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Stream: theory: category theory

Topic: free cat w/ duals on a point

Daniel Teixeira (Mar 26 2024 at 17:28):

Let X be the folowing coloured PRO: it has two generators A and B , a morphism ev:A⊗B -> 1, a morphism coev:1 -> B ⊗ A, such that the triangle identities are satisfied. Models of X in a monoidal category are dualizable objects with duality data. On a hand, this gives it right to be called "the free monoidal category with duals on a point". However, it is not $\text{Bord}_{1,0}$, as it would have to be by the Cobordism Hypothesis (for instance, it doesn't have circles).

If we make this PRO a PROP (i.e. add symmetry), then our left dual in X is a right dual, and I'm hopeful that the resulting PROP is $\text{Bord}_{1,0}$. But why would you call this rather than X "the free monoidal category with duals on a point"?

John Baez (Mar 26 2024 at 19:12):

For me the "free monoidal category with duals for objects on a point" is the monoidal category generated by an object x which has an ambidextrous adjoint, i.e. another object that is both a left and right dual to x. I like it because it is the category of oriented tangles in 2 dimensions. I don't think it's either of the categories you mentioned.

Amar Hadzihasanovic (Mar 26 2024 at 19:45):

I would not call the first one a "free monoidal category with duals" because it is not a monoidal category with duals (interpreting this as "every object has a left and a right dual"). I would call it the "walking duality".

Amar Hadzihasanovic (Mar 26 2024 at 19:48):

Pedantically I would expect the "free monoidal category with duals on a point" to be a coloured pro whose colours are the integers, with $k$ a left dual of $k+1$ for every $k \in \mathbb{Z}$, but I would not be surprised if someone meant what John suggests, which I would more precisely call the "free monoidal category with two-sided (or ambidextrous) duals on a point"...

Daniel Teixeira (Mar 26 2024 at 20:42):

I'm starting to believe that the PROP I described is $\text{Bord}^{fr}_{1,0}$, while the category described by John is the oriented $\text{Bord}^{or}_{1,0}$. I don't know what the category described by Amar be, although these chains of left-right duals have often appeared before...

John Baez (Mar 26 2024 at 20:54):

There's no real difference between framed 1-tangles in 2d and oriented 1-tangles in 2d, since in 2d you can use the standard orientation of the plane and the orientation of the tangle to determine a framing on the tangle.

John Baez (Mar 26 2024 at 20:56):

It doesn't work like that in any higher dimension.

Daniel Teixeira (Mar 27 2024 at 17:23):

The reason for my intuition was coming from 1-framed 1-manifolds and 0-manifolds, namely that if you frame the circle as a 1-manifold, the corresponding immersion in R^2 is a figure 8 in the plane, not an unentangled circle. The crossing in the figure eight is like a symmetry operation in the monoidal category

hum, I sould ponder longer over this over the next few days and come back

John Baez (Mar 27 2024 at 22:21):

I think by bringing in 'crossings' like this you are now starting to talk about either:

1. The braided monoidal category of framed 1-tangles in 3d space

or

2. The symmetric (?) monoidal category of immersed rather than embedded 1-tangles in 2d space.

The first one has been studied a lot; the second one much less, and it suggests all sorts of interesting questions and generalizations.