Waldhausen category
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| Waldhausen category | |
|---|---|
| Name | Waldhausen category |
| Introduced | 1985 |
| Introduced by | Friedhelm Waldhausen |
| Field | Algebraic K-theory |
| Notable for | S-construction, algebraic K-theory of spaces |
Waldhausen category
A Waldhausen category is a categorical framework introduced to generalize algebraic K-theory of rings, schemes, and spaces. It provides axioms for categories with cofibrations and weak equivalences suitable for constructions linking algebraic K-theory, homotopy theory, and higher category theory. Developed in the context of work on pseudo-isotopies and algebraic K-theory, it connects to many figures and institutions in topology and algebra.
Definition
A Waldhausen category is a pointed category C equipped with subcategories of cofibrations and weak equivalences satisfying axioms that permit pushouts along cofibrations and cylinder constructions; this setup was formalized by Friedhelm Waldhausen in 1985. The definition was motivated by problems tied to the Waldhausen S-construction and later related to developments by Daniel Quillen, Michael Atiyah, John Milnor, William Browder, Jean-Pierre Serre, André Joyal, Jacob Lurie, Graeme Segal, George Whitehead, and Dennis Sullivan. Foundational axioms reference constructions used in work at institutions such as the Institute for Advanced Study, Princeton University, Massachusetts Institute of Technology, Clay Mathematics Institute, University of Chicago, Harvard University, and University of Bonn.
Examples
Standard examples include the category of chain complexes of projective modules over a ring considered by Alexander Grothendieck, Jean-Pierre Serre, and Pierre Deligne; the category of retractive spaces over a fixed base space studied by William Browder, Hyman Bass, and Isadore M. Singer; and categories of perfect complexes on schemes related to work by Grothendieck and Alexander Beilinson. Other examples arise from stable model categories developed by Daniel Quillen, Max Kelly, Mark Hovey, Philip S. Hirschhorn, J. Peter May, J. Frank Adams, and Elmendorf-Kriz-Mandell-May collaborators. Further instances connect to exact categories of Daniel Quillen and Waldhausen-type structures appearing in studies by Bokstedt, Neeman, Thomason, Trobaugh, Christensen, Verdier, Seidel, Kontsevich, Toen, Voevodsky, Suslin, Weibel, Cortiñas, and Tabuada.
S-construction and K-theory
The S-construction produces a connective spectrum whose homotopy groups define algebraic K-theory; this construction extended ideas from Quillen's Q-construction and interacts with spectra studied by J. H. C. Whitehead, Adams, May, Boardman, Vogt, and Peter May. Waldhausen's S•C yields a simplicial category leading to a delooping machine related to the work of Graeme Segal, Thomason, Lewis-May-Steinberger, Elmendorf, Mandell, and Lurie. Computations using the S-construction informed breakthroughs by Friedhelm Waldhausen, Bokstedt, and Madsen-Tillmann-Weiss and link to the study of pseudo-isotopy theories investigated by Igor M. Krichever, Sullivan, and Hatcher. Algebraic K-theory spectra from Waldhausen categories connect to concepts advanced at the Simons Center for Geometry and Physics and in seminars at MSRI.
Exact functors and weak equivalences
Exact functors between Waldhausen categories preserve cofibrations and pushouts and send weak equivalences to weak equivalences; this notion parallels exactness in Quillen's framework and was influenced by categorical exactness concepts from Alexander Grothendieck, Jean Giraud, Pierre Deligne, and Jean-Louis Loday. Weak equivalences generalize homotopy equivalences from algebraic topology studied by Henri Poincaré, Henri Cartan, and Maurice H. A. Newman and interact with model category weak equivalences from Quillen, Bousfield, Friedlander, and Hovey. Natural transformations between exact functors produce maps of K-theory spectra, a procedure used in analysis by Waldhausen, Thomason, Trobaugh, Neeman, Weibel, Cortiñas, and Suslin in various localization and comparison theorems.
Homotopy invariance and localization
Homotopy invariance in the Waldhausen setting asserts that suitably homotopy equivalent input categories yield equivalent K-theory spectra; this connects to localization sequences and dévissage arguments reminiscent of results by Quillen, Bass, Gersten, Bloch, Thomason, Trobaugh, Neeman, Gabriel, and Zisman. Localization theorems for Waldhausen K-theory mirror localization in triangulated categories developed by Verdier and were applied in contexts studied at IHES, University of Oxford, Cambridge University Press seminars, and by scholars such as Ravenel, Hesselholt, Madsen, Weibel, and Cortiñas. These results underlie comparisons between algebraic K-theory of rings, schemes, and ring spectra appearing in work by Connes, Loday, Goodwillie, McCarthy, Dundas, Hesselholt, Nikolaus, and Scholze.
Relation to other categorical frameworks
Waldhausen categories relate to model categories of Quillen, exact categories of Quillen and Gabriel, and stable infinity-categories developed by Jacob Lurie and Carlos Simpson. Connections link Waldhausen constructions to derived categories studied by Grothendieck, Verdier, and Neeman; to A-infinity categories researched by Jim Stasheff, Bernard Keller, and Kontsevich; and to motives investigated by Voevodsky, Beilinson, and Deligne. Recent work ties Waldhausen-type K-theory to structured ring spectra and higher algebra as in research by J. Peter May, Mandell, Schwede, Shipley, Lurie, Angeltveit, Brave New World contributors, and teams at Institute for Advanced Study and Princeton University.