introduced by Martin Hyland (1982) captures the mathematical idea of effectivity within the category theoretical framework.
However in general, this process assigns more data to a proposition than just a binary truth value.
free variables will give rise to a map in
These are a class of elementary topoi with an intuitionistic internal logic and fulfilling a form of dependent choice.
Other realizability topos construction can be said to abstract away the some aspects played by
, representing a form of equality predicate, but taking values that are subsets of
with just one argument to denote the so called existence predicate, expressing how an
This may be empty, expressing the relation is not generally reflexive.
Arrows amount to equivalence classes of functional relations appropriately respecting the defined equalities.
The pair (or, by abuse of notation, just that underlying powerset) may be denoted as
Such a context allows to define the appropriate lattice-like logic structure, with logical operations given in terms of operations of the realizer sets, making use of pairs and computable functions.
The finite product respects the equality appropriately.
Some objects exhibit a rather trivial existence predicate depending only on the validity of the equality relation "
This gives rise to a full and faithful functor
out of the category of sets, which has the finite limits preserving global sections functor
This factors through a finite-limit preserving, full and faithful embedding
are exactly the recursively realized sentences of Heyting arithmetic
may be understood as the total recursive functions and this also holds internally for
The latter is a subset of the naturals but not just a singleton, since there are several indices computing the same recursive function.
So here the second entry of the objects represent the realizing data.
and functions from and to it, as well as with simple rules for the equality relations when forming finite products
, one may now more broadly define the hereditarily effective operations.
as given by indices and their equality is determined by the objects that compute the same function.
(and a second-order variant thereof), which come down to simple statement about object such as
related to Brouwerian weak continuity fails.
is effective in a formal sense and from it one may define computable Cauchy sequences.
Through a quotient, the topos has a real numbers object which has no non-trivial decidable subobject.
With choice, the notion of Dedekind reals coincides with the Cauchy one.
Analysis here corresponds to the recursive school of constructivism.
A Specker sequence exists and then Bolzano-Weierstrass fails.