The n-Category Café

Here are the short courses being taught as part of the Geometric Applications of Homotopy Theory program at the Fields Institute from January to June 2007. Many of them are related to n-categories!

This schedule is a bit tentative.

Eugenia Cheng: Higher categories (6 lectures, January)

Andre Joyal: Basic aspects of quasi-categories (6 hours, late January)

The fundamental category of a simplicial set. Categorical equivalences. Adjoint maps. Quasi-categories. Weak categorical equivalences. Inner fibrations, isofibrations. Join and slice. Left and right fibrations. The model structure for quasi-categories. Relation with the classical model structure on simplicial sets. Equivalence with complete Segal spaces, Segal categories and simplicial categories. The fibered model structures. Homotopy factorisation systems. The covariant and contravariant model structures over a base. Initial and terminal objects. Initial and final maps. Fully faithful maps, dominant maps. Minimal quasi-categories. Minimal fibrations. Morita equivalences. Localisation.

Reading list:

J.E. Bergner, A survey of $(\infty,1)$-categories. In preparation.

A. Joyal, Quasi-categories and Kan complexes, JPAA vol 175 (2002), 207-222.

A. Joyal, The theory of quasi-categories I. In preparation.

A. Joyal and M. Tierney, Quasi-categories vs Segal spaces. To appear.

A. Joyal, Quasi-categories vs simplicial categories. In preparation.

André Joyal: Extension of category theory to quasi-categories (6 hours, early February)

Adjoint maps. Diagrams. Limits and colimits. Large quasi-categories. The quasi-category K of Kan complexes. Complete and cocomplete quasi-categories. Grothendieck fibrations. Proper and smooth maps. Kan extensions. Cylinders, distributors and spans. Duality. Yoneda Lemma. The universal left fibration over K. Trace and cotrace. Factorisation systems in quasi-categories. Quasi-algebra. Locally presentable quasi-categories. Quasi-varieties. Internal categories. Descent. Exact quasi-categories. Quasi-topos. Higher quasi-categories.

Reading list:

A. Joyal, Quasi-categories and Kan complexes, JPAA vol 175 (2002), 207-222.

A. Joyal, The theory of quasi-categories II. In preparation.

A. Joyal, The theory of quasi-categories in perspective. To appear.

J. Lurie, Higher topos theory.

J.F. Jardine: Simplicial presheaves (6 lectures, late January)

Simplicial sheaves and presheaves, homology and cohomology, descent, localization and motivic homotopy theory, sheaves and presheaves of groupoids, stacks, cocycle categories, torsors, non-abelian cohomology.

Reading list:

P.G. Goerss and J.F. Jardine, Simplicial Homotopy Theory, Progress in Mathematics Vol. 174, Birkhäuser Basel-Boston-Berlin (1999).

J.F. Jardine, Stacks and the homotopy theory of simplicial sheaves, Homology, Homotopy and Applications 3(2) (2001), 361-384.

J.F. Jardine, Cocycle categories.

J.F. Jardine, Lectures on simplicial presheaves.

Fabien Morel and Vladimir Voevodsky, $A^{1}$-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math., 90:45-143 (2001), 1999.

Max Karoubi: Hermitian K-theory (6 lectures, early March)

Hermitian forms and quadratic forms, positive and negative hermitian K-theory, various versions of the periodicity theorem, homotopy invariance, topological analogs, the case when 2 is not invertible, a new homology theory on rings : the stabilized Witt group.

Reading list :

M. Karoubi and O. Villamayor, K-théorie algébrique et K-théorie topologique II. Math. Scand. 32, pp. 57-86 (1973).

A. Bak, K-theory of forms, Annals of Math. Studies 98. Princeton University Press (1981).

M. Karoubi, Théorie de Quillen et homologie du groupe orthogonal. Le théoreme fondamental de la K-théorie hermitienne. Annals of Math. 112, pp. 207-282 (1980).

F.J-B.J Clauwens, The K-theory of almost hermitian forms, Topological structures II, Mathematical Centre Tracts 115, pp. 41-49 (1979).

M. Karoubi, Stabilization of the Witt group, C.R. Math. Acad. Sci. Paris 342, pp. 165-168 (2006)

Fabien Morel: Structure and computations of $A^1$-homotopy sheaves, with applications (6 lectures, early March)

Basic structure of $A^1$-homotopy sheaves as unramified sheaves on the category of smooth $k$-schemes. $A^1$-motives and $A^1$-homology of spaces. The Hurewicz theorem. Description of the $H_0$ of smash powers of $G_m$’s in terms of Milnor-Witt K-theory. Consequences and examples of computations. The first non-trivial $A^1$-homotopy sheaf of algebraic spheres. The Brouwer degree.

$A^1$-homotopy classification of rank n vector bundles. Application to the Euler class for rank n vector bundles over dim n affine smooth n-dimensional schemes and to stably free vector bundles.

