202508310108

Tags : Category Theory

Left Adjoint in a monadic adjunction create coequalizers

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Theorem

If $F,U$ for a monadic adjunction over $C,D$, then

1. $U:D\to C$ creates coequalizer of $U$-split pairs
2. For any $d\in D$, there is a coequalizer diagram involving the counit of the adjunction,

We use the equivalence of the category $D$ with the category $C^T$, where $T$ is the monad and $K$ is the equivalence.

If $f,g:A\rightrightarrows B$ is a $U$-split pair in $D$, then commutativity $U=U^K$ implies $Kf,Kg:KA\rightrightarrows KB$ us a $U^T$ split pair in $C^T$. Then Monadic forgetful functors strictly create coequalizers of Usplit pairs, then any inverse equivalence to $K$ maps the data to the coequalizer of $f,g$. This works because $K$ presreves and reflects all coequalizers.

For the second part we can make the Split Coequalizer diagram: and we are done.

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References

• Equivalence of Categories
• Equivalences Reflect, Preserve and Create Colimits
• Monadic forgetful functors strictly create coequalizers of Usplit pairs
• Split Coequalizer
• Monads and Comonads