Contents

# Contents

## Definition

A bisimplicial set is a bisimplicial object in Set.

## Properties

### Diagonal

###### Definition

(diagonal)

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

$d(X)_n \;\coloneqq\; X_{n,n} \,.$
###### Definition

(realization)

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

$|X| = \int^{[n] \in \Delta} X_{n, *} \times \Delta[n]$

in sSet.

###### Proposition

(diagonal is realization)

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

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

This is for instance exercise 1.6 in in chapter 4 Goerss-Jardine. For a derivation see the examples at homotopy colimit.

###### Proposition

(diagonal is homotopy colimit)

The diagonal of a bisimplicial set $X_{\bullet,\bullet}$ is also (up to weak equivalence) the homotopy colimit of $X$ regarded as a simplicial diagram in the model structure on simplicial sets

$diag X \simeq hocolim (X : \Delta^{op} \to sSet_{Quillen}) \,.$

This appears for instance as theorem 3.6 in (Isaacson).

###### Proof

This follows with the above equivalence to the coend $diag X \simeq \int^{[k] \in \Delta} \Delta[k] \cdot X_k$ and general expression of homotopy colimits by coends (as discussed there) in terms of the Quillen bifunctor

$\int^\Delta (-) \cdot (-) : [\Delta, sSet_{Quillen}]_{Reedy} \times [\Delta, sSet_{Quillen}]_{Reedy} \to sSet_{Quillen}$

in Reedy model structures (as discussed there) by using that $\Delta[-] : \Delta \to sSet_{Quillen}$ is a Reedy cofibrant resultion of the point in $[\Delta, sSet_{Quillen}]$ and that every object in $[\Delta^{op}, sSet_{Quillen}]_{Reedy}$ is cofibrant.

###### Proposition

(degreewise weak equivalences)

Let $X,Y : \Delta^{op} \times \Delta^{op} \to Set$ be bisimplicial sets. A morphism $f : X \to Y$ which is degreewise in one argument a weak equivalence $f_{n,\bullet} : X(n,\bullet) \to Y(n,\bullet)$ 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 sets).

###### Proof

This is prop 1.9 in chapter 4 of

• Goerss-Jardine, Simplicial Homotopy Theory (dvi)

### Total décalage and total simplicial sets

There is a functor called ordinal sum (see also at simplex category)

$+ : \Delta^\op \times \Delta^{op} \to \Delta^{op} \,.$
$+ : [k], [l] \mapsto [k+l+1] \,.$

$ssSet \stackrel{\overset{+_!}{\longrightarrow}}{\stackrel{\overset{+^*}{\longleftarrow}}{\underset{+_*}{\longrightarrow}}} sSet \,.$
###### Definition

Here

• $T \coloneqq +_*$ is called the total simplicial set functor or Artin-Mazur codiagonal (we will use the first of these as codiagonal also has another accepted meaning, see codiagonal);

• $Dec \coloneqq +^*$ is called the total décalage functor (inside which is plain décalage);

###### Proposition

$T$ preserves degreewise weak equivalences of simplicial sets.

###### Proposition

For $X$ any bisimplicial set

These statements are for instance in (CegarraRemedios) and (Stevenson). They may be considered as a non-additive versions of the Eilenberg-Zilber theorem.

###### Remark

By prop. and the usual Eilenberg-Zilber theorem it follows that under forming chain complexes for simplicial homology, total simplicial sets correspond to total complexes of double complexes.

###### Remark

After geometric realization these spaces are even related by a homeomorphism.

(This seems to be due to Berger and Hübschmann, but related results were known to Zisman as they are so mentioned by Cordier in his work on homotopy limits.)

