Free Internal Groups

Abstract

In the first part of this note an elementary proof is given of the fact that algebraic functors, that is, functors induced by morphisms of Lawvere theories, have left adjoints provided that the category $\mathcal{K}$ in which the models of these theories take their values is locally presentable. The main focus however lies on the special cases of the underlying functor of the category $\mathsf{Grp}(\mathcal{K})$ of internal groups in $\mathcal{K}$ and the embedding of $\mathsf{Grp}(\mathcal{K})$ into $\mathsf{Mon}(\mathcal{K})$, the category of monoids in $\mathcal{K}$: Here a unifying construction of the respective left adjoints is provided which not only works in case $\mathcal{K}$ is a locally presentable category but also when $\mathcal{K}$ is, for example, a particular category of topological spaces such as the category of Hausdorff or Tychonoff spaces or a cartesian closed topological category.