functorial action of ${\sc\mbox{C}\hspace{1.0pt}}$. Given a finitary $2$-representation $\mathbf{M}$ of ${\sc\mbox{C}\hspace{1.0pt}}$, the [*rank*]{} of $\mathbf{M}$ is the number of isomorphism classes of indecomposable objects in $$\coprod_{\mathtt{i}\in{\scc\mbox{C}\hspace{1.0pt}}}\mathbf{M}(\mathtt{i}).$$

For simplicity, we will often use the “module” notation $\mathrm{F}\, X$ instead of the corresponding “representation” notation $\mathbf{M}(\mathrm{F})\, X$.

Combinatorics {#s2.4}
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Let ${\sc\mbox{C}\hspace{1.0pt}}$ be a finitary $2$-category and $\mathcal{S}[{\sc\mbox{C}\hspace{1.0pt}}]$ the corresponding [*multisemigroup*]{} in the sense of [@MM2 Section 3]. The objects in $\mathcal{S}[{\sc\mbox{C}\hspace{1.0pt}}]$ are isomorphism classes of indecomposable $1$-morphisms in ${\sc\mbox{C}\hspace{1.0pt}}$. For $\mathrm{F},\mathrm{G}\in \mathcal{S}[{\sc\mbox{C}\hspace{1.0pt}}]$, we set $\mathrm{F}\leq_L\mathrm{G}$ provided that $\mathrm{G}$ is isomorphic to a summand of $\mathrm{H}\circ \mathrm{F}$, for some $1$-morphism $\mathrm{H}$. The relation $\leq_L$ is called the [*left*]{} (pre)-order on $\mathcal{S}[{\sc\mbox{C}\hspace{1.0pt}}]$. The [*right*]{} and [*two-sided*]{} (pre)-orders are defined similarly using multiplication on the right or on both sides. Equivalence classes with respect to these orders are called [*cells*]{} and the corresponding equivalence relations are denoted $\sim_L$, $\sim_R$ and $\sim_J$, respectively. As usual, for simplicity, we say “cells of ${\sc\mbox{C}\hspace{1.0pt}}$” instead of “cells of $\mathcal{S}[{\sc\mbox{C}\hspace{1.0pt}}]$”.

Given a two-sided cell $\mathcal{J}$ in ${\sc\mbox{C}\hspace{1.0pt}}$, we call ${\sc\mbox{C}\hspace{1.0pt}}$ [*$\mathcal{J}$-simple*]{} provided that every non-zero two-sided $2$-ideal in ${\sc\mbox{C}\hspace{1.0pt}}$ contains the identity $2$-morphisms for some (and hence for all) $1$-morphisms in $\mathcal{J}$, see [@MM2 Subsection 6.2] for details.

