  We theoretically study the inception and propagation of positive and negative
streamers in CO$_2$. Our study is done in 3D, using a newly formulated kinetic
Monte Carlo discharge model where the electrons are described as drifting and
diffusing particles that adhere to the local field approximation. Our emphasis
lies on electron attachment and photoionization. For negative streamers we find
that dissociative attachment in the streamer channels leads to appearance of
localized segments of increased electric fields, while an analogous feature is
not observed for positive-polarity discharges. Positive streamers, unlike
negative streamers, require free electrons ahead of them in order to propagate.
In CO$_2$, just as in air, these electrons are supplied through
photoionization. However, ionizing radiation in CO$_2$ is absorbed quite
rapidly and is also weaker than in air, which has important ramifications for
the emerging positive streamer morphology (radius, velocity, and fields). We
perform a computational analysis which shows that positive streamers can
propagate due to photoionization in CO$_2$. Conversely, photoionization has no
affect on negative streamer fronts, but plays a major role in the coupling
between negative streamers and the cathode. Photoionization in CO$_2$ is
therefore important for the propagation of both positive and negative
streamers. Our results are relevant in several applications, e.g., CO$_2$
conversion and high-voltage technology (where CO$_2$ is used in pure form or
admixed with other gases).
