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A bisimplicial set is a bisimplicial object in Set.
(diagonal)
For $X_{\bullet,\bullet}$ a bisimplicial set, its diagonal is the simplicial set that is the precomposition with $(Id, Id) : \Delta^{op} \to \Delta^{op} \times \Delta^{op}$, i.e. the simplicial set with components
(realization)
The realization $|X|$ of a bisimplicial set $X_{\bullet,\bullet}$ is the simplicial set that is given by the coend
in sSet.
(diagonal is realization)
For $X$ a bisimplicial set, its diagonal $d(X)$ is (isomorphic to) its realization $|X|$:
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 $X_{\bullet,\bullet}$ is also (up to weak equivalence) the homotopy colimit of $X$ regarded as a simplicial diagram in the model structure on simplicial sets
This appears for instance as theorem 3.6 in (Isaacson).
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
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.
(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).
There is a functor called ordinal sum (see also at simplex category)
This induces an adjoint triple
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);
$T$ preserves degreewise weak equivalences of simplicial sets.
For $X$ any bisimplicial set
the canonical morphism
from the diagonal to the total simplicial set is a weak equivalence in the model structure on simplicial sets;
the adjunction unit
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.
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.
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.)
The standard delooping functor for simplicial groups
is the composite
We have the following explicit formula for $T X$, attributed to John Duskin:
For $X$ a bisimplicial set the total simplicial set $T X$ is in degree $n$ the equalizer
where the components of the two morphisms on the right are
and
The face maps $d_i : (T X)_n \to (T X)_{n-1}$ are given by
and the degeneracy maps are given by
The $(Dec \dashv T)$-adjunction unit $\eta : X \to T Dec X$ is given in degree $n$ by
See geometric realization of simplicial topological spaces.
There are various useful model category structures on the category of bisimplicial sets.
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)
The transferred model structure on $ssSet$ along the total simplicial set functor $T$ exists. And for it
is a Quillen equivalence.
Every diag-fibration is also a $T$-fibration.
This is (CegarraRemedios, theorem 9).
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.
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.
This is Lemma 2.7 in chapter 4 of (GoerssJardine)
Some standard material is for instance in
Rick Jardine, Lecture 008 (2010) (pdf)
Samuel Isaacson, Excercises in homotopy colimits (pdf)
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 $\bar W$-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
Dordrecht (1989)
The behaviour of fibrations under geometric realization of bisimplicial sets is discussed in
Discussion of respect of $\bar W$ for fibrant objects is discussed in fact 2.8 of
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