Could not include topos theory - contents
It is an environment for higher order recursion theory, where, in the internal logic, it is provable that every total function from natural numbers to natural numbers is recursive (furthermore, the functor from the effective topos into Set, the inverse image part of a geometric morphism preserves the natural numbers object, giving a suitable version of this result in the external logic as well).
The effective topos construction alluded in the above paragraph can be performed more generally, in every topos with a natural numbers object, replacing Set. To every such topos one constructs the corresponding “external” effective topos and the correspondence extends to a functor admitting a fully faithful right adjoint. Kleene’s first algebra can also be replaced by any partial combinatory algebra, or even some more general types of gadgets; toposes obtained in this way are called realizability toposes.
The effective topos is the category obtained from the category of sets by first freely adjoining recursively-indexed coproducts (but being careful to preserve the empty set), and then adding quotients of (pseudo-)equivalence relations. (RobinsonRosolini).
J.M.E. Hyland, The effective topos in A. S. Troelstra (ed.) D. van Dalen (ed.) , The L.E.J. Brouwer Centenary Symposium, North-Holland (1982) pp. 165–216.
In the context of triposes:
Edmund Robinson, Giuseppe Rosolini, Colimit completions and the effective topos, The Journal of symbolic logic, 55, no 2 (1990) (JSTOR)