Effective topos

The topos is based on the partial combinatory algebra given by Kleene's first algebra ${\displaystyle {\mathcal {K}}_{1}}$. In Kleene's notion of recursive realizability, any predicate is assigned realizing numbers, i.e. a subset of ${\displaystyle {\mathbb {N} }}$. The extremal propositions are ${\displaystyle \top }$ and ${\displaystyle \bot }$, realized by ${\displaystyle {\mathbb {N} }}$ and ${\displaystyle \{\}}$. However in general, this process assigns more data to a proposition than just a binary truth value.

A formula with ${\displaystyle k}$ free variables will give rise to a map in ${\displaystyle ({\mathcal {P}}{\mathbb {N} })^{{\mathbb {N} }^{k}}}$ the values of which is the subset of corresponding realizers.

${\displaystyle {\mathsf {Eff}}}$ is a prime example of a realizability topos. These are a class of elementary topoi with an intuitionistic internal logic and fulfilling a form of dependent choice. They are generally not Grothendieck topoi.

In particular, the effective topos is ${\displaystyle {\mathsf {RT}}({\mathcal {K}}_{1})}$. Other realizability topos construction can be said to abstract away the some aspects played by ${\displaystyle {\mathbb {N} }}$ here.

The objects are pairs ${\displaystyle \langle X,E_{X}\rangle }$ of sets together with a symmetric and transitive relation in ${\displaystyle ({\mathcal {P}}{\mathbb {N} })^{X\times X}}$, representing a form of equality predicate, but taking values in subsets of ${\displaystyle \mathbb {N} }$. One writes ${\displaystyle E_{X}(x)}$ with just one argument to denote the so called existence predicate, expressing how an ${\displaystyle x}$ relates to itself. This may be empty and so the relation is not generally reflexive. Arrows amount to equivalence classes of functional relations appropriately respecting the defined equalities.

The classifier amounts to ${\displaystyle {\mathcal {P}}{\mathbb {N} }}$. The pair (or, by abuse of notation, just that underlying powerset) may be denoted as ${\displaystyle \Omega }$. An entailment relation ${\displaystyle \vdash _{X}}$ on ${\displaystyle {\mathcal {P}}{\mathbb {N} }^{X}}$ makes it into a Heyting pre-algebra. 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 terminal object is a singleton ${\displaystyle \langle \{*\},E_{\{*\}}\rangle }$ with trivial existence predicate (${\displaystyle {\mathbb {N} }}$). The finite product respect the equality appropriately. The classifier's equality ${\displaystyle E_{{\mathcal {P}}{\mathbb {N} }}}$ is given through equivalences in its lattice.

Some objects exhibit a rather trivial existence predicate depending only on the validity of the equality relation "${\displaystyle =}$" of sets, so that valid equality maps to the top set ${\displaystyle \mathbb {N} }$ and rejected equality maps to ${\displaystyle \{\}}$. This gives rise to a full and faithful functor ${\displaystyle \nabla \colon {\mathsf {Sets}}\to {\mathsf {Eff}}}$ out of the category of sets, which has the finite limits preserving global sections functor ${\displaystyle \Gamma }$ as its left-adjoint. This factors through a finite-limit preserving, full and faithful embedding ${\displaystyle \omega }$-${\displaystyle {\mathsf {Sets}}\to {\mathsf {Eff}}}$.

The topos has a natural numbers object ${\displaystyle N=\langle {\mathbb {N} },E_{\mathbb {N} }\rangle }$ with simply ${\displaystyle E_{\mathbb {N} }(n)=\{n\}}$. Sentences true about ${\displaystyle N}$ are exactly the recursively realized sentences of Heyting arithmetic ${\displaystyle {\mathsf {HA}}}$.

Now arrows ${\displaystyle N\to N}$ may be understood as the total recursive functions and this also holds internally for ${\displaystyle N^{N}}$. The latter is the pair given by total recursive functions ${\displaystyle \mathrm {TR} }$ and a relation such that ${\displaystyle E_{\mathrm {TR} }(f)}$ is the set of codes ${\displaystyle e\in {\mathbb {N} }}$ for ${\displaystyle f}$. 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.

With ${\displaystyle N}$ and functions from and to it, as well as with simple rules for the equality relations when forming finite products ${\displaystyle \times }$, one may now more broadly define the hereditarily effective operations. Again one may think of functions in ${\displaystyle N^{N}}$ as given by indices and their equality is determined by the objects that compute the same function. This equality clearly puts a constraint on ${\displaystyle N^{(N^{N})}}$, as these functions come out to be only those computable functions that also properly respect the mentioned equality in their domain. Et cetera. The situation for general ${\displaystyle \langle X,E_{X}\rangle \to \langle Y,E_{Y}\rangle }$, equality (in the sense of the ${\displaystyle E}$'s) in domain and image must be respected.

With this, one may validate Markov's principle ${\displaystyle {\mathrm {MP} }}$ and the Extended Church's principle ${\displaystyle {\mathrm {ECT} }_{0}}$ (and a second-order variant thereof), which come down to simple statement about object such as ${\displaystyle N^{N}}$ or ${\displaystyle (1+1)^{N}}$. These imply ${\displaystyle {\mathrm {CT} }_{0}}$ and ${\displaystyle {\mathrm {IP} }_{0}}$.

A choice principle ${\displaystyle N^{N}}$ related to Brouwerian weak continuity fails. From any object, there are only countably many arrows to ${\displaystyle N}$. ${\displaystyle \Omega ^{N}}$ fulfills a uniformity principle. ${\displaystyle N}$ is not the countable coproduct of copies of ${\displaystyle 1}$. This topos is not a category of sheaves.

The object ${\displaystyle \langle {\mathbb {Q} }^{\mathbb {N} },E_{{\mathbb {Q} }^{\mathbb {N} }}\rangle }$ 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. It rejects the claim that ${\displaystyle x\leq 0\lor 0\leq x}$ would hold for all reals ${\displaystyle x}$. Formulations of the intermediate value theorem fail and all functions from the reals to the reals are provenly continuous. A Specker sequence exists and then Bolzano-Weierstrass fails.