nLab partial map classifier

Idea

A partial function $f \colon A\rightharpoonup B$ between two sets can be turned into a total function $\overline{f} \colon A\to B_\bot$ by adjoining a further element $\bot$ to $B$ and sending all $a\in A$ that are not in the domain of definition of $f$ to this new value $\bot$ = “undefined” and the rest just to their value under $f$. Conversely, every function with codomain $B_\bot$ corresponds to a unique partial function to $B$ whence $B_\bot$ is a ‘classifying object’ for these.

More generally, a partial function classifier or partial map classifier of an object $B$ in a category $\mathcal{C}$ is a representing object for partial maps with codomain $B$.

Definition

Let $\mathcal{C}$ be a category with pullbacks. Recall that a partial map $A \rightharpoonup B$ in $\mathcal{C}$ is a span $A \leftarrow D \to B$ in which the map $D\to A$ is a monomorphism. The subobject $D \hookrightarrow A$ is called the domain of the partial map. Two partial maps are considered equal if they are related by an isomorphism of spans; in this way we obtain a set of partial maps $Par_{\mathcal{C}}(A,B)$.

We can compose a partial map $A\rightharpoonup B$ with a map $B\to B'$ in an obvious way. We can also compose it with a map $A'\to A$ by pulling back the mono $D\hookrightarrow A$ to $A'$. In this way $Par_{\mathcal{C}}(-,-)$ becomes a profunctor from $\mathcal{C}$ to itself. (In fact, it is the hom-set of another category with the same objects as $\mathcal{C}$.)

A partial map classifier for $B$ is an object $B_\bot$ together with an isomorphism

$$\mathcal{C}(A,B_\bot) \cong Par_{\mathcal{C}}(A,B)$$

natural in $A$. By the Yoneda lemma this means there is a universal partial map $B_\bot \rightharpoonup B$.

This can be unpacked into the following condition:

Proposition

A partial map classifier for $B$ is uniquely determined by a map $B \to B_\bot$ with the property that, for every partial map $A\rightharpoonup B$ depicted below, there is a unique $A \to B_\bot$ making a pullback square

Proof

For any monomorphism $B \to B_\bot$, there is a natural transformation $\mathcal{C}(-,B_\bot) \to Par_{\mathcal{C}}(-,B)$ given by forming pullbacks along $B \to B_\bot$. It is a natural bijection iff it satisfies the condition of the proposition.

If $B \to B_\bot$ satisfies the condition of the proposition, then it is a monomorphism, by considering the (total) partial map $\id_B$.

The existence of partial map classifiers $B_\bot$ for all objects $B$ in $\mathcal{C}$ amounts to the existence of a right adjoint for the inclusion $\mathcal{C}\hookrightarrow Par(\mathcal{C})$ where the latter is the usual category of partial maps of $\mathcal{C}$.

Constructions

In a Boolean extensive category (such as a Boolean topos), we can define the partial map classifier as $B_\bot = B + 1$, where $1$ is the terminal object. This is because in an extensive category, a map $A\to B+1$ is equivalent to a decomposition of $A$ as a coproduct $D+E$ together with a map $D\to B$ (the map $E\to 1$ being unique), and in a Boolean category every subobject of $A$ is complemented and hence induces such a coproduct decomposition. The universal partial map $B+1 \rightharpoonup B$ has domain the summand $B$, on which it is the identity. Note that $B\mapsto B+1$ is also known as the maybe monad.

Partial map classifiers also exist in every elementary topos, but in the non-Boolean case they are harder to construct. Letting $t : 1 \to \Omega$ denote the top element, the dependent product $t_*(B)$ is precisely the map $B_\bot \to \Omega$ classifying $B \subseteq B_\bot$. To wit, with this definition,

$$\begin{aligned} \mathcal{C}(A, B_\bot) &\cong \coprod_{p : A \to \Omega} \mathcal{C}/\Omega(p, t_*(B)) \\&\cong \coprod_{p : A \to \Omega} \mathcal{C}(t^*(p), B) \\&\cong \coprod_{A' \subseteq A} \mathcal{C}(A', B) \\&\cong Par_{\mathcal{C}}(A, B) \end{aligned}$$

Alternatively, in the internal logic, two ways of defining $B_\bot$ that more closely follow the classical construction are:

• $B_\bot$ is a subobject of the power object $\mathcal{P}B$ consisting of the subsingleton subobjects of $B$. In this case, the universal partial map $B_\bot \rightharpoonup B$ has domain the set of singleton subobjects of $B$, on which it is an isomorphism.

