Mathematics really never proves anything real, it just pulls together some theory, facts and reasoning within an agreed on frame of reasoning - ergo a body of knowledge. The better the theory the more comprehensive the body of knowledge and perhaps the closer the theory is capable of describing and predicting "reality". Thus understood reality is defined by our ability to construct it.
CT provides us with a number of constructs that prove useful for describing conceptual structures in the problem (requirements) domain during the analysis phase of developing an information system.
The purpose of the talk is to illustrate these constructs with some examples, to request a little patience if I miss some important subtleties, and ask some questions. The early illustrations will be skipped over quickly.
The constructs are: Categories themselves (arrows, objects, composition and identity) and specification using sketches. Commuting Diagrams (Consistent Dependency) which appear absolutely fundamental to information system structure Products Limits including the pullback in particular always seem to mean something useful A 1-Cat to 2-Cat refinement which enables dynamics to be specified - see also work by Katis, Walters and others. This also reveals that consistent dependency requires consistent role and vice versa, where role is defined by a permitted pattern of actions. Specifying constraints with some additional structure. Fibration, really opfibrations (equivalently indexed categories) as a means of abstracting clusters of object types