A continuing theme in "controlled topology" over the last fifteen years has been to use geometric modules to define and investigate algebraic invariants of geometric problems. One impetus to this approach was the work of Quinn [Qn1], [Qn2], [Qn3], and [Qn4] although the idea of a geometric module preceeded it by several years. Subsequently, many authors have studied different forms of geometric module theory and explored their properties. Among these are Pedersen [P] and Pedersen and Weibel [PW1] and [PW2], who studied the additive categories of what we now call geometric modules with bounded control and constant coefficients and showed how to use their algebraic K-theory to construct a homology theory on the category of compact Lipschitz spaces. The present authors [AM2] developed these ideas further by using geometric modules with bounded control and variable coefficients to construct a homology theory on a certain category of maps T OP/LIP c . More recently Anderson, Connolly, Ferry, and Pedersen [ACFP] introduced the notion of geometric modules with continuous control at infinity and used the constant coefficient case of this theory to construct a homology theory on the category of compact metrizable spaces. This point of view was subsequently used in [AC] and [CP].Although it has been clear for some time that there should be a "pushout" of these three approaches in which a homology theory is constructed by using geometric modules with continuous control at infinity and variable coefficients, it has been less clear whether such a theory would have any new applications and, at a more technical level, what the domain of such a theory should be. The former question, however, is resolved in [ACM]. That paper completes a series of papers by the present authors [AM2] and [AM3] that compare their approach to K-theory in [AM1] to the approaches of other researchers by examining the relationship between their work and the "controlled" K-theory developed by Quinn [Qn3] and [Qn4], Yamasaki [Y] and Ranicki and Yamasaki [RY]. The K-theory defined and developed in this paper plays a central and natural role in [ACM].In this paper we introduce the category T OP cc /LC in which an object, (X, B; p), is a map p : E → X where X = X − B, X is a locally compact, Hausdorff space, and B is a compact subspace. We then define a functor GM cc : T OP cc /LC → ADDCAT , the category of additive categories. It assigns to the object (X, B; p) in T OP cc /LC, the category GM cc (X, B; p) whose objects are geometric modules in E and whose morphisms are path matrix morphisms having continuous control at B.