# Graphical calculus in braided G crossed fusion categories: Explanation request and a question

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:

http://arxiv.org/pdf/0906.0620.pdf

brought me to this paper by Alexander Kirillov Jr:

http://de.arxiv.org/PS_cache/math/pdf/0104/0104242v1.pdf

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.

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 $(\mathrm{id}_A \otimes \epsilon_X) \circ (\beta_{X,A}^{-1} \otimes \mathrm{id}_X)$.
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".