The concept of an elementary topos abstracts from categories of sheaves and the category of sets. It is well known that the effective topos, which is an example of an elementary topos, has many applications in mathematical logic and programming language semantics.
In this talk, we will start with a brief introduction to topos theory and focus on the concept of a Lawvere–Tierney topology: it coincides with a covering in sheaf toposes. In contrast, Lawvere–Tierney topologies in the effective topos are related to oracle computations and techniques for the investigation of intuitionistic logic.
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