5

Given the definition of a category $\mathbb{C}$ (that I rewrite just to agree on the notation), that consists of

  • a collection of objects $\mathsf{Obj} ( \mathbb{C} )$;
  • a collection of $\mathsf{Arr} ( \mathbb{C} )$, with $f \in \mathsf{Arr} ( \mathbb{C} )$,

I have a problem with the formal definition of forgetful functor. What I mean is that I always find this concept only – so to speak – informally defined. This is a problem to me, because begin self-thaught, I prefer to have formal definitions, where my bad intuition can fail less frequently (...in principle!).

Thus, here there is my definition.

A forgetful functor is a functor $U: \mathbb{X} \to \mathbb{Y}$ that assigns to each $A \in \mathbb{X}$ a corresponding $U(A) \in \mathsf{Obj}(\mathbb{Y})$, and assigns to each morphism $f : A \to A'$ in $\mathbb{X}$ the same function $f$, regarded as a function between elements of $\mathsf{Obj}(\mathbb{Y})$.

Is this definition correct?

Thanks as always for your time.
Any feedback (or improvement, or comment on conceptual typos) is more than welcome.

PS: I am aware that this question could come as a duplicate (and see reference therein). However, I do think it is not the case. My question has not deep mathematical or conceptual implications: it is more a question in order to get some intuition behind this concept. In other words, I would find a reasonable answer, one where somebody writes "This is wrong, and you cannot do that, but in principle this is what roughly speaking we all have in mind (being aware it is not completely right!), when we find that expression".

Community
  • 1
Kolmin
  • 4,083
  • 1
    Hi Kolmin, I am also a self-taught. To me forgetful functor is just that we forget additional structure on objects of $\mathbb X$. Formally, it is known as the right-adjoint of the free functor. I find this book a little bit helpful. – Troy Woo Feb 12 '15 at 13:59
  • Hi Troy. Thanks a lot for the reference. I do see what you mean, and I think (hope?!) that the case of forgetful functor is exactly one in which my intuition does not fail when facing the informal definition. However, as I wrote, I was simply looking for something that looks more technical, without necessarily being THE definition (if we really cannot have a formal definition). – Kolmin Feb 12 '15 at 14:08
  • 1
    You certainly know the functor axioms. You can check that the forgetful functor is a functor, from ((the category in discussion)) to ((set)). – Troy Woo Feb 12 '15 at 14:11
  • 1
    Looking at the definition in your question I notice that this is for so called constructs. This because 'morphism' $f$ is linked with 'function' $f$. There is a wider definition and for this you can take a look here especially definition 5.1. Essential is that a forgetful functor is faithful. – drhab Feb 12 '15 at 14:35
  • Thanks for the hints, and – in particular – for the reference with the definition (even if now I have to admit that I don't see why it is then such a problem to define it, if Adamek et al. do it smoothly!?). – Kolmin Feb 12 '15 at 14:51
  • 2
    I don't know what "the same function" means in your definition. – Qiaochu Yuan Feb 12 '15 at 16:32

2 Answers2

9

You should think of "forgetful functor" not as a class of functors that you can hope to isolate by some property but as a particularly common method of producing functors, which roughly goes like this: many categories are defined as having objects which are tuples of things $(a, b, c, ...)$ (e.g. the tuple $(M, e, m)$ consisting of a set $M$, an element $e \in M$, and a binary operation $m : M \times M \to M$) which satisfy some axioms (e.g. the identity axiom, associativity), and morphisms which are maps of tuples in some way. A forgetful functor is a functor you get by literally forgetting one or more of the things in the tuple (e.g. the identity $e$ and the multiplication $m$, which gets you the forgetful functor from monoids to sets).

In particular, this notion of forgetful functor is not invariant under equivalence of categories: it depends on a choice of a presentation of a category in terms of objects made up of certain kinds of data, so that you can then forget some of the data.

