Finite semisimple 2-category

Let $\mathcal{B}$ be a weak linear $2$-category. A monad on $A$ in $\mathcal{B}$ is a $1$-morphism $T:A\to A$ with $2$-morphisms $\eta :1_A\Rightarrow T$ and $\mu :T\circ T\Rightarrow T$ satisfying $\mu \cdot (\mu \ast 1_T)=\mu \cdot (1_T\ast \mu ),\mu \cdot (\eta \ast 1_T)=1_T=\mu \cdot (1_T\ast \eta ),$ where $\cdot$ is vertical composition, $\ast$ is horizontal composition, and the associator and unitors of $\mathcal{B}$ are inserted. Equivalently it is an algebra object in the monoidal endomorphism category $\mathcal{B}(A,A)$. This extends the ordinary-category definition in Monad on a category. It is separable if there is a $2$-morphism $\sigma :T\Rightarrow T\circ T$ with $\mu \cdot \sigma =1_T,\sigma \cdot \mu =(\mu \ast 1_T)\cdot (1_T\ast \sigma )=(1_T\ast \mu )\cdot (\sigma \ast 1_T),$ again with coherent associators. The last two equalities say that $\sigma$ is a map of regular $T$-bimodules.

An adjunction $L⊣U$ in $\mathcal{B}$, for $L:A\to B$ and $U:B\to A$, consists of $2$-morphisms $\eta :1_A\Rightarrow UL$ and $\epsilon :LU\Rightarrow 1_B$ with $(\epsilon \ast 1_L)\cdot (1_L\ast \eta )=1_L,(1_U\ast \epsilon )\cdot (\eta \ast 1_U)=1_U,$ with associators and unitors inserted. These are the same triangle equations as in Adjunction by unit, counit, and the triangle identities, now for $1$- and $2$-morphisms of $\mathcal{B}$. Its induced monad is $(UL,\eta ,1_U\ast \epsilon \ast 1_L)$.

A finite semisimple linear 2-category is a weak linear 2-category in which:

• every hom category is a finite semisimple $k$-linear category (it has finitely many simple isomorphism classes, finite-dimensional hom spaces, and every object is a finite direct sum of simples);
• there is a zero object, and each pair $A,B$ has a biproduct $A\oplus B$ with inclusions and projections ${\iota }_A,{\iota }_B,{\pi }_A,{\pi }_B$ satisfying ${\pi }_i\circ {\iota }_j\cong 0$ for $i\mathrm{\ne }j$, ${\pi }_i\circ {\iota }_i\cong 1$, and ${\iota }_A\circ {\pi }_A\oplus {\iota }_B\circ {\pi }_B\cong 1_{A\oplus B}$; finite biproducts are preserved by composition;
• every 1-morphism has both a left and a right adjoint (Adjunction by unit, counit, and the triangle identities), with unit and counit 2-morphisms satisfying the two triangle identities;
• every separable monad has a separable splitting: a separable monad $T:A\to A$ is represented by an object $A_T$ and an adjunction $L⊣U$, where $L:A\to A_T$ and $U:A_T\to A$, in the sense of Adjunction by unit, counit, and the triangle identities. Its induced monad $UL$ is isomorphic to $T$ as a monad, and its counit $\epsilon :LU\Rightarrow 1_{A_T}$ has a section $s:1_{A_T}\Rightarrow LU$ as a 2-morphism in $\mathcal{B}(A_T,A_T)$, with $\epsilon \circ s=1_{1_{A_T}}$ (in a bimodule presentation this section is a bimodule map);
• there are finitely many equivalence classes of simple objects, where an object $A$ is simple when its identity 1-morphism is simple in $\mathcal{B}(A,A)$.

The splitting condition concerns monads in the 2-category; it is stronger than the existence of finite direct sums in each hom category. A choice of separable splitting is not part of the property.