Let $\mathbf{COB}_2$ denote the 2-category given by

$\bullet$ objects are finite sets of points

$\bullet$ 1-morphisms between these are 1d cobordisms

$\bullet$ 2-morphisms are 2d cobordisms with edges between these.

Let $\mathbf{Cob}_2$ denote the 1-category given by

$\bullet$ objects are disjoint unions of circles

$\bullet$ morphisms between these are 2d cobordisms.

We clearly have:

$(*)$ $\mathbf{Cob}_2=\text{End}_{\mathbf{COB}_2}(\varnothing)$.

We also have the following universal characterizations.

By the cobordism hypothesis:

$(\mathbf{1})$ $\mathbf{COB}_2$ is the symmetric monoidal 2-category with duals freely generated by one object.

By a well known result:

$(\mathbf{2})$ $\mathbf{Cob}_2$ is the symmetric monoidal category freely generated by a comm. Frobenius monoid.

My question:

Can one, in a purely algebraic/categorical manner, derive $(\mathbf{2})$ by applying $(*)$ to $(\mathbf{1})$?

I know that a symmetric monoidal 2-category with one object should simply be a symmetric monoidal category.

I'm guessing that the duality structure maps for objects and 1-morphisms in $\mathbf{COB}_2$ ``collapse" into the structure maps for the commutative Frobenius monoid generating $\mathbf{Cob}_2$.

  • $\begingroup$ Not as far as I know. This would imply that we know how to derive universal classifications of non-fully-extended TFTs from the cobordism hypothesis, and e.g. I already think we don't know the classification of 3-2 or 4-3-2-1 theories. Also, in the first definition you need framings everywhere. $\endgroup$ – Qiaochu Yuan Sep 20 '16 at 17:51
  • 2
    $\begingroup$ The statement "COB2 is the symmetric monoidal 2-category with duals freely generated by one object." is not quite correct, because you didn't put framings on (the stabilised tangent bundles of) your points/1-manifolds/2-manifolds. @Qiaochu: the the classification of 3-2 theories is discussed here: arxiv.org/abs/1408.0668 $\endgroup$ – André Henriques Sep 20 '16 at 20:27

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