Cubical Monad

A Cubical Monad on a category $C$ consists of:

• An endofunctor $\mathbb{I}:C\to C$
• Face natural transformations $\partial^{0},\partial^{1}:\mathrm{Id}\to\mathbb{I}$
• A degeneracy natural transformation $\sigma:\mathbb{I}\to\mathrm{Id}$
• Connections $\land,\lor:\mathbb{I}^{2}\to\mathbb{I}$

Such that the following equations hold:

• $\sigma\circ\partial^{0}=\mathrm{Id}=\sigma\circ\partial^{1}$
• $\sigma\circ\mathbb{I}\sigma=\sigma\circ\lor=\sigma\circ\sigma\mathbb{I}$
• $\sigma\circ\mathbb{I}\sigma=\sigma\circ\land=\sigma\circ\sigma\mathbb{I}$
• $\lor\circ\mathbb{I}\partial^{0}=\mathrm{Id}=\lor\circ\partial^{0}\mathbb{I}$
• $\land\circ\mathbb{I}\partial^{1}=\mathrm{Id}=\land\circ\partial^{1}\mathbb{I}$
• $\partial^{0}\circ\sigma=\land\circ\mathbb{I}\partial^{0}$
• $\partial^{0}\circ\sigma=\land\circ\partial^{0}\mathbb{I}$
• $\partial^{1}\circ\sigma=\lor\circ\mathbb{I}\partial^{1}$
• $\partial^{1}\circ\sigma=\lor\circ\partial^{1}\mathbb{I}$

The resolutions of cubical monads give rise to cubical diagrams (with connections). A cubical semimonad consists of only face and degeneracy natural transformations, and lacks connections.

Examples

• Cubical semimonads often arise from cylinder functors: IE, the functor $-\times I$ induced by a Interval Object; The two face transformations are given by $\langle id,i_{0}\rangle$ and $\langle id,i_{1}\rangle$, and the degeneracy is given by $\pi_{1}$. If $I$ has some sort of further structure that describes connections, then we can get a cubical monad.

References

• Grandis, M. “Cubical Monads and Their Symmetries,” 1993. https://www.semanticscholar.org/paper/Cubical-monads-and-their-symmetries-Grandis/02b28c63868fd9d04bc8f2c73923d0b6640329e0.