Physlib.Relativity.Tensors.UnitTensor

The unit tensor associated with a color $c \in C$. It defines a tensor of shape $(\tau(c), c)$ in a tensor species $S$, where $\tau$ denotes the duality map on the set of colors $C$.

Let $S$ be a tensor species and $C$ be the set of colors. For any colors $c, c_1 \in C$, if $c = c_1$, then the unit tensor associated with $c$ is equal to the unit tensor associated with $c_1$ under the identity permutation of its indices. Formally, if $c = c_1$, then $\text{unitTensor}(c) = \text{permT}(\text{id}, \text{unitTensor}(c_1))$, where $\text{permT}$ denotes the permutation of tensor indices and $\text{id}$ is the identity permutation.

For any $c \in C$, the application of the unit of $S$ at $c$ is equal to the application of the unit at $S.\tau(c)$ composed with the braiding and the isomorphism induced by $S.\tau(S.\tau(c)) = c$. Expressed as a formula: $$S.\text{unit}_c = S.\text{unit}_{S.\tau(c)} \gg \beta_{S.FD(S.\tau^2(c)), S.FD(S.\tau(c))} \gg (S.FD(S.\tau(c)) \triangleleft S.FD(S.\tau^2(c) \to c))$$ where $\gg$ denotes diagrammatic categorical composition, $\beta$ is the braiding, $\triangleleft$ is the left whiskering, $S.FD$ is the functor mapping colors to objects, and $S.\tau$ is the duality map on $C$.

For any color $c \in C$ in a tensor species $S$, the unit tensor associated with $c$ is equal to the permutation of indices $(1, 0)$ applied to the unit tensor associated with the dual color $\tau(c)$. Mathematically: $$S.\text{unitTensor}(c) = \text{permT}_{[1, 0]}(S.\text{unitTensor}(\tau(c)))$$ where $S.\text{unitTensor}(c)$ is a tensor of shape $(\tau(c), c)$, $\tau$ is the duality map on the set of colors $C$, and $\text{permT}_{[1, 0]}$ denotes the operation that swaps the first and second indices of the tensor.

Let $S$ be a tensor species with a set of colors $C$ and a duality map $\tau: C \to C$. For any color $c \in C$, the unit tensor associated with the dual color $\tau(c)$ is equal to the unit tensor associated with $c$ after swapping its first and second indices. Mathematically: $$S.\text{unitTensor}(\tau(c)) = \text{permT}_{[1, 0]}(S.\text{unitTensor}(c))$$ where $S.\text{unitTensor}(c)$ is a tensor of shape $(\tau(c), c)$ and $\text{permT}_{[1, 0]}$ denotes the permutation operation that swaps the two indices of the tensor.

For any $c \in C$ and $x \in S.FD(\text{Discrete.mk}(c))$, the contraction of $x$ with the unit morphism of $S$ at $c$ evaluated at $1 \in k$ is equal to `fromSingleT x`. That is, $$\text{fromSingleTContrFromPairT}(x, (S.\text{unit.app}(\text{Discrete.mk}(c))).\text{hom}(1)) = \text{fromSingleT}(x).$$

Let $S$ be a tensor species with a set of colors $C$ and a duality map $\tau: C \to C$. For any color $c \in C$ and any rank-1 tensor $x$ of shape $(c)$, the contraction of the tensor product of $x$ and the unit tensor $\text{unitTensor}(c)$ (which has shape $(\tau(c), c)$) over the index of $x$ and the first index of the unit tensor is equal to $x$. That is, $$\text{contrT}_{1, 0, 1}(x \otimes \text{unitTensor}(c)) = x$$ where the contraction is performed on the $0$-th and $1$-st indices of the product tensor.

Let $S$ be a tensor species with a set of colors $C$ and a duality map $\tau: C \to C$. For any color $c \in C$ and any rank-1 tensor $x$ of shape $(\tau(c))$, the contraction of the tensor product of the unit tensor $\text{unitTensor}(c)$ (which has shape $(\tau(c), c)$) and $x$ over the second index of the unit tensor and the index of $x$ is equal to $x$. That is, $$\text{contrT}_{1, 1, 2}(\text{unitTensor}(c) \otimes x) = x$$ where the contraction is performed on the $1$-st and $2$-nd indices of the product tensor.

Let $S$ be a tensor species defined over a set of colors $C$, and let $G$ be a group acting on the tensors of $S$. For any color $c \in C$ and any group element $g \in G$, the unit tensor $S.\text{unitTensor } c$ (which is a tensor of shape $(\tau(c), c)$) is invariant under the action of $g$, such that $g \bullet S.\text{unitTensor } c = S.\text{unitTensor } c$.