###### Remark
$\overline{W} \;\colon\; sGrp \longrightarrow sSet_*$

is the composite

$\bar W \colon sGrp \stackrel{\mathbf{B}}{\longrightarrow} sGrpd \stackrel{N}{\longrightarrow} ssSet \stackrel{T}{\longrightarrow} sSet \,.$

We have the following explicit formula for $T X$, attributed to John Duskin:

###### Lemma

For $X$ a bisimplicial set the total simplicial set $T X$ is in degree $n$ the equalizer

$(T X)_n \to \prod_{i = 0}^n X_{i, n-i} \stackrel{\longrightarrow}{\longrightarrow} \prod_{i = 0}^{n-1} X_{i, n-i-1}$

where the components of the two morphisms on the right are

$\prod_{i = 0}^n X_{i,n-i} \stackrel{p_i}{\to} X_{i, n-i} \stackrel{d_0^v}{\to} X_{i, n-i-1}$

and

$\prod_{i = 0}^n X_{i,n-i} \stackrel{p_{i+1}}{\to} X_{i+1,n-i-1} \stackrel{d_{i+1}^h}{\to} X_{i,n-i-1} \,.$

The face maps $d_i : (T X)_n \to (T X)_{n-1}$ are given by

$d_i = (d_i^v p_0, d_{i-1}^v p_1, \cdots, d_1^v p_{i-1}, d_i^h p_{i+1}, d_i^h p_{i+2}, \cdots, d_i^h p_n )$

and the degeneracy maps are given by

$s_i = (s_i^v p_0, s_{i-1}^v p_1, \cdots, s_0^v p_i, s_i^h p_{i+1}, \cdots, s_i^h p_n) \,.$

The $(Dec \dashv T)$-adjunction unit $\eta : X \to T Dec X$ is given in degree $n$ by

$\eta : x \mapsto (s_0(x), s_1(x), \cdots, s_n(x)) \,.$

## Model structures

There are various useful model category structures on the category of bisimplicial sets.

### Induced from the diagonal

$(L \dashv diag) : ssSet \stackrel{\overset{L}{\longleftarrow}}{\underset{diag}{\longrightarrow}} sSet \,.$

The transferred model structure along this adjunction of the standard model structure on simplicial sets exists and with respect to it the above Quillen adjunction is a Quillen equivalence.

This is due to (Moerdijk 89)

### Induced from codiagonal $T$.

The transferred model structure on $ssSet$ along the total simplicial set functor $T$ exists. And for it

$(Dec \dashv T) : ssSet \stackrel{\overset{Dec}{\longleftarrow}}{\underset{T}{\longrightarrow}} sSet$

is a Quillen equivalence.

###### Proposition

Every diag-fibration is also a $T$-fibration.

This is (CegarraRemedios, theorem 9).

#### Remark on notation

There are two uses of $\bar W$ in this area, one is as used in (CegarraRemedios) where it is used for the codiagonal (sometimes denoted “$\nabla$” or $T$ as above), the other is for the classifying space functor for a simplicial group. This latter is not only the older of the two uses, but also comes with a related $W$ construction. The relationship between the two is that given a simplicial group or simplicially enriched groupoid, $G$, applying the nerve functor in each dimension gives a bisimplicial set and $\bar{W}G = T Ner G$. Because of this, some care is needed when using these sources.

## Bisimplicial abelian groups

###### Proposition

Let $A,B : \Delta^{op} \times \Delta^{op} \to Ab$ be bisimplicial abelian groups. A morphism $f : A \to B$ which is degreewise in one argument a weak equivalence $f_{n,\bullet} : A(n,\bullet) \to B(n,\bullet)$ 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 (GoerssJardine)

Some standard material is for instance in

The total simplicial set functor goes back to

The diagonal, total décalage and total simplicial set constructions are discussed in

The diagonal-induced model structure on $ssSet$ is discussed in

• Ieke Moerdijk, Bisimplicial sets and the group completion theorem in Algebraic K-Theory: Connections with Geometry and Topology, pp 225–240. Kluwer, Dordrecht (1989)

On the behaviour of fibrations under geometric realization of bisimplicial sets:

Discussion of the simplicial classifying space-construction $\overline{W}$ respecting fibrant objects is in Fact 2.8 of: