  The purpose of this paper is twofold: 1. we prove the triangulability of
smooth orbifolds with corners, generalizing the same statement for orbifolds.
2. based on 1, we propose a new homology theory. We call it geometric homology
theory (GHT for abbreviaty). GHT is a natural and flexible generalization of
singular homology. It has some advantages overcoming the unpleasant
combinatoric rigidity of singular homology, e.g. ill-defined pullbacks of
singular chains along fiber bundles. The method we use are mainly based on the
celebrated stratification and triangulation theories of Lie groupoids and their
orbit spaces, as well as the extension to Lie groupoids with corners by us. We
illustrate a simple application of GHT in categorical Gromov-Witten theory,
initiatied by Costello. We will develop further of this theory in our sequel
paper.
