Definition: Declared edge-path presentation

A finite presentation consists of a connected finite cell structure $K$, a marking identifying its based edge-path homotopy group with the fundamental group of the punctured-disk domain, and:

1. invertible transition values on oriented edges, with inverse values on reversed edges;

2. regular two-cell boundary relators;

3. a marked primitive positive peripheral edge word;

4. the frame transports used to compare based words.

One may use a triangulated compact annular truncation and a finite refinement with all required boundary paths. A dual structure is allowed only when its marking actually gives this presentation.