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