Title: All Concepts are Kan Extensions… and Kan Extensions are Concepts: A Categorical Theory of Vector Symbolic Architectures
Abstract: It is widely understood by category theorists that Kan extensions subsume all other notions in category theory (limits, ends, initial objects, adjoints, etc.). In this talk, I introduce vector symbolic architectures (VSAs)—a distributed model of data storage and manipulation that have gained popularity alongside the neuromorphic hardware upon which engineers hope to implement VSAs. Broadly, VSAs populate a vector space with a collection of “primitive” vectors, then use two operations, ‘binding’ and ‘bundling’, to create new vectors (ex. “red” and “car” bind as “red car”). Despite their many implementations, VSAs have little unifying theory. I discuss my recent work generalising and unifying VSAs. I focus on extending from vectors to co-presheaves and demonstrating that bind and bundle arise as right Kan extensions of the external tensor product and direct sum.
