Homotopically trivial toposes

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.