Double Categories, Category of Categories and Supercategories
A higher level concept is thus defined as a category of categories, or super-category, which generalises to higher dimensions the notion of category – regarded as any structure which is an interpretation of Lawvere's axioms of the elementary theory of abstract categories (ETAC). Thus, a supercategory and also a super-category, can be regarded as natural extensions of the concepts of meta-category, multicategory, and multi-graph, k-partite graph, or colored graph (see a color figure, and also its definition in graph theory).
Double groupoids were first introduced by Ronald Brown in 1976, in ref. and were further developed towards applications in nonabelian algebraic topology. A related, 'dual' concept is that of a double algebroid, and the more general concept of R-algebroid.
Read more about this topic: Higher-dimensional Algebra
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