Definition of forgetful functor

Question

Is there an actual definition of "forgetful functor?" Wolfram MathWorld defines it in terms of functors from algebraic categories to the category of sets, but then says, "Other forgetful functors..." which seems an odd way to "define" something. In any event, it seems to exclude, say, the functor which takes topological groups to topological spaces (or groups.)

Wikipedia contains no definition, just lots of examples.

Is this one of those cases where we don't need a strict definition, we just recognize it when we see it?

Answer 1

The treatment on the nLab seems likely to be the state of the art here. There's no generally accepted definition, and to get one you might have to generalize to the stuff-structure-property perspective in which every functor can be seen as forgetful. This makes functors such as the projections $\mathcal C\times\mathcal D\to\mathcal C$ "forget stuff," namely, the $\mathcal D$ object of a pair.

If the idea of forgetting stuff feels unnatural, then maybe you want forgetful functors only to forget properties and structure-but this includes all faithful functors.

I think it's clear that every fully faithful functor $F:\mathcal D\to \mathcal C$ can reasonably be called forgetful, since it forgets the property of being in the subcategory $F(\mathcal D)$.

I don't see that there's such a natural description of what $F$ forgets when $F$ is merely faithful, but neither do I know of any way to specify when $\mathcal D$ is a category of $\mathcal C$-objects with structure, which might describe the most natural class of forgetful functors, better than the existence of a faithful functor $\mathcal D\to \mathcal C$. The comments under this question discuss the case of defining categories structured over sets, though not conclusively-but I'm at least convinced that there's no extant definition better than faithfulness.

So, it appears your options are "all functors," "faithful functors," or this.

Answer 2

I think that calling a functor forgetful is making a choice of a particular perspective on the source category, rather than declaring that the functor has some property.

When you choose to think of a functor $F : C \to D$ as forgetful, you're choosing to think of the objects in $C$ as objects in $D$ with extra structure in some vague sense. After all, from the perspective of abstract category theory $C$ is some opaque collection of objects and morphisms, which have no internal structure except what internal structure you choose to provide them, e.g. by choosing suitable forgetful functors.

(Incidentally, this is also an answer to a different question that one might want to ask about category theory, namely "what's the point of objects if all of the structure of a category is encoded in its morphisms?" One answer is that how you choose to describe the objects of a category can be thought of as declaring an intent to study a particular collection of forgetful functors out of your category. For example, choosing to describe the objects as sets with extra structure corresponds to declaring an intent to study a particular forgetful functor to $\text{Set}$.)

"Faithful functor" is not a terrible approximation but let me point out one example where it doesn't apply: many people would be happy to say that there is a forgetful functor from schemes to topological spaces, but this functor fails to be faithful. From my perspective, though, all that people mean when they say this is that they're declaring an intent to study schemes as topological spaces with extra structure (namely the structure sheaf).