论文标题
在物体系统中具有系数的立方组的同源性和共同体学
Homology and cohomology of cubical sets with coefficients in systems of objects
论文作者
论文摘要
本文继续对立方体和半阵容集的同源性进行研究，并在对象系统中具有系数。主要结果是定理认为，在Abelian类别中具有与精确相关的Abelian类别中具有系数的立方体集合的同源性是与Colimit Fuctor的左卫星同构的同构。这使得可以证明本文中提出的以下许多新主张，即在物体系统中具有系数的立方组的同源和共同体。这些同源性在传递到系数系统的直接图像时，在立方组之间的形态下是不变的。传递到反图像时，这些同源物的不变性有一个标准。这些同源性通过局部系数和违反系统中系数的半立方组合的奇异立方同源性概括了奇异的立方体。在逆向系统中具有系数的立方组的结论性同源物具有光谱序列。立方组的弱等效性引起与局部系统同源性的同构。对于立方体的形态，其反纤维形态是弱等效的，存在与局部系统同源的光谱序列，这些系统融合到了这种形态的域的同源性。图中具有系数的小类别的同源性可以计算为立方同源性。在自然系统中具有系数的Baues-Wirsching共同体是同构与立方体共同体，在协方差系统中具有系数。
This paper continues the research of the author on the homology of cubical and semi-cubical sets with coefficients in systems of objects. The main result is the theorem that the homology of cubical sets with coefficients in contravariant systems in an Abelian category with exact coproducts is isomorphic to the left satellites of a colimit functor. This made it possible to prove a number of the following new assertions, presented in the paper, about the homology and cohomology of cubical sets with coefficients in systems of objects. These homology are invariant under morphism between cubical sets when passing to the direct image of the system of coefficients. There is a criterion for the invariance of these homologies when passing to the inverse image. These homology generalize the singular cubical homology with local coefficients and the homology of semi-cubical sets with coefficients in contravariant systems. There is a spectral sequence for colimit homologies of cubical sets with coefficients in contravariant systems. The weak equivalence of cubical sets induces an isomorphism of homology with local systems. For a morphism of cubical sets whose inverse fiber morphisms are weak equivalences, there exists a spectral sequence for homology with local systems converging to the homology of the domain of this morphism. The homology of small category with coefficients in a diagram can be calculated as cubical homology. The Baues-Wirsching cohomologies with coefficients in natural systems are isomorphic to cubical cohomologies with coefficients in covariant systems.
