. This is useful for lots of things, but tends to be too restrictive in other contexts.

Another is, recursively: an (n-1)-algebra object (or its (n-1)-category of modules) in the n-category of (n-1)-bimodules. For higher n this is mentioned towards the end of Topological Quantum Field Theories from Compact Lie Groups.

For n=2 this subsumes various definitions that are in the literature, such as notably Kapranov-Voevodsky.
- CommentRowNumber3.
- CommentAuthordomenico_fiorenza
- CommentTimeMay 3rd 2010
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Author: domenico_fiorenza
Format: MarkdownItexnow we also have [[n-vector space]] :) (just a stub)

now we also have n-vector space :)

(just a stub)
- CommentRowNumber4.
- CommentAuthordomenico_fiorenza
- CommentTimeMay 12th 2010
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Author: domenico_fiorenza
Format: MarkdownItex> One notion is: an n-vector space is a chain complex of vector spaces in degrees 0 to n shouldn't that be "in degrees 0 to $n-1$? I'm asking to eventually correct at [[n-vector space]].

One notion is: an n-vector space is a chain complex of vector spaces in degrees 0 to n

shouldn’t that be “in degrees 0 to $n-1$ ? I’m asking to eventually correct at n-vector space.
- CommentRowNumber5.
- CommentAuthorUrs
- CommentTimeMay 12th 2010
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Author: Urs
Format: MarkdownItexRight. Or would prefer: in degree 1 through n. So that the degree-numbering matches the categorical degree numbering. (For instance in the underlying n-vector space of a Lie n-algebra the vectors in degree k are the tangent to the space of k-morphisms of some Lie n-groupoid).

Right. Or would prefer: in degree 1 through n.

So that the degree-numbering matches the categorical degree numbering. (For instance in the underlying n-vector space of a Lie n-algebra the vectors in degree k are the tangent to the space of k-morphisms of some Lie n-groupoid).
- CommentRowNumber6.
- CommentAuthorHarry Gindi
- CommentTimeMay 12th 2010
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Author: Harry Gindi
Format: MarkdownWhy do we privilege vector spaces over abelian groups/R-modules or at the very worst free abelian groups/R-modules.

Why do we privilege vector spaces over abelian groups/R-modules or at the very worst free abelian groups/R-modules.
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeMay 12th 2010
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Author: Urs
Format: MarkdownItexTrue. In fact the focus on "vector spaces" here is more a reflection of how we like to think of these beast, than of their true nature. You'll notice that the definition "an $n$-vector space is an algebra object in the symmetric monoidal category of (n-1)-vector spaces" goes trough no matter if we start the induction with $Vect_k$ or with $R Mod$ for some commutative ring $R$. So, yeah, maybe one should think of another term. But then, somehow for lots of things that one wants to do with "n-vector spaces" it is suggestiv to think of them as vector spaces. Such as when talking about representations on n-groups, of n-vector spaces of states in an extended QFT and so on.

True. In fact the focus on “vector spaces” here is more a reflection of how we like to think of these beast, than of their true nature.

You’ll notice that the definition “an $n$ -vector space is an algebra object in the symmetric monoidal category of (n-1)-vector spaces” goes trough no matter if we start the induction with $Vect_k$ or with $R Mod$ for some commutative ring $R$ .

So, yeah, maybe one should think of another term. But then, somehow for lots of things that one wants to do with “n-vector spaces” it is suggestiv to think of them as vector spaces. Such as when talking about representations on n-groups, of n-vector spaces of states in an extended QFT and so on.
- CommentRowNumber8.
- CommentAuthordomenico_fiorenza
- CommentTimeMay 12th 2010
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Author: domenico_fiorenza
Format: MarkdownItex> Or would prefer: in degree 1 through n. not sure about this: I would like $n$Vect to be an $n$-category. so 1Vect would be the ordinary category of vector spaces; each object of 1Vect is a 0-category, so I would say that a 1-vector space has categorical degree 0, it is the category of 1-vector spaces as a whole to have categorical degree 1. also, it would be nice to have a bit of negative thinking here. namely, it would be convenient to have $0Vect_\mathbb{C}$ to be the set $\mathbb{C}$. this would fit the pappern we are discussing in the thread on Dijkgraaf-Witten model: for any $n$ we have the defining representation $\mathbf{B}^n U(1)\to n{Vect}_\mathbb{C}$. for $n=0$ this is the map of sets $U(1)\hookrightarrow\mathbb{C}$.

