I do like that Rosetta Stone paper you linked, thanks for that. And I also recently finished going through a set of applied category theory lectures based on that book you linked. That’s exactly the sort of thing which informs my intuitions about where the field is headed, although it’s also exactly the sort of thing which informs my intuition that some key foundational pieces are still missing. Problem is, these “applications” are mostly of the form “look we can formalize X in the language of category theory”… followed by not actually doing much with it. At this point, it’s not yet clear what things will be done with it, which in turn means that it’s not yet clear we’re using the right formulations. (And even just looking at applied category theory as it exists today, the definitions are definitely too unwieldy, and will drive away anyone not determined to use category theory for some reason.)
I’m the wrong person to write about the differences in how mathematicians and physicists approach group theory, but I’ll give a few general impressions. Mathematicians in group theory tend to think of groups abstractly, often only up to isomorphism. Physicists tend to think of groups as matrix groups; the representation of group elements as matrices is central. Physicists have famously little patience for the very abstract formulation of group theory often used in math; thus the appeal of more concrete matrix groups. Mathematicians often use group theory just as a language for various things, without even using any particular result—e.g. many things are defined as quotient groups. Again, physicists have no patience for this. Physicists’ use of group theory tends to involve more concrete objectives—e.g. evaluating integrals over Lie groups. Finally, physicists almost always ascribe some physical symmetry to a group; it’s not just symbols.
So your path-based approach to category theory would be analogous to the matrix-based approach of group theory in physics? That is, removing the abstraction that made us stumble into theses concepts in the first place, and keeping only what is of use for our applications?
I would like to see that. I’m not sure that your own proposition is the right one, but the idea is exciting.
I do like that Rosetta Stone paper you linked, thanks for that. And I also recently finished going through a set of applied category theory lectures based on that book you linked. That’s exactly the sort of thing which informs my intuitions about where the field is headed, although it’s also exactly the sort of thing which informs my intuition that some key foundational pieces are still missing. Problem is, these “applications” are mostly of the form “look we can formalize X in the language of category theory”… followed by not actually doing much with it. At this point, it’s not yet clear what things will be done with it, which in turn means that it’s not yet clear we’re using the right formulations. (And even just looking at applied category theory as it exists today, the definitions are definitely too unwieldy, and will drive away anyone not determined to use category theory for some reason.)
I’m the wrong person to write about the differences in how mathematicians and physicists approach group theory, but I’ll give a few general impressions. Mathematicians in group theory tend to think of groups abstractly, often only up to isomorphism. Physicists tend to think of groups as matrix groups; the representation of group elements as matrices is central. Physicists have famously little patience for the very abstract formulation of group theory often used in math; thus the appeal of more concrete matrix groups. Mathematicians often use group theory just as a language for various things, without even using any particular result—e.g. many things are defined as quotient groups. Again, physicists have no patience for this. Physicists’ use of group theory tends to involve more concrete objectives—e.g. evaluating integrals over Lie groups. Finally, physicists almost always ascribe some physical symmetry to a group; it’s not just symbols.
So your path-based approach to category theory would be analogous to the matrix-based approach of group theory in physics? That is, removing the abstraction that made us stumble into theses concepts in the first place, and keeping only what is of use for our applications?
I would like to see that. I’m not sure that your own proposition is the right one, but the idea is exciting.
Yup, that’s basically the idea.