Does forgetting colimits preserve colimits?

Question

For each regular cardinal $\kappa$ let $\operatorname{Cat}_{\kappa}$ be the $(2,1)$-category of small categories with $\kappa$-small colimits, and functors that preserve those colimits. For each pair of regular cardinals $\tau > \kappa$ there is an evident forgetful functor $U_{\tau, \kappa}: \operatorname{Cat}_{\tau} \rightarrow \operatorname{Cat}_\kappa$. It can be seen that $U_{\tau,\kappa}$ preserves $\kappa$-small coproducts, as these can be computed in terms of $\kappa$-small products, which are preserved by virtue of the fact that $U_{\tau,\kappa}$ is a right adjoint.

Question: Does $U_{\tau,\kappa}$ preserve all $\kappa$-small colimits?

Feel free to assume $\kappa$ to be uncountable or even bigger if it helps. I'm ultimately interested in the $\infty$-categorical version of this question so answers in that context are welcome as well.

Answer 1

Here is an example showing that $U_{\tau, \kappa}$ does not preserve pushouts for any pair of regular cardinals $\tau > \kappa$. Let $C = \tau$, where here $\tau$ is thought of as an ordinal (i.e. the smallest ordinal of cardinality $\tau$) and $C$ is regarded as a category where there is an arrow $x \rightarrow y$ if and only if $x \leq y$. Using the ordinal identity $2 \cdot \tau = \tau$ we may construct full subcategories $C_0$ and $C_1$ of $C$ where $C_i$ contains those objects that correspond to elements of $2 \cdot \tau$ of the form $(x, i)$. The inclusions $C_i \rightarrow C$ admit left adjoints, so we have a span $C_0 \leftarrow C \rightarrow C_1$ inside $\operatorname{Cat}_{\tau}$. Let $D_\tau$ be its pushout, and let $D_\kappa$ be the pushout of its image under $U_{\tau, \kappa}$. We will show that $D_\tau \neq D_\kappa$.

The object $D_\tau$ in $\operatorname{Cat}_{\tau}$ is defined by the following property: for each $E$ in $\operatorname{Cat}_{\tau}$, the category of $\tau$-small colimit preserving functors $D_\tau \rightarrow E$ is equivalent to the category of $\tau$-small colimit preserving functors $C \rightarrow E$ which factor through the localizations $C \rightarrow C_i$. The latter is the same as the category of $\tau$-small colimit preserving functors which invert the arrow $x \rightarrow x+1$ for every $x$ in $C$. A functor has this property if and only if it is the left Kan extension of its restriction along $\emptyset \rightarrow C$. It follows that $D_\tau$ is the initial object of $\operatorname{Cat}_\tau$, so that $D_\tau$ is the singleton category.

Arguing as above, to show that $D_\kappa \neq D_\tau$ it now suffices to show that there is an object $E$ in $\operatorname{Cat}_\kappa$ and a non-constant $\kappa$-small colimit preserving functor $F: C \rightarrow E$ which inverts the arrow $x \rightarrow x + 1$ for every $x$ in $C$. We may take $E = 2$ to be the walking arrow, and $F$ to be the functor sending $x$ to $0$ if and only if $x < \kappa$.

Answer 2

Here's an example to play around with, with $\kappa = \omega$ and $\tau = \omega_1$ and $Vect = Vect_k$ for $k$ a finite field, look at the pushout of $Vect \leftarrow Set \to Vect$, where both functors are the free functor. The question becomes whether $$Ind(Vect_{\omega_1} \times_{Set_{\omega_1}} Vect_{\omega_1}) \to Ind(Vect_{\omega_1}) \times_{Ind(Vect_{\omega_1})} Ind(Vect_{\omega_1})$$ is an equivalence. I think maybe it's not essentially surjective: the domain consists of Ind-systems of countable-sets-equipped-with-two-vector-space-structures, whereas the codomain consists of ind-countable-sets-equipped-with-two-ind-countable-vector-space-structures. If we restrict to $\omega$-indexed ind-systems, it already looks problematic to try to rectify the latter to the former.

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(EDIT: The following argument is problematic --see the comments below.) Maybe the answer is actually yes?

Translating across the equivalence $Ind_\kappa : Cat_\kappa {}^\to_\leftarrow Pr^L_\kappa : (-)_\kappa$ (and similarly for $\tau$), we are asking whether the functor $\Phi: \mathcal K \mapsto Ind_\kappa(U^\tau_\kappa(\mathcal K_\tau)) : Pr^L_\tau \to Pr^L_\kappa$ preserves pushouts. Since $\Pr^L_\kappa$ and $Pr^L_\tau$ are closed under colimits in $Pr^L$, we may compute our pushouts there.

So consider a span $B \xleftarrow F A \xrightarrow G C$ in $Pr^L_\tau$, and let $F^\ast, G^\ast$ be the right adjoints of $F, G$ respectively. The pushout of this span in $Pr^L$ is the pullback $B \times_A C$ computed in $Cat$ using $F^\ast, G^\ast$. For some regular cardinal $\rho \geq \tau$ which is large enough that $F^\ast, G^\ast$ preserve $\rho$-compact objects, consider the fully faithful inclusion $Y^\ast : A \to A' = Ind_\tau(A_\rho)$, with left adjoint $Y$. Then the pullback is equally $B \times_{A'} C$, since $Y^\ast : A \to A'$ is fully faithful. Moreover, $(FY)^\ast$ and $(GY)^\ast$ preserve $\tau$-compact objects.

Now, the pushout of $\Phi B \xleftarrow{\Phi F} \Phi A \xrightarrow{\Phi G} \Phi C$ in $Pr^L$ is the pullback in $Cat$ $\Phi B \times_{\Phi A} \Phi C$, computed using the right adjoints $(\Phi F)^\ast, (\Phi G)^\ast$. Since $\Phi$ is a left 2-adjoint, it preserves localizations. Since $Y$ is a localization, so is $\Phi Y$, and thus $(\Phi Y)^\ast$ is fully faithful. So this pullback is equally $\Phi B \times_{\Phi A'} \Phi C$.

In other words, we have reduced to the case where $F^\ast$ and $G^\ast$ preserve $\tau$-compact objects. Note that $(-)_\tau$ is in fact 2-functorial in all functors preserving $\tau$-compact objects, and $U^\tau_\kappa$ and $Ind_\kappa$ are 2-functorial in all functors (in the sense that they carry functors to functors and natural transformations to natural transformations coherently). Moreover, these functors commute with pullbacks as computed in $Cat$ -- $(-)_\tau$ and $U^\tau_\kappa$ preserve all limits in this sense while $Ind_\kappa$ preserves $\kappa$-small limits as a functor $Cat \to Cat$. So $\Phi$, as the composite of these functors, likewise preserves the relevant $Cat$-pullback as desired.