nLab symmetric comonad

Given a comonad $\mathbf{G}=(G,\delta,\epsilon)$ on a category $A$, a natural transformation $t : G G\rightarrow G G$ is a symmetry of $\mathbf{G}$ if it sastisfies the quantum Yang-Baxter equation (transposed braid relation)

$$G(t) t_G G(t) = t_G G(t) t_G$$

and also

(1)$$t^2 = 1_{G G},\,\,\,\,t\delta = \delta,\,\, \,\,\,\epsilon_G t = G(\epsilon),\,\,\, \delta_G t = G(t) t_G G(\delta)$$

In that case, the pair $(G,t)$ is called a symmetric comonad.

The simplicial set obtained by the standard construction from a symmetric comonad has always a structure of a symmetric simplicial set.

• M. Grandis, Finite sets and symmetric simplicial sets, Theory of Applications of Categories, Vol. 8, No. 8, pp. 244–253, link (to come?).

• Z. Škoda, Cyclic structures for simplicial objects from comonads, math.CT/0412001.