nLab diagonal of a bisimplicial set

Definition

Definition

(diagonal of a bisimplicial set)

For $X_{\bullet,\bullet}$ a bisimplicial set, its diagonal is the simplicial set that is the precomposition with the diagonal functor $(Id, Id) \colon \Delta^{op} \to \Delta^{op} \times \Delta^{op}$ on the opposite of the simplex category, i.e. the simplicial set with components:

$$\Delta(X)_n \;\coloneqq\; X_{n,n} \,.$$

Properties

See at Bisimplicial set – Properties – Diagonal.

Related concepts

• diagonal morphism

• total complex, totalization

• total simplicial set, décalage

• Eilenberg-Zilber theorem

References

• Antonio Cegarra, Josué Remedios, The relationship between the diagonal and the bar constructions on a bisimplicial set, Topology and its applications, volume 153 (1) (2005) (pdf, doi:10.1016/j.topol.2004.12.003)

• Danny Stevenson, pages 2 & 11 of: Décalage and Kan’s simplicial loop group functor, Theory and Applications of Categories, Vol. 26, 2012, No. 28, pp 768-787 (arXiv:1112.0474, tac:26-28)