The maximal decomposition of the Turaev-Viro TQFT

In a previous work arXiv:0903.4512, we have built an homotopical Turaev-Viro invariant and an HQFT from the universal graduation of a spherical category. In the present paper, we show that every graduation $(G,p)$ of a spherical category $\C$ defines an homotopical Turaev-Viro invariant $HTV_{\C}^{(G,p)}$ and an HQFT $\m{H}_{\C}^{(G,p)}$. Furthermore we show that the Turaev-Viro TQFT will be split into blocks coming the HQFT $\m{H}_{\C}^{(G,p)}$. We show that this decomposition is maximal for the universal graduation of the category, which means that for every graduation $(G,p)$ the HQFT $\m{H}_{\C}^{(G,p)}$ is split into blocks coming from the HQFT obtained from the universal graduation.