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].