Natural disasters increasingly threaten communities worldwide, creating an urgent need for rapid, reliable building damage assessment to guide emergency response and recovery efforts. Current methods typically classify damage in binary (damaged/undamaged) or ordinal severity terms, limiting their practical utility. In fact, the determination of damage typology is crucial for response and recovery efforts. To address this important gap, this paper introduces DamageCAT, a novel framework that provides typology-based categorical damage descriptions rather than simple severity ratings. Accordingly, this study presents two key contributions: (1) the BD-TypoSAT dataset containing satellite image triplets (pre-disaster, post-disaster, and damage masks) from Hurricane Ida with four damage categories (partial roof damage, total roof damage, partial structural collapse, and total structural collapse), and (2) a hierarchical U-Net-based transformer architecture that effectively processes pre-post disaster image pairs to identify and categorize building damage. Despite significant class imbalances in the training data, our model achieved robust performance with overall metrics of 0.7921 Intersection over Union (IoU) and 0.8835 F1 scores across all categories. The model's capability to recognize intricate damage typology in less common categories is especially remarkable. The DamageCAT framework advances automated damage assessment by providing actionable, typological information that better supports disaster response decision-making and resource allocation compared to traditional severity-based approaches.