**ARTICLE VII**

**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* = *d*_{0}, *d*_{1}*.*,…, *d _{n}* =

arrows *a*_{1},…, *a _{n}* such that

for each *i* either ...

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