Understanding the Infinite II: Coalgebra

In this paper we give an account of the rise and development of coalgebraic thinking in mathematics and

computer science as an illustration of the way mathematical frameworks may be transformed. Originating

in a foundational dispute as to the correct way to characterise sets, logicians and computer scientists

came to see maximizing and minimizing extremal axiomatisations as a dual pair, each necessary to represent

entities of interest. In particular, many important infinitely large entities can be characterised in

terms of such axiomatisations. We consider reasons for the delay in arriving at the coalgebraic framework,

despite many unrecognised manifestations occurring years earlier, and discuss an apparent asymmetry

in the relationship between algebra and coalgebra.