Could not include topos theory - contents
constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
The effective topos $Eff$ is an example of an elementary topos with a natural numbers object which is not a Grothendieck topos (and doesn’t even have a geometric morphism to Set).
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 $Hom(1, -)$ from the effective topos into Set, the inverse image part of a geometric morphism $Set \to Eff$ preserves the natural numbers object, giving a suitable version of this result in the external logic as well).
It can be specified as the realizability topos for Kleene's first algebra.
The effective topos construction alluded in the above paragraph can be performed more generally, in every topos $E$ with a natural numbers object, replacing Set. To every such topos one constructs the corresponding “external” effective topos $e E$ and the correspondence $E \mapsto e E$ 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).
Reviews include
Stijn Vermeeren, section 3 of Realizability Toposes, 2009 (pdf)
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:
Then
Sori Lee, Jaap van Oosten, Basic subtoposes of the effective topos, arxiv/1201.2571
Edmund Robinson, Giuseppe Rosolini, Colimit completions and the effective topos, The Journal of symbolic logic, 55, no 2 (1990) (JSTOR)