AXIOMATIC COHESION
AXIOMATIC COHESION
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公理内聚
DOI：
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发表时间：
2007
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通讯作者：
F. Lawvere
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文献类型：
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作者：
F. Lawvere
经典分析和现代连续介质物理理论的空间背景的性质需要的不仅仅是局部不变量和上同环的描述。正如麦克斯韦所强调的那样，根据研究的需要，这种描述具有不同程度的精确度。这些层次对应着不同的空间范畴，都直观地具有衔接的特征。在此，我们的目的是继续对这些范畴进行公理化的研究，这涉及到以下几个方面：一、作为衔接背景的空间范畴；衔接与非衔接；质量类型广泛的质量;精练的品质：精练和浓缩的品质；形式与实体的标准性质IV.通过无穷小上的恒定性进行内聚的非内聚V.自反图及其原子数的例子VI.充分内聚与格罗滕迪克条件VII。我期待着在这些方面的进一步工作，以及动态定律、本构关系和其他自然存在于内聚范畴中的数学结构的范畴的发展。一、作为数学结构内聚背景的空间类别需要一门明确的内聚科学来解释动态数学理论的各种背景模型。这样一门科学需要有足够的表现力来解释这些背景如何与其他数学类别如此不同，彼此之间也不同，但又如此统一，以至于它们可以相互转化。这种相互转换的一个日常例子是天气预报员将有限元方法（可以看作是组合拓扑中的分析）应用于连续统热力学方程（可以看作是光滑拓扑中的分析，其中存在光滑函数和分布）。2。衔接与非衔接；我将衔接与非衔接进行对比分析。在这一点上，我追随康托尔，他接近他的梦根，把它们否定为卡迪纳伦；收到编辑2007-01-16和修订版2007-05-31。由p·t·约翰斯通传送。出版日期：2007-06-052000数学学科分类：18A40、18B25、18B30、74A60、74A99。
The nature of the spatial background for classical analysis and for modern theories of continuum physics requires more than the partial invariants of locales and cohomology rings for its description. As Maxwell emphasized, this description has various levels of precision depending on the needs of investigation. These levels correspond to different categories of space, all of which have intuitively the feature of cohesion. Our aim here is to continue the axiomatic study of such categories, which involves the following aspects: I. Categories of space as cohesive backgrounds II. Cohesion versus non-cohesion; quality types III. Extensive quality; intensive quality in its rarefied and condensed aspects; the canonical qualities form and substance IV. Non-cohesion within cohesion via constancy on infinitesimals V. The example of reflexive graphs and their atomic numbers VI. Sufficient cohesion and the Grothendieck condition VII. Weak generation of a subtopos by a quotient topos I look forward to further work on each of these aspects, as well as development of categories of dynamical laws, constitutive relations, and other mathematical structures that naturally live in cohesive categories. I. Categories of space as cohesive backgrounds for mathematical structures An explicit science of cohesion is needed to account for the varied background models for dynamical mathematical theories. Such a science needs to be sufficiently expressive to explain how these backgrounds are so different from other mathematical categories, and also different from one another and yet so united that they can be mutually transformed. An everyday example of such mutual transformation is the weatherman’s application of the finite element method (which can be viewed as analysis in a combinatorial topos) to equations of continuum thermomechanics (which can be viewed as analysis in a smooth topos, where smooth functions and distributions live). II. Cohesion versus non-cohesion; quality types I analyze cohesion by contrasting it with non-cohesion. In that I follow Cantor, who approached his Mengen by negating them into Kardinalen; the latter are (not isomorphism Received by the editors 2007-01-16 and, in revised form, 2007-05-31. Transmitted by P. T. Johnstone. Published on 2007-06-05. 2000 Mathematics Subject Classification: 18A40, 18B25, 18B30, 74A60, 74A99.