  We provide, with proofs, a complete description of the authors' construction
of state-sum invariants announced in [CY], and its generalization to an
arbitrary (artinian) semisimple tortile category. We also discuss the
relationship of these invariants to generalizations of Broda's surgery
invariants [Br1,Br2] using techniques developed in the case of the semi-simple
sub-quotient of $Rep(U_q(sl_2))$ ($q$ a principal $4r^{th}$ root of unity) by
Roberts [Ro1]. We briefly discuss the generalizations to invariants of
4-manifolds equipped with 2-dimensional (co)homology classes introduced by
Yetter [Y6] and Roberts [Ro2], which are the subject of the sequel. (citations
refer to bibliography in the paper)
