Categories of inverse systems of compacta with upper semi-continuous bonding functions

We define a category GΛ) of inverse systems of compacta with upper semi-continuous bonding functions indexed by a given directed set Λ so that the topological generalisation of inverse limits proposed by Mahavier can be characterised by a certain universal property in this category. Our defining new category is justified by generating a class of inverse systems that do not have a categorical limit in the usual sense. We also construct a category G_pro in analogy with pro-categories in an attempt to derive generalisations of some results on classical inverse limits.