nLab
symmetric comonad

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

G(t)t GG(t)=t GG(t)t G G(t) t_G G(t) = t_G G(t) t_G

and also

(1)t 2=1 GG,tδ=δ,ϵ Gt=G(ϵ),δ Gt=G(t)t GG(δ) 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)(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.

Revised on December 18, 2009 03:15:15 by Toby Bartels (173.60.119.197)