A folk model structure on omega-cat

The finiteness property extends in fact to all dimensions [14], but the above theorem may also be refined in another direction: the same hypothesis implies that M has finite derivation type [25], a property of homotopical nature. We claim that these ideas are better expressed in terms of ω-categories (see [10,11,17]). Thus we work in the category ωCat, whose objects are the strict ω-categories and the morphisms are ω-functors (see Section 3). In fact, when considering the interplay between the monoid itself and the space of computations attached to any presentation of it, one readily observes that both objects support a structure of ω-category in a very direct way: this was the starting point of [19], which introduces a notion of resolution for ω-categories, based on computads [26,21] or polygraphs [6], the terminology we adopt here. Recall that a polygraph S consists of sets of cells of all dimensions, determining a freely generated ω-category S * . A resolution of an ω-category C by a polygraph S is then an ω-functor p : S * → C satisfying a certain lifting property (see Section 5 below); [19] also defines a homotopy relation between ω-functors and shows that any two resolutions of the same ω-category are homotopically equivalent in this sense. This immediately suggests looking for a homotopy theory on ωCat in which the above resolutions become trivial fibrations: the model structure we describe here does exactly that. Notice, in addition, that polygraphs turn out to be the cofibrant objects (see [20] and Section 5 below). On the other hand, our model structure generalizes in a very precise sense the “folk” model structure on Cat (see [13]) as well a model structure on 2Cat in a similar spirit (see [15,16]). Incidentally, there is also a quite different, Thomason-like, model structure on 2Cat (see [27]). Its generalisation to ωCat remains an open problem. Since [22], the notion of model structure has been gradually recognized as the appropriate abstract framework for developing homotopy theory in a category C: it consists in three classes of morphisms, weak equivalences, fibrations, and cofibrations, subject to axioms whose exact formulation has somewhat evolved in time. In practice, most model structures are cofibrantly generated. This means that there are sets I of generating cofibrations and J of generating trivial cofibrations which determine all the cofibrations and all the fibrations by lifting properties. Recall that, given a set I of morphisms, I-injectives are the morphisms which have the right lifting property with respect to I. They build a class denoted by I-inj. Likewise, I-cofibrations are the morphisms having the left lifting property with respect to I-inj (see Section 2.2). The class of I-cofibrations is denoted by I-cof. Now, our

Section 2 reviews combinatorial model categories, with special emphasis on our version of Smith’s theorem (Section 2.4), while Section 3 recalls the basic definitions of globular sets and ω-categories, and sets the notations. Section 4 is the core of the paper, that is the derivation of our model structure by means of a set I of generating cofibrations and a class W of weak equivalences, satisfying conditions (S1) to (S4) .

We first define the set I of generating cofibrations, and establish closure properties we shall use later in the proof of condition (S3) . We then define the class W of ω-weak equivalences, which are at this stage our candidates for the rôle of weak equivalences (Section 4.3). For this purpose, we first need a notion of ω-equivalence between parallel cells (Section 4.2), together with crucial properties of this notion. We then prove condition (S2) , and part of (S1) (Section 4.3), as well as additional closure properties contributing to (S3) . At this stage, just one point of (S1) remains unproved, namely the assertion if f : X → Y and g • f : X → Z belong to W, then so does g : Y → Z.

This requires an entirely new construction: we define an endofunctor Γ of ωCat, which to each ω-category X associates an ω-category Γ(X) of reversible cylinders in X. Section 4.4 summarizes the main features of Γ, whereas the more technical proofs are given in Appendix A. This eventually leads to an alternative characterization of weak equivalences and to a complete proof of (S1) . As for condition (S3) , the difficult point is to prove the closure of I-cof ∩ W by pushout, which does not follow from the previously established properties. The main obstacle is that W itself is definitely not closed by pushout. What we need instead is a new class Z of immersions such that: i. Z is closed by pushout;

ii. I-cof ∩ W ⊆ Z ⊆ W, which completes the proof of (S3) . Immersions are defined in Section 4.6, by using again the functor Γ in an essential way.

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