This is the first of two papers whose main purpose is to prove a generalization of the Seifert-Van Kampen theorem on the fundamental group of a union of spaces. This generalisation (Theorem C of [8]) will give information in all dimensions and will include as special cases not only the above theorem (without the usual assumptions of path-connectedness) but also

Crossed complexes have longstanding uses, explicit and implicit, in homotopy theory and the cohomology of groups. It is here shown that the category of crossed complexes over groupoids has a symmetric monoidal closed structure in which the internal Hom functor is built from morphisms of crossed complexes, nonabelian chain homotopies between them and similar higher homotopies. The tensor product involves non-abelian constructions related to the commutator calculus and the homotopy addition lemma. This monoidal closed structure is derived from that on the equivalent category of ω-groupoids where the underlying cubical structure gives geometrically natural definitions of tensor products and homotopies.