Or would prefer: in degree 1 through n.

not sure about this: I would like $n$ Vect to be an $n$ -category. so 1Vect would be the ordinary category of vector spaces; each object of 1Vect is a 0-category, so I would say that a 1-vector space has categorical degree 0, it is the category of 1-vector spaces as a whole to have categorical degree 1.

also, it would be nice to have a bit of negative thinking here. namely, it would be convenient to have $0Vect_\mathbb{C}$ to be the set $\mathbb{C}$ . this would fit the pappern we are discussing in the thread on Dijkgraaf-Witten model: for any $n$ we have the defining representation $\mathbf{B}^n U(1)\to n{Vect}_\mathbb{C}$ . for $n=0$ this is the map of sets $U(1)\hookrightarrow\mathbb{C}$ .
- CommentRowNumber9.
- CommentAuthorUrs
- CommentTimeMay 12th 2010
- (edited May 12th 2010)
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Author: Urs
Format: MarkdownItexDomenico, I guess you are right, I was thinking more of oo-Lie algebras. Right, whatever the counting is. > it would be convenient to have $0 Vect_{\mathbb{C}}$ to be the set $\mathbb{C}$. Yes, I think that's the usual way to think about it. > for $n = 0$ this is the map of sets $U(1) \hookrightarrow \mathbb{C}$ very good point, yes.

Domenico,

I guess you are right, I was thinking more of oo-Lie algebras. Right, whatever the counting is.

it would be convenient to have $0 Vect_{\mathbb{C}}$ to be the set $\mathbb{C}$ .

Yes, I think that’s the usual way to think about it.

for $n = 0$ this is the map of sets $U(1) \hookrightarrow \mathbb{C}$

very good point, yes.
- CommentRowNumber10.
- CommentAuthordomenico_fiorenza
- CommentTimeMay 12th 2010
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Author: domenico_fiorenza
Format: MarkdownItex> I was thinking more of oo-Lie algebras I think this is correct: namely, as we think of a Lie group $G$ as the delooped groupoid $\mathbf{B}G$, it is natural to think of $\mathfrak{g}$ as something concentrated in degree 1 (and which maybe we should denote $\mathfrak{b}\mathfrak{g}$).

I was thinking more of oo-Lie algebras

I think this is correct: namely, as we think of a Lie group $G$ as the delooped groupoid $\mathbf{B}G$ , it is natural to think of $\mathfrak{g}$ as something concentrated in degree 1 (and which maybe we should denote $\mathfrak{b}\mathfrak{g}$ ).
- CommentRowNumber11.
- CommentAuthorUrs
- CommentTimeMay 12th 2010
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Author: Urs
Format: MarkdownItexYes, right. I quickly started adding <a href="http://ncatlab.org/nlab/show/n-vector+space">a defnition-section</a> to [[n-vector space]]. But will have to do something else now...

Yes, right.

I quickly started adding a defnition-section to n-vector space. But will have to do something else now…
- CommentRowNumber12.
- CommentAuthordomenico_fiorenza
- CommentTimeJun 8th 2010
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Author: domenico_fiorenza
Format: MarkdownItexin the Lab we now have a recursive definition of [[n-vector space]] such that $2Vect$ is the 2-category of $k$-algebras, bimodules and bimodule homomorphisms. I would like to have something less restrictive like: $2Vect$ is the 2-category whose objects are additive categories over $k$ endowed with a structure of $Vect$-module; morphism are $Vect$-linear additive functors and 2-morphisms are natural transformations. This seems a quite natural generalization of the notion of vector space, and the 2-category of $k$-algebras, bimodules and bimodule homomorphisms would naturally embed into this. iterating this definition seems to bring us back to post #1 above.

