n-RELATIVE CATEGORIES: A MODEL FOR THE HOMOTOPY THEORY OF n-FOLD HOMOTOPY THEORIES

We introduce, for every integer n ≥ 1, the notion of an n-relative category and show that the category of the small n-relative categories is a model for the homotopy theory of n-fold homotopy theories, i.e. homotopy theories of . . . of homotopy theories. 1. Background and motivation In this introduction we • recall some results of (higher) homotopy theory, and • explain how they led to the current manuscript. We start with 1.1. Rezk and re-Rezk. In [R] Charles Rezk constructed a left Bousfield localization of the Reedy structure on the category sS of small simplicial spaces (i.e. bisimplicial sets) and showed it to be a model for the homotopy theory of homotopy theories. Furthermore it was noted in [B] (and a proof thereof can be found in [Lu, §1]) that iteration of Rezk’s construction yields, for every integer n > 1, a left Bousfield localization of the Reedy structure on the category sS of small n-simplicial spaces (i.e. (n + 1)-simplicial sets) which is a model for the homotopy theory of n-fold homotopy theories, i.e. homotopy theories of . . . of homotopy theories. We will call the weak equivalences in these left Bousfield localization (which are often referred to as complete Segal equivalences) just Rezk equivalences. Rezk’s original result also gave rise to the following result on 1.2. Relative categories. Recall that a relative category is a pair (C, wC) consisting of a category C and a subcategory wC ⊂ C which contains all the objects of C and of which the maps are called weak equivalences. Then it was shown in [BK] that Rezk’s model structure on sS (1.1) can be lifted to a Quillen equivalent Rezk structure on the category RelCat of the small relative categories, the weak equivalences of which will also (1.1) be called Rezk equivalences. The categoryRelCat is connected to sS by a simplicial nerve functor N : RelCat → sS with the property that a map f ∈ RelCat is a Rezk equivalence iff the map Nf ∈ sS is so. Moreover if we denote by Rk the subcategories of the Rezk equivalences in both RelCat and sS, then the simplicial nerve functor has the property that Date: December 14, 2010. 1 2 C. BARWICK AND D. M. KAN (i) the relative functor N : (RelCat,Rk) −→ (sS,Rk) is a homotopy equivalence of relative categories, in the sense that there exists a relative functor M : (sS,Rk) −→ (RelCat,Rk) called a homotopy inverse of N such that the compositions MN and NM can be connected to the identity functors of RelCat and sS by finite zigzags of natural weak equivalences. This in turn implies that (ii) the relative category (RelCat,Rk) is, just like (sS,Rk), a model for the homotopy theory of homotopy theories. The proof of all this is essentially a relative version of the proof of the following classical result of Bob Thomason. 1.3. Thomason’s result. In [T] Bob Thomason lifted the usual model structure on the category S of small spaces (i.e. simplicial sets) to a Quillen equivalent one on the category Cat of small categories and noted that these two categories were connected by the nerve functor N : Cat → S which has the property that a map f ∈ Cat is a weak equivalence iff Nf ∈ S is so. It follows that, if W denotes the categories of weak equivalences in both Cat and S, then (i) the relative functor N : (Cat,W ) → (S,W ) is a homotopy equivalence of relative categories (1.2(i)) which in turn implies that (ii) the relative category (Cat,W ) is, just like (S,W ) a model for the theory of homotopy types. His proof was however far from simple as it involved notions like two-fold subdivision and so-called Dwyer maps. We end with recalling 1.4. A result of Dana Latch. In [La] Dana Latch noted that, if one just wanted to prove 1.3(i) and 1.3(ii), one could do this by an argument that was much simpler than Thomason’s and that, instead of the cumbersome two-fold subdivisions and Dwyer maps, involved the rather natural notion of the category of simplices of a simplicial set. Now we can finally discuss 1.5. The current paper. The results mentioned in 1.1 and 1.2 above suggest that, for every integer n > 1, there might exist some generalization of the notion of a relative category such that the category of such generalized relative categories admits a model structure which is Quillen equivalent to the Rezk structure on sS. As however we did not see how to attack this question we turned to a much simpler one suggested by the result of Dana Latch that was mentioned in 1.4 above, namely to prove 1.2(i) directly by showing that n-RELATIVE CATEGORIES 3 • the simplicial nerve functor N : (RelCat,Rk) −→ (sS,Rk) has an appropriately defined relative category of bisimplices functor ∆rel : (sS,Rk) −→ (RelCat,Rk) as a homotopy inverse. It turned out that not only could we do this, but the relative simplicity of our proof suggested that a similar proof might work for appropriately generalized relative categories. And indeed, after the necessary trial and error and frustration, we discovered a notion of what we will call n-relative categories which fitted the bill. Hence the current manuscript. 2. An overview 2.1. Summary. There are five more sections. • In the first (§3) we introduce n-relative categories. • In the second (§4) we investigate an adjunction K : sS ←→ RelCat :N between the category sS of small n-simplicial spaces and the category RelCat of small n-relative categories, in which the right adjoint N is the n-simplicial nerve functor. • Next (in §5 and 6) we formulate and prove our main result. • In an appendix (§7) we mention two relations between the categories Rel n Cat and RelCat. In more detail: 2.2. n-Relative categories. Motivated by the fact that in an n-simplicial space (i.e. an (n+ 1)-simplicial set), just like in a simplicial space, the “space direction” plays a different role than “the n simplicial directions”, we define (in §3) an nrelative category C as an (n+ 2) tuple C = (aC, v1C, . . . , vnC, wC) consisting of a category aC and subcategories v1C, . . . , vnC and wC ⊂ aC subject to the following conditions: (i) Each of the subcategories contains all the objects of aC and together with aC they form a commutative diagram with 2n arrows of the form wC