A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on
A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on
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对自由代数、自由幺半群、余极限、关联滑轮等超限构造的统一处理
DOI：
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发表时间：
1980
影响因子：
0.7
通讯作者：
G. Kelly
中科院分区：
文献类型：
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作者：
G. Kelly
许多问题导致考虑“代数”，由范畴A的对象A加上A的一个或多个内函子在A上的“作用”TkA → A，服从方程公理。这些问题包括自由单子和自由幺半群、单子范畴和幺半群范畴的余完备性、正交子范畴（=广义层范畴）、连续函子范畴等等;除了涉及代数本身的问题。代数范畴的理想性质--自由范畴的存在性、余完备性、代数函子的伴随的存在性--如果这个范畴可以在某个行为良好的范畴中被证明是反射的，那么所有这些性质都成立：我们证明了自反的存在性，并给出了一个简单超限序列的上极限，如果A是余完备的，并且Tk保持适当长的子代数链的余极限或并。本文大量借鉴了早期作者的工作，统一和简化了这一点，并将其扩展到新的问题。此外，T/A中的反射性比任何早期的结果都要强，并且将在以后的文章中以一个丰富的版本应用于研究具有结构的范畴。
Many problems lead to the consideration of “algebras”, given by an object A of a category A together with “actions” TkA → A on A of one or more endofunctors of A, subjected to equational axioms. Such problems include those of free monads and free monoids, of cocompleteness in categories of monads and of monoids, of orthogonal subcategories (= generalized sheaf-categories), of categories of continuous functors, and so on; apart from problems involving the algebras for their own sake. Desirable properties of the category of algebras - existence of free ones, cocompleteness, existence of adjoints to algebraic functors - all follow if this category can be proved reflective in some well-behaved category: for which we choose a certain comma-category T/A We show that the reflexion exists and is given as the colimit of a simple transfinite sequence, if A is cocomplete and the Tk preserve either colimits or unions of suitably-long chains of subobjects. The article draws heavily on the work of earlier authors, unifies and simplifies this, and extends it to new problems. Moreover the reflectivity in T/A is stronger than any earlier result, and will be applied in forthcoming articles, in an enriched version, to the study of categories with structure.