in the Lab we now have a recursive definition of n-vector space such that $2Vect$ is the 2-category of $k$ -algebras, bimodules and bimodule homomorphisms. I would like to have something less restrictive like: $2Vect$ is the 2-category whose objects are additive categories over $k$ endowed with a structure of $Vect$ -module; morphism are $Vect$ -linear additive functors and 2-morphisms are natural transformations. This seems a quite natural generalization of the notion of vector space, and the 2-category of $k$ -algebras, bimodules and bimodule homomorphisms would naturally embed into this. iterating this definition seems to bring us back to post #1 above.
- CommentRowNumber13.
- CommentAuthorzskoda
- CommentTimeJun 9th 2010
- (edited Jun 9th 2010)
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Author: zskoda
Format: MarkdownSo what is the analogue of "additive" at level n ? Another thing is about the "bimodule" at every higher level. Namely the actions do not need to commute strictly but in a pseudosense, what is difficult to define at level n. Second the tensor product of bimodules is certain bicoequalizer in dim 2 and gets more complicated at higher levels. Cf. my 2006 manuscript [link](http://www.irb.hr/korisnici/zskoda/biact.pdf) from about page 6 on (there are some wrong things in first 2-3 pages which are not useful anyway).

So what is the analogue of "additive" at level n ?

Another thing is about the "bimodule" at every higher level. Namely the actions do not need to commute strictly but in a pseudosense, what is difficult to define at level n. Second the tensor product of bimodules is certain bicoequalizer in dim 2 and gets more complicated at higher levels. Cf. my 2006 manuscript link from about page 6 on (there are some wrong things in first 2-3 pages which are not useful anyway).
- CommentRowNumber14.
- CommentAuthordomenico_fiorenza
- CommentTimeJun 9th 2010
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Author: domenico_fiorenza
Format: MarkdownItexThanks, Zoran. My naive idea is that an $n$-additive category over $k$ should be a category enriched in $n$-vector spaces over $k$ with finite direct sums. so one should have: $0Vect_k=k$ $1Vect_k=0Vect_k-modules$ $2Vect_k=1Vect_k-modules$ and so on. but to say "and so on" there are a lot of details to be fixed. for instance, is 2Vect_k a ring in some sense (so that saying $2Vect_k$-modules is not meaningless)? in any case what I'm really interested at the moment is the $n=2$ case.

Thanks, Zoran.

My naive idea is that an $n$ -additive category over $k$ should be a category enriched in $n$ -vector spaces over $k$ with finite direct sums. so one should have:

$0Vect_k=k$

$1Vect_k=0Vect_k-modules$

$2Vect_k=1Vect_k-modules$

and so on. but to say “and so on” there are a lot of details to be fixed. for instance, is 2Vect_k a ring in some sense (so that saying $2Vect_k$ -modules is not meaningless)? in any case what I’m really interested at the moment is the $n=2$ case.
- CommentRowNumber15.
- CommentAuthorzskoda
- CommentTimeJun 9th 2010
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Author: zskoda
Format: MarkdownThat is what I was saying, to have a ring you have a particular kind of tensor product. At each level that is certain pseudo-n-colimit. In the case of n=2 that is what I have sent you. More recently an alternative approach to the same topic is by a student of Nikshych, se ereferences at [[biactegory]].

That is what I was saying, to have a ring you have a particular kind of tensor product. At each level that is certain pseudo-n-colimit. In the case of n=2 that is what I have sent you. More recently an alternative approach to the same topic is by a student of Nikshych, se ereferences at biactegory.
- CommentRowNumber16.
- CommentAuthordomenico_fiorenza
- CommentTimeJun 10th 2010
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Author: domenico_fiorenza
Format: MarkdownItexok, now I see what you are saying. thanks.

ok, now I see what you are saying. thanks.
- CommentRowNumber17.
- CommentAuthorzskoda
- CommentTimeJun 10th 2010
- (edited Jun 10th 2010)
- PermaLink
Author: zskoda
Format: MarkdownI hear algebraic geometers and representation theorists talk similar tower of infinity categories. At level 0 one has a scheme. At level 1 one has an enhancement of a derived category of qcoh sheaves on the scheme. Then one looks at modules over that stable infinity category and so on. Notice that Lurie's stuff with infinity categories is already at level 1. But then one goes on. So I guess at level two one has stable (infinity,2) category whatever it is.

