Conditions for adjoints to preserve monos?

Question

Given an adjoint pair $L \dashv R$ and a mono $f \in \text{Hom}(X, RY)$, what are some conditions which will guarantee $\tilde{f} \in \text{Hom}(LX, Y)$ is still a mono? Obviously this becomes easy if we restrict our attention to the subcategories which are equivalent by the adjunction, but I want a more general condition. This feels like something people would have thought about, but I can't seem to find any references for it.

Since $\tilde{f} = \epsilon_Y \circ Lf$ (where $\epsilon$ is the counit of the adjunction), it would be enough to show that $Lf$ and $\epsilon_Y$ are both monos. However this condition is both restrictive and hard to meet. As $L$ is a left adjoint, there's no reason for it to play nice with monos (unless that mono is split, etc). Similarly, $\epsilon_Y$ tends to be an epi, not a mono.

One thing that comes to mind is $\epsilon_Y$ doesn't all have to be mono. It suffices to have $\epsilon_Y \upharpoonright \text{im}(Lf)$ a mono (provided your category is rich enough for this to make sense). That said, I'm not sure if we can do better. I'm looking for as many ways as possible, because I'm not sure which (if any) will be helpful for the problem that gave rise to this question.

Thanks in advance!

Answer 1

For a general adjoint pair $L\dashv R$, given a monic $f:X\to RY$, its adjunct $\tilde f=\epsilon_Y\circ Lf$ being monic requires $Lf$ to be monic also (indeed, in general if $p\circ q$ is monic, then $q$ must be monic). At this point, checking that $\epsilon_Y\circ Lf$ is mono is probably done most easily by having $\epsilon_Y$ be monic also (otherwise, you're just as well off checking the adjuncts of the $f:X\to RY$ you care about on a case-by-case basis).

In fact, suppose $L\dashv R$ preserve all monos in the sense you defined, then in particular $\epsilon_Y:LRY\to Y$ will have to be monic because it is the adjunct of the identity $\def\id{\operatorname{id}}\id_{RY}$, which is monic. (More generally, once one of the $f:X\to RY$ you care about becomes an isomorphism, you're forced to take $\epsilon_Y$ monic.) Combining these, we find that

$L\dashv R$ preserves monos in the sense you define if and only if the counits are monic and $L$ preserves monos.

so in some sense you really can't do better than this, except to find sufficient conditions for $L$ to preserve monos and for the counits to be monic.

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For example, proposition 2.4 here tells us that the counits are split monic if and only if $R$ is full, which is a relatively easy-to-check condition. As for $L$ preserving monos, a strong sufficient condition would be that $L$ preserves limits (e.g., if $L$ is also a right adjoint) so you have for example the following sufficient condition:

$L\dashv R$ preserves monos in the sense you define whenever it is a part of an adjoint triple $F\dashv L\dashv R$ and $R$ is full.

For example, say $L:\mathbf{Top}\to\mathbf{Set}$ is the forgetful functor, then it has a left and a right adjoint, and the right adjoint $R:\mathbf{Set}\to\mathbf{Top}$ endows sets with the codiscrete topology. The right adjoint is also fully faithful simply because maps of sets are automatically continuous as maps between codiscrete spaces. (However, in this case it is already easy to check that $L$ preserves monics, and $\epsilon_Y:LRY\to Y$ is just the identity on the set $Y$ without much effort.)