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nLab > Latest Changes: Definition of extended TQFT

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  1. I apologize in case this discussion is already open and I have been unable to find it.

    There is something I am unable to undrstand in the definition of extended TQFT as on the nLab page http://ncatlab.org/nlab/show/extended+topological+quantum+field+theory

    Namely, it seems to me that the recursive definition should rather end with "smooth compact oriented (n-m+1)-manifolds to R-linear (m?2)-categories"
  2. also the definition "nCob m is an n-category with smooth compact oriented n-manifolds as objects and cobordisms of cobordisms up to m-cobordisms, up to diffeomorphism, as morphisms." I guess should read as

    "nCob m is an n-category with smooth compact oriented (m-n)-manifolds as objects and cobordisms of cobordisms up to n-cobordisms, up to diffeomorphism, as morphisms."

    with this choice one should interchange the role of n and m in the definition of extended TQFT a few lines below, which seems reasonable, if one wants that n refers to the degree of high-categoricality and m to the dimension of the manifolds (by the way, maybe nCob d would be a more self-explaining notation).

    in the definition of extended TQFT (even leaving it at a non fully detailed level) I would add the "vacuum axioms", namely what is Z(emptyset) considering the empty set as a closed manifold of various dimensions.

    by the way, the mysterious "m?2" in my post above should have been "m-2"
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2009

    I think you are right, there is somethig wrong with the n-s and m-s in that enty. Do you want to fix it?

    Concerning the "vacuum": since functor is required to be monoidal, it needs to send monoidal units to monoidal units. Emptyset is the monoidal unit for nCob, so its value under the functor is fixed to be the tensor unit in the codomain.

  4. I'll try to fix n's and m's within a few days.

    Concerning the vacuum, I absolutely agree, but if one really insists on monoidal functoriality, then the first three lines in the definition, namely,

    # smooth compact oriented n-manifolds to elements of R
    # smooth compact oriented (n-1)-manifolds to R-modules
    # cobordisms of smooth compact oriented (n-1)-manifolds to R-linear maps between R-modules

    can be reduced to the single third line. but then one notices that reasoning in the same way the fourth and fifth lines

    # smooth compact oriented (n-2)-manifolds to R-linear additive categories
    # cobordisms of smooth compact oriented (n-2)-manifolds to functors between R-linear categories

    should imply the third one (at least with the "right assumptions" on the functor), and since the foruth is clearly contained in the fifth, one sees that the first five lines in the definition should reduce to the fifth, and so on, till one is rediced to a single line: the last one, prescribing the value oz Z on a point. That is, the cobordism hypotesis.

    So my feeling is that vaccum axioms would fit with an "explicit" definition of extended TQFT. Maybe a good place to put them would be just below the definition, as a remark:

    "since functor is required to be monoidal, it needs to send monoidal units to monoidal units. Emptyset is the monoidal unit for nCob, so its value under the functor is fixed to be the tensor unit in the codomain, e.g., n-dimensional vacuum is mapped to 1, "(n-1)-dimensional vacuum to the R-module R, (n-2)-dimensional vacuum to teh category of R-modules, etc."

    But I also agree this is redundant, so I won't make this particular correction now and I'll wait for feedback from the forum on this.
    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeDec 4th 2009

    I think that sort of remark is good.

  6. I added the remark and tried to use d and n consistently thoughout the whole entry. I also changed

    "decomposing $\Sigma$ into pieces of arbitrary codimension d, i.e. into (n?d)-dimensional pieces"

    with

    "decomposing $\Sigma$ into $d$-dimensional pieces with piecewise smooth boundaries, whose boundary strata are of arbitrary codimension $k$"

    which seems to be more correct to me.

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