An almost full embedding of the category of graphs into the category of abelian groups
An almost full embedding of the category of graphs into the category of abelian groups
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图类别几乎完全嵌入到阿贝尔群类别中
DOI：
10.1016/j.aim.2014.02.027
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
A. Przeździecki
中科院分区：
文献类型：
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作者：
A. Przeździecki
我们构造了一个嵌入图类G: G图→A b到阿贝尔群的范畴，使得对于G图中的X和Y，我们有hm (G X, G Y) = Z [hm G图（X， Y）]，自由阿贝尔群的基是集合hm G图（X， Y）。同构在X和y上是泛函的。这种嵌入的存在意味着，与通常的信念相反，阿贝尔群的范畴与任何其他具体范畴一样复杂和全面。我们利用这种嵌入解决了Isbell的一个老问题，即阿贝尔群的范畴的每一个在极限下闭合的满子范畴是否都是反射的。一个肯定的答案被证明相当于弱沃普涅卡原理，这是一个无法证明但被认为与标准集合论一致的大基本公理。我们得到了关于稳定同伦范畴中任意局域存在的若干结果。作为嵌入的快速应用，得到了阿贝尔群范畴内的几个已知结构。
We construct an embedding G: G raphs→ A b of the category of graphs into the category of abelian groups such that for X and Y in G raphs we have Hom (G X, G Y)≅ Z [Hom G raphs (X, Y)], the free abelian group whose basis is the set Hom G raphs (X, Y). The isomorphism is functorial in X and Y. The existence of such an embedding implies that, contrary to a common belief, the category of abelian groups is as complex and comprehensive as any other concrete category. We use this embedding to settle an old problem of Isbell as of whether every full subcategory of the category of abelian groups, which is closed under limits, is reflective. A positive answer turns out to be equivalent to weak Vopěnka's principle, a large cardinal axiom which is not provable but believed to be consistent with standard set theory. We obtain some consequences to the Hovey–Palmieri–Strickland problem about existence of arbitrary localizations in a stable homotopy category. Several known constructions in the category of abelian groups are obtained as quick applications of the embedding.