partialization

Partialization is a categorical construction that enlarges a given category by adjoining partial morphisms between its objects. In a typical setup, objects of the resulting category Par(C) are the same as in C, while a morphism from A to B is represented by a span A <- D -> B, where the left leg is a monomorphism i: D -> A and the right leg is a morphism f: D -> B. The left leg encodes the domain of definition of the partial map, and the right leg gives its value on that domain.

Composition in Par(C) is defined by pullbacks. Given A <- D1 -> B and B <- D2 -> C, one forms the pullback of D1 -> A with D2 -> B to obtain a new domain object, and then composes to obtain a partial map from A to C. This construction preserves identities and yields associativity under the usual categorical pullback conditions. There is a natural embedding C -> Par(C) that sends a total morphism f: A -> B to the span A <- A -> B with the left leg being the identity on A.

Partialization typically requires C to have suitable pullbacks, and often that monos are stable under pullback. Under these hypotheses Par(C) has a rich structure for studying partial maps and their compositions. The category Par(Set) is the classical example, where morphisms A -> B are partial functions defined on a subset of A.

Par(C) is related to restriction categories and other frameworks for partiality. It provides a canonical way to formalize partial maps within a categorical context, enabling the study of undefined or domain-restricted computations, subobject-based domains, and their algebraic properties in a uniform setting.