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Explore a mathematical colloquium lecture introducing Cartesian Fermat categories as a new framework within differential category theory. Discover how these categories bridge the gap between Cartesian differential categories and Fermat theories, providing enhanced categorical foundations for differential calculus, differentiable programming, and automatic differentiation techniques used in machine learning. Learn about the theoretical development that extends beyond the limitations of traditional Fermat theories by removing dependencies on Lawvere theory structures and map multiplication requirements. Understand the main theorem demonstrating that every Cartesian Fermat category is indeed a Cartesian differential category, establishing this new class as a proper analogue to existing frameworks. Benefit from an accessible introduction to Cartesian differential categories that welcomes newcomers to the field, making complex mathematical concepts approachable for those interested in differential categories and their applications to modern computational methods.
