A Quillen theorem Bn for homotopy pullbacks

This was generalized in [DKS,§6] where it was shown that increasingly weaker properties B n (n > 1) allowed for increasingly less simple descriptions of these homotopy fibres. Moreover it was noted that a sufficient condition for a functor f : X → Y to have property B n (n > 1) was that the category Y has a certain property C n .

1.2. The current paper. We show that for a zigzag f : X → Y ← Z : g in which f has property B n (1.1) (and in particular if Y has property C n (1.1)), its homotopy pullback admits a description rather similar to the ones mentioned in 1.1.

Moreover the pullback X × Y Z of this zigzag comes with a monomorphism into this homotopy pullback and hence is itself a homotopy pullback if the monomorphism is a weak equivalence.

1.3. The motivation. Our result (1.2) and in particular its second half is what really motivated us to write the present note, and well for the following reasons.

In [R, 8.3] Charles Rezk proved that

• for every simplicial model category one (and hence every) Reedy fibrant replacement of its simplicial nerve is a complete Segal space.

Although the proof of the Segal part of this result relied heavily on the simplicial structure, it seemed that this result would also hold without the assumption of a simplicial structure.

In fact as we will show in [BK], most of the model structure is superfluous. All that is needed is that there is a category of weak equivalences with three rather simple properties. More precisely we will show that • Charles Rezk’s result holds for every relative category which has the two out of six property and admits a 3-arrow calculus.

Date: October 29, 2018.

It turns out that in that situation the category of the weak equivalences has property C 3 with the result that, in view of our result (1.2) for n = 3, the verification of the Segal property, i.e. showing that certain fibre products (which are iterated pullbacks) are homotopy fibre products (which are iterated homotopy pullbacks), is reduced to a rather simple calculation.

1.4. The proof. The homotopy fibre results of (1.1) were obtained by an induction on n which at each stage used Quillen’s Theorem B.

To prove our homotopy pullback results (1.2) it turns out to be convenient to go one step further back to the lemma that Quillen used to prove his Theorem B and which can be summarized as follows:

• If F : D → Cat is a D-diagram of categories and weak equivalences between them, Gr F its Grothendieck construction and π : Gr F → D the associated projection functor, then, for every object D ∈ D, the fibre

of π over D is also a homotopy fibre. Using this result we first give a different non-inductive proof of the results of (1.1) and then note that this proof almost effortless extends to a proof of the homotopy pullback results of (1.2).

1.5. Organization of the paper. There are three more sections.

In the first ( §2) we discuss various Grothendieck constructions and give a precise formulation of what we will call Quillen’s lemma.

In the next section ( §3) we recall the properties B n and C n and state the Theorems B n for homotopy fibres and for homotopy pullbacks.

The last section ( §4) then is devoted to a proof of these two Theorems B n .

In preparation for the formulation and the proofs of our results we here • briefly discuss Grothendieck constructions,

• formulate, in terms of Grothendieck constructions, a categorical version of the lemma that Quillen used in his proof of Theorem B, and • describe three Grothendieck constructions which will be used in our proofs. But first a comment on 2.0. Terminology. We will work in the category Cat of small categories with the Thomason model structure [T2] in which a map is a weak equivalence iff its nerve is a weak equivalence of simplicial sets and in which homotopy fibres and homotopy pullbacks have a similar meaning. (i) The Grothendieck construction is a homotopy colimit construction on the category Cat, and hence (ii) it is homotopy invariant in the sense that every natural weak equivalence

Next we note that Quillen’s key observation in the lemma that he used to prove Theorem B was that certain functors D → Cat had what we will call 2.3. Property Q. Given a small category D, a functor F : D → Cat will be said to have property Q if it sends all maps of D to weak equivalences in Cat.

A categorical version of the lemma that Quillen used in the proof of Theorem B (a proof of which can be found in [GJ,IV,5.7]) then becomes in view of 2.2(i) above 2.4. Quillen’s lemma. If, given a small category D, a functor F : D → Cat has property Q (2.3), then, for every object D ∈ D, the fibre

It remains to construct the promised three Grothendieck constructions. We start with 2.5. Two Grothendieck constructions associated with a functor X → Y . Given a