topos
category admitting all finite limits and power objects, that can be viewed as either as a mathematical universe (replacing the category of sets) or as a generalized space (considered as a generalization of the category of sheaves on a space)
category of sets
category in mathematics where the objects are sets and the morphisms are the total functions between the sets
effective topos
topos that captures the idea of effectivity in mathematics
coherent topos
topos generated by a collection of quasi-compact quasi-separated objects closed under finite products