Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: "Dual braided structure"?

Amar Hadzihasanovic (Dec 29 2020 at 09:53):

If $B$ is a braided monoidal category with braidings $\{\sigma_{x,y}: x \otimes y \to y \otimes x\}$, then there is another braided structure on the same underlying monoidal category, given by $\{\sigma^*_{x,y} := \sigma^{-1}_{y,x}\}$. Call the resulting braided monoidal category $B^*$.

The underlying functor of a braided monoidal functor $f: B \to C$ is also a braided monoidal functor $B^* \to C^*$, so this defines an involution on the category of small braided monoidal categories and braided monoidal functors, whose fixpoints are exactly the symmetric monoidal categories.

Do you know if this simple duality appears with a special name or notation in the literature?

Amar Hadzihasanovic (Dec 29 2020 at 09:55):

Graphically it is the one that e.g. turns a knot into its dual knot (mirror reflection).
With the identification of braided monoidal categories with doubly degenerate tricategories, it comes from $-^{\mathrm{op(1)}}$, the duality reversing the direction of 1-cells in a tricategory.

Fabrizio Genovese (Dec 29 2020 at 11:56):

I sounds very unintuitive to me that the same monoidal structure can admit more than one braiding. Does this vanish when one considers symmetries?

Amar Hadzihasanovic (Dec 29 2020 at 13:11):

No, you can have lots of symmetric structures on the same monoidal category.

Reid Barton (Dec 29 2020 at 13:11):

No, for example on graded vector spaces (in characteristic not equal to 2), you can put a factor of $(-1)^{ij}$ in degree $(i,j)$ or not.

Amar Hadzihasanovic (Dec 29 2020 at 13:12):

I was about to give the same example (well, just the case of $\mathbb{Z}/2\mathbb{Z}$-graded vector spaces).

Amar Hadzihasanovic (Dec 29 2020 at 13:32):

Another systematic source of examples is categories of representations of bialgebras (or Hopf algebras).
A triangular bialgebra is a bialgebra $H$ together with an element $R \in H \otimes H$ that satisfies certain equations. In general, $R$ is not unique with those properties. But every choice of $R$ that makes $H$ triangular gives a different symmetric monoidal structure on the category $\mathrm{Rep}(H)$ of representations of $H$.

Fabrizio Genovese (Dec 29 2020 at 13:49):

Amar Hadzihasanovic said:

No, you can have lots of symmetric structures on the same monoidal category.

This really fucks up my intuition about string diagrams, when you interpret objects as wires it seems weird that you can define different ways in which they can cross, that are represented in the same way topologically. Maybe this has to do with the fact that your space truly is higher dimensional and taking different ways of defining symmetries amounts to different ways in which you project the space on the plane?

Amar Hadzihasanovic (Dec 29 2020 at 13:54):

It's rather the opposite: different ways of picking symmetries gives you different ways in which your planar diagrams will be “opened up” to become higher-dimensional diagrams.

Amar Hadzihasanovic (Dec 29 2020 at 13:54):

(Whereas the projection is “deterministic”, there's only one way to project...)

Amar Hadzihasanovic (Dec 29 2020 at 13:56):

From the point of view of the underlying monoidal category, a braiding is just a cell with two inputs and two outputs like many others. To “open up” the diagrams, you have to “tell” it “Hey, this one we actually break into two wires”.

Fabrizio Genovese (Dec 29 2020 at 14:03):

Amar Hadzihasanovic said:

(Whereas the projection is “deterministic”, there's only one way to project...)

Yeah, it's what I was trying to say somehow

Nathanael Arkor (Dec 29 2020 at 14:52):

Is there a notion of monoidal category with multiple compatible braidings/symmetries?

John Baez (Dec 29 2020 at 18:38):

Fabrizio Genovese said:

I sounds very unintuitive to me that the same monoidal structure can admit more than one braiding. Does this vanish when one considers symmetries?

No - otherwise people wouldn't bother to specify the symmetry! It's not just a useless ornament, it's a knob you can twist.

Take your category to consist of vector bundles on a group $G$: that is, ways of picking a vector space $V_g$ for each $g \in G$. Use the "convolution" tensor product:

$(V \otimes W)_g = \bigoplus_{h, h' \; \text{s.t.} \; hh' = g} V_h \otimes V_{h'}$

Then there are typically lots of different associators you can use - and for each choice of associator there are typically lots of braidings and lots of symmetries. If your associator $\alpha$ consists of multiplying by a "phase" (unit complex number) that depends on $g,g',g'' \in G$ then the pentagon identity implies that

$\alpha_{g,g',g''} \in \mathbb{C}$

defines a "3-cocycle" on the group $G$. And if you don't know what a 3-cocycle on a group is, this is a good way to define it and learn about it! Group cohomology is secretly all about this stuff. Braidings and symmetries correspond to other subtler cocycles. All this was explained in the very first paper on braided monoidal categories.

John Baez (Dec 29 2020 at 18:43):

The most famous case is when $G = \mathbb{Z}/2$. Then if you use the trivial associator you get two braidings, which are both symmetries: the "boring" symmetry and the one that sticks in a phase of -1 when you switch two vectors living in "odd" vector spaces (called "fermions"). Nature uses the second one: when you switch two "fermions" their phase gets multiplied by -1.

This is why two electrons can't be in the same state at the same time!

If nature had chosen the other symmetry, chemistry would not exist because all electrons in an atom would fall to the lowest energy level - there wouldn't be "shells" or the periodic table.

John Baez (Dec 29 2020 at 18:44):

So be grateful that there's more than one way to make a monoidal category symmetric monoidal!

Matteo Capucci (he/him) (Dec 29 2020 at 19:58):

I love this!!

Matteo Capucci (he/him) (Dec 29 2020 at 19:59):

I'm ready for the next time someone asks me what is CT doing for the world