• $B_\bot$ is the object of partial maps $1 \rightharpoonup B$. In this case, the universal partial map $B_\bot \rightharpoonup B$ has domain the set of partial maps which are total, on which it is an isomorphism.

Note that neither of these constructions is predicative. The second makes more sense in a higher category (or in its internal logic such as homotopy type theory).

Remark

In an intuitionistic context, $B + 1$ still classifies something: namely partial maps $A \rightharpoonup B$ whose domain of definition is a decidable subobject of $A$.

In type theory one uses the partiality monad to treat general recursion as an algebraic effect. In this we we obtain a classifier for recursively enumerable subsets. Is this the first appearance? It is clear that this idea can be generalized to other classes of propositions.

Relative partial maps

Given objects $p : X \to S$ and $q : Y \to S$ of $\mathcal{C}/S$, a partial map from $p$ to $q$ can be described as a commutative diagram in $\mathcal{C}$.

A partial map classifier $Opt_S(q)$ for $q$ in $\mathcal{C}/S$ can be described as a representing object for the presheaf on $\mathcal{C}$ sending $X$ to the set of diagrams of this form (modulo choice of representatives for subobjects), together with its structure map $Opt_S(q) \to S$, which represents the natural transformation sending this diagram to $p \in \mathcal{C}(X, S)$.

Proposition

If $\mathcal{C}$ is a topos and $p : X \to S$ is an object of $\mathcal{C}/S$, there is a pullback in $\mathcal{C}$

where $Opt_S(q) \to X_\bot$ corresponds to forgetting the base of a relative partial map, and $Opt_S(q) \to S$ is the structure map, and $Opt_S(q) \to \Omega$ corresponds to giving the domain of definition.

Proof

Let $A$ be an object of $\mathcal{C}$. It suffices to show this square is a pullback after applying $\mathcal{C}(A, -)$ and the isomorphism to the presheaves these objects represent.

An object of the pullback can be described as an arrow $A \to S$, a subobject $B \hookrightarrow A$, and a partial map $A \hookleftarrow B' \to X$ with the property that the induced partial maps $A \hookleftarrow B \to S$ and $A \hookleftarrow B' \to S$ are the same partial map (up to choice of representative for the subobject).

This is bijective with the set of commutative diagrams of the form

and the projection to $\mathcal{C}(A, S)$ takes the bottom horizontal arrow.

Properties

• In a topos, the partial map classifier $B_\bot$ of $B$ is injective. The canonical embedding $B\rightarrowtail B_\bot$ shows accordingly that a topos has enough injectives!

• In a topos, the subobject classifier $true:1\rightarrow\Omega$ coincides with the monomorphism $\eta_1:1\rightarrowtail 1_\bot$.

• In a topos, the assignment of $B_\bot$ to $B$ is functorial.

References

• Francis Borceux, sec.5.5 in: Handbook of Categorical Algebra vol. 3, Cambridge UP (1994)

• Peter Johnstone, Section 1.2 of: Topos theory, London Math. Soc. Monographs 10, Acad. Press 1977, xxiii+367 pp. (Dover reprint 2014)

• Peter Johnstone, Section A2.4 in: Sketches of an Elephant

For related topics in homotopy type theory:

• Thorsten Altenkirch, Nils Anders Danielsson, Nicolai Kraus, Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type [abs:1610.09254]

• Martín Escardó, Cory Knapp, Partial Elements and Recursion via Dominances in Univalent Type Theory [doi:10.4230/LIPIcs.CSL.2017.21]

• Leoni Pugh, Jonathan Sterling, When is the partial map classifier a Sierpiński cone? (arXiv:2504.06789)

• Fredrik Bakke, Jonathan Sterling, Mark Damuni Williams, Lingyuan Ye, The Synthetic Sierpiński Cone (arXiv:2605.00773)

Discussion for topological spaces:

• Peter May, Johann Sigurdsson, Def. 1.3.6 in: Parametrized Homotopy Theory, Mathematical Surveys and Monographs, vol. 132, AMS 2006 (ISBN:978-0-8218-3922-5, arXiv:math/0411656, pdf)