Structure of algebras
Structure of algebras
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代数的结构
DOI：
10.1007/bf01707315
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发表时间：
1941
期刊：
影响因子：
--
通讯作者：
Hofreiter
中科院分区：
文献类型：
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作者：
Hofreiter
数学对象描述的主要活动之一是通过比较嵌入在这些对象中的代数结构来表征。最好的情况是这些结构是自然的，因为它们通常带有关于对象的新信息，有时需要新的工具来处理。在代数结构中编码，关于对象的信息，在这种情况下是拓扑空间，将是必要的，以便仅从代数部分重建空间。所得到的空间将是原空间的近似，也就是说，它们是相同的，但在同调上是同构的。这种类型的关系称为拟同构。在这个意义上，A∞-代数是与拓扑空间相关联的代数结构，以刻画它们的同伦类型。为了更好地理解这个断言，我们将看到一个循环空间的例子。取一个点拓扑空间（X，X）。现在考虑X上的循环，其开始和结束于X，或者用更专业的话来说，从单位区间I = [0，1]到X的连续应用，使得每个应用f满足f（0）= f（1）= X。循环空间带有一个自然的循环产品：循环的串联。假设一个循环是区间I = [0，1]和一个有限的开放区间I1，. . .，嵌入在I.其思想是，开放区间的补表示循环中具有基点的值的部分。请注意，不同的循环可以具有相同的表示。现在，取区间I与（0，1 2）和（1 2，1），两个循环的乘积是嵌入每个开放区间（0，12）和（1 2，1）的结果，我们想要乘以的循环。圈积的一个主要性质是结合性公理丢失了：只有在同调中，积才是结合的。因此，对于循环空间，我们处理的运算不是结合的，而是同伦结合的。美国数学家Jon Stasheff在他的博士论文中详细描述了同伦结合论的这种新结构。
One of the principal activities in the description of mathematical objects is the characterization by comparing the algebraic structures embedded in these objects. The best situation is when these structures are natural, because usually they come with new information about the object, and sometimes it needs new tools to be handled. Coded in the algebraic structure, the information about the object, in this case a topological space, will be necessary in order to reconstruct the space only from the algebraic part. The spaces obtained will be approximations to the original space, that is, they will be the same but isomorphisms in homology. This type of relation is called quasiisomorphism. In this sens, the A∞-algebras, are algebraic structures associated to topological spaces in order to characterize their type of homotopy. To better understand this affirmation we will see an example with loop spaces. Take a pointed topological space (X, ∗). Now consider the loops over X which begin and end in ∗, or in more technical words, the continuous applications from the unit interval I = [0, 1] to X, such that each application f satisfies f(0) = f(1) = ∗. The space of loops comes with a natural loop product: the concatenation of loops. Imagine a loop as the interval I = [0, 1] together with a finite collection of open intervals I1, . . . , In embedded in I. The idea is that the complement of the open intervals represents the parts of the loop with value the base point ∗. Note that different loops can have the same representation. Now, take the interval I with (0, 1 2 ) and ( 1 2 , 1), and the product of two loops is the result of embed in each open interval (0, 12 ) and ( 1 2 , 1), the loops we want to multiply. One of the principal properties of loop product is that the axiom of associativity is lost: only in homology the product is associative. So, with loop spaces, we are dealing with an operation which is not associative but homotopy associative. In his PhD thesis, the American mathematician Jon Stasheff, describe in detail this new structure of homotopy associativity.