Is there a fusion subcategory in sphericalization tensor equivalent to the original one? Let $C$ be a fusion category. Then $C$ is not necessary spherical. But its sphericalization $\tilde{C}$ has a canonical spherical structure $i:Id\to **$. The  simple objects of $\tilde{C}$ are pairs $(V,\alpha)$ where $V$ is a simple object of $C$ and $\alpha: V\simeq V^{**}$ satisfies $\alpha^{∗∗}\alpha =\gamma$, where $\gamma:Id\to ****$ is an  canonical isomorphism of tensor functor.
For each simple object $V$ of $C$, we have two such $\alpha$. Fixing one, we write $(V, \alpha) = V+$ and $(V, −\alpha) = V−$. If we set $d=dim(V+)=Tr_{X+}(i)=Tr_{X}(\alpha)$ then $dim(V-)=-d$. Reference [sectionII.2.3, 1],[2].
My questions are:

*

*Assume $C$ is already spherical.  Does there exist a fusion subcategory of $\tilde{C}$ tensor equivalent to $C$? If it exists, how to build such equivalence?


*Assume the Frobenius-Perron dimension of $C$ is integral. Then the Frobenius-Perron dimension of $\tilde{C}=2FPdim(C)$ is also integral. Hence $\tilde{C}$ is pseudo-unitary by [Proposition 8.24, 2]. Then $\tilde{C}$ admits a unique spherical structure, with respect to which the categorical dimensions of all simple objects are positive, and coincide with their Frobenius- Perron dimensions, see [Proposition 8.23, 2]. I am confused that this may contradicts the construction of $\tilde{C}$ in which $(V, \alpha) = V+$ or  $(V, −\alpha) = V−$ has negative categorical dimension.
Thank you very much!
References:
[1] J.E. Thornton, Generalized near-group categories, PhD thesis, University of Oregon, 2012.
[2]Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik. On fusion categories. Ann. Math.,
162(2):581–642, 2005.
 A: 
Assume C is already spherical. Does there exist a fusion subcategory of C~ tensor equivalent to C? If it exists, how to build such equivalence?

If $k$ is the extant spherical structure on $\mathcal C$, the the embedding should be something like
$$
X\mapsto (X,k_X)\hspace{10pt}\&\hspace{10pt} (f:X\to Y)\mapsto \big(f:(X,k_X)\to(Y,k_Y)\big)
$$
I've never seen it written down anywhere, but I suspect that monoidal embeddings $\mathcal C\hookrightarrow\tilde{\mathcal C}$ should be in bijective correspondence with spherical structures on $\mathcal C$.  For the converse direction, start with an embedding and restrict the canonical spherical structure in $\tilde{\mathcal C}$ to the embedded copy of $\mathcal C$.

Assume the Frobenius-Perron dimension of C is integral. Then the Frobenius-Perron dimension of C~=2FPdim(C) is also integral. Hence C~ is pseudo-unitary by [Proposition 8.24, 2]. Then C~ admits a unique spherical structure, with respect to which the categorical dimensions of all simple objects are positive, and coincide with their Frobenius- Perron dimensions, see [Proposition 8.23, 2]. I am confused that this may contradicts the construction of C~ in which (V,α)=V+ or (V,−α)=V− has negative categorical dimension.

I believe that the canonical spherical structure alluded to for pseudo-unitary categories is not the same as the canonical spherical structure that exists by construction on the sphericalization.  The spherical structure $i$ on $\tilde{\mathcal C}$ is given by
$$
i_{(V,\alpha)}:=\alpha\,,
$$
whereas the canonical spherical structure produced from the theorem on pseudounitary categories picks out $\pm\alpha$ according to which one produces the positive dimension.  In other words,
$$
i_{V-}=-i_{V+}\hspace{10pt}\text{but}\hspace{10pt}j_{V+}=j_{V-}
$$
