Representation theory of groupoids and its applications to complex cobordism theory

首先，我们考虑了纤维类和2类概念之间的关系，并证明了在一个范畴ε上由纤维类组成的2类等价于从ε的相反范畴到由2类组成的2类由2函子组成的2类。然后，利用纤维范畴的概念，给出了群类群表示的一般定义。此外，我们通过构造群类群表示的各种例子，并定义平凡表示和正则表示来收集基本事实。我们用F_p表示特征p的素域，用TopVect^*表示F_p上具有邻域为0的由子向量空间组成的基本系统的梯度拓扑向量空间的范畴。通过给每个模p上同调群一个合适的拓扑，我们观察到普通模p上同调理论是一个从拓扑空间范畴到TopVect^*的函子。在适当地定义了线性映射的空间hm （V^*, W^*）之后，我们证明了一个函子Z^*→hm （W^*, Z^*）可以代替由完备张量积给出的函子V^*→V^*【叉乘】^^ W^*的右伴随，并研究了这些函子的性质。在此基础上，我们证明了空间的模p上同调理论是由对偶Steenrod代数表示的仿射群格式的表示，并尝试在Steenrod代数上重构了Lanne的不稳定模理论，提出了几个表征Steenrod代数的公理。另一方面，为了研究复k论的形式群律与具有Hopf代数体但不面向复的实k论之间的关系，我们确定了复投影空间CP^l及其乘积CP^l × CP^m的实k上同调的KO^*-代数结构。此外，我们还确定了无限维复射影空间的实k上同调的KO_*-代数结构。少

First, we considered the relations between the notions of fibered categories and 2-categories and showed that the 2-category consisting of fibered categories over a category ε is equivalent to the 2-category consisting of 2-functors from the opposite category of ε to the 2-category cat consisting of categories. Then, we gave a general definition of the representation of groupoid using the notion of fibered category. Moreover, we collected basic facts by constructing various examples of representations of groupoids and defining the trivial representations and regular representations.We denote by F_p the prime field of characteristic p and by TopVect^* the category of graded topological vector spaces over F_p which have fundamental systems of the neighborhood of 0 consisting of sub vector spaces. We observed that the ordinary mod p cohomology theory is regaded as a functor from the category of topological spaces to TopVect^* by giving a suitable topology to each mod p cohomology group of … More a space. After defining the space Hom(V^*, W^*) of linear maps suitably, we showed that a functor Z^* → Hom(W^*, Z^*) is a substitute for the right adjoint of the functor V^* → V^*【cross product】^^^W^* given by the completed tensor product and investigated properties of these functors. With the preparations above, we showed that the mod p cohomology theory of spaces are regarded as representations of the affine group scheme represented by the dual Steenrod algebra and tried to reconstruct the Lanne's theory of unstable modules over the Steenrod algebras and submit several axioms which characterize the Steenrod algebras.On the other hand, in order to investigate the relations between the formal group law of the complex K-theory and the real K-theory which has the associated Hopf algebroid though it is not complex oriented, we determined the KO^*-algebra structures of the real K-cohomology of complex projective spaces CP^l and their product CP^l × CP^m. Moreover, we determined the KO_*-algebra structure of the real K-cohomology of infinite dimensional complex projective space. Less