The Connected Components Functor
1. Connectedness versus discreteness
Besides map spaces and the truth space, another construction that is characterized by a ‘higher universal mapping property’ objectifies the counting of connected components. Reflexive graphs and discrete dynamical systems, though very different categories, support this ‘same’ construction. For example, we say that dots d and d′ in a reflexive graph are connected if for some n 0 there are
dots d = d0, d1.,…, dn = d′ and
arrows a1,…, an such that
for each i either ...