Physlib.Relativity.Tensors.ComponentIdx.Single

For a given color $c \in C$, let $\text{ComponentIdx}([c])$ be the type of component indices for a tensor with a single index of color $c$. This definition establishes a canonical equivalence between $\text{ComponentIdx}([c])$ and the type of basis indices for that color, $\text{basisIdx}(c)$. Specifically, it maps a single-index component $b$ (which is a function from a singleton set to the basis indices) to its unique component $b_0$, and vice-versa.

For a given color $c \in C$, let $b \in \text{ComponentIdx}([c])$ be a component index for a tensor with a single index of color $c$. The canonical equivalence $\text{ComponentIdx}([c]) \simeq \text{basisIdx}(c)$ maps the multi-index $b$ to the basis index $b_0$ at its only index position. That is, $\text{ComponentIdx.single}(b) = b_0$.

For any color $c \in C$ and any basis index $b \in \text{basisIdx}(c)$, let $\text{single}^{-1} : \text{basisIdx}(c) \to \text{ComponentIdx}([c])$ be the inverse of the canonical equivalence between the basis indices of color $c$ and the component indices of a rank-1 tensor with index structure $[c]$. For the unique index $i$ in the domain $\{0\}$, the $i$-th component of the multi-index $\text{single}^{-1}(b)$ is equal to the original basis index $b$.