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

本文的研究命题分为四个部分。它们按重要性递减的顺序列出 (1)拟范畴理论。拟范畴是Boardman和Vogt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，并且我发现它们可以在更高范畴理论和抽象同伦理论中发挥重要作用。我们的主要目标是从拟2-范畴入手，发展更高的拟范畴理论。为此，我们将使用我们引入的theta-范畴的概念。另一个目标是进一步发展准范畴理论，例如通过加强它与Quillen模型范畴的联系。 (2)BAR-Cobar对偶的一般理论。BAR和Cobar结构是代数拓扑学的主要工具之一。它们最初被用来理解拓扑群的同调和它的分类空间的同调之间的关系，以及一个空间的同调和它的圈空间的同调之间的对偶关系。它们在代数的Koszul对偶中起着重要的作用。根据康采维奇的思想，这种二元性扩展到金兹堡和卡普拉诺夫的歌剧中。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学、代数几何和K-理论中都是重要的。$E_inty$-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，但知之甚少。利用p为奇素数的q-环的概念来组织mod p同调运算的代数，p为奇素数。 (4)范畴的自由双补。范畴C上的预留格范畴是C在并列限制下的自由补全。C在限制和列限制下的自由补全包含了对预堆概念的广泛概括。

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.