Left adjoint of pullback $cat^1$-profinite groups

Abstract

In this paper, we present a brief review crossed modules [9], $cat^1$-groups[7], profinite crossed modules [6], $cat^1$-profinite groups[6], pullback profinite crossed modules [6] and also the pullback $cat^1$- profinite groups [2] We prove that the pulback $cat^1$-profinite group has a left adjoint which is the induced $cat^1$-group.