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An introduction to \(\mathbb{A}^1\)-homotopy theory. (English) Zbl 1081.14029
Karoubi, M. (ed.) et al., Contemporary developments in algebraic \(K\)-theory. Proceedings of the school and conference on algebraic \(K\)-theory and its applications, ICTP, Trieste, Italy, July 8–19, 2002. Dedicated to H. Bass on the occasion of his 70th birthday. Trieste: ICTP - The Abdus Salam International Centre for Theoretical Physics (ISBN 92-95003-21-7/pbk). ICTP Lecture Notes 15, 361-441 (2003).
The article by Fabien Morel is part of a collection of lecture notes of lecture series on new developments in algebraic \(K\)-theory, which were given during the summer school on algebraic \(K\)-theory and its applications at the International Centres for Theoretical Physics IQTP in Trieste, Italy from July 8 to July 19, 2002.
These lecture notes by Morel give a very clear and thorough introduction into motivic homotopy theory as invented by F. Morel and V. Voevodsky [Publ. Math., Inst. Hautes √Čtud. Sci. 90, 45–143 (1999; Zbl 0983.14007)]. After recollecting all the basic definitions and theorems of simplicial homotopy theory and especially of the homotopy theory for simplicial presheaves and sheaves using Quillen’s language of closed model categories [D. Quillen, Homotopical Algebra (Lecture Notes in Mathematics 43, Berlin: Springer) (1967; Zbl 0168.20903)] the unstable \(\mathbb{A}^1\)-homotopy category of Morel and Voevodsky is constructed and the main constructions and theorems of F. Morel and V. Voevodsky (loc. cit.) are discussed. The next two section cover stable motivic homotopy theory, namely the construction of the stable \(\mathbb{A}^1\)-homotopy category of \(S^1\)-spectra (\(S^1= \Delta^1/\partial\Delta^1\) is the simplicial circle) and of the stable \(\mathbb{A}^1\)-homotopy category of \(\mathbb{P}^1\)-spectra. Also their triangulated structures are examined in detail and the construction of generalized motivic cohomology theories is described and examples of calculations are given. In the last section the motivic \(\pi_0(S^0)\) is calculated and its relation to the Grothendieck-Witt ring of quadratic forms over a field is discussed in detail as well as the higher degree generalization obtained in the joint work by Hopkins and Morel which relates Milnor-Witt \(K\)-theory with stable motivic homotopy classes of maps from \(S^0\) to \((\mathbb{G}_m)^n\).
These lecture notes will serve as an excellent introduction into motivic homotopy theory and are warmly recommended to everyone who likes to get a feeling for this fasciniating circle of ideas and future directions of research.
For the entire collection see [Zbl 1050.19001].

14F35 Homotopy theory and fundamental groups in algebraic geometry
14F42 Motivic cohomology; motivic homotopy theory
19E08 \(K\)-theory of schemes
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)