Calculating Simplicial Localizations*

1.1. Summary. This paper is essentially a continuation of [3], where we introduced a (standard) simplicial localization functor, which assigned to every category C and subcategory W c C, a simpficiaf category LC with in each dimension the same objects as C (i.e. for every two objects X, YE C, the maps X -+ YE LC form a simplicial set LC(X, Y)). This simplicial localization has all kinds of nice general properties, but, except in a few extreme cases [3, Section 51, it is difficult to get a hold on the homotopy type of the simplicial sets LC(X, Y). In this paper we therefore consider a homotopy variation on the standard simplicial localization LC, the hammock localization LHC (Section 2), which (Section 3) has some of the nice properties of the standard localization only up to homotopy, but is in other respects considerably better behaved. In particular (Sections 4 and 5) the simplicial sets LHC(X, Y) are much more accessible; each simplicialsetLHC(X, Y) is the direct limit of a diagram of simplicial sets which are nerves ofcategories and (Section 6) if the pair (C, W) admits a “homotopy calculus of fractions,” then several of these nerves already have the homotopy type of LHC(X, Y). When W satisfies a mild closure condition this happens, for instance, if (Section 7) the pair (C, W) admits a calculus of feft fractions in the sense of Gabriel-Zisman [5] or if (Section 8) W is closed under push outs, in which case LHC(X, Y) has the homotopy type of the nerve of the category which has as objects the sequences X + C t Y in C for which the second map is in W and which has as maps the commutative diagrams