Introducing post-modern algebra according to

Sjoerd Crans, CTRC Seminar 29 September 1998

Then came Modern Algebra (see the book ``Moderne algebra'' [B. L. van der Waerden, Springer 1930-31] (Barr: ``Oh, come on, you've got to mention Emmy Noether.'' I mention Emmy Noether.)), which is about structures, or more precisely, sets with operations/relations and axioms.

Then came Category Theory (see Eilenberg and Mac Lane's paper ``General theory of natural equivalences'' [Trans. Amer. Math. Soc. 58 (1945), 231-294]), which realizes/emphasizes the importance of morphisms between structures.

See also the book ``Modern algebra and the rise of mathematical structures'' [L. Corry, Birkhäuser-Verlag, 1996].

Now comes Post-Modern Algebra, which realizes/emphasizes the importance of further structure on morphisms, ad infinitum. The term was introduced by Ross Street, in his paper ``Descent theory'': ``Higher descent theory, non-abelian cohomology, and higher-order category theory are all one subject which might be called post-modern algebra [...]''. Alternative terms are higher-dimensional algebra, introduced by Ronnie Brown in his paper ``Higher-dimensional group theory'' [in Low-dimensional topology, London Math. Soc. Lecture Notes 48, 215-238, Cambridge U. Press, 1982] and popularized by John Baez in his and Dolan's paper ``Higher-dimensional algebra and topological quantum field theory'' [Jour. Math. Phys. 36 (1995), 6073-6105], and higher-dimensional category theory, which has the two disadvantages that it -wrongly, in my opinion- suggests a more restricted subject and a being part of category theory.

Mathematics is a reflexive subject, so post-modern algebra should give rise to a new mathematical theory. This theory should have direct use in other branches of mathematics, and in fact major mathematical motivation for developing post-modern algebra comes from:

• homotopy theory, where there are paths, homotopies, homotopies of homotopies, ad infinitum,
• non-abelian cohomology, where sheaves glue matching families, stacks (of groupoids) glue families matching up to iso, 2-stacks glue families that match up to equivalences that match up to iso, ad infinitum,
• quantum field theory, where there are (n+1)-manifolds (cobordisms) between n-manifolds, ad infinitum,
• Hopf algebroid stuff.

I don't want weak n-categories, nor tri- or quadricategories, but a coherence theorem ``all the way'' which allows one to disregard coherence issues, as is done by Joyal and Street in their paper ``The geometry of tensor calculus, I'' [Adv. Math. 88 (1991), 55-112].

I defined NI 3-categories, Gray-categories. Gray-categories are the appropriate 3-dimensional post-modern algebraic structure, viz. classification of homotopy 3-types (see the paper ``Algebraic homotopy types'' [A. Joyal and M. Tierney, in preparation], and the paper ``Coherence for tricategories'' [R. Gordon, A. J. Power, and R. Street, Memoirs Amer. Math. Soc. 117 (1995), no. 558]. I then defined 4-dimensional teisi as NI 4-categories with dimension raising horizontal compositions satisfying functoriality and naturality axioms.
References for teisi: [gcbs, tpgc, obss, lotr].

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