Locales are dense in toposes
Errol Yuksel, Stockholm University
Locales and toposes are the geometric analogues of propositional and first-order logic, and in both instances the former is contained in the latter. As in the logical case, there is a non-negligible jump in complexity from locales to toposes. In this talk, I will provide a way to overcome some of the added complexity by sketching a proof that locales form a dense subcategory of toposes, meaning that toposes can be expressed in terms of locales in a canonical way. This result can be understood as a categorical justification for studying first-order logic through the lens of propositional logic, and, time permitting, we will see examples.