Edit: I found a serious flaw in the question and my answer, and I had to change a lot. The basic question is still there, but the details are a lot different.


Premodular categories
---------------------


In braided spherical fusion (=premodular) categories $\mathcal{C}$ with braiding $c\_{-,-}$, we can define the double braiding:
$$\gamma\_{A,X}\colon A \otimes X \to A \otimes X$$
$$\gamma\_{A,X} := c\_{X,A} \circ c\_{A,X}$$


Now we can define *encirclings*, where we braid an object around another and close the loop on the right hand side:


$$\Delta\_{A,X}\colon A \to A$$
$$\Delta\_{A,X} = \operatorname{tr}\_X(\gamma\_{A,X})$$


(The trace is defined via the pivotal structure.)


The **"killing" lemma** is quite well known.
Assume that $A$ is simple. Then:


$$\Omega := \bigoplus\_{X \text{ simple}} d(X) X$$
$$\Delta\_{A,\Omega} = \left\{ \begin{matrix} d(\Omega) \cdot 1\_A && \qquad A \text{ is in the symmetric centre} \\ 0 && \qquad \text{otherwise} \end{matrix}\right.$$
$A$ is in the *symmetric centre* if braids trivially with every object. One says that $A$ is killed if it's not in the symmetric centre, hence the name of the lemma.


$G$-crossed braided fusion categories
-------------------------------------


Now consider [$G$-crossed braided fusion categories](https://ncatlab.org/nlab/show/%24G%24-crossed+braided+fusion+category). If you think of fusion categories as some sort of categorified group (algebra), then a $G$-crossed braided fusion category is a categorified crossed module. It is, amongst other things, a $G$-graded fusion category:


$$\mathcal{C} = \bigoplus\_{g \in G} \mathcal{C}\_g$$


It also has a monoidal $G$-action:


$${}^g(-)\colon \mathcal{C}\_h \to \mathcal{C}\_{ghg^{-1}} \quad \forall g, h \in G$$


And it has a crossed braiding:


$$ c\_{A,X}\colon A \otimes X \to {}^gX \otimes A\quad \forall A \in \mathcal{C}\_g$$


This means that the double braiding is now *not an endomorphism anymore*:


$$ \gamma\_{A,X}\colon A \otimes X \to {}^{ghg^{-1}}A \otimes {}^g X \quad\forall A \in \mathcal{C}\_g, X \in \mathcal{C}\_h$$


You can only take the trace on the right hand side, and thus define encircling, if $X \cong {}^gX$.
The only canonical way I can think of is to demand that the grade of $A$ is trivial, i.e. $g = e$, since there is a coherence isomorphism $\epsilon\_X\colon {}^eX \to X$.
Then we can define:


$$\Omega\_h := \bigoplus\_{X \text{ simple } \in \mathcal{C}\_h} d(X) X$$
$$\Delta\_{A, \Omega\_h} = \operatorname{tr}\_{\Omega\_h}\left((1\_A \otimes \epsilon\_{\Omega\_h}) \circ \gamma\_{A,\Omega\_h} \right) \qquad \forall A \in \mathcal{C}\_e$$


But now source and target aren't equal:
$$\Delta\_{A,\Omega\_h}\colon A \to {}^hA$$


In some cases we have $A \cong {}^hA$, and then one can ask whether the endomorphism is 0 or not, which leads me to my question:


**Is there a similar lemma to the standard killing lemma in the $G$-crossed case?**



Is there a "killing" lemma for G-crossed braided fusion categories?
Edit:
I used to believe that there is a possible generalisation stemming from work of Altschüler and Bruguières. (See Appendix C in [Drinfeld, Gelaki, Nikshych, Ostrik - On braided fusion categories I](http://arxiv.org/abs/0906.0620)). But that was based on a flawed assumption. (I didn't realise that $A$ must be in the trivial degree for it to work.) With the correct assumptions, the following can be said:


Sliding lemma
-------------


Like in braided spherical fusion categories, the sliding lemma is closely related to the killing lemma.


**Lemma**


Let $Y \in \mathcal{C}\_g, A \in \mathcal{C}\_e$. Up to coherences in the $G$-crossed fusion category, we have:


$$\Delta\_{A,\Omega\_h} \otimes 1\_Y = c\_{Y,A} \circ (\epsilon\_Y \otimes \Delta\_{A,\Omega\_{hg^{-1}}}) \circ c\_{A,Y}$$


In words, we can *slide* the $Y$-strand over the $\Omega\_h$-encircling so that it links with the $A$-strand. If you draw a picture, it will become clear.


Killing lemma
-------------


Recall that the trivial degree $\mathcal{C}\_e$ is a braided spherical fusion category, so the classical killing lemma applies to $\Delta\_{A,\Omega\_e}$.


**Lemma**


Let $A \in \mathcal{C}\_e$ be a simple object, and $\mathcal{C}\_h \not\simeq 0$. Then:


$$\Delta\_{A,\Omega\_h}\colon A \to {}^hA$$ is nonzero, and thus an isomorphism, iff $A$ is in the symmetric centre of $\mathcal{C}\_e$.


*Proof*


The statement is equivalent to the statement with $- \otimes 1\_{\Omega\_{h^{-1}}}$ applied to it (recall that $\mathcal{C}\_h \not\simeq 0 \implies \Omega\_h \neq 0 \implies \Omega\_{h^{-1}} \cong \Omega\_h^\* \neq 0)$. By the sliding lemma, the left hand side is then equal to $c\_{\Omega\_{h^{-1}}, A} \circ (\epsilon\_{\Omega\_{h^{-1}}} \otimes \Delta\_{A,\Omega\_e}) \circ c\_{A, \Omega\_{h^{-1}}}$. Apply the killing lemma for $\Delta\_{A,\Omega\_e}$, yielding:


$$\Delta\_{A,\Omega\_h} \otimes 1\_{\Omega\_{h^{-1}}} = \left\{ \begin{matrix} d(\Omega\_e) \cdot \gamma\_{A,\Omega\_{h^{-1}}} && \qquad A \text{ is in the symmetric centre of } \mathcal{C}\_e \\ 0 && \qquad \text{otherwise} \end{matrix}\right.$$


### Observation


This seems to prove that if all degrees of $\mathcal{C}$ are nonzero (that is, the grading is "*faithful*"), all objects in the symmetric centre of $\mathcal{C}\_e$ are automatically *equivariant*, i.e. possess a family of isomorphisms $A \cong {}^gA, \forall g \in G$. Surely, that result must have been known before?


**Edit** This has been commented on briefly by Etingof, Nikshych, Ostrik and Meir in [Fusion categories and homotopy theory](https://arxiv.org/abs/0909.3140), (39).


