nLab
bisimplicial object

Definition

A bisimplicial object in a category $C$ is a functor

$$F:{\Delta }^{\mathrm{op}}\times {\Delta }^{\mathrm{op}}\to C$$F : \Delta^{op} \times \Delta^{op} \to C

where $\Delta$ is the simplex category.

This is the same as a simplicial object in the category of simplicial objects in $C$.

Special cases

Bisimplicial sets

Definition

A bisimplicial set is a bisimplicial object in Set.

Properties

Proposition

(degreewise weak equivalences)

Let $X,Y:{\Delta }^{\mathrm{op}}\times {\Delta }^{\mathrm{op}}\to \mathrm{Set}$ be bisimplicial sets. A morphism $f:X\to Y$ which is degreewise in one argument a weak equivalence $f_{n,•}:X(n,•)\to Y(n,•)$ induces a weak equivalence $d(f):d(X)\to d(Y)$ of the associated diagonal simplicial sets (with respect to the standard model structure on simplicial setss).

Proof

This is prop 1.9 in chapter 4 of

• Goerss-Jardine, Simplicial Homotopy Theory (dvi)

Definition

(diagonal)

For $X_{bulet,•}$ a bisimplicial set, its diagonal is the simplicial set that this 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} \,.

Definition

(realization)

The realization $\mid X\mid$ of a bisimplicial set $X_{•,•}$ is the simplicial set that is given by the coend

$$\mid X\mid ={\int }^{[n]\in \Delta }{X}_{n,•}\times \Delta [n{]}_{•}$$|X| = \int^{[n] \in \Delta} X_{n,\bullet} \times \Delta[n]_\bullet

in sSet.

Definition

(diagonal is realization)

For $X$ a bisimplicial set, its diagonal $d(X)$ is (isomorphic to) its realization $\mid X\mid$:

$$\mid X\mid \simeq d(X).$$|X| \simeq d(X) \,.

Proof

This is exercise 1.6 in in chapter 4 of

• Goerss-Jardine, Simplicial Homotopy Theory (dvi)

Bisimplicial abelian groups

Proposition

Let $A,B:{\Delta }^{\mathrm{op}}\times {\Delta }^{\mathrm{op}}\to \mathrm{Ab}$ be bisimplicial abelian groups. A morphism $f:A\to B$ which is degreewise in one argument a weak equivalence $f_{n,•}:A(n,•)\to B(n,•)$ induces a weak equivalence $d(f):d(A)\to d(B)$ of the associated diagonal complexes.

Proof

This is Lemma 2.7 in chapter 4 of

• Goerss-Jardine, Simplicial Homotopy Theory (dvi)