Suppose $V$ is a $\mathbb{Z}$-graded vector space and $T(V):=\oplus_{j\in \mathbb{Z}}\oplus_{p+q=j}V_p\otimes V_q$ the graded tensor power (as a vector space).
1) Is then $T(T(V))\simeq T(V)$?
2) What about the various quotients?
Suppose $\odot V$ is the 'graded symmetric' quotient. Is then $\odot(\odot V)\simeq \odot V$, too? And similarly for the graded anti-symmetric quotient $\wedge V$.