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 the source of ai is di-1 and the target of ai is di.
or the source of ai is di and the target of ai is di-1.
The graph as a whole is connected if it has at least one dot and any two dots in it are connected; it is noteworthy that to prove a given graph to be connected may involve arbitrarily long chains of elementary connections ai, even though the structural operators s, t, i are finite in number (Sessions 13 to 15).
By contrast, this aspect of steps without limit does not arise in the same way for dynamical systems, even though the dynamical systems themselves involve infinitely many structural operators αn, effecting evolution of a system for n units of time. We say that the states x, y are connected if there are n, m such that αnx = αmy, and that the system is connected if it has at least one state and every two states are connected.
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