nLab bisimplicial set

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)

$$+ : \Delta^\op \times \Delta^{op} \to \Delta^{op} \,.$$

$$+ : [k], [l] \mapsto [k+l+1] \,.$$

This induces an adjoint triple

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

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

• the canonical morphism

  $$diag X \to T X$$

  from the diagonal to the total simplicial set is a weak equivalence in the model structure on simplicial sets;

• the adjunction unit

  $$X \to T Dec X$$

  is a weak equivalence.

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

The simplicial classifying space-functor for simplicial groups

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

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

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

Geometric realization

See geometric realization of simplicial topological spaces.

Model structures

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

Induced from the diagonal

There is an adjunction

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

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)

Related concepts

• π-Kan condition

• decalage

• bicubical set

References

Some standard material is for instance in

• Paul Goerss and Rick Jardine, Simplicial Homotopy Theory

• Rick Jardine, Lecture 008 (2010) (pdf)

• Samuel Isaacson, Excercises in homotopy colimits (pdf)

The total simplicial set functor goes back to

• M. Artin, B. Mazur, On the Van Kampen theorem , Topology 5 (1966) 179–189.

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

• 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)

• Antonio Cegarra, Josué Remedios, The behaviour of the $\bar W$-construction on the homotopy theory of bisimplicial sets, Manuscripta Mathematica, volume 124 (4) Springer (2007)

• Danny Stevenson, Décalage and Kan’s simplicial loop group functor (arXiv:1112.0474)

• Viktoriya Ozornova, Martina Rovelli, The unit of the total décalage adjunction, Journal of Homotopy and Related Structures (2020) 15:333–349, doi.org/10.1007/s40062-020-00257-1

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:

• Donald Werner Anderson, Fibrations and geometric realization, Bull. Amer. Math. Soc. Volume 84, Number 5 (1978), 765-788. (euclid:1183541139)

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

• Antonio Martínez Cegarra, Benjamın A. Heredia, Josué Remedios, Double groupoids and homotopy 2-types, Appl. Categ. Struct. 20, No. 4, 323-378 (2012), see also arXiv:1003.3820.