We give a number of equivalent
conditions for a topos to be homotopically trivial and then relate these conditions to
the logic of the topos. This is accomplished by constructing a family of intervals that
can detect complemented, regular subobjects of the terminals. It follows that these
conditions generally are weaker than the Stone condition but are equivalent to it if
they hold locally. As a consequence we obtain an extension of Johnstone’s list of
conditions equivalent to DeMorgan’s law. Thus, for example, the fact that there is
no nontrivial homotopy theory in the category of sets is equivalent to the
fact, among others, that maximal ideals in commutative rings are prime.
Moreover, any topos has a ’best approximation’ by a locally homotopically trivial
topos.
