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

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.