Physlib.Relativity.Tensors.Color.Discrete

Given a functor $F: \text{Discrete}(C) \to \text{Rep}(k, G)$ from a discrete category of colors $C$ to the category of representations of a group $G$ over a field $k$, the functor `pair` (defined as $F \otimes F$) maps each object $c \in C$ to the tensor product representation $F(c) \otimes_k F(c)$.

Given a functor $F$ from the discrete category $\text{Discrete } C$ to the category of representations $\text{Rep}_k(G)$ and a map $\tau: C \to C$, this definition constructs a functor from $\text{Discrete } C$ to $\text{Rep}_k(G)$ that maps each color $c \in C$ to the tensor product of representations $F(c) \otimes F(\tau(c))$. Here, $\text{Rep}_k(G)$ denotes the category of representations of a group $G$ over a field $k$, and $\text{Discrete } C$ is the category whose objects are the elements of $C$ with only identity morphisms.

Let $C$ be a set of colors, $k$ a field, and $G$ a group. Let $F: \text{Discrete } C \to \text{Rep}_k(G)$ be a functor and $\tau: C \to C$ be a map. Let $\text{pair}\tau(F, \tau)$ be the functor that maps each color $c \in C$ to the tensor product representation $F(c) \otimes_k F(\tau(c))$. For any colors $c, c_1 \in C$ such that $c = c_1$, let $h$ be the unique morphism $c \to c_1$ in the discrete category. For any elements $x \in F(c)$ and $y \in F(\tau(c))$, the linear map induced by the functor $\text{pair}\tau(F, \tau)$ from the equality $h$ satisfies: $$((\text{pair}\tau(F, \tau)).\text{map}(h))(x \otimes y) = (F.\text{map}(h))(x) \otimes (F.\text{map}(h'))(y)$$ where $h': \tau(c) \to \tau(c_1)$ is the morphism induced by the equality $c = c_1$.

Given a functor $F: \text{Discrete } C \to \text{Rep}_k(G)$ and a map $\tau: C \to C$, this functor maps each object (color) $c \in C$ to the tensor product of representations $F(\tau(c)) \otimes_k F(c)$.