On the category of bisimplicial sets there are different Quillen closed model structures associated to various definitions of fibrations. In one of them, which is due to Bousfield and Kan and that consists of seeing a bisimplicial set as a simplicial object in the category of simplicial sets, fibrations are those bisimplicial set maps such that each of the induced simplicial set maps is a Kan fibration, that is, the pointwise fibrations. In another of them, introduced by Moerdijk, a bisimplicial map is a fibration if it induces a Kan fibration of associated diagonal simplicial sets, that is, the diagonal fibrations. In this note, we prove that every diagonal fibration is a pointwise fibration.
There are several (Quillen) closed model structures on the category of bisimplical sets, see [3,IV,§3]. This paper concerns two of them, namely, the so-called Bousfield-Kan and Moerdijk structures, that we briefly recall below:
On the one hand, in the closed model structure by Bousfield-Kan, bisimplicial sets are regarded as diagrams of simplicial sets and then fibrations are the pointwise Kan fibrations and weak equivalences are the pointwise weak homotopy equivalences. To be more precise, a bisimplicial set X : ∆ op × ∆ op → Set, ([p], [q]) → X p,q , is seen as a “horizontal” simplicial object in the category of “vertical” simplicial sets, X : ∆ op → S, [p] → X p, * and then, a bisimplicial map f : X → Y is a fibration (resp. a weak equivalence) if all simplicial maps f p, * : X p, * → Y p, * , p 0, are Kan fibrations (resp. weak homotopy equivalences).
On the other hand, the Moerdijk closed model structure on the bisimplicial set category is transferred from the ordinary model structure on the simplicial set category through the diagonal functor, X → diag X : [n] → X n,n . Thus, in this closed model structure, a bisimplicial map f : X → Y is a fibration (resp. a weak equivalence) if the induced diagonal simplicial map diag f : diag X → diag Y is a Kan fibration (resp. a weak homotopy equivalence).
Several useful relationships between these two different homotopy theories of bisimplicial sets have been established and, perhaps, the best known of them is the following:
Theorem. (Bousfield-Kan) Let f : X → Y be a bisimplicial map such that f p, * : X p, * → Y p, * is a weak homotopy equivalence for each p 0. Then diag f : diag X → diag Y is a weak homotopy equivalence.
The purpose of this brief note is to state and prove a suitable counterpart to Bousfield-Kan’s theorem for fibrations, namely:
Note that the converse of Theorem 1 is not true in general. A counterexample is given in the last section of the paper.
Acknowledgements. The authors are much indebted to the referees, whose useful observations greatly improved our exposition. The second author is grateful to the Algebra Department in the University of Granada for the excellent atmosphere and hospitality.
We use the standard conventions and terminology which can be found in texts on simplicial homotopy theory, e. g. [3] or [6]. For definiteness or emphasis we state the following.
We denote by ∆ the category of finite ordered sets of integers [n] = {0, 1, . . . , n}, n 0, with weakly order-preserving maps between them. The category of simplicial sets is the category of functors X : ∆ op → Set, where Set is the category of sets. If X is a simplicial set and α : [m] → [n] is a map in ∆, then we write X n = X[n] and α * = X(α) : X n → X m . Recall that all maps in ∆ are generated by the injections
i n, which miss out the ith element and the surjections σ i : [n + 1] → [n] (codegeneracies), 0 i n, which repeat the ith element (see [5, VII, §5, Proposition 2]). Thus, in order to define a simplicial set, it suffices to give the sets of n-simplices X n , n 0, together with maps
satisfying the well-known basic simplicial identities such as [5, p. 175]). In addition, we shall write down a list of other identities between some iterated compositions of face and degeneracy maps, which will be used latter. The proof of these equalities is straightforward and left to the reader. Lemma 1. On any simplicial set, the following equalities hold:
Let f : X → Y be a simplicial map. A collection of simplices
where I ⊆ [n] is any subset, is said to be f -compatible whenever the following equalities hold:
The map f is said to be a Kan fibration whenever for every given collection of f -compatible simplices
there is a simplex x ∈ X n such that d i x = x i for all i = k and f x = y. The next lemma (cf. [6, Lemma 7.4]) will be very useful in our development. For I any finite set, |I| denotes its number of elements.
Lemma 2. Let f : X → Y be a Kan fibration. Suppose that there are given a subset I ⊆ [n] such that 1 |I| n and an f -compatible family of simplices
Then, there exists x ∈ X n such that d i x = x i for all i ∈ I and f x = y.
Proof. Suppose |I| = r. If r = n, the statement is true since f is a Kan fibration. Hence the statement holds for n = 1. We now proceed by induction: Assume n > 1 and the result holds for n ′ < n and assume r < n and the result holds for r ′ > r.
∈ I}, we wish to find a simplex x k ∈ X n-1 , such that the collection of simplices
be f -compatible, since then an application of the induction hypothesis on r gives the claim. To find such an x k , let I ′ ⊆ [n -1] be the subset
and let
be the family of simplices defined by
The category of bisimplicial sets is the category of functors X : ∆ op × ∆ op → Set. It is often convenient to see a bisimplicial set X as a (horizontal) simplicial object in the category of (vertical) simplicial sets. If α : [p] → [p ′ ] and β : [q] → [q ′ ] are any two maps in ∆, then we will write α * h : X p
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