#coalgebra

The basic observation of category theory that /universal/ constructions always /come in pairs/, has motivated research on the duality between algebras and coalgebras, which provides a bridge between models of /static/ (constructive, data-oriented) and /dynamical/ (observational, behaviour-oriented) systems. At the /programming/ level, the intuitive symmetry between /data/ and /behaviour/ provides evidence of such a duality, in its canonical /initial-final/ specialisation.

The two most important things to keep in mind about the difference between functions and distributions is the following.    1.   Functions pull back, while distributions push forward.    2.   Functions form commutative algebras, while distributions form cocommutative coalgebras.

[I]magine you want to connect two knowledge bases that describe different aspects of transport patterns. The first knowledge base describes travel activities in relation to individuals. The second knowledge base circumstantiates the volume of traffic on public and individual transport in temporal terms. [...] In a seemingly simple piece of knowledge such as ‘Normally, the likelihood of road congestion is smaller on weekends’, one implicitly makes use of default logics (‘normally’), probabilistic reasoning (‘the likelihood’) and temporal knowledge (‘weekends’) under a quantitative regime (‘smaller’). If we link this in with both knowledge bases that we seek to combine, we will moreover encounter spatial reasoning (to cater for distances), epistemic principles (knowledge of individuals), deontic constructs (obligations that arise e.g. through social norms) adorned with constructs that formalise the joint behaviour of individual agents.

The theory of universal algebras was well-developed in the twentieth century. [It] provides a basis for model theory, and [provides] an abstract understanding of familiar principles of induction, recursion, and freeness. The theory of coalgebras is considerably [less] developed. Coalgebras arise naturally, as Kripke models for modal logic, as automata and objects for object oriented programming languages in computer science, and more.

Paul Finsler believed that sets could be viewed as generalised numbers. Generalised numbers, like numbers, have finitely many predecessors. Numbers having the same predecessors are identical. We can obtain a directed graph for each generalised number by taking the generalised numbers as points and directing an edge from a generalised number toward each of its immediate predecessors. It has been shown that these generalised numbers can be "added" and "multiplied" in a natural way by combining the associated graphs. The sum a+b is obtained by "hanging" the diagram of b onto that of a so the bottom point of a coincides with the top point of b. The product a·b is obtained by replacing each edge of the graph of a with the graph of b where the graphs are similarly oriented.