hom approach of tensor of type p q

[jjaassoonn]

1. a tensor of type $(p, q)$ is a $k$ linear map $V^{\otimes^q} \to V^{\otimes^p}$.
2. a $k$-object of type $(p,q)$ is a $k$ vector space $V$ with a tensor $\Phi$ of type $(p, q)$.
3. a $k$-object morphism between $(V, \Phi) \to (W, \Psi)$ is a $k$ linear map $f$ such that

V^q -f^q-> W^q
  |\Phi             |\Psi
  v                   v
V^p -f^p-> W^p

5. if $K/k$ is field extension, there is functor from $k$-object to $K$-object sending $(V, \Phi)$ to $(V_K, \Phi_K)$.
6. if $\sigma \in \mathrm{Gal}(K/k)$ and $f$ is a $K$-morphism between $(V_K, \Phi_K)$ and $(W_K, \Psi_K)$, $\sigma$ induces another $K$-morphism by conjugation

[jjaassoonn]

This is the action of $\mathrm{Gal}(K/k)$ on $K$-morphism between $V_K$ and $W_K$

[Whysoserioushah]

牛啊wk