Rewriting the elements in the intersection of the kernels of two morphisms between free groups

Pub Date : 2021-03-08 DOI:10.36045/j.bbms.210310
Franccois Renaud
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引用次数: 1

Abstract

Let F be the free group functor, left adjoint to the forgetful functor between the category of groups GRP and the category of sets SET. Let f from A to B, and h from A to C be two functions in SET and let Ker(F(f)) and Ker(F(h)) be the kernels of the induced morphisms between free groups. Provided that the kernel pairs Eq(f) and Eq(h) of f and h permute (such as it is the case when the pushout of f and h is a double extension in SET), this short article describes a method to rewrite a general element in the intersection of Ker(F(f)) and Ker(F(g)) as a product of generators in A which is (f,h)-symmetric in the sense of the higher covering theory of racks and quandles.
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重写自由群间两个态射核的交点上的元素
设F是自由群函子,左伴随在群的范畴GRP和集合的范畴SET之间的遗忘函子。设从A到B的f和从A到C的h是SET中的两个函数,设Ker(f (f))和Ker(f (h))是自由群间诱导态射的核。假设f和h的核对Eq(f)和Eq(h)是置换的(例如当f和h的推出是SET中的双扩展时),本文描述了一种将Ker(f (f))和Ker(f (g))交点上的一般元素改写为a中发生器的乘积的方法,该乘积在架和堆的高覆盖理论意义上是(f,h)对称的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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