Homological Algebra for Superalgebras of Differentiable Functions
Homological Algebra for Superalgebras of Differentiable Functions
复制标题
可微函数超代数的同调代数
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
Dmitry Roytenberg
中科院分区：
文献类型：
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作者：
D. Carchedi;Dmitry Roytenberg
这是为导出微分几何（以及特征为零的其他几何）的微分分级方法奠定基础的系列论文中的第二篇。在本文中，我们扩展了 dg 代数的经典概念，特别是定义了 C 无穷环世界中微分分级代数的概念。微分分级C-无穷代数范畴的相反包含微分分级流形范畴作为完整的子范畴。更一般地说，微分分级代数的概念对于任何（超）费马理论的代数都是有意义的，因此我们还得出了适合研究导出的实数和复数解析流形和其他变体的微分分级代数的定义。我们继续证明，对于任何允许积分的超费马理论 S，我们定义和展示的概念都满足所有重要的例子，微分分级 S 代数的范畴支持自然扩展微分分级代数上的经典模型结构的 Quillen 模型结构，无论是在有界和无界情况下（以及无分级的微分代数）。最后，我们表明，在相同的假设下，任何这些类别的微分分级 S 代数都具有单纯丰富性，在适当的意义上与模型结构兼容。
This is the second in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we extend the classical notion of a dg-algebra to define, in particular, the notion of a differential graded algebra in the world of C-infinity rings. The opposite of the category of differential graded C-infinity algebras contains the category of differential graded manifolds as a full subcategory. More generally, this notion of differential graded algebra makes sense for algebras over any (super) Fermat theory, and hence one also arrives at the definition of a differential graded algebra appropriate for the study of derived real and complex analytic manifolds and other variants. We go on to show that, for any super Fermat theory S which admits integration, a concept we define and show is satisfied by all important examples, the category of differential graded S-algebras supports a Quillen model structure naturally extending the classical one on differential graded algebras, both in the bounded and unbounded case (as well as differential algebras with no grading). Finally, we show that, under the same assumptions, any of these categories of differential graded S-algebras have a simplicial enrichment, compatible in a suitable sense with the model structure.