Many of the higher-dimensional algebraic structures are noncommutative and, therefore, their study is a very significant part of nonabelian category theory, and also of Nonabelian Algebraic Topology (NAAT),[1] which generalises to higher dimensions ideas coming from the fundamental group.[2] Such algebraic structures in dimensions greater than 1 develop the nonabelian character of the fundamental group, and they are in a precise sense ‘more nonabelian than the groups'.[1][3] These noncommutative, or more specifically, nonabelian structures reflect more accurately the geometrical complications of higher dimensions than the known homology and homotopy groups commonly encountered in classical algebraic topology.
A fundamental result in NAAT is the generalised, higher homotopy van Kampen theorem proven by R. Brown, which states that "the homotopy type of a topological space can be computed by a suitable colimit or homotopy colimit over homotopy types of its pieces''. A related example is that of van Kampen theorems for categories of covering morphisms in lextensive categories.[5] Other reports of generalisations of the van Kampen theorem include statements for 2-categories[6] and a topos of topoi [1].
Important results in higher-dimensional algebra are also the extensions of the Galois theory in categories and variable categories, or indexed/'parametrized' categories.[7] The Joyal–Tierney representation theorem for topoi is also a generalisation of the Galois theory.[8]
Thus, indexing by bicategories in the sense of Benabou one also includes here the Joyal–Tierney theory.[9]
References
Brown, Ronald (Bangor University, UK); Higgins, Philip J. (Durham University, UK); Sivera, Rafael (University of Valencia, Spain) (2010). Non-Abelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids. Tracts in Mathematics. Vol. 15. European Mathematical Society. p. 670. ISBN978-3-03719-083-8.{{cite book}}: CS1 maint: multiple names: authors list (link)[1]
^Baianu, I. C. (2007). "A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity". Axiomathes. 17 (3–4): 353–408. doi:10.1007/s10516-007-9012-1. S2CID3909409.
^Ronald Brown and George Janelidze, van Kampen theorems for categories of covering morphisms in lextensive categories, J. Pure Appl. Algebra. 119:255–263, (1997)