  This matter is further complicated by the fact that the foundations of category theory itself have yet to be clarified. For there may be many different ways to think of a universe of higher-dimensional categories as a foundations for mathematics. It is safe to say that we now have a good understanding of what are called \((\infty, 1)\)-categories and important mathematical results have been obtained in that framework. (See, for instance, Cisinski 2019 for a presentation.) An adequate language for the universe of arbitrary higher-dimensional categories still has to be presented together with definite axioms for mathematics. (See Makkai 1998 for a short description of such a language. A different approach based on homotopy theory but with closed connections with higher-dimensional categories has been proposed by Voevodsky et al. and is being vigorously pursued. See the book Homotopy Type Theory, by Awodey et al. 2013.) 