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bisimplicial set

Contents

Definition

A bisimplicial set is a bisimplicial object in Set.

Properties

Diagonal

Definition

(diagonal)

For X ,X_{\bullet,\bullet} a bisimplicial set, its diagonal is the simplicial set that this the precomposition with (Id,Id):Δ opΔ op×Δ op(Id, Id) : \Delta^{op} \to \Delta^{op} \times \Delta^{op}, i.e. the simplicial set with components.

d(X) n=X n,n. d(X)_n = X_{n,n} \,.
Definition

(realization)

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

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

in sSet.

Proposition

(diagonal is realization)

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

Xd(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 ,X_{\bullet,\bullet} is also (up to weak equivalence) the homotopy colimit of XX regarded as a simpliciall diagram in the model structure on simplicial sets

diagXhocolim(X:Δ opsSet Quillen). 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 diagX [k]ΔΔ[k]X kdiag 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

Δ()():[Δ,sSet Quillen] Reedy×[Δ,sSet Quillen] ReedysSet Quillen \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 Δ[]:ΔsSet Quillen\Delta[-] : \Delta \to sSet_{Quillen} is a Reedy cofibrant resultion of the point in [Δ,sSet Quillen][\Delta, sSet_{Quillen}] and that every object in [Δ op,sSet Quillen] Reedy[\Delta^{op}, sSet_{Quillen}]_{Reedy} is cofibrant.

Proposition

(degreewise weak equivalences)

Let X,Y:Δ op×Δ opSetX,Y : \Delta^{op} \times \Delta^{op} \to Set be bisimplicial sets. A morphism f:XYf : X \to Y which is degreewise in one argument a weak equivalence f n,:X(n,)Y(n,)f_{n,\bullet} : X(n,\bullet) \to Y(n,\bullet) induces a weak equivalence d(f):d(X)d(Y)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 (discussed at simplicial category)

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

This induces an adjoint triple

ssSet+ *+ *+ !sSet. ssSet \stackrel{\overset{+_!}{\to}}{\stackrel{\overset{+^*}{\leftarrow}}{\underset{+_*}{\to}}} sSet \,.
Definition

Here

  • T:=+ *T := +_* 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:=+ *Dec := +^* is called the total décalage functor;

Proposition

TT preserves degreewise weak equivalences of simplicial sets.

Proposition

For XX any bisimplicial set

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

Remark

By prop. 5 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 standard delooping functor for simplicial groups

W¯:sGrpsSet * \bar W : sGrp \to sSet_*

is the composite

W¯:sGrpBsGrpdNssSetTsSet. \bar W : sGrp \stackrel{\mathbf{B}}{\to} sGrpd \stackrel{N}{\to} ssSet \stackrel{T}{\to} sSet \,.

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

Lemma

For XX a bisimplicial set the total simplicial set TXT X is in degree nn the equalizer

(TX) n i=0 nX i,ni i=0 n1X i,ni1 (T X)_n \to \prod_{i = 0}^n X_{i, n-i} \stackrel{\to}{\to} \prod_{i = 0}^{n-1} X_{i, n-i-1}

where the components of the two morphisms on the right are

i=0 nX i,nip iX i,nid 0 vX i,ni1 \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

i=0 nX i,nip i+1X i+1,ni1d i+1 hX i,ni1. \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:(TX) n(TX) n1d_i : (T X)_n \to (T X)_{n-1} are given by

d i=(d i vp 0,d i1 vp 1,,d 1 vp i1,d i hp i+1,d i hp i+2,,d i hp n) 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 vp 0,s i1 vp 1,,s 0 vp i,s i hp i+1,,s i hp n). 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 (DecT)(Dec \dashv T)-adjunction unit η:XTDecX\eta : X \to T Dec X is given in degree nn by

η:x(s 0(x),s 1(x),,s n(x)). \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

(Ldiag):ssSetdiagLsSet. (L \dashv diag) : ssSet \stackrel{\overset{L}{\leftarrow}}{\underset{diag}{\to}} 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 \nabla.

The transferred model structure on ssSetssSet along the total simplicial set functor TT exists. And for it

(DecT):ssSetTDecsSet (Dec \dashv T) : ssSet \stackrel{\overset{Dec}{\leftarrow}}{\underset{T}{\to}} sSet

is a Quillen equivalence.

Proposition

Every diag-fibration is also a TT-fibration.

This is (CegarraRemedios, theorem 9).

Remark on notation

There are two uses of W¯\bar W in this area, one is as used in (CegarraRemedios) where it is used for the codiagonal (denoted ”\nabla” 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 WW construction. The relationship between the two is that given a simplicial group or simplicially enriched groupoid, GG, applying the nerve functor in each dimension gives a bisimplicial set and W¯G=NerG\bar{W}G = \nabla Ner G. Because of this, some care is needed when using these sources.

Bisimplicial abelian groups

Proposition

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

Proof

This is Lemma 2.7 in chapter 4 of (GoerssJardine)

References

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

  • 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)

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

The diagonal-induced model structure on ssSetssSet 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)

The behaviour of fibrations under geometric realization of bisimplicial sets is discussed in

  • D. Anderson, Fibrations and geometric realization , Bull. Amer. Math. Soc. Volume 84, Number 5 (1978), 765-788. (ProjEuclid)

Discussion of respect of W¯\bar W for fibrant objects is discussed in fact 2.8 of

Revised on November 24, 2013 23:46:29 by Tim Porter (2.26.15.29)