Even then, there are things that people call forgetful functors that aren't at least obviously described by this construction. For example, there is a functor from associative algebras to Lie algebras given by sending an algebra $A$ to the Lie algebra whose underlying vector space is $A$ and whose Lie bracket is the commutator bracket, and I often call this a forgetful functor even though I haven't, say, forgotten the entire multiplication on $A$. In order to describe this forgetful functor in terms of the above construction it's necessary to talk about every operation you get on algebras coming from composing scalar multiplication, addition, and multiplication (these form a Lawvere theory, or if you like an operad) and then forgetting some of them (the ones that you can get by composing scalar multiplication, addition, and the commutator bracket).

Sometimes it's just not worth trying to come up with a general definition. An analogous term it's not worth (in my experience) coming up with a general definition for is "geometry." Any definition you come up with will probably either exclude some important example or be so general that you can't do anything useful with it.

Qiaochu Yuan
  • 419,620
  • 2
    People have mentioned faithfulness as a criterion. Here is a functor that I might be happy to call a forgetful functor, but that is not faithful: take the category $\text{Set} \times \text{Set}$ of pairs of sets, and consider the functor which forgets the first set! – Qiaochu Yuan Feb 12 '15 at 16:33
  • See also the nLab: http://ncatlab.org/nlab/show/forgetful+functor, http://ncatlab.org/nlab/show/stuff%2C+structure%2C+property, where the perspective is "every functor forgets something, but we can try to describe what it's forgetting." – Qiaochu Yuan Feb 12 '15 at 16:35
  • Just seven words (plus six symbols): great! +1 – Kolmin Feb 12 '15 at 16:37
  • 3
    @Kolmin: here's a more explicit example for this lack of invariance under equivalence of categories. There is, of course, a standard forgetful functor $\text{Vect} \to \text{Set}$ sending a vector space to its underlying set. It has a left adjoint $\text{Set} \to \text{Vect}$ sending a set to the free vector space on it. Let me try to convince you that this "is" a forgetful functor. The reason is that $\text{Set}$ is equivalent to the category of pairs consisting of a vector space and a basis of it, with the obvious choice of morphisms. And the forgetful functor just forgets the basis! – Qiaochu Yuan Feb 12 '15 at 16:46
4

There isn't a formal definition of forgetful functors. It is a general name for functors $\mathcal C \to \mathcal D$ that takes an object $c$ of $\mathcal C$ to its underlying objects in $\mathcal D$ and does nothing on morphisms. The words in italic are expected to have obvious definition in working examples.

The only things that people tends to agree on is that a forgetful functor must be faithful (this is the best we can hope of do nothing in full generality). Beyond that, it is whatever you want it to be in the examples you are interesting it. For example, algebraists will probably require that the functor has also a left adjoint in order to have a free algebraic construction.

Pece
  • 11,584
  • Thus, does my "definition" look close to what you have in mind, even very informally, or you would simply dismiss it? Because I see that it is whatever I want it to be in the various situations at hand, but still category theory is a very general and abstract way to talk about structures and in some way, maybe naively, it should be able to give a precise definition of it. Or, is the take-home lesson that not even category theory can address the generality of a forgetful functor? – Kolmin Feb 12 '15 at 15:34
  • Well, I don't think the "does nothing to morphisms" is agreed upon. Some consider that the functor from abelian groups to abelian monoids is forgetful, in that it forgets inverses; but inverses certainly are morphisms (to and from the same object), so this forgetful functor takes $\mathbb{Z}$ to $\mathbb{N}$ and also has a left adjoint; source. – the gods from engineering Feb 12 '15 at 19:01
  • Also the 1967 edition of Mac Lane and Birkhoff's Algebra calls (on p. 514 or so) the functor from affine spaces to vectors spaces "forgetful". This functor changes almost every morhpism of the form $Ax+B$ to $Ax$; it only does nothing when $B=0$. Note that in the more recent edition of the book (1999) this material has been reworked and the statement is no longer there, but then neither is the functor mentioned. – the gods from engineering Feb 12 '15 at 19:32
  • @RespawnedFluff, interesting. So its forgetting the "translatedness" of affine maps, thereby making them linear. It probably would not have occurred to me that this was a genuine functor had you not pointed it out. In fact, its even an $\mathbf{Aff}_\mathbb{R}$-enriched functor. Incredible. – goblin GONE Apr 09 '15 at 01:07