I hear algebraic geometers and representation theorists talk similar tower of infinity categories. At level 0 one has a scheme. At level 1 one has an enhancement of a derived category of qcoh sheaves on the scheme. Then one looks at modules over that stable infinity category and so on. Notice that Lurie's stuff with infinity categories is already at level 1. But then one goes on. So I guess at level two one has stable (infinity,2) category whatever it is.
- CommentRowNumber18.
- CommentAuthordomenico_fiorenza
- CommentTimeJun 16th 2010
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Author: domenico_fiorenza
Format: MarkdownItexconcerning #2 above: One notion of $n$-vector space is a chain complex of vector spaces in degrees 0 to $n-1$; but then a notion of $\infty$-vector space is "a complex concentrated in nonnegative degrees". and with this notion we have a very simple notion of $\infty$-flat bundle on a topological space $X$: it is simply a functor $\Pi_\infty(X)\to Ch_+$. In particular, a morphism of $\infty$-flat bundles on $X$ (i.e. a natural transformation of functors) should be the datum of a morphism of complexes for each point of $x$, of an equvalence of morphims for each path in $X$, and so on. This is reminiscent of what happen in a [[TCFT]], so I'm now wondering whether one can think of TCFTs in terms of $\infty$-flat bundles on moduli spaces. this would not be to surprising, since at the level of TQFTs one has a descripion in terms of flat vector bundles on modul spaces (conformal blocks), but it would be nice :)

concerning #2 above:

One notion of $n$ -vector space is a chain complex of vector spaces in degrees 0 to $n-1$ ; but then a notion of $\infty$ -vector space is “a complex concentrated in nonnegative degrees”. and with this notion we have a very simple notion of $\infty$ -flat bundle on a topological space $X$ : it is simply a functor $\Pi_\infty(X)\to Ch_+$ . In particular, a morphism of $\infty$ -flat bundles on $X$ (i.e. a natural transformation of functors) should be the datum of a morphism of complexes for each point of $x$ , of an equvalence of morphims for each path in $X$ , and so on. This is reminiscent of what happen in a TCFT, so I’m now wondering whether one can think of TCFTs in terms of $\infty$ -flat bundles on moduli spaces. this would not be to surprising, since at the level of TQFTs one has a descripion in terms of flat vector bundles on modul spaces (conformal blocks), but it would be nice :)
- CommentRowNumber19.
- CommentAuthorUrs
- CommentTimeJun 16th 2010
- (edited Jun 16th 2010)
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Author: Urs
Format: MarkdownItexconcerning Zoran's remark about the difficulty of making sense of the recursive definition of n-vector spaces: yes, indeed, it is clear that thee definition should exist as stated, but making this explicit is understood in detail only for low n at the moment. At least to ordinary mortals. Judging from the discussion at the end of FHLT, they (or at least one of them...) know how to make it precise for all n.

concerning Zoran’s remark about the difficulty of making sense of the recursive definition of n-vector spaces:

yes, indeed, it is clear that thee definition should exist as stated, but making this explicit is understood in detail only for low n at the moment. At least to ordinary mortals. Judging from the discussion at the end of FHLT, they (or at least one of them…) know how to make it precise for all n.
- CommentRowNumber20.
- CommentAuthorUrs
- CommentTimeJun 16th 2010
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Author: Urs
Format: MarkdownItexConcerning Domenico's remark on oo-vector bundles, part 2: I think we should be speaking of $(n,r)$-vector spaces and vector-bundles. The category of Baez-Crans n-vector spaces is really that of $(n,1)$-vector space. Unbounded chain complexes are $(\infty,1)$-vector spaces. More generally, the latter are modules over a ring spectrum other than the Eilenberg-MacLabe spectrum for the ground field. Then an $(\infty,2)$-vector space is given by a dg-algebra, a morphism of these by a dg-bimodule, etc.

Concerning Domenico’s remark on oo-vector bundles, part 2:

I think we should be speaking of $(n,r)$ -vector spaces and vector-bundles. The category of Baez-Crans n-vector spaces is really that of $(n,1)$ -vector space. Unbounded chain complexes are $(\infty,1)$ -vector spaces. More generally, the latter are modules over a ring spectrum other than the Eilenberg-MacLabe spectrum for the ground field.

