A semi-simplicial complex (see Moore's 1958 paper: Semi-simplicial complexes and Postnikov systems, available at http://www-math.mit.edu/~hrm/kansem/moore-semi-simplicial-complexes.pdf, cf MathSciNet) consists of the following
A set $X=\cup_{q\ge0}X_q$, where the $X_q$ are disjoint sets (an element of $X_q$ is called a $q$-simplex of $X$),
functions $\partial_i:X_{q+1}\to X_q$, $i=0,\dots,q+1$, called face operators, and
functions $s_i:X_q\to X_{q+1}$, $i=0,\dots,q$, called degeneracy operators.
The face and degeneracy operators are assumed to satisfy the relations $$\partial_i\partial_j=\partial_{j-1}\partial_i, i<j,$$ $$s_is_j=s_{j+1}s_i, i\le j,$$ $$\partial_is_j=\partial_{j+1}s_j=\text{identity},$$ $$\partial_is_j=s_{j-1}\partial_i, i<j,$$ and $$\partial_is_j=s_j\partial_{i-1}, i>j+1.$$
Let $\underline{C}$ be a small category. Its nerve (see Quillen's 1973 paper: Higher algebraic K-theory I), denoted $N\underline{C}$, is the (semi-) simplicial set whose $p$-simplices are the diagrams in $\underline{C}$ of the form $$X_0\to X_1\to\cdots\to X_p.$$ The $i$-th face (resp. degeneracy) of this simplex is obtained by deleting the object $X_i$ (resp. replacing $X_i$ by $id: X_i\to X_i$) in the evident way.
Question: what is the nerve of the Ising category?
Here we take the Ising category as the fusion category part of Ising modular tensor category (MTC), see 5.3.4 in p.37 of On classification of modular tensor categories. We only demand the same fusion rule as the Ising MTC.