A correspondence between racks and g-digroups

Abstract The aim of this paper is to establish a correspondence between the categories given by the structures of racks and g-digroups, by defining three functors, denoted similarly to those in the case of rack-groups, $$Conj: \mathbf {g-dig}\rightarrow {\textbf {Rack}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>o</mml:mi> <mml:mi>n</mml:mi> <mml:mi>j</mml:mi> <mml:mo>:</mml:mo> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>-</mml:mo> <mml:mi>dig</mml:mi> </mml:mrow> <mml:mo>→</mml:mo> <mml:mi>Rack</mml:mi> </mml:mrow> </mml:math> , $$As: {\textbf {Rack}}\rightarrow \mathbf {g-dig}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>s</mml:mi> <mml:mo>:</mml:mo> <mml:mi>Rack</mml:mi> <mml:mo>→</mml:mo> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>-</mml:mo> <mml:mi>dig</mml:mi> </mml:mrow> </mml:mrow> </mml:math> , and $$Inn: {\textbf {Rack}}\rightarrow \mathbf {g-dig}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>I</mml:mi> <mml:mi>n</mml:mi> <mml:mi>n</mml:mi> <mml:mo>:</mml:mo> <mml:mi>Rack</mml:mi> <mml:mo>→</mml:mo> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>-</mml:mo> <mml:mi>dig</mml:mi> </mml:mrow> </mml:mrow> </mml:math> . Additionally, we establish certain properties of these functors and prove, as in the case of rack-groups, that the functor As is left adjoint to the functor Conj .