Then an $(\infty,2)$ -vector space is given by a dg-algebra, a morphism of these by a dg-bimodule, etc.
- CommentRowNumber21.
- CommentAuthorUrs
- CommentTimeJun 16th 2010
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Author: Urs
Format: MarkdownItexconcerning Domenico's remark about TCFT: I think a TCFT gives not so much a bundle of chain complexes on moduli space than an action of the homology of moduli space on certain things. Notice that there is not really a functor $\Pi_\infty(X) \to Ch_\bullet$ in the game, but rather the enriched functor consists of morphisms $$ F(a) \otimes C_\bullet(\Pi_\infty(M_{a,b})) \to F(b) $$ for $M_{a,b}$ a moduli space with incoming punctures $a$ and outgoing punctures $b$.

concerning Domenico’s remark about TCFT:

I think a TCFT gives not so much a bundle of chain complexes on moduli space than an action of the homology of moduli space on certain things.

Notice that there is not really a functor $\Pi_\infty(X) \to Ch_\bullet$ in the game, but rather the enriched functor consists of morphisms
$F(a) \otimes C_\bullet(\Pi_\infty(M_{a,b})) \to F(b)$
for $M_{a,b}$ a moduli space with incoming punctures $a$ and outgoing punctures $b$ .
- CommentRowNumber22.
- CommentAuthordomenico_fiorenza
- CommentTimeJun 16th 2010
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Author: domenico_fiorenza
Format: MarkdownItexthis is exactly what I'm saying: a TCFT as a morphism of flat bundles over moduli. in your notation: $$ C_\bullet(\Pi_\infty(M_{a,b}))\to Hom(F(a),F(b)). $$ but looking back at what I wrote I see it is not clear at all the digression on morphisms I make at a certain point should be relevant to what I write son after that. sorry for this.

this is exactly what I’m saying: a TCFT as a morphism of flat bundles over moduli. in your notation:
$C_\bullet(\Pi_\infty(M_{a,b}))\to Hom(F(a),F(b)).$
but looking back at what I wrote I see it is not clear at all the digression on morphisms I make at a certain point should be relevant to what I write son after that. sorry for this.
- CommentRowNumber23.
- CommentAuthorUrs
- CommentTimeMay 17th 2011
- (edited May 17th 2011)
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Author: Urs
Format: MarkdownItexI have added a few more details to [[n-vector space]], but somewhat in a rush. Should eventually be polished and expanded.

I have added a few more details to n-vector space, but somewhat in a rush. Should eventually be polished and expanded.
- CommentRowNumber24.
- CommentAuthorUrs
- CommentTimeJun 8th 2011
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Author: Urs
Format: MarkdownItexI have boosted the definition at [[n-vector space]] to that of $(\infty,n)$-vector spaces, where instead of starting the iteration with modules over a field we start with module spectra over a ring spectrum $k$. the original definition sits in the new one as a sub-$(\infty,n)$-category for the case that $k$ happens to be an ordinary ring (that happens to be a field).

I have boosted the definition at n-vector space to that of $(\infty,n)$ -vector spaces, where instead of starting the iteration with modules over a field we start with module spectra over a ring spectrum $k$ .

the original definition sits in the new one as a sub- $(\infty,n)$ -category for the case that $k$ happens to be an ordinary ring (that happens to be a field).
- CommentRowNumber25.
- CommentAuthorUrs
- CommentTimeAug 1st 2015
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Author: Urs
Format: MarkdownItexI have added pointer to * [[Rune Haugseng]], _The higher Morita category of $E_n$-algebras_ ([arXiv:1412.8459](http://arxiv.org/abs/1412.8459)) to the entries _[[(infinity,n)-module]]_ and _[[(infinity,1)-bimodule]]_
I have added pointer to
- Rune Haugseng, The higher Morita category of $E_n$ -algebras (arXiv:1412.8459)
to the entries (infinity,n)-module and (infinity,1)-bimodule

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nLab > Latest Changes: n-vector spaces