Grothendieck homotopy hypothesis
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| Grothendieck homotopy hypothesis | |
|---|---|
| Name | Grothendieck homotopy hypothesis |
| Field | Algebraic topology |
| Proposed by | Alexander Grothendieck |
| Year | circa 1983 |
Grothendieck homotopy hypothesis is a conjectural identification between homotopy types and infinity-groupoids proposed within the work of Alexander Grothendieck and later popularized in algebraic topology and higher category theory. The hypothesis connects ideas from Alexander Grothendieck, René Thom, Henri Cartan, Jean-Pierre Serre, André Weil and Raoul Bott to developments by William Thurston, John Milnor, Michael Boardman, J. Peter May, Vladimir Voevodsky and Jacob Lurie, and has influenced research at institutions such as Institut des Hautes Études Scientifiques, Clay Mathematics Institute, Max Planck Institute for Mathematics, Mathematical Sciences Research Institute and Courant Institute.
Statement
The Grothendieck homotopy hypothesis asserts that the homotopy theory of CW complexes, as studied by Solomon Lefschetz and Eduard Čech, is equivalent to the theory of infinity-groupoids as envisioned by Alexander Grothendieck in the Pursuing Stacks manuscript; this equivalence was articulated in dialogues among Jean Bénabou, Gilles Deleuze comments in philosophy aside, André Joyal and Ross Street, and later formalized in frameworks developed by Clark Barwick, Chris Schommer-Pries, Tom Leinster and Emily Riehl. The claim equates topological homotopy types of spaces considered by Henri Poincaré and Eilenberg–MacLane with algebraic structures appearing in the work of Samuel Eilenberg, Saunders Mac Lane, Daniel Quillen and Quentin Stafford.
Background and motivation
Grothendieck motivated the hypothesis in correspondence with Pierre Deligne and discussions with Jean-Pierre Serre about pursuing foundations connecting algebraic geometry from Grothendieck classrooms at University of Montpellier to homotopical algebra as developed by Daniel Quillen and Jean-Louis Verdier. The idea draws on precedents in the classification of coverings by Hermann Weyl and Élie Cartan and on the conception of groupoids in the work of Alonzo Church-adjacent logic circles and later codified by Grothendieck himself. Motivating examples include the homotopy classification results of René Thom, stratified spaces studied by William Thurston and the categorical perspectives of Mac Lane and Galois influenced approaches used at École Normale Supérieure.
Models and formulations
Multiple models realize the hypothesis: weak Kan complexes popularized by André Joyal, quasi-categories championed by Jacob Lurie, complete Segal spaces introduced by Charles Rezk, Segal categories advanced by Hiroshi Kihara-adjacent researchers and model categories developed by Daniel Quillen and refined by Mark Hovey and Paul Goerss. Other approaches include simplicial categories used by Dwyer–Kan school with William Dwyer and Dan Kan, homotopical algebraic geometry frameworks by Bertrand Toën and Gabriele Vezzosi, and higher topos theory as systematized by Jacob Lurie and extended at Institut des Hautes Études Scientifiques. Comparisons among these models draw on equivalences proven by Carlos Simpson, J. Michael Boardman, Ralph Cohen and categorifications studied by Hugh M. Curry.
Consequences and applications
If taken as a principle, the hypothesis yields bridges between classical results by Élie Cartan, Charles Ehresmann, Hermann Weyl and modern higher categorical formulations used in Algebraic geometry-adjacent programs like motivic homotopy theory by Vladimir Voevodsky and applications in Topological quantum field theory explored by Michael Atiyah and Edward Witten. It underpins aspects of descent theory central to work by Grothendieck and Pierre Deligne, informs obstruction theory developed by J. H. C. Whitehead and Marshall Cohen, and is relevant to computation methods used by Daniel Quillen in higher K-theory as pursued at Institute for Advanced Study. Connections have been drawn to the cobordism categories studied by M. A. Madsen and Isadore Singer and to the axiomatization of moduli problems in the style of Alexander Grothendieck and David Mumford.
Examples and key results
Concrete validations of the hypothesis appear as equivalences between homotopy categories of spaces considered by J. H. C. Whitehead, Serre and combinatorial models like Kan complexes by Daniel Kan and quasi-categories by André Joyal and Jacob Lurie. Key theorems include comparisons by J. Peter May and Vladimir Voevodsky in motivic contexts, model equivalences proven by Charles Rezk and Pedro Boavida de Brito and classification results akin to Eilenberg–Mac Lane space computations used by Samuel Eilenberg and Saunders Mac Lane. Work by Clark Barwick and Chris Schommer-Pries on the cobordism hypothesis provides a template for translating between categorical and topological invariants as envisaged in Grothendieck's program.
Variations and related conjectures
Variants include the n-groupoid versions related to the homotopy n-type problems studied by Henri Poincaré and René Thom and conjectures connecting to higher topos theory proposed by Jacob Lurie, Bertrand Toën, Gabriele Vezzosi and Marc Hoyois. Related proposals include the Baez–Dolan stabilization hypothesis by John Baez and James Dolan, the cobordism hypothesis by Jacob Lurie and Christopher Schommer-Pries and formalisms in derived algebraic geometry developed by Bertrand Toën, Jacob Lurie and Maxim Kontsevich. Ongoing research links these ideas to advances by Pierre Deligne, Vladimir Voevodsky, Edward Witten and institutions such as Clay Mathematics Institute and Mathematical Sciences Research Institute.