nLab
diagonal of a bisimplicial set

Definition

(diagonal)

For $X_{•,•}$ a bisimplicial set, its diagonal is the simplicial set that is the precomposition with $(\mathrm{Id},\mathrm{Id}):{\Delta }^{\mathrm{op}}\to {\Delta }^{\mathrm{op}}\times {\Delta }^{\mathrm{op}}$, i.e. the simplicial set with components.

$$d(X{)}_n=X_{n,n}.$$d(X)_n = X_{n,n} \,.

See also diagonal.