The Clowder Project

The core of $\mathcal{C}$ is the wide subcategory of $\mathcal{C}$ spanned by the isomorphisms of $\mathcal{C}$, i.e. the category $\mathsf{Core}\webleft (\mathcal{C}\webright )$ where¹

1. Objects. We have
   \[ \text{Obj}\webleft (\mathsf{Core}\webleft (\mathcal{C}\webright )\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{Obj}\webleft (\mathcal{C}\webright ). \]
2. Morphisms. The morphisms of $\mathsf{Core}\webleft (\mathcal{C}\webright )$ are the isomorphisms of $\mathcal{C}$.