# Talk:Forgetful functor

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Field: Foundations, logic, and set theory

## §Faithfulness ?

I have seen "forgetful functor" defined as equivalent to "faithful functor" (e.g. Abstract and Concrete Categories A/H/S), so it seems that the real distinction between the terms is in usage -- "forgetful" is simply evocative of the times when faithful functors "seem forgetful". The article gives the impression that the notion of "forgetful functor" is an actual mathematical definition, and that 3 days after the fact, we "realise" that all forgetful functors are faithful, or that all concrete categories have faithful functors into Set (how else would a concrete category be defined??) I think it's important to be clear about what is fuzzy intuitive explanation and what is a definition. Mistaking the former for the latter leads one to make crazy statements like, "in fact, one can define, ..." Revolver 09:22, 27 Apr 2005 (UTC)

Really, now that I think about it, there IS a distinction between faithful and forgetful functor. A faithful functor is just that -- a functor that is faithful. But a forgetful functor is the second piece of data that comprises the structure of a concrete category, i.e. it's a faithful functor, but it's not just any faithful functor, it's the one we have in mind when we're thinking of a concrete category. So, it seems this article should have a lot of overlap and connection with concrete category. Revolver 10:16, 27 Apr 2005 (UTC)
For example, the forgetful functor from the category of rings to the category of abelian groups assigns to each ring R the underlying additive abelian group of R. To each morphism of rings is assigned the same function considered merely as a morphism of addition between the underlying groups.
Sure, this is just another way of saying that the category of rings is a concrete category over the category of abelian groups, with the obvious forgetful functor. Revolver 10:18, 27 Apr 2005 (UTC)
That seems right to me. A forgetful functor is some particular faithful functor that defines a concrete category. And every concrete category has a defining forgetful functor. Since we are used to working with concrete categories concretely, the defining forgetful functor is the "obvious" one. -- Fropuff 14:23, 2005 Apr 27 (UTC)
FYI, Note that forgetful+functor in nLab makes the argument that all functors are forgetful, it just depends on what they forget. linas (talk) 02:56, 23 August 2012 (UTC)

## §forgetful functors are not always faithful

For example, the forgetful functor from the category of manifolds with spin structure to the category of manifolds is not faithful; there may be many bundle morphisms of the spin structure which restrict to the same morphism of the manifold (or even bundle morphism of the frame bundle with Riemannian structure. this can be two to one). -lethe talk 08:45, 22 November 2005 (UTC)

If that is the case, it would be nice if you could provide a precise definition of "forgetful functor". I can't find any such precise definition in the article. There is a vague universal algebra definition given, and there is a description of "3 types" of forgetful functors, but nowhere do I find exactly what a forgetful functor is! Definitions, please. (Apart from the one I gave above.) Revolver 22:35, 25 January 2006 (UTC)
It's tricky. It may be the case that there is no precise definition. I want to do a moderate rewrite of this article to accommodate that view. For concrete categories, we might just define a forgetful functor as a functor to Set. But can we make a definition which catches all cases? I think to be completely general, we might find that any functor can be a forgetful functor. I think the best strategy would be to avoid giving a precise fully general definition, and just mention all the possibilities. -lethe talk 22:48, 25 January 2006 (UTC)
FYI, note that forgetful+functor in nLab agrees that there is no 'precise definition' and makes the argument that all functors are forgetful, it just depends on what they forget. It gives a reference where John Baez expounds on this at length. The three things that can be forgotten are listed as -- structure, stuff, properties. Here: stuff,+structure,+property in nLab linas (talk) 02:56, 23 August 2012 (UTC)

## §Relation between from… and to… categories

How is it referred to? If a functor acts from the "complex" category to the "simple" one, then how to say "complex" is a … of "simple", or versa? Incnis Mrsi (talk) 13:13, 14 December 2012 (UTC)