The reaction system can be described as:
$\ce{H2O <=> H+ + OH-}$
$\ce{H+ + CO3^{2-} <=> HCO3-}, \mathrm{p}K_\mathrm{a2}=10.33$
Note, the net of the first two reactions imply a rise in pH in the presence of carbonate. Further, with time and carbon dioxide exposure:
$\ce{H2O + CO2 <=> HCO3- + H+}$
$\ce{H2CO3 <=> H2O + CO2}$
And, with the introduction of bicarbonate, the $\ce{Na2CO3}{/}$NaHCO3 equilibrium apparently progresses to a pKa of 10.33, which makes it a poor pH 9 buffer.
The reaction system, upon reaching equilibrium, has been succintly presented as:
$\ce{H2CO3 <=> HCO3- + H+ <=> H2O + CO2(g)}$
which could move to the right with warming.
[EDIT] A more quantitative (and useful) illustrative tool of the pH effect is given, in logarithmic form, as the Henderson-Hasselbalch Equation, see (4) in this source, presented below:
$\ce{pH = pK + log([CO_2]/[HCO3^-])}$
However, the above assumes an equilibrium situation, and in the case of sodium carbonate added to a swimming pool with continuing exposure to air containing carbon dioxide, in time perhaps, such an equilibrium could be established measuring at the same temperature.
@hahaha
to 'contact' the user setting the bounty. (It will not autofill though.) I believe this is possible because s..he is an editor of the post, see help center, but please don't put a gun to my head if that information is not accurate. You can only notify one (additional) person per comment though. $\endgroup$@HahaHahah
which offered an autofill where I asked him / her about the motivation for this bounty. We will see «what happenz». Thank you. $\endgroup$