CONVENIENT CATEGORIES OF SMOOTH SPACES
CONVENIENT CATEGORIES OF SMOOTH SPACES
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DOI：
10.1090/s0002-9947-2011-05107-x
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发表时间：
2011-11-01
影响因子：
1.3
通讯作者：
Hoffnung, Alexander E.
中科院分区：
文献类型：
--
作者：
Baez, John C.;Hoffnung, Alexander E.
"Chen空间“是配备有”情节“的集合的集合X，即，从凸集映射到X，满足三个简单公理。虽然一个单独的Chen空间可能比一个光滑流形差得多，但所有Chen空间的范畴比光滑流形的范畴表现得好得多。例如，陈空间的任何子空间或商空间都是陈空间，而陈空间之间的光滑映射空间也是陈空间。Souriau的“空间学”也有这些便利的性质。在这里，我们给一个统一的治疗这两种形式主义。根据Penon和Dubuc的思想，我们证明了Chen空间、拓扑空间甚至单纯复形都是“具体场地上的具体层”的例子。因此，这类空间的范畴是局部笛卡尔闭的，具有所有极限、所有共极限和弱子对象分类器。为了微分几何学家的利益，我们的处理解释了我们使用的大部分范畴理论。
A 'Chen space' is a set X equipped with a collection of 'plots', i.e., maps from convex sets to X, satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace or quotient space of a Chen space is a Chen space, and the space of smooth maps between Chen spaces is again a Chen space. Souriau's 'diffeological spaces' share these convenient properties. Here we give a unified treatment of both formalisms. Following ideas of Penon and Dubuc, we show that Chen spaces, diffeological spaces, and even simplicial complexes are examples of 'concrete sheaves on a concrete site'. As a result, the categories of such spaces are locally Cartesian closed, with all limits, all colimits, and a weak subobject classifier. For the benefit of differential geometers, our treatment explains most of the category theory we use.