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Suppose that the diagram

$$\require{AMScd} \begin{CD}A @>{f}>> X \\ @V{g}VV @VVV \\ Y @>>> Z\end{CD}$$

where $A,X$ and $Y$ are based spaces and $f$ and $g$ are based maps. Verify that the image of the basepoint of $A$ in $Z$ endows $Z$ with a basepoint and the maps $X \longrightarrow Z$ and $Y \longrightarrow Z$ are based maps. Now prove that for any based space $W,$

$$\require{AMScd} \begin{CD}A \wedge W @>{f\ \wedge\ \text {id}}>> X \wedge W \\ @V{g\ \wedge\ \text {id}}VV @VVV \\ Y \wedge W @>>> Z \wedge W \end{CD}$$

is also a pushout.

I am trying use universal property of pushout. For a given space $V$ and maps $F : X \wedge W \longrightarrow V$ and $G : Y \wedge W \longrightarrow V$ such that $F \circ (f\ \wedge\ \text {id}) = G \circ (g\ \wedge\ \text {id})$ we need to find out a map $H : Z \wedge W \longrightarrow V$ such that $$H \circ (\overline {g}\ \wedge\ \text {id}) = F$$ and $$H \circ (\overline {f}\ \wedge\ \text {id}) = G$$

where $\overline {f} : Y \longrightarrow Z$ and $\overline {g} : X \longrightarrow Z.$

But I am unable to find a map $H.$ Could anyone please help me in this regard?

Thanks for reading.

Anacardium
  • 2,716
  • It is not true for all $W$. Similar questions have recently been asked for unbased spaces. See https://math.stackexchange.com/q/4158852 and https://math.stackexchange.com/q/4150491 . – Paul Frost Jun 10 '21 at 22:22
  • @Paul Frost Sorry I forgot to mention that I am implicitly assuming that the space $W$ is locally compact and Hausdorff. Can it be done then? – Anacardium Jun 11 '21 at 05:01
  • Then you should add the assumptions on $W$ to your question. - I guess it is true in that case (for unbased spaces yes!), but there are slight differences between the unbased and based cases. Try to check whether the proof of the unbased exponential law $Z^{X \times Y} \approx (Z^Y)^X$ carries over to $Z^{X \wedge Z} \approx (Z^Y)^X$ for based spaces. Perhaps there are also questions in this forum. – Paul Frost Jun 11 '21 at 10:09

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