# nLab symmetric comonad

Given a comonad $G=\left(G,\delta ,ϵ\right)$ on a category $A$, a natural transformation $t:GG\to GG$ is a symmetry of $G$ if it sastisfies the quantum Yang-Baxter equation (transposed braid relation)

$G\left(t\right){t}_{G}G\left(t\right)={t}_{G}G\left(t\right){t}_{G}$G(t) t_G G(t) = t_G G(t) t_G

and also

(1)${t}^{2}={1}_{GG},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}t\delta =\delta ,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{ϵ}_{G}t=G\left(ϵ\right),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\delta }_{G}t=G\left(t\right){t}_{G}G\left(\delta \right)$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 $\left(G,t\right)$ 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.