  An alternative approach, that of Lawvere (1963, 1966), begins by characterizing the category of categories, and then stipulates that a category is an object of that universe. This approach, under active development by various mathematicians, logicians and mathematical physicists, lead to what are now called “higher-dimensional categories” (Baez 1997, Baez &amp; Dolan 1998a, Batanin 1998, Leinster 2002, Hermida et al. 2000, 2001, 2002). The very definition of a category is not without philosophical importance, since one of the objections to category theory as a foundational framework is the claim that since categories are defined as sets, category theory cannot provide a philosophically enlightening foundation for mathematics. We will briefly go over some of these definitions, starting with Eilenberg’s &amp; Mac Lane’s (1945) algebraic definition. However, before going any further, the following definition will be required.