3.1.4. Category Theory (CT)

Together with the DPF-example, this example should demonstrate the potential of our framework to support a shift of paradigm from string-based to diagrammatic specification formalisms.

Located on the same abstraction (modeling) level as the examples FOL and ALC and reflecting the viewpoint that a category is a graph equipped with composition and identities, we outline a diagrammatic version of the theory of small categories.

In fact, we take a step back from Ehresmann's sketches. We restrict ourselves to the language of graphs, composition and identities and reconstruct the concepts commutative diagram, limit and colimit, respectively, by means of diagrammatic first-order statements formulated in this restricted language. The universal properties defining the different kinds of limits and colimits, respectively, do have a uniform and very simple logical structure; thus, we need only a very restricted form of first-order statements to express them. In the light of this observation, our envisioned diagrammatic version of the theory of small categories goes beyond Ehresmann's sketches in the sense that we allow for utilizing arbitrary first-order statements. Even if we do not need the full "first-order power" to define limits and colimits, this power will be probably useful (or even necessary) to formulate category theoretic statements and to prove them.
