ct.category theory - Definition of forgetful functor

This is very difficult to do. The Stuff, Structure, Property approach coined by Jim Dolan, John Baez and Toby Bartels is the best formalization I've seen. Here is the synopsis from the nLab page:

Category theory frequently allows to give precise and useful formalized meanings to “everyday” terms, at least terms used everyday by practicing mathematicians.

It was indeed introduced originally in order to formalize the use of the notion “natural” in mathematics. Another frequently recurring pair of terms in math are “extra structure” and “extra properties”, to which we add the more general concept of “extra stuff”. In discussion among Jim Dolan, John Baez and Toby Bartels, the following useful formalization of these concepts in category theoretic terms was established.

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Here is a counterpoint to some of the other responses. It's true that the abstract approach of Stuff, Structure, Property may seem sketchy (especially the passage to the core groupoid) but it is in fact surprisingly correct for some very broad classes of concrete categories with very rich notions of forgetfulness.

One such class (one that I am more familiar with) are the categories Mod(T) of models of a first-order theory T. There are three very distinct ways of forgetting things in Mod(T):

• Forgetting axioms of T (Forgetting Properties)


• Forgetting parts of the language (Forgetting Structure)


• Restricting to definable substructures (Forgetting Stuff)

In the absence of evil and under other ideal conditions, the functorial characterizations of Properties, Structure, and Stuff translate to important results in model theory (various definability, interpolation, and consistency theorems). The translations are sometimes a little on the weak side, but I think that with adjustments to account for type information not captured by the theory alone, the translation can be made broader and even more precise.

To me, this is strong evidence that Stuff, Structure, Property is indeed the correct way to translate these notions of forgetfulness from the concrete to the abstract. While it's true that talking about forgetfulness without knowing what you're forgetting is nonsensical in when looking at particular instances, this approach provides a way of abstracting and even reasoning about forgetfulness in a completely general setting.

PS: Note that I am not a category theorist, I'm just a very impressed outsider. I suspect that category theorists have even stronger intuitions for Stuff, Structure, Property approach.