We establish the foundations of categorical weave calculus, developing the diagrammatic calculus of weaves and braid varieties within the study of Calabi-Yau triangulated categories and cluster tilting theory. This is achieved by associating a perverse sheaf of triangulated categories to each Demazure weave. A central contribution is the construction and study of the categorical Lusztig cycles and their duals, which we show form simple-minded and silting collections in the category of global sections of such a sheaf of categories. These categorical collections are built using the diagrammatics of weaves and we study their behavior under changes of weaves. For instance, we show that they undergo tilts under weave mutations. En route, we develop the study of categorical weighted braid words, as canonical rigid filtered dg modules over derived preprojective algebras, and the categorical incarnation of the tropical Lusztig rules, as a gluing mechanism for such filtered objects. Appendix A contains homological results, providing a novel construction of simple-minded and silting collections from full exceptional collections, and characterizing when these arise from a highest weight structure on an abelian category.