Paths and cylinders
A bisimplicial set is a bisimplicial object in Set.
For a bisimplicial set, its diagonal is the simplicial set that this the precomposition with , i.e. the simplicial set with components.
The realization of a bisimplicial set is the simplicial set that is given by the coend
(diagonal is realization)
For a bisimplicial set, its diagonal is (isomorphic to) its realization :
This is for instance exercise 1.6 in in chapter 4 Goerss-Jardine. For a derivation see the examples at homotopy colimit.
(diagonal is homotopy colimit)
The diagonal of a bisimplicial set is also (up to weak equivalence) the homotopy colimit of regarded as a simpliciall diagram in the model structure on simplicial sets
This appears for instance as theorem 3.6 in (Isaacson).
This follows with the above equivalence to the coend and general expression of homotopy colimits by coends (as discussed there) in terms of the Quillen bifunctor
in Reedy model structures (as discussed there) by using that is a Reedy cofibrant resultion of the point in and that every object in is cofibrant.
(degreewise weak equivalences)
Let be bisimplicial sets. A morphism which is degreewise in one argument a weak equivalence induces a weak equivalence of the associated diagonal simplicial sets (with respect to the standard model structure on simplicial sets).
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)
This induces an adjoint triple
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);
is called the total décalage functor;
For 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.
(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.)
We have the following explicit formula for , attributed to John Duskin:
For a bisimplicial set the total simplicial set is in degree the equalizer
where the components of the two morphisms on the right are
The face maps are given by
and the degeneracy maps are given by
The -adjunction unit is given in degree by
See geometric realization of simplicial topological spaces.
There are various useful model category structures on the category of bisimplicial sets.
Induced from the diagonal
There is an adjunction
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 .
The transferred model structure on along the total simplicial set functor exists. And for it
is a Quillen equivalence.
Every diag-fibration is also a -fibration.
This is (CegarraRemedios, theorem 9).
There are two uses of in this area, one is as used in (CegarraRemedios) where it is used for the codiagonal (denoted ”” 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 construction. The relationship between the two is that given a simplicial group or simplicially enriched groupoid, , applying the nerve functor in each dimension gives a bisimplicial set and . Because of this, some care is needed when using these sources.
Bisimplicial abelian groups
Let be bisimplicial abelian groups. A morphism which is degreewise in one argument a weak equivalence induces a weak equivalence of the associated diagonal complexes.
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 -construction on the homotopy theory of bisimplicial sets, Manuscripta Mathematica, volume 124 (4) Springer (2007)
The diagonal-induced model structure on 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 for fibrant objects is discussed in fact 2.8 of