I’m not familiar with category theory in depth, but homotopy type theory uses category theory for its semantic model and HoTT’s a big deal in the automated reasoning world.
> Others thought it was pretty and useful for identifying similar structures in different fields but wasn’t much use for making significant new mathematical discoveries. A sentence in Emily Riehl’s book—“The category-theoretic perspective can function as a simplifying abstraction, isolating propositions that hold for formal reasons from those whose proofs require techniques particular to a given mathematical discipline”—seems to align with the latter opinion.
Two points:
1. Being able to “lift and shift” techniques is a big deal in mathematics — so the two cases are the same. A lot of recent innovations are along those lines, where techniques in one area were applied to a new area. In that sense, category theory is wonderful because it gives us a map for how to move techniques around. You could even go so far as to say the algebra-geometry correspondence is category theory’s first “big win”, as it came about as a way to formalize that body of work.
2. Category theory is the native language of data fusion, while categorical syntax is easy to represent in an image-completion kind of way... so category theory is useful for labs working on using machine learning to fuse data and extract semantically meaningful information. Or labs working on an AI model which can suggest improvements to itself.
> Others thought it was pretty and useful for identifying similar structures in different fields but wasn’t much use for making significant new mathematical discoveries. A sentence in Emily Riehl’s book—“The category-theoretic perspective can function as a simplifying abstraction, isolating propositions that hold for formal reasons from those whose proofs require techniques particular to a given mathematical discipline”—seems to align with the latter opinion.
Two points:
1. Being able to “lift and shift” techniques is a big deal in mathematics — so the two cases are the same. A lot of recent innovations are along those lines, where techniques in one area were applied to a new area. In that sense, category theory is wonderful because it gives us a map for how to move techniques around. You could even go so far as to say the algebra-geometry correspondence is category theory’s first “big win”, as it came about as a way to formalize that body of work.
2. Category theory is the native language of data fusion, while categorical syntax is easy to represent in an image-completion kind of way... so category theory is useful for labs working on using machine learning to fuse data and extract semantically meaningful information. Or labs working on an AI model which can suggest improvements to itself.