CatDat

• notation: $U_{\Grp,\Mon}$ : $\Grp$ → $\Mon$
• Source: category of groups
• Target: category of monoids
• Left adjoint: $F_{\Mon,\Grp}$
• Right adjoint: $(-)^{\times}$
• Related functors: $U_{\Ab,\Grp}$
• nLab Link

This functor maps a group to its underlying monoid. We view groups as structured sets $(X,m,e,i)$ (consisting of a set, a multiplication, a neutral element, and an inverse operation), and monoids as structured sets $(X,m,e)$. This forgetful functor precisely maps $(X,m,e,i)$ to $(X,m,e)$. From this point of view, it does not merely forget a property; it forgets an operation. This perspective is useful in contexts where the inverse operation is no longer reducible to a property, for example, the forgetful functor from topological groups to topological monoids.