# A right adjoint preserves Phi-colimits if and only if the left adjoint does what?

Let $$\Phi$$ be a class of categories (e.g. filtered categories), and consider an adjunction $$L : \mathbf C \rightleftarrows \mathbf D : R$$. A $$\Phi$$-colimit is a colimit whose diagram is in $$\Phi$$. We should like to characterise when $$R$$ preserves $$\Phi$$-colimits (e.g. filtered colimits) in terms of $$L$$. We can certainly do this in some cases.

Here is a prototypical setting (which I shall spell out as it most likely informs the general setting).

Say that a $$\Phi$$-compact object is an object $$x$$ for which $$\mathbf C(x, {-}) : \mathbf C \to \mathbf{Set}$$ preserves $$\Phi$$-colimits. If $$R$$ preserves $$\Phi$$-colimits, then $$L$$ preserves $$\Phi$$-compact objects, since \begin{align*} \mathbf D(Lx, \mathrm{colim}_\phi y_i) & \cong \mathbf C(x, R(\mathrm{colim}_\phi y_i)) \\ & \cong \mathbf C(x, \mathrm{colim}_\phi (R y_i)) \\ & \cong \mathrm{colim}_\phi \mathbf C(x, R y_i) \\ & \cong \mathrm{colim}_\phi \mathbf D(Lx, y_i) \end{align*} Conversely, assume the full subcategory of $$\Phi$$-compact objects in $$\mathbf C$$ is dense. If $$L$$ preserves $$\Phi$$-compact objects, then $$R$$ preserves $$\Phi$$-colimits, since for a $$\Phi$$-compact object $$x$$, we have \begin{align*} \mathbf C(x, R(\mathrm{colim}_\phi y_i)) & \cong \mathbf D(Lx, \mathrm{colim}_\phi y_i) \\ & \cong \mathrm{colim}_\phi \mathbf D(Lx, y_i) \\ & \cong \mathrm{colim}_\phi \mathbf C(x, R y_i) \\ & \cong \mathbf C(x, \mathrm{colim}_\phi (R y_i)) \end{align*} after which we use density.

A consequence, for instance, is that a right adjoint between locally finitely presentable categories is finitary (preserves filtered colimits) if and only if its left adjoint preserves finitely presentable objects.

However, the restriction that $$\mathbf C$$ have a dense subcategory of $$\Phi$$-compact objects is rather strong. Can we relax this assumption at all? In other words, are there natural conditions on $$L$$ that hold if and only if $$R$$ preserves $$\Phi$$-colimits, without (m)any assumptions about $$\mathbf C$$ and $$\mathbf D$$?

• The dual situation of a left adjoint functor that preserves finite limits has been considered a lot (The geometric morphisms between topos) and I have never seen a condition only involving the right adjoint functors. Commented Jul 13, 2022 at 15:38
• @SimonHenry: thanks for pointing that out. I found essentially this question in the context of geometric morphisms. An answer to one will most likely answer the other, but I think it's useful to phrase it more generally, so I'll keep my question open. Commented Jul 13, 2022 at 15:44
• Would an answer in the case where C and D are presentable be satisfactory ? Or is that too restrictive ? Commented Jun 24, 2023 at 16:21
• @MaximeRamzi: the case where $\mathbf C$ is presentable is already spelled out in my question. Commented Jun 24, 2023 at 16:31
• Not exactly - the case where $C$ is $\omega$-presentable and $\Phi$ = filtered colimits is spelled out. For example there is a more general criterion for the same $\Phi$ but when $C$ is only compactly assembled, which is weaker than compactly generated. But you might also be interested in different $\Phi$ Commented Jun 24, 2023 at 16:37