In the groupoid survey listed below, the term higher dimensional algebra was introduced in the provocative form: n-dimensional phenomena require for their description n-dimensional algebra. Since then the term has come to be used in various senses, and to include most forms of research into higher order categories, often of a lax nature. A more specific definition is to say that Higher Dimensional Algebra means the study of systems of partial algebraic structures whose domains of definition are given by geometric conditions.
Of course the partial composition of paths or of functions was used if not formalised early on. The earliest formal use of partial operations in algebra was that of a groupoid as defined by Brandt in 1926, in connection with the laws for the classification of quaternary quadratic forms, generalising Gauss’ work on binary quadratic forms. Curiously, groupoids were not given as an example in Eilenberg and Mac Lane’s fundamental paper on category theory, although they were well known by the Chicago algebraists of the time. The general theory of such partial algebra was developed by Philip Higgins.
The next easiest to understand example in dimension 2 is possibly that of maps where is a topological space, called squares in . Such a map determines four paths for given by . Such squares in have 2 obvious partial compositions where for example is defined if and only if .
The problem is to obtain a strict homotopy double groupoid from such squares. Brown and Higgins realised in 1974 that this could be achieved fairly easily in a relative situation, i.e. if we are given a triple where , and then consider maps which take the edges into and the vertices into , and then form of homotopy classes of such maps rel vertices. It is not quite trivial to prove that the partial compositions of such squares are inherited by to make it a double groupoid, and that, quite importantly, there is an extra structure of connections. This extra structure makes the category of such objects equivalent to the category of crossed modules but of groupoids, rather than just groups.
Under this equivalence, the double groupoid becomes the crossed module consisting of the family of relative homotopy groups , , with the boundary to the fundamental groupoid and the operations of this groupoid. Hence a van Kampen type theorem for yields a van Kampen type theorem for and so previously unobtainable determinations of some nonabelian second relative homotopy groups.
Notice that the compositions in require a choice of direction, and so is unaesthetic, whereas is a symmetric construction. Also allows for convenient multiple compositions appropriate to algebraic inverses to subdivision, which are essential for the proof of the 2-dimensional van Kampen type theorem.
P.J. Higgins, Algebras with a scheme of operators, Math. Nachr. 27 1963 115–132.
R. Brown and P.J. Higgins, On the connection between the second relative homotopy groups of some related spaces, Proc. London Math. Soc. (3) 36 (1978) 193-212.