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Stream: theory: category theory

Topic: adjoints are maps of adjoints?

Christian Williams (Feb 07 2021 at 00:12):

For a right adjoint $G:A\to B$ to preserve limits means that $G$ is a morphism of the diagonal-limit adjunction. Is there a proof of this preservation purely in terms of co/units rather than the hom?

I am wondering whether more generally we have that often or always adjoints commute with co/units, hence are maps of adjunctions. Does anyone have ideas or know a reference for this topic?

Morgan Rogers (he/him) (Feb 07 2021 at 11:25):

This is most easily expressed in terms of *Beck-Chevalley* conditions (and conversely it's the best way I've found for getting a hold of what those conditions mean)! You can easily construct a square out of the diagonal/constant functors, $G$ and the lifting of $G$. There is a canonical transpose map involving $G$, the lifting of $G$ and the adjoints of the constant functors. The relevant Beck-Chevalley condition says that this comparison is an isomorphism.

Christian Williams (Feb 07 2021 at 16:48):

ah yes, that is helpful. thanks. I'll think more and come back.