This article assesses how people reason with categories whose features are related in causal cycles. Whereas models based on causal graphical models (CGMs) have enjoyed success modeling category-based judgments as well as a number of other cognitive phenomena, CGMs are only able to represent causal structures that are acyclic. A number of new formalisms that allow cycles are introduced and evaluated. "Dynamic Bayesian networks" (DBNs) represent cycles by unfolding them over time. "Chain graphs" augment CGMs by allowing the presence of undirected links that model feedback relations between variables. "Unfolded chain graphs" are chain graphs that unfold over time. An existing model of causal cycles ("alpha centrality") is also evaluated. Four experiments in which subjects reason about categories with cyclically related features provided evidence against DBNs and alpha centrality and for the two types of chain graphs. Chain graphs--a mechanism for representing the equilibrium distribution of a dynamic system--may thus be good candidates for modeling how people reason causally with complex systems. Applications of chain graphs to areas of cognition other than category-based judgments are discussed.