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WikiProject Mathematics (Rated B-class, High-importance)
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 Field: Foundations, logic, and set theory

The rant about the terminology 'cofunctor' is a bit unnecessary, isn't it? Sam Staton 13:24, 9 January 2007 (UTC)

I removed it. We can discuss it here if there is a problem. Sam Staton 12:35, 20 May 2007 (UTC)


An anonymous editor has included the introductory sentence "Functors and constructions relate to each other roughly as morphisms to maps". Does "construction" mean something precise? I am not aware that it does, and find the sentence misleading, so I propose to undo. Sam Staton 17:59, 24 September 2007 (UTC)

No response, so I've undone it. Sam Staton 15:00, 20 October 2007 (UTC)

That anonymous editor was probably me… The problem I was trying to address is that texts introducing category theory, for example this article, have a tendency to disappear into the hyper-abstract about the time they get to functors: "morphisms in the category of small categories" is not wrong, but nor is it likely to be helpful to anyone but those already versed in category theory, and these already know what a functor is! An introduction should give those who don't already know it a feeling for the concept.

If one looks at elementary examples of categories, then it often turns out that objects are sets (possibly with some structure), morphisms are maps/functions, and functors are constructions — e.g. the construction of the set of all formal linear combinations of elements from some given set is in fact a functor (most likely also the left adjoint of some forgetful functor, but that's not so relevant here). To put it another way, just like the function concept may be considered a formal/abstract counterpart of "formula of some variable", so may the concept of functor be considered a formal/abstract counterpart of "construction". Not all constructions are functorial, just like not every formula defines a function — consider \arcsin(x^2+2) — but it's a very good starting point for someone trying to understand roughly what it all is about. (talk) 13:21, 27 October 2009 (UTC)

problem with definition[edit]

If for each X \in C and there is a unique F(X) \in D, then wouldn't

 associates to each morphism f:X\rightarrow Y \in C a morphism F(f):F(X) \rightarrow F(Y) \in D

automatically mean that

  • X_1, X_2 \in C must be projected into different elements in D, or F(Y_1) = F(Y_2)
  • F(id_{X}) = id_{F(X)} for every object X \in C
  • F(X \xrightarrow{f} Y \xrightarrow{g} Z) = F(X) \xrightarrow{F(f)} F(Y) \xrightarrow{F(g)} F(Z) for all morphisms f:X \rightarrow Y and g:Y\rightarrow Z., that is the same as, F(g \circ f) = F(g) \circ F(f) for all morphisms f:X \rightarrow Y and g:Y\rightarrow Z.

so what would the two conditions be there for, then?

I'm no expert on the subject but I just have a queasy feeling that there's something wrong with fixing the morphism F(f) in D to be exactly from F(X) to F(Y)... that would mean there's really no "topological" difference between them.

--Sigmundur (talk) 15:41, 2 October 2008 (UTC)

