Skeletal Ising fusion-data package

Let the scalar field be $\mathbb{C}$ and let the simple labels be $\mathbf{1}$, $\psi$, and $\sigma$. Define a $\mathbb{C}$-linear semisimple category candidate whose objects are finite direct sums of these labels, whose simple-label Hom space is $\mathbb{C}$ when the labels agree and $0$ otherwise, and whose Hom spaces between direct sums are the corresponding finite matrices. Give the simple-label fusion product by

$$\mathbf{1}\otimes a=a\otimes \mathbf{1}=a,\psi \otimes \psi =\mathbf{1},\psi \otimes \sigma =\sigma \otimes \psi =\sigma ,\sigma \otimes \sigma =\mathbf{1}\oplus \psi \mathrm{.}$$

Extend the product bilinearly to direct sums and on morphisms by the induced tensor product of scalar maps. The unit is $\mathbf{1}$ and the unitors are strict. Fusion spaces are $V_{ab}^c=\mathrm{Hom}(a\otimes b,c)$; each is one-dimensional when the displayed fusion rule allows $c$ and is zero otherwise. In every allowed one-dimensional fusion space choose its displayed basis vector. On three simple labels, define the associator in these bases by the matrix $[F_d^{abc}{]}_{xy}$, where $x$ is the intermediate label in $(a\otimes b)\otimes c\to d$ and $y$ is the intermediate label in $a\otimes (b\otimes c)\to d$. Set

$${\left[F_{\sigma }^{\sigma \sigma \sigma }\right]}_{xy}=\frac{1}{\sqrt{2}}{\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)}_{xy},x,y\in \{\mathbf{1},\psi \},$$ $${\left[F_{\sigma }^{\psi \sigma \psi }\right]}_{\sigma \sigma }=-1,{\left[F_{\psi }^{\sigma \psi \sigma }\right]}_{\sigma \sigma }=-1.$$

Every other coefficient is $1$ when all four fusion vertices are allowed and $0$ when any required fusion vertex is forbidden. Extend each associator linearly over direct sums. The candidate dual of each simple label is itself. For $\mathbf{1}$ and $\psi$, take evaluation and coevaluation to be the unit-channel basis maps with coefficient $1$. For $\sigma$, take evaluation $\sigma \otimes \sigma \to \mathbf{1}$ and coevaluation $\mathbf{1}\to \sigma \otimes \sigma$ to have coefficient $2^{1\mathrm{/}4}$ in the $\mathbf{1}$ channel and zero in the $\psi$ channel. This item specifies the data; the separate coherence result checks that they satisfy the fusion-category axioms. No braiding or physical realization is included in this definition.

Simon, Topological Quantum, §§9.2–9.3

Aasen, Mong, and Fendley, §§4.1–4.3