A characterization of simplicial localization functors

1.1. Relative categories. As in [BK] we denote by RelCat the category of (small) relative categories and relative functors between them, where by a relative category we mean a pair (C, W ) consisting of a category C and a subcategory W ⊂ C which contains all the objects of C and their identity maps and of which the maps will be referred to as weak equivalences and where by a relative functor between two such relative categories we mean a weak equivalence preserving functor.

1.2. Homotopy equivalences between relative categories. A relative functor f : X → Y between two relative categories (1.1) is called a homotopy equivalence if there exists a relative functor g : Y → X (called a homotopy inverse of f ) such that the compositions gf and f g are naturally weakly equivalent (i.e. can be connected by a finite zigzag of natural weak equivalences) to the identity functors of X and Y respectively. 1.3. DK-equivalences. A map in the category SCat of simplicial categories (i.e. categories enriched over simplicial sets) is [Be] called a DK-equivalence if it induces weak equivalences between the simplicial sets involved and an equivalence of categories between their homotopy categories, i.e. the categories obtained from them by replacing each simplicial set by the set of its components.

Furthermore a map in RelCat will similarly be called a DKequivalence if its image in SCat is so under the hammock localization functor [DK2] L H : RelCat -→ SCat (or of course the naturally DK-equivalent functors RelCat → SCat considered in [DK] and [DHKS,35.6]).

We will denote by both DK ⊂ SCat and DK ⊂ RelCat the subcategories consisting of these DK-equivalences.

Date: May 3, 2019.

Next we define what we mean by 1.4. Simplicial localization functors. In defining DK-equivalences in RelCat (1.3) we used the hammock localization functor and not one of the other DKequivalent functors mentioned because, for our purposes here it seemed to be the more convenient one. However in other situations the others are more convenient and it therefore makes sense to define in general a simplicial localization functor as any functor RelCat → SCat which is naturally DK-equivalent to the functors mentioned above (1.3).

We also need 1.5. The relativization functor. In contrast with the situation mentioned in 1.4 there is a preferred choice for a relativization functor

which is a kind of inverse of the simplicial localization functor, namely the delocalization mentioned in [DK3,2.5] which assigns to an object A ∈ SCat its relative flattening which is the relative category which consists of (i) a category which is the Grothendieck construction on A, where A is considered as a simplicial diagram of categories, and (ii) its subcategory obtained by applying the same construction to the subobject of A which consists of its objects only.

Our main result then is 1.6. Theorem. A relative functor

is a simplicial localization functor (1.5) iff it is a homotopy inverse (1.2) of the realization functor (1.5)

Rel : (SCat, DK) -→ (RelCat, DK) .

We end with some 1.7. Comments on the proof of 1.6. The proof of theorem 1.6 heavily involves some of the results of [DK] and [DK3, 2.5] and we therefore first (in §2) review some of the results of these papers.

In §3 we then actually prove theorem 1.6. It turns out however that in addition to the results mentioned in §2 we need a property of the hammock localization of which we will give two proofs. The first is a very short one based on a remark of Toen and Vezzosi [TV,2.2.1] involving the homotopy category of SCat. The other, which is due to Bill Dwyer, relies heavily on [DK] and [DK2] and is longer, but has the “advantage” of taking place in the model category itself.

In preparation for the proof (in §3) of theorem 1.6 we review here some of the results of [DK], [DK2] and [DK3,2.3] which will be needed.

2.1. The hammock localization. In the proof of 1.6 we will make extensive use of the hammock localization L H of [DK2] because, unlike the other simplicial localization functors, it has the property that every relative category (C, W ) comes with a natural embedding C → L H (C, W ).

2.2. The category RelSCat. This will be the category which has as its objects the pairs (A, U ) where A ∈ SCat and U ⊂ A is a subobject which contains all the objects of A.

One then can consider RelCat as a full subcategory of RelSCat and [DK2, 2.5] extend the functor L H : RelCat → SCat to a functor L H : RelSCat → SCat by sending an object of RelSCat to the diagonal of the bisimplicial set obtained from it by dimensionwise application of the hammock localization.

To deal with [DK] and [DK3, 2.5] it will be convenient to introduce a notion of 2.3. Neglectable categories. Given an object (A, U ) ∈ RelSCat (2.2) we will say that U is neglectable in A if every map of U goes to an isomorphism in π 0 A.

[DK, 3.4 and 5.1] imply (i) Let A be a category, let U and V ⊂ A be subcategories which contain

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