Applications de la théorie des catégories à la topologie et à l'algèbre

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The research proposition has four parts. They are listed in order of decreasing importance (1) The theory of quasi-categories. Quasi-categories were introduced by Boardman and Vogt in their work on homotopy invariant algebraic structures. I have proved fundamental results about them and I have discovered that they can play an important role in higher category theory and abstract homotopy theory. Our main goal is to develop the theory of higher quasi-categories, starting with quasi-2-categories. For this, we will use the notion of theta-categories introduced by us. Another goal is to further develop the theory of quasi-categories, for example by tightening its connection with Quillen model categories. (2) A general theory of the bar-cobar duality. The bar and cobar constructions are among the main tools of algebraic topology. They were initially developed to understand the relation between the homology of a topological group and the homology of its classifying space, and dually between the homology of a space and the homology of its loop space. They are playing an important role in Koszul duality of algebras. The duality was extend to operads by Ginzburg and Kapranov, following an idea of Kontsevich. Our main goal is to gives the bar-cobar duality a general categorical framework and to extend its applications. (3) The algebra of homology operations. Homology and cohomology operations are important in algebraic topology, algebraic geometry and K-theory. The algebra of (unstable) homology operations in $E_\infty$-spaces is the Dyer-Lashof algebra. It is related to the Steenrod algebra but less understood. We wish to organise the algebra of mod p homology operation, p an odd prime, by using the notion of Q-ring for p an odd prime. (4) Free bicompletion of categories. The category of presheaves on a category C is the free completion of C under colimits. The free completion of C under limits and colimits contains a vast generalisation of the notion of presheaf.