What chemical compound exists at the cathode of a lithium-ion battery when the lithium is not there?
The cathode is usually described as $\ce{LiCoO2}$, so does it become $\ce{CoO2}$?
$\ce{CoO2}$ puts cobalt in an oxidation state of +4, which is possible but rare and unstable. I think $\ce{CoO2}$ doesn't even have a Wikipedia page.
The general textbook formula for the Li ion battery is
$\ce{Li_{1-a}Ni_{1-x-y}Mn_{x}Co_{y}O2}$
, and the simplest/earliest type has $y=1$, i.e. cobalt-only. These "mixed (lithium transition metal) oxides" have a layered structure (see e.g. wikipedia), where you can relatively easily electrochemically remove (and later reinsert) a part $a$ of the Li atoms, without destroying the whole structure. You cannot increase $a$ to much more than 0.1 - 0.3 or so, depending on the exact formulation/grain size/quality/temperature/etc., before the structure does break down.
Even the "uncharged" $a=0$ Li-cobalt(III) oxide gives off oxygen (!) if heated above 180°C. (It is produced by tempering at ~800°C in a pure oxygen atmosphere.) With $a>0$, this happens earlier already, and basically always leads to a runaway destruction of your whole battery setup. With higher contents of Ni and Mn, this danger becomes less pronounced. I´m not sure the Co-only variant was ever commercialised. (?)
Simply speaking, a cobalt-rich battery blows up when overcharged, and in ones with more nickel and manganese the layered structure still breaks down, and you get somewhat stable Ni/Mn(III/IV) species (and still quite a lot of overheating/general destruction). In any case this happens long, long before you reach the ideal $\ce{CoO2}$ (or generally $\ce{MO2}$) stochiometry.
If you look closely (e.g. via XRD) at the crystal structure during charging, you seemingly find that there are a number of intermediate deformed structures, and even more mixed/disordered states in between. And that of course becomes even more complicated (or washed out) with a (more or less) random arrangement of different transition metals in the structure, which will depend also on the specific synthesis route.
