Alain Badiou's Mistake: Two Postulates of Dialectic Materialism

I discuss how Alain Badiou's Logics of Worlds attempts to rephrase his material dialectic philosophical project in terms of topos theory. It turns out that his account restricts to the so called local topos theory. In particular, his claim that categorical change is not genuine is based on a constrained understanding of topos theory. We then discuss his own 'postulate of materialism' and demonstrate that it has two different interpretations depending on whether it is articulated in local or elementary topos theory. While the main concerns in this paper are technical, we also address the serious consequences of topos theory that weigh Badiou's philosophical project.