Cultural attractor landscapes describe the time-evolution of cultural variants (i.e. behaviors, artifacts) over successive transmission events. Because cultural attractors are emergent products of dynamic populations of \textit{cognitive} landscapes, which are in turn emergent products of individual experience within a culture, stable cultural attractor landscapes cannot be taken for granted. Yet, little is known about how cultural attractors form, change, or stabilize. We present an agent-based model of cultural attractor dynamics, which adapts a cognitive model of unsupervised category learning to a multi-agent sociocultural setting wherein individual learners provide the training input to each other. We highlight three interesting behaviors exhibited by our model that are not accounted for in other models of cultural evolution: First, we find that some noise is beneficial to stabilizing cognitive alignment. Second, we find that long learning times may destabilize and limit the complexity of cultural repertoires, while critical or sensitive periods of learning enhance stability. Third, we find that larger populations develop less complex, but more stable patterns of alignment, and that this effect can be moderated by network structure. These results suggest that additional complexity may be needed in models of cultural evolution to adequately understand how human-level culture develops.