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In this chapter we present our basic actors: topological spaces, simplicial sets, simplicial abelian groups, spectra, and chain complexes. We concentrate on the formal structures and the connections between them, and postpone most technicalities.

In this chapter we will develop some further properties necessary to understand simplicial sets. In order to control the weak equivalences we introduce two classes of maps: fibrations and cofibrations. These maps formalize “obstruction theory”, or rather they tell us when existence of liftings can be expected. This is intimately connected with the fact that weak equivalences are not isomorphisms, although they become so in the homotopy category.

We are interested in homotopy theory in a variety of categories, and we want to compare these. There is an efficient machinery due to Quillen, which encodes this structure. We have used this language in our discussion of simplicial sets. In addition to weak equivalences (which is all that is needed to form the homotopy category) we have fibrations and cofibrations satisfying certain axioms. This structure ensures that the homotopy category actually exists, but more importantly it encodes the deeper homotopical structures, making a large class of arguments formal. It also makes comparison between different homotopical structures more transparent.

As our last application of the machinery discussed in these talks, we come to an approach to the main topic of this summer school: motivic spaces and spectra.

In this first part, we give a quick overview of some of the foundational material of elementary algebraic geometry needed for a study of motivic homotopy theory. All of this material is well-known and excellently discussed in numerous texts; our goal is to collect the main facts to give the reader a convenient first introduction and quick reference. For this reason, many of the proofs will be only sketched or completely omitted. For further details, we suggest the reader take a look at [4], [10], [14] for the commutative algebra and [7], [9] or [12] for the algebraic geometry; for a more analytic point of view, we suggest [5].

A presheaf on a topological space is a contravariant functor on the category Op() of open subsets of ; a sheaf is just a presheaf satisfying a patching condition and a locality condition, namely, the exactness of

Motivic homotopy theory is a new and in vogue blend of algebra and topology. Its primary object is to study algebraic varieties from a homotopy theoretic viewpoint. Many of the basic ideas and techniques in this subject originate in algebraic topology.