Dwyer–Kan hammock localization
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| Dwyer–Kan hammock localization | |
|---|---|
| Name | Dwyer–Kan hammock localization |
| Field | Algebraic topology |
| Introduced | 1980s |
| Authors | William Dwyer; Daniel Kan |
| Related | Homotopical algebra; Model category; Simplicial localization |
Dwyer–Kan hammock localization is a construction in algebraic topology and homotopical algebra that produces a simplicial category associated to a category with weak equivalences, providing a setting for derived mapping spaces and homotopy categories. It was introduced by William Dwyer and Daniel Kan and has been influential in work linking Quillen, Grothendieck-style homotopical ideas to explicit models used by Boardman and Vogt. The construction connects with developments by Joyal, Lurie, Hovey, Smith, and Rezk in higher category theory and model categories.
Definition and overview
The hammock localization assigns to a relative category (a category C with a subcategory W of weak equivalences) a simplicial category L^H(C,W) whose homotopy category is obtained by formally inverting W and whose mapping spaces model derived maps. The founders William Dwyer and Daniel Kan framed this within the tradition of Quillen’s model categories and the simplicial techniques pioneered by Moore, Eilenberg, Mac Lane, and Segal. The output is compatible with constructions by Sullivan, Brown, Boardman, Vogt, and later comparisons by Hirschhorn, Hovey, and Bousfield. This device permits comparisons with the simplicial localization of Dwyer–Kan and the nerve constructions of Grothendieck and Thomason.
Construction of the hammock localization
Given a relative category (C,W), the hammock localization L^H(C,W) is a simplicial category with the same objects as C; for objects x and y the simplicial set of maps is built from diagrams—“hammocks”—involving zigzags of morphisms in C with designated weak equivalences in W. The combinatorial patterns used recall methods of Boardman, Vogt, and Stasheff; the indexing simplicial directions reflect techniques from Eilenberg–Mac Lane bar constructions and from the cosimplicial machinery of May. Dwyer and Kan encoded higher simplices as families of string diagrams resembling compositions appearing in work by Milnor and Atiyah, while ensuring functoriality under maps of relative categories treated by Grothendieck and Gabriel–Zisman.
Key steps mirror steps in model-category localization by Quillen: one forms multi-span diagrams reminiscent of calculations in Serre’s homotopy spectral sequences and then collects them into simplicial sets of homotopy classes as in constructions by Adams and Cartan. The simplicial enrichment aligns with the simplicial model category frameworks developed by Hovey and Hirschhorn, and the hammock localization admits comparisons with rigidification techniques of Joyal and Lurie.
Properties and homotopical significance
Hammock localization yields mapping spaces whose connected components correspond to morphisms in the localized homotopy category, paralleling classical results by Quillen and Brown. It preserves equivalences of relative categories induced by Dwyer–Kan equivalences studied by Dwyer, Kan, and later analyzed by Toën, Vezzosi, and Rezk. Functoriality and homotopical invariance follow patterns related to model structures investigated by Hovey, Hirschhorn, Smith, and Bousfield; homotopy limits and colimits computed in the hammock localization reflect the behavior in derived categories of Grothendieck-style contexts and in triangulated contexts linked to Verdier.
The hammock localization is particularly robust: mapping spaces are simplicial sets with the correct homotopy type for derived mapping spaces used in obstruction theory by Toda and in computations by May and Boardman. The construction interacts with spectral sequences and cohomology operations traced back to Serre, Eilenberg, and Mac Lane.
Relation to other localizations and model categories
Comparisons exist between the hammock localization and the simplicial localization of Dwyer and Kan as well as the homotopy coherent nerve of Cordier and Porter, and the nerve constructions of Thomason and Grothendieck. Through these comparisons, L^H(C,W) is related to model categories of SSet-enriched categories studied by Bergner and to complete Segal spaces introduced by Rezk. The construction corresponds under suitable equivalences to the simplicial mapping spaces in combinatorial model categories due to Lurie and Hovey, and it is compatible with left Bousfield localization techniques developed by Bousfield and Hirschhorn.
Dwyer–Kan equivalences between simplicial categories correspond to Quillen equivalences between appropriate model structures considered by Bergner and Bousfield; these correspondences are essential in work by Toën and Vezzosi linking homotopical algebraic geometry to model category presentations. The hammock localization also interfaces with localization of differential graded categories studied by Keller and with infinity-categorical localizations in the frameworks of Joyal and Lurie.
Examples and computations
For an ordinary category C with weak equivalences W given by isomorphisms, the hammock localization recovers the discrete mapping simplicial sets equivalent to hom-sets of C, echoing calculations in Gabriel–Zisman. For simplicial model categories such as simplicial sets or chain complexes over a ring studied by Quillen and Keller, the hammock localization produces mapping spaces weakly equivalent to derived mapping spaces computed via cofibrant and fibrant replacements as in work by Hovey and Hirschhorn. Computations in algebraic K-theory contexts by Waldhausen employ related localization ideas, and explicit hammock-type computations appear in analyses by Dwyer, Kan, Schwede, and Shipley.
Concrete examples include localizing the category of topological spaces with respect to weak homotopy equivalences analyzed by Serre and Whitehead, or localizing chain complexes with quasi-isomorphisms as in Grothendieck and Keller; in these cases the hammock localization recovers classical mapping spaces and Ext-complexes studied by Cartan, Eilenberg, and Mac Lane.
Applications in homotopy theory and higher category theory
Hammock localization is used to model derived mapping spaces in homotopical algebra, to compare different models for (infinity,1)-categories by Joyal, Lurie, Rezk, and Bergner, and to formulate invariants in derived algebraic geometry developed by Toën and Vezzosi. It underpins constructions in obstruction theory examined by Toda and provides tools for categorical localization techniques used by Thomason and Neeman in triangulated settings. Applications extend to algebraic K-theory methods of Waldhausen, to Morita theory in dg-categories by Keller, and to modular interpretations seen in work by Deligne and Drinfeld.