Delay-Inverse Systems

In Shape Theory (1982), S. Mardesic and J. Segal presented abstract shape theory as an application of the theory of inverse systems. In the first step of their work, they defined inverse systems in any category C and formed a category C-Inv of such C-inverse systems by defining morphisms between pairs of them. More recently, in our study of Cech systems of topological ˇ spaces, we accidentally discovered a new class of systems which we defined to be delay-inverse systems. They are generalized inverse systems in the sense that commutative diagrams of morphisms that appear in inverse systems only (and always) appear, but with some “delay.” As with inverse systems, for any category C, there is a class of C-delay-inverse systems. Every C- inverse system is a C-delay-inverse system, but the reverse is not true. Is  there a category of C-delay-inverse systems?

In our recent work we have in fact defined two distinctly different categories of  C-delay-inverse systems, C-delay1 and C-delay2. The relationship among the three categories is that C-Inv is a full subcategory of C-delay1 and of C-delay2. Moreover, C-delay1 a not necessarily full subcategory of C-delay2. In this presentation we will go into some detail about how to form the morphisms needed for these 3 categories. We will also provide some discussion about Cech systems.