nLab
diagonal of a bisimplicial set

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Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

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Contents

Definition

Definition

(diagonal of a bisimplicial set)

For X ,X_{\bullet,\bullet} a bisimplicial set, its diagonal is the simplicial set that is the precomposition with the diagonal functor (Id,Id):Δ opΔ op×Δ op(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:

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

Properties

See at Bisimplicial set – Properties – Diagonal.

References

Last revised on October 24, 2021 at 07:10:53. See the history of this page for a list of all contributions to it.