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Explore CAP (Categories, Algorithms and Programming), a multi-package open-development software project for algorithmic category theory in this colloquium talk. Discover how CAP organizes computer algebra through the 3-category of "doctrines" - defined as 2-categories of structured categories, structure-preserving 2-functors, and natural transformations between them. Learn about "categorical towers," a highly economic and modular approach to creating various computational contexts in computer algebra through compositions of 2-functors applied to specific categories. Understand the computational challenges that arise from this modular architecture, where each layer treats the layer below as a blackbox, making hand-optimization impossible without breaking modularity. Examine the solution through CompilerForCAP, a category-theory-aware compiler that eliminates categorical abstraction to produce efficient low-level code that would be nearly impossible to write by hand while maintaining error-free execution.
