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Why did CO2 go up during the interglacials over the last 800,000 years?
The release of stored CO2 can be both a cause and an effect of global warming.
As for the interglacials, they were started by the Milankovic cycles. Regularly, the changes in Earth's orbit and axial tilt lead to conditions in which high latitudes in the Northern Hermisphere - where there are continents that warm quicker than oceans - get a maximum amount of energy which starts a strong melting.
This melting then starts to set free stored CO2 (and also adds more water vapor to the atmosphere, which is also a greenhouse gas), which makes the melting stronger.
This melting lasts for thousands of years. During this time, the phase of optimal conditions in the Milankovich cycles has already ceased, and the process is no longer driven by them.
Eventually, the warming stops due to a new equilibrium. And since the conditions are no longer the optimal ones, a very slow cooling begins during which CO2 is stored away again. But this storing away takes a much longer time than setting it free. So after the melting at the start of the interglacial, we've had thousands of years of extremely slow cooling, and it was so slow that human civilisations developed their agriculture in this relative stability.
So the greenhouse effect of CO2 did not start the interglacial, but it kept it going long after optimal conditions has passed.