I’m not a researcher, but I would guess the generating process is quite iterative. Someone writes down the first crappy definition that comes to mind, they slog through some attempts to formulate theorems and prove them, gain some insights that help them refine the definition a bit, then repeat the process. I would also guess this kind of iteration makes the process more tractable than coming up with the “right” definition on the first try.
Also as additional theorems about a given category arise, and various equivalencies are proven, one often ends up with definitions that are much “neater” than the original. But there is sometimes value in learning the historical definitions.
I like this model.
I’m not a researcher, but I would guess the generating process is quite iterative. Someone writes down the first crappy definition that comes to mind, they slog through some attempts to formulate theorems and prove them, gain some insights that help them refine the definition a bit, then repeat the process. I would also guess this kind of iteration makes the process more tractable than coming up with the “right” definition on the first try.
Like in an examination for most of us. That’s why we fail in the time allowed.
Also as additional theorems about a given category arise, and various equivalencies are proven, one often ends up with definitions that are much “neater” than the original. But there is sometimes value in learning the historical definitions.