Pathabempty

Pathabempty is a term used in some category-theoretic and homotopical formalizations to denote a property of path-object constructions relative to the initial (empty) object. It is not a universal axiom with a single, canonical formulation; instead, its exact statement can vary by framework, but it generally expresses compatibility between path objects and the initial object in a category with finite limits or a model-category-like setting.

In broad terms, a path object Path(A) for an object A comes equipped with two endpoint maps ev0, ev1: Path(A) → A, playing the role of the start and end of a path. The pathabempty condition asserts that when A is the initial object (often denoted 0 or ∅), the associated path object behaves in a predictable way with respect to the unique map from the initial object to itself. In many familiar categories (such as topological spaces or simplicial sets), Path(0) is isomorphic to 0, and the endpoint maps are the unique maps to 0. Under such circumstances, the path-object construction interacts trivially with the initial object, aligning with the intended notion of a “path object over an empty base.”

Because the concept is framework-dependent, pathabempty is mainly used as a diagnostic or compatibility condition in specific formalizations rather than as a widely adopted standard theorem. It can help ensure that homotopies or path-based factorizations behave coherently when the base object is empty, which in turn supports constructions and proofs in model-categorical or higher-categorical contexts.

See also: path object, homotopy, model category, initial object, abelianization (where relevant in a given framework).