Abstract
Let $F_g$ be the free group functor, left adjoint to the forgetful functor between the category of groups $\mathsf{Grp}$ and the category of sets $\mathsf{Set}$. Let $f\colon A \to B$ and $h\colon A \to C$ be two functions in $\mathsf{Set}$ and let $\mathrm{Ker}(\mathrm{F}_g(f))$ and $\mathrm{Ker}(\mathrm{F}_g(h))$ 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 $\mathsf{Set}$), this short article describes a method to rewrite a general element in the intersection $\mathrm{Ker}(\mathrm{F}_g(f)) \cap \mathrm{Ker}(\mathrm{F}_g(g))$ as a product of generators in $A$ which is $\langle f,h \rangle$-symmetric in the sense of the higher covering theory of racks and quandles.
Citation
François Renaud. "Rewriting the elements in the intersection of the kernels of two morphisms between free groups." Bull. Belg. Math. Soc. Simon Stevin 28 (4) 547 - 559, may 2022. https://doi.org/10.36045/j.bbms.210310
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