The theory of $A^1$-coverings, $A^1$-universal coverings and $A^1$-fundamental group. Examples of computations (for surfaces for instance). Some perspectives towards a “surgery classification” of smooth projective $A^1$-connected varieties and more.

References:

F. Morel, An introduction to $A^1$-homotopy theory, In Contemporary Developments in Algebraic K-theory, ICTP Lecture notes, 15 (2003), pp. 357-441, M. Karoubi, A.O. Kuku , C. Pedrini (ed.).

F. Morel, On the structure of $A^1$-homotopy sheaves, part I and part II.

F. Morel, $A^1$-homotopy classification of vector bundles over smooth affine schemes.

F. Morel and V. Voevodsky, $A^1$-homotopy theory of schemes, Publications Mathématiques de l’I.H.E.S, volume 90.

J.F. Jardine: Presheaves of spectra (6 lectures, early May)

Presheaves of spectra and symmetric spectra, stable categories, homology and cohomology, motivic stable categories, chain complexes and simplicial abelian presheaves, derived categories, Voevodsky’s category of motives (maybe).

Reading list:

J.F. Jardine, Generalized Etale Cohomology Theories, Progress in Mathematics Vol. 146, Birkhäuser, Basel-Boston-Berlin (1997).

J.F. Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000), 445-552.

J.F. Jardine, Presheaves of chain complexes, K-Theory 30(4) (2003), 365-420.

J.F. Jardine, Generalised sheaf cohomology theories, in Axiomatic, Enriched and Motivic Homotopy Theory, NATO Science Series II 131 (2004), 29-68.

J.F. Jardine, Lectures on presheaves of spectra.

Jacob Lurie (6 hours, late May - early June)

Derived algebraic geometry, representability of derived moduli functors, virtual fundamental classes, derived group schemes and equivariant cohomology theories, elliptic cohomology, topological modular forms.

Reading list:

M. Hopkins, Topological modular forms, the Witten genus, and the theorem of the cube, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 554–565, Birkäuser, Basel, 1995.

B. Toen and G. Vezzosi, Algebraic geometry over model categories (a general approach to derived algebraic geometry).

B. Toen, Higher and derived stacks: a global overview.

J. Lurie, Higher topos theory.

J. Lurie, Derived algebraic geometry. Being rewritten; old version available at http://www.math.harvard.edu/~lurie/.

J. Lurie. A survey of elliptic cohomology. Available at http://www.math.harvard.edu/~lurie/.

Paul Goerss: The moduli stack of formal groups and homotopy theory (late May - early June)

The interplay between the geometry of formal groups and homotopy theory has been a guiding influence since the work of Morava in ’70s, and it has been a thread in homotopy theory ever since. The current language of algebraic geometry allows us to make very concise and natural statements about this relationship.

In these lectures, I will discuss how the geometry of the moduli stack M of smooth one-dimensional formal groups dictates the chromatic structure of stable homotopy theory. At a prime, there is an essentially unique filtration of M by the open substacks U(n) of formal groups of height no more than n and the resulting decomposition of coherent sheaves on M gives exactly the chromatic filtration. Topics may include formal groups, the relationship between quasi-coherent sheaves and complex cobordism, the role of coordinates, the height filtration, closed points and Lubin-Tate deformation theory, and algebraic chromatic convergence. Since M is in some sense very large and cumbersome, I hope also to give some discussion of small (i.e. Deligne-Mumford) stacks over M; these include the moduli stack of elliptic curves and certain Shimura varities, thus bringing us to the current research of Hopkins, Miller, Lurie, Behrens, Lawson, Naumann, Ravenel, Hovey, Strickland, and many others.

Let me remark that I am merely the expositor here, building on the work of many people, especially Jack Morava and Mike Hopkins.

Suggested reading:

Behrens, Mark, A modular description of the $K(2)$-local sphere at the prime 3, Topology 45 No. 2 (2006), 343–402.

Goerss, Paul, (Pre-)sheaves of ring spectra over the moduli stack of formal group laws, Axiomatic, Enriched and Motivic Homotopy Theory, NATO Sci. Ser. II Math. Phys. Chem., 131, 101-131, Kluwer Acad. Publ., Dordecht 2004.

Hopkins, M. J., Algebraic topology and modular forms, Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 291–317, Higher Ed. Press, Beijing, 2002.

Hopkins, M. J. and Gross, B. H., The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory, Bull. Amer. Math. Soc. (N.S.), 30 No. 1 (1994), 76-86.

Hovey, Mark and Strickland, Neil, Comodules and Landweber exact homology theories, Adv. Math., 192 No. 2 (2005), 427-456.

Naumann, Niko, Comodule categories and the geometry of the stack of formal groups.

Ezra Getzler: Lie n-groupoids (June)

Kai Behrend (June)