nLab separable monad

Idea

A separable monad is a monad that categorifies idempotents by replacing the equation $p^2 = p$ with a retract of $p^2$ onto $p$.

This is to distinguish from idempotent monads, for which the equation $p^2 = p$ is replaced with an invertible 2-cell.

Definition

Note that for a monad $(t:a\to a,\mu:t\circ t\to t,\eta:1_a\to t)$, associativity and unitality turn $(t,\mu)$ into a $t,t$-bimodule.

Definition

A separable monad is a monad $(t,\mu,\eta)$ in a bicategory such that the multiplication $\mu:t\circ t\to t$ admits a $t,t$-bilinear section.

Examples

• (Separable adjunctions) An adjunction in a 2-category is separable if the counit $\varepsilon:l\circ r \to 1_a$ admits a section, i.e. a 2-cell $\sigma:1_a\to l\circ r$ such that $\varepsilon\circ\sigma = 1_{1_a}$. The monad? $r\circ l$ is canonically separable.

• (Split separable monads) Separable monads that arise from a separable adjunction are called split. The name derives from similarities between the behaviour of split separable monads and split idempotents. A 2-category is Karoubi complete? if it is locally idempotent complete and all separable monads split. This is one of the conditions for a linear 2-category to be fusion.

References

• Christopher L. Douglas, David J. Reutter, Fusion 2-categories and a state-sum invariant for 4-manifolds [arXiv:1812.11933]

• Theo Johnson-Freyd, David J. Reutter, Minimal nondegenerate extensions [arXiv:2105.15167]

• Xiao-Wu Chen?, A note on separable functors and monads [arXiv:1403.1332]