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Stream: learning: questions

Topic: composing monadic and non-monadic adjunctions

Ralph Sarkis (Apr 20 2021 at 17:25):

I have a monadic adjunction and an adjunction that is not monadic as depicted below.
image.png

Can I say things about the composite adjunction being monadic or not without studying the composite? For example, if both adjunctions were crudely monadic, I could infer the monadicity of the composite without studying it.

Possibly helpful information: in my setting, the adjunction $L \dashv R$ is idempotent but $R$ is full and not faithful (hence the adjunction is not monadic).

John Baez (Apr 20 2021 at 22:55):

For example, if both adjunctions were crudely monadic, I could infer the monadicity of the composite without studying it.

You can do a bit better than that. The nLab says:

A further advantage of crude monadicity is this: while in general the composite of monadic functors need not be monadic, if $U_1\colon D \to C$ satisfies the hypotheses of the crude monadicity theorem and $U_2\colon C \to B$ is any monadic functor then $U_2 \circ U_1$ is monadic. See Barr and Wells, Toposes, Triples and Theories Proposition 3.5.1 for this and further results.