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Mathlib.CategoryTheory.Monad.Coequalizer

Special coequalizers associated to a monad #

Associated to a monad T : C ⥤ C we have important coequalizer constructions: Any algebra is a coequalizer (in the category of algebras) of free algebras. Furthermore, this coequalizer is reflexive. In C, this cofork diagram is a split coequalizer (in particular, it is still a coequalizer). This split coequalizer is known as the Beck coequalizer (as it features heavily in Beck's monadicity theorem).

Show that any algebra is a coequalizer of free algebras.

The top map in the coequalizer diagram we will construct.

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    The bottom map in the coequalizer diagram we will construct.

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      The cofork map in the coequalizer diagram we will construct.

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        Construct the Beck cofork in the category of algebras. This cofork is reflexive as well as a coequalizer.

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          The cofork constructed is a colimit. This shows that any algebra is a (reflexive) coequalizer of free algebras.

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            This is the Beck cofork. It is a split coequalizer, in particular a coequalizer.

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