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  1. 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 nn-vector space is an (n1)(n-1)-category of modules over a (n1)(n-1)-Vect enriched symmetric monoidal (n1)(n-1)-category.

    how far is this from the correct notion?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2010
    • (edited May 3rd 2010)

    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.

  2. now we also have n-vector space :)

    (just a stub)

  3. 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 n1n-1? I’m asking to eventually correct at n-vector space.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 12th 2010

    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

    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

    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 nn-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 kVect_k or with RModR Mod for some commutative ring RR.

    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.

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

    not sure about this: I would like nnVect to be an nn-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 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 nn we have the defining representation B nU(1)nVect \mathbf{B}^n U(1)\to n{Vect}_\mathbb{C}. for n=0n=0 this is the map of sets U(1)U(1)\hookrightarrow\mathbb{C}.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 12th 2010
    • (edited May 12th 2010)

    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 0Vect 0 Vect_{\mathbb{C}} to be the set \mathbb{C}.

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

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

    very good point, yes.

  5. I was thinking more of oo-Lie algebras

    I think this is correct: namely, as we think of a Lie group GG as the delooped groupoid BG\mathbf{B}G, it is natural to think of 𝔤\mathfrak{g} as something concentrated in degree 1 (and which maybe we should denote 𝔟𝔤\mathfrak{b}\mathfrak{g}).

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMay 12th 2010

    Yes, right.

    I quickly started adding a defnition-section to n-vector space. But will have to do something else now…

  6. in the Lab we now have a recursive definition of n-vector space such that 2Vect2Vect is the 2-category of kk-algebras, bimodules and bimodule homomorphisms. I would like to have something less restrictive like: 2Vect2Vect is the 2-category whose objects are additive categories over kk endowed with a structure of VectVect-module; morphism are VectVect-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 kk-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)

    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).

  7. Thanks, Zoran.

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

    0Vect k=k0Vect_k=k

    1Vect k=0Vect kmodules1Vect_k=0Vect_k-modules

    2Vect k=1Vect kmodules2Vect_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 k2Vect_k-modules is not meaningless)? in any case what I’m really interested at the moment is the n=2n=2 case.

    • CommentRowNumber15.
    • CommentAuthorzskoda
    • CommentTimeJun 9th 2010

    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.

  8. ok, now I see what you are saying. thanks.

    • CommentRowNumber17.
    • CommentAuthorzskoda
    • CommentTimeJun 10th 2010
    • (edited Jun 10th 2010)

    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.

  9. concerning #2 above:

    One notion of nn-vector space is a chain complex of vector spaces in degrees 0 to n1n-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 XX: it is simply a functor Π (X)Ch +\Pi_\infty(X)\to Ch_+. In particular, a morphism of \infty-flat bundles on XX (i.e. a natural transformation of functors) should be the datum of a morphism of complexes for each point of xx, of an equvalence of morphims for each path in XX, 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)

    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

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

    I think we should be speaking of (n,r)(n,r)-vector spaces and vector-bundles. The category of Baez-Crans n-vector spaces is really that of (n,1)(n,1)-vector space. Unbounded chain complexes are (,1)(\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 (,2)(\infty,2)-vector space is given by a dg-algebra, a morphism of these by a dg-bimodule, etc.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeJun 16th 2010

    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 Π (X)Ch \Pi_\infty(X) \to Ch_\bullet in the game, but rather the enriched functor consists of morphisms

    F(a)C (Π (M a,b))F(b) F(a) \otimes C_\bullet(\Pi_\infty(M_{a,b})) \to F(b)

    for M a,bM_{a,b} a moduli space with incoming punctures aa and outgoing punctures bb.

  10. this is exactly what I’m saying: a TCFT as a morphism of flat bundles over moduli. in your notation:

    C (Π (M a,b))Hom(F(a),F(b)). 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)

    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

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

    the original definition sits in the new one as a sub-(,n)(\infty,n)-category for the case that kk happens to be an ordinary ring (that happens to be a field).

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeAug 1st 2015

    I have added pointer to

    to the entries (infinity,n)-module and (infinity,1)-bimodule