We describe a geometric and homotopical perspective on higher categories: higher categories are built by iterated attachment of directed cells in the spirit of CW-complexes and admit Postnikov towers giving rise to homotopy posets. These carry the obstruction classes controlling extensions from one skeleton to the next. The associated graded is given by wedges of spheres and directed spheres, leading to stabilization phenomena and categorical analogues of spectra. This is joint work with David Gepner.