 Connected component of a category

Two objects of a category lie in the same connected component if a finite zigzag of morphisms, with either orientation allowed, joins them. This is an equivalence relation; its classes give full subcategories, each a connected category. For a small category, the collection of components is a set denoted ${\pi }_0(\mathcal{C})$. A functor induces a function on components. Empty categories have no components.