Sigmundur, there is no problem with the definition, but perhaps there is something misleading about the wording. I'm not sure. The first two lines of your message do not imply the three bullet points, no. NB (in case this is what is confusing you), a functor need not be bijective on objects or morphisms. (talk) 17:20, 2 October 2008 (UTC)
I shared Sigmundur's confusion at first, until I looked up another definition of functors with slightly different wording. What I overlooked at first was, for example, that the arrows F(f):F(X) \rightarrow F(Y) aren't necessarily part of category D for any map F from objects in C to objects in D. Having seen this, I can imagine the other properties are non-trivial, too, but I can't come up with counter-examples. So I feel some counter-examples that show why all conditions are necessary, would be welcome. For example, 2 categories C and D with a map F: C \rightarrow D which maps morphisms to morphisms, but for which the compositions of morphisms is not mapped to the composition of the maps. Or is wikipedia not the right place for this? (talk) 13:48, 16 September 2009 (UTC)
I'm stuck on this same problem, so I agree that it might be useful to clarify it in the article! Either the article's definition has changed since 2009 or Sigmundur paraphrased it, but the word unique (which may have suggested the bijection bit to the IP user two posts up) is not part of the definition at present, and I agree that it should not be. Like Sigmundur, I feel that if C and D are categories and if a functor F:C->D is defined by the object mapping from X in C to F(X) in D and by the morphism mapping that f:X->Y (X,Y in C) is mapped to F(f):F(X)->F(Y), then it seems like automatically the identity is mapped to the identity and that composition must be preserved. Perhaps Sigmundur and I are misunderstanding what F(f):F(X)->F(Y) means. I take it to mean that if F really is a functor from C to D, then if there is an arrow from X to Y, there must be an arrow from F(X) to F(Y), which automatically produces the identity and composition results. On the other hand, if the definition of functor were "to each X in C there is an F(X) in D and to each f:X->Y in C there is an F(f) in D" (omitting that F(f) must go from F(X) to F(Y)), then I can see why it would be required to state that identities and composition are preserved. Can an expert weigh in on whether the current definition is indeed redundant? (talk) 23:45, 19 September 2011 (UTC)
Someone else should chime in, but I think the confusion is that arrows f:X->Y actually have structure or "data" in addition to having a domain X and codomain Y. In particular, the identity arrow must map X->X, but not every map from X->X is the identity. For instance consider a simple category (in fact, this category is basically the group of integers mod 2) where the object is {1,2} and there are two arrows: an identity arrow i:{1,2}->{1,2} that maps pairs i(1) = 1, i(2) = 2, and an arrow f:{1,2}->{1,2} that flips the elements f(1) = 2, f(2) = 1 (and is its own inverse, hence this category being a group). A functor F from this category to itself must map object {1,2} to {1,2}, but it could either map F(i) = i, F(f) = f, which would make it a true functor since it maps the identity to the identity, or F(i) = f, F(f) = i, which fails to map the identity to the identity and so would not be a functor. As for the composition of arrows, imagine a category with three elements X, Y, and Z, with non-identity arrows: f:X->Y, g:X->Y, h:Y->Z, j:Y->Z (and their four compositions). A mapping F from this category to itself would not be a functor if, for example, F(f) o F(h) was mapped to F(g o j), even though those arrows all agree in their domains and codomains. Maybe adding a couple simple examples like this will make the definition clearer. (talk) 17:26, 20 September 2011 (UTC)
Oops! At the end I meant "A mapping F from this category to itself would not be a functor if, for example, g o j was mapped to F(f) o F(h), even though those arrows all agree in their domains and codomains." (talk) 17:28, 20 September 2011 (UTC)


Endofunctor redirects here but it is not explainded in the article. However, it is mentioned that Identity functor is an endofunctor, which suggests that endofunctors are functors within the same category. It would be nice if someone could clear that up (in the article). -- PyroPi (talk) 23:51, 21 June 2011 (UTC)

As I see it is explained. --Beroal (talk) 19:51, 20 September 2011 (UTC)

Not Clear[edit]

Despite having studied math and computer science at a university, and being one who is interested in this article, I feel after reading it carefully that I have no clear idea what functors are or their typical applications. Like too many mathematics articles, this one is nearly impenetrable, perhaps to anyone who does not already know the subject matter well. The identity functor seems easy enough to comprehend; but if someone were to add a "peanut butter functor" defined as "a functor that serves you peanut butter as a side effect" it would make exactly as much sense. -- Canistota (talk) 05:58, 18 May 2013 (UTC)

Prgramming Languages/Prolog[edit]

A Functor is a poorly documented feature in Prolog. Editors are invited to disambig and clarify. Cheers.--Connection (talk) 14:16, 16 June 2014 (UTC)

"Functor" as a synonym for "higher-order function"[edit]

See the discussion here: Talk:Higher-order_function#Is_.22higher-order_function.22_synonymous_with_.22functor.22.3F Jarble (talk) 18:01, 10 November 2014 (UTC)