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Register a new userCoproducts of Monads on Set
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Added by
Jiri Adamek
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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Coproducts of monads on Set have arisen in both the study of computational effects and universal algebra. We describe coproducts of consistent monads on Set by an initial algebra formula, and prove also the converse: if the coproduct exists, so do the required initial algebras. That formula was, in the case of ideal monads, also used by Ghani and Uustalu. We deduce that coproduct embeddings of consistent monads are injective; and that a coproduct of injective monad morphisms is injective. Two consistent monads have a coproduct iff either they have arbitrarily large common fixpoints, or one is an exception monad, possibly modified to preserve the empty set. Hence a consistent monad has a coproduct with every monad iff it is an exception monad, possibly modified to preserve the empty set. We also show other fixpoint results, including that a functor (not constant on nonempty sets) is finitary iff every sufficiently large cardinal is a fixpoint.
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The category of all monads over many-sorted sets (and over other set-like categories) is proved to have coequalizers and strong cointersections. And a general diagram has a colimit whenever all the monads involved preserve monomorphisms and have arbitrarily large joint pre-fixpoints. In contrast, coequalizers fail to exist e.g. for monads over the (presheaf) category of graphs. For more general categories we extend the results on coproducts of monads from [2]. We call a monad separated if, when restricted to monomorphisms, its unit has a complement. We prove that every collection of separated monads with arbitrarily large joint pre-fixpoints has a coproduct. And a concrete formula for these coproducts is presented.
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