For a fairly general algebraic category C (possible interpretations of C include the categories of groups, rings (associative, commutative), algebras (associative,
commutative, Lie or Jordan)) we give various alternative descriptions of an n-fold category internal to C. One of these descriptions we call a "crossed n-cube in C".
Crossed 1-cubes are better known as "crossed modules" (this latter term being .due to Whitehead [Wl]). Crossed 2-cubes in the category of groups are originally due to Loday [L].
We give a combinatorial description of crossed n-cubes for n = 1,2,3 and C equal to the category of groups, Lie algebras, commutative algebras and (n = 1,2) associative algebras.
The study of certain universal crossed 2-cubes leads us to notions of non-abelian tensor, exterior and antisymmetric products of groups and of Lie and commutative algebras. The tensor and exterior products of groups are originally due to Brown and Loday [B-L]. We also look at the crossed 3-cube analogue of the tensor product of groups.
We study the relevance of crossed modules and crossed 2-cubes to the homology of groups and Lie algebras. In particular we prove I~ OI.
THEOREM If a:M ~ PLprojective crossed P-rnodule (of groups) with im a• N, then H2(N) e ker an [M,M].
THEOREM If M,N are normal subgroups of a group G such that G • MN, then there is an exact sequence "3(M h N) ◄ H2(G) ◄ H2(G/M) © H2(G/N) ◄ MnN/[M,N] ◄
➔ H1(G) ➔ H1(G/M) ©H1(G/N) ➔ 1 where "3(M h N) is the kernel of a map MAN ➔ M from the exterior product of Mand N.
THEOREM If, in the preceding theorem, G • N, then we can extend the exact sequence by two terms:H3(G) ➔ H3(G/M) ➔ "3(M h N).
The second two theorems are originally due to Brown and Loday [B-L] who obtained them as a corollary to their van Kampen type theorem for squares of maps. The proofs in this thesis are purely algebraic.
We give the analogue of the second theorem in which H2(G) is replaced by the group H2(G), this group !:!2(G) being the one introduced by Dennis [DJ as a kind of "second homology group suitable for algebraic K-theory".
We give Lie algebra versions of the first two theorems.