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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. Assume that every object of $D$ is a $\Phi$-colimit. 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 assumptions that $\mathbf C$ have a dense subcategory of $\Phi$-compact objects and that objects of $\mathbf D$ be $\Phi$-colimits are rather strong. Can we relax these 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 $\mathbf D$?

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 $\mathbf D$?

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. Assume that every object of $D$ is a $\Phi$-colimit. 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 $\mathbf D$?

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. Assume that every object of $D$ is a $\Phi$-colimit. 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 assumptions that $\mathbf C$ have a dense subcategory of $\Phi$-compact objects and that objects of $\mathbf D$ be $\Phi$-colimits are rather strong. Can we relax these 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 $\mathbf D$?

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 $\mathbf D$?

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. Assume that every object of $D$ is a $\Phi$-colimit. 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 $\mathbf D$?