I know that finite limits do not commute with filtered colimits in general in $\mathbf{CGWH}$, nevertheless, do colimits commute with pullbacks, when we consider diagrams of the form $$\begin{matrix}&&B\\&&\downarrow\\colim_{\mathcal{C}}F&\rightarrow&A\end{matrix}$$ when $\mathcal{C}$ is a pushout, and the map $B\rightarrow A$ is a closed inclusion?
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$\begingroup$ I am not sure I am parsing your question correctly, but since I linked Strickland's paper on your other question let me ask: is what you want contained in Strickland's Lemma 3.9 or is it different? Also, you might want to ask the moderators to merge your new account with the old one: mathoverflow.net/users/128856/user09127 $\endgroup$– David WhiteSep 26, 2018 at 21:44
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$\begingroup$ Also, Strickland's 6.9 shows that some pullbacks don't commute with coequalizers. Aren't coequalizers a special case of pushouts, so this would be a negative answer to your question? $\endgroup$– David WhiteSep 26, 2018 at 21:46
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$\begingroup$ @DavidWhite In the presence of an initial object pushouts imply colimits and consequently coequalizers, but without an initial object I don't think this necessarily holds. $\endgroup$– Alec RheaSep 27, 2018 at 5:30
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$\begingroup$ @AlecRhea yes, I know. I'm trying to help the OP a bit, as you can see from other answers and comments I've given him/her. My sense is that the OP is just starting, and needs to read this Strickland paper carefully. Certainly CGWH has an initial object - it's complete and cocomplete! $\endgroup$– David WhiteSep 27, 2018 at 13:20
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$\begingroup$ You are definitely right for filtered colimit part. (It is used rather heavily in a paper I'm reading at the moment, but it really is wrong). Nevertheless, I am also interested in the pushout part (which I don't think is treated in Strickland.) $\endgroup$– user09127Sep 29, 2018 at 20:53
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