Pushouty

Pushouty is an informal, nonstandard term used in discussions of category theory and related areas to describe a diagram, construction, or object that behaves like a pushout or results from gluing data along a common part. It is not a standard technical term found in textbooks or formal literature.

In category theory, a pushout of morphisms f: A → B and g: A → C is a universal object P equipped with morphisms B → P and C → P making the square commute, such that any other pair of arrows from B and C agreeing on A factors uniquely through P. A construction is described as pushouty when it either is a pushout square or closely mimics this universal gluing behavior, potentially up to isomorphism or equivalence in a given category.

The term is typically used informally to signal that a gluing construction yields, or is intended to yield, a pushout-like result without asserting a formal pushout property in every detail. This can arise when a diagram is observed to function as a pushout in practice, or when one works with data that is equivalent to a pushout up to a chosen notion of equivalence.

Examples or contexts where pushouty might be invoked include: gluing topological spaces along a common subspace, which yields a pushout in the category of spaces; amalgamated products in algebra, which are pushouts in the category of groups; and constructions in sets or modules that identify subobjects.

Because pushouty is informal, precise statements should rely on explicit universal properties: state that a square is a pushout, or that a construction satisfies the pushout universal property in the relevant category. See also pushout, universal property, coproduct.