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Well Founded Coalgebras

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Joint with Tallinn.

Categorical set theory explores ideas taken from set theory to develop mathematics using category theoretic tools. It began in the 1970s when Mikkelsen and Osius interpreted recursion and epsilon-structures in an elementary topos. Well founded coalgebras generalise epsilon-structures to give approximations to the free algebra for a functor even when this does not exist.

The main recursion theorem is based on the one of von Neumann for ordinals. Originally that was based on fixed points in complete lattices, but in order to consider more general categories and functors, we must use Pataraia’s Theorem for dcpos instead.

However, for our more complicated constructions, we need to find a scalpel not a sledgehammer, so a more subtle form of Pataraia’s Theorem is developed.

The paper develops analogues of the recursion theorem and Mostowski extensional quotient potentially in much more general categories, with factorisation systems intead of 1-1 functions.

The obvious first application of this generalisation replaces Set with Pos to study the different forms of intuitionistic ordinals that were introduced in the 1990s. This in turn leads to a formulation of transfinite iteration of functors, based on a categorical axiom instead of the set-theoretic axiom-scheme of replacement.

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