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In overcategory, it is shown that the forgetful functor a↓C→C reflects limits, and it is mentioned that this is a consequence of the undercategory being the category of algebras for the monad b↦a∐b. What about the comma category a↓R, where R:C→D and a∈D? On the one hand, it seems like the diagrammatic proof still goes through and the forgetful functor a↓R→C which takes (a→Rx)↦x reflects limits. On the other hand, I cannot see a monad (or even just an endofunctor) for which a↓R comprises the algebras. Seems like it wants to be the functor x↦a∐Rx, but this is not an endofunctor. Is there a better way to understand how a↓R→C behaves with respect to limits?
To answer the title question: no, the comma category need not be monadic. One can easily contrive an example where C is non-empty but (a↓R) is empty. (For instance, take C to be the category of non-trivial rings, D the category of all rings, R the inclusion, and a a trivial ring.)
Ok yes, that’s excellent. Thank you. So a↓R is definitely not monadic. But even in this case, the functor a↓R→C reflects limits (vacuously).
If R doesn’t preserve limits, then I don’t see how to show that (a↓R)→C reflects them.
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