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

研究命题分为四个部分。它们是按重要性递减的顺序列出的 (1)拟范畴理论。拟范畴是由博德曼和沃格特在他们关于同伦不变代数结构的工作中引入的。我已经证明了关于它们的基本结果，并且发现它们在高级范畴理论和抽象同伦理论中可以发挥重要作用。 我们的主要目标是发展高级拟范畴理论，从拟2范畴开始。为此，我们将使用我们引入的θ-范畴的概念。另一个目标是进一步发展准范畴理论，例如通过加强其与奎伦模型范畴的联系。 (2)bar-cobar对偶的一般理论。 bar和cobar构造是代数拓扑学的主要工具之一。它们最初是为了理解拓扑群的同调与其分类空间的同调之间的关系，以及空间的同调与其回路空间的同调之间的对偶关系而发展起来的。它们在代数的Koszul对偶中起着重要的作用。二元性被扩展到由金兹伯格和Kapranov操作，以下的想法Kontsevich。我们的主要目标是给bar-cobar对偶一个一般的范畴框架，并扩展它的应用。 (3)同调运算的代数。同调和上同调运算在代数拓扑、代数几何和K-理论中具有重要的意义。E_\infty$-空间中的（不稳定）同调运算的代数是Dyer-Lashof代数。它与Steenrod代数有关，但较少被理解。我们希望组织模p同调运算的代数，p是奇素数，通过使用的概念，Q-环的p是奇素数。 (4)范畴的自由双完备性。范畴C上的预层范畴是C在余极限下的自由完备化。C在极限和余极限下的自由完备化包含了对预层概念的广泛推广。

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.