n-categories, internal n-categories
Segal spaces, see the references there
category object in an (infinity,1)-category, category object
‘’…with all adjoints’‘
(directed) homotopy type theory
Daniel R.Licata, Robert Harper, 2-dimensional directed type theory, pdf
Marco Grandis, Directed homotopy theory, I. The fundamental category (arXiv)
Tim Porter, Enriched categories and models for spaces of evolving states, Theoretical Computer Science, 405, (2008), pp. 88–100.
Tim Porter, Enriched categories and models for spaces of
dipaths. A discussion document and overview of some techniques_ (pdf)
cobordism hypothesis and topological quantum field theories