nLab category of PERs

Idea

The category of PERs over a partial combinatory algebra $A$ is a particularly concrete and simple subcategory of its realizability topos, which is especially well-suited to modelling “computable” objects relative to $A$.

It is based on the idea that a “computable set” $X$ relative to $A$ is specified by saying which elements of $A$ are “codes” representing which elements of $X$, such that no element of $A$ can code for more than one element of $X$ (but not all elements of $A$ need to code for any element of $X$). Thus this sort of “computable set” (sometimes called a modest set) is equivalently a partition of a subset of $A$, i.e. a partial equivalence relation on $A$.

Definition

Let $A$ be a partial combinatory algebra. If $E$ is a partial equivalence relation on $A$, we write $A/E$ for its set of equivalence classes. Now the category $PER(A)$ has:

• as objects, the partial equivalence relations on $A$.

• as morphisms $E\to E'$, for PERs $E$ and $E'$, the set-theoretic functions $\phi:A/E \to A/E'$ with the property of being “tracked”, i.e. such that there exists an element $f\in A$ such that whenever $(a,a)\in E$ we have $\phi([a]) = [f\cdot a]$ (hence in particular $(f\cdot a,f\cdot a)\in E'$). Equivalently, it is the quotient of the set of elements $f\in A$ such that $(a,b)\in E$ implies $(f\cdot a,f\cdot b)\in E'$ by the relation $f\sim g$ meaning that $(f\cdot a,g\cdot a)\in E'$ whenever $(a,a)\in E$.

Properties

• $PER(A)$ is a locally cartesian closed category and a Heyting category.

• $PER(A)$ is a full subcategory of the category of assemblies over $A$, which is in turn a full subcategory of the realizability topos over $A$.

In first-order logic

Given a theory in first-order logic with equality (categorically, this can be taken as a first-order hyperdoctrine), its category of partial equivalence relations represents all the subquotients the theory can “see” of its given sorts and all the functions the theory can “see” between them.

Specifically, the objects of this category are pairs $(X, R)$ where $X$ is a sort in the first-order theory and $R$ is a binary predicate which the theory proves to be a partial equivalence relation on $X$. A morphism between objects $(X, R)$ and $(Y, S)$ can be taken to be a binary predicate $F \in P(X \times Y)$ such that the theory proves $F$ is essentially the graph of a function between the specified subquotients (In detail: …). Composition is given in the obvious way (In detail: …).

In the case where the original hyperdoctrine was in fact a tripos, the resulting category of PERs will be a topos (while the converse isn’t quite true, the condition of being a tripos is just slightly stronger than necessary for this to happen).

Relationship with exact completions

The construction of partial equivalence relations is part of the tripos to topos construction. The following universal property is noted by Maietti and Rosolini (2012).

Let $\mathbf{Ex}$ be the (bi-)category of exact categories and regular functors between them. There is also a $\mathbf{EED}$ of elementary existential doctrines (aka regular hyperdoctrines). There is a forgetful functor $\mathbf{Ex} \to \mathbf{EED}$, mapping an exact category to its doctrine of subobjects. The construction of PERs forms a left adjoint to this functor.

References

• Maria Maietti and Giuseppe Rosolini, Unifying exact completions. Applied Categorical Structures, 2012. arxiv:1212.0966