I am trying to understand the equivalence between the 2 category of braided G crossed categories and the 2 category of braided categories containing Rep(G) as a symmetric category. The references in this important paper:


brought me to this paper by Alexander Kirillov Jr:


which also has references to some previous papers by the same author.

I am not familiar with graphical calculus since I don't work a lot with modular tensor categories. I would really appreaciate if one can write down for me the formula (not graphical calculus) for the morphism T_X from Formula 4.5 on page 14 of this paper.

The horizontal circle probably as usually means the dimension dim X and the punctured vertical line is the identity of A.The punctured curve between the circle and the line should be the structure \mu_X of X as an A-module in C. What I don't understand is how the evaluation and coevaluation are apllied to end up again in A.

I also have the following question:

What is the G crossed braiding $X\otimes_A Y \rightarrow \;^gY\otimes_A X$ if $X \in Rep_g(A)$ in the settings of this paper?

The braiding is given in the first paper as the unique morphism that extends the initial braiding from C. I was thinking maybe in the settings of Kirillov's paper one can write directly a formula ffor the braiding.
Thank you in advance for your help!


I recommend that you become familiar with graphical notation. As you will see from my answer, other ("formula") approaches are sub-par.

Formula (4.1) on page 14 of the linked paper by Kirillov defines a morphism $T_X : A \to A$ as follows. First, $A$ is a rigid algebra object in a modular category $\mathcal C$ (with all associators suppressed, and braiding denoted $\beta_{M,N} : M\otimes N \to N\otimes M$), and $X$ is a rigid $A$-module. I will write the multiplication on $A$ by $m_A$ and the left-action of $A$ on $X$ by $m_X$. Not having read the paper carefully, I believe that "rigid" means that $A$ and $X$ are each isomorphic to their own duals, and that this isomorphism is chosen to have good compatibility properties. In particular, $A$ should in fact be a Frobenius algebra for this isomorphism, and perhaps a symmetric one at that. I will write the unit as $u_X : 1 \to X\otimes X$ and the counit as $\epsilon_X : X\otimes X \to 1$, and similarly for $A$. Then we can consider the following composition: $$ \begin{eqnarray} A & \to & X \otimes X \otimes A & \quad\quad & (u_X \otimes \mathrm{id}_A) \\ & \to & X \otimes A \otimes X && (\mathrm{id}_X \otimes \beta_{X,A}) \\ & \to & X \otimes A \otimes A \otimes A \otimes X && (\mathrm{id}_A \otimes \mathrm{id}_X \otimes u_A \otimes \mathrm{id}_{X}) \\ & \to & X \otimes A \otimes X && (\mathrm{id}_X \otimes m_A \otimes m_X) \\ & \to & X \otimes X \otimes A && (\mathrm{id}_X \otimes \beta_{A,X}) \\ & \to & A && (\epsilon_X \otimes \mathrm{id}_A) \end{eqnarray} $$ Or, to put it another way, $$ T_X = (\epsilon_X \otimes \mathrm{id}_A) \circ (\mathrm{id}_X \otimes \beta_{A,X}) \circ (\mathrm{id}_X \otimes m_A \otimes m_X) \circ (\mathrm{id}_A \otimes \mathrm{id}_X \otimes u_A \otimes \mathrm{id}_{X}) \circ (\mathrm{id}_X \otimes \beta_{X,A}) $$

The point is, this is a mess, and is entirely unenlightening what's really happening. Moreover, the coherency axioms assure that there are many equivalent ways to write the above map. For example, I could have replaced the last two steps $(\epsilon_X \otimes \mathrm{id}_A) \circ (\mathrm{id}_X \otimes \beta_{A,X})$ with $(\mathrm{id}_A \otimes \epsilon_X) \circ (\beta_{X,A}^{-1} \otimes \mathrm{id}_X)$.

As for your second question, skimming did not for me reveal any place in the paper where that notation is used. I could make guesses, but perhaps someone else will have studied this paper more closely.

A final, non-math remark: The phrase "punctured curve" has a technical meaning in various areas, including areas close to this paper, to mean a compact Riemann surface with finitely many points removed (or variations on this notion). The standard term for "font" in which Kirillov draws his $A$ strands is "dashed", as opposed to "solid" for $X$. And a good term for the edges in such diagrams is "strands" — I have also seen "edges" and "strings", but the latter in particular is problematic because to a physicist a "string" is something that through time traces out a surface ("worldsheet"), whereas one meaning of these graphical calculi is some "particles" $A$ and $X$ traveling through time and thereby tracing out "worldlines".

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  • $\begingroup$ @Theo: Thank you very much for your answer. It is exactly what I needed. Now, I see it is much easier to write the things down with diagrams. Knowing the morphism from your answer I was able to construct back the diagram up to homotopy. Basically the „trick” is that one can introduce evaluations at some stages. Yes, the crossed braiding does not appear in this paper but it is explained in the first paper I mentioned. $\endgroup$ – math user Oct 29 '12 at 8:30

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