Talk:Category theory

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A question on functors[edit]

Must a functor be one-to-one? Can it assign the same object in D to many objects in C, or many objects in D to a single object in C? I can't see that this is ruled out by the requirements on a functor, but maybe I'm just not being smart enough.
-- Stuart Presnell 28 Mar 2003 — Preceding unsigned comment added by (talk) 19:00, 28 March 2003‎

A functor can be many-to-one but not one-to-many, so much like a function really. It isn't actually a function because functions are defined on sets and categories are generally 'bigger' than sets. Pcb21 14:06 31 May 2003 (UTC)

Actually, it isn't actually a function because it is required to preserve structure (objects, functions, and their pattern of connections). — Preceding unsigned comment added by (talk) 07:20, 8 September 2003‎

Structure of the article[edit]

Something else: I think it would be useful for the structure of the page to export some of the topics (the numerous examples are great but the presentation becomes not very concise). Especially "Natural transformations" could go to Natural transformation and there should be an extra page for "Equivalence of Categories", since there is really much more to say about this. The page would be clearer by just pointing to these topics (and giving very short explanations).

Comments welcome --- Markus 21 Nov 2003

OK, no comments, but I did it anyway. I did not destroy any information but added some (especially on equivalence of categories). I think it is a much better structure now, also in the light of further extensions. Still the most basic definitions remain in category theory, but important further topics are now more easily recognized (not just by reading the examples...). I also included the suggested literature. Still comments are welcome.

--- Markus 25 Nov 2003

Mor(-,-) vs. Hom(-,-)[edit]

This section's heading was created in Sep '04 by Charles Stewart, in commenting on terms mentioned months earlier by Markus and Revolver. 8 hours later, 14:09, 22 September 2004, Fropuff made a talk contrib (lower on this talk page) and in the same edit, also moved M's and R's heading-less contribs at the top of the page into the head of this later-begun section that then had, and still has, related contribs that were made after the (previously) heading-less contribs were, and after C.S.'s presumable reactions to those.--Jerzyt 03:18, 9 May 2015 (UTC)

The set of morphisms is first introduced as Mor(-,-) and later called Hom(-,-). I consider Hom to be the better notation (its kind of standard, isn't it?). Or do we want to change the name of this functor depending on whether the considered set are actually homomorphisms in an algebraic sense? The later would be quite strange in my oppinion, since in the abstract setting of category theory, one usually does not emphasize the internal structure of the objects/morphisms.

-- Markus 25 Nov 2003

I've seen both, but have seen Hom(-,-) much more often than Mor(-,-). I suppose it's a matter of taste, my personal opinion would be that Hom(-,-) be given as the standard notation, with the mention that Mor(-,-) is an acceptable alternative notation. Revolver 01:39, 17 Mar 2004 (UTC)

In general a category C consists of objects and morphisms between objects. Morphisms are called homomorphisms if the category has more structure. If I remember it right, then morphisms of additive categories are called homomorphisms.— Preceding unsigned comment added by Fluch (talkcontribs) 17:19, 18 June 2004

(MacLane 1971) talks about hom-sets wrt. Set (p60), so the nearest we have to a canonical reference doesn't recognise the distinction. The filters " inurl:catlist" gets google to search postings to the categories mailing list from June 1994 to December 1999: the additional term "mor" gets 0 hits, while "hom" gets 42 hits (this of course doesn't get all postings, obnly those linked to from elsewhere, but the message is the same): the the Mor(-,-) usage is definitely an oddity. I propose we switch to Hom(-,-). ---- Charles Stewart 06:12, 22 Sep 2004 (UTC)

I'd vote for switching. Though I have seen Mor(-,-) used occasionally in texts, it's very rare. And I've never heard the phrases "mor-set" or "mor-functor". This will involve a lot of page edits though. I'm not volunteering. -- Fropuff 14:09, 2004 Sep 22 (UTC)

Actually I've seen C(A,B) a lot for the set of morphisms in C from A to B. I think it's nice, but I don't know how common it really is. Bgohla 23:20, 2005 May 3 (UTC)

Category theory is not only "a language" !![edit]

Sorry, if my English is too bad you can...

- There are books much more readable than "categories for the working mathematician". I propose, like does "super" John Báez (UCR), this book: "William Lawvere and Steve Schanuel, Conceptual Mathematics: A First Introduction to Categories, Cambridge U. Press, Cambridge, 1997."

- The theory is not only a major challenge for the "serious mathematicians" (a serious mathematician is also interested on "foundations", he not only is working in concrete "mathematical" problems), the theory is also a great event in philosophy, and it is a beautiful and powerful tool for "thinking". The article in Wikipedia is too cold and "mathematical". Mathematics depends on a decision, and then... deduction and more deduction. But today mathematics and science in general, have -within it- a medium that can "to bring" "consciousness" and have a marvellous "compass" neccesary for their interminable work. Also I hope that the theory can enormously help in "learning/teaching" about anything.

- I think that phrases like "abstract-nonsense" are signs of "reactionary will". All maths are "abstract".

Thank you very much. --ivian 09:29, 20 May 2004 (UTC) ivan.domingo(arroba)

Well, no. You could say abstract nonsense is rather a dated phrase. But mathematics contains an entire range of abstraction, from a hands-on tiling question to topos theory.

There is another, more technical point, about whether category theory is really more like Mac Lane or Grothendieck's visions; or more combinatorial stuff (cf. De Bruijn notation). Again it is both, really.

Charles Matthews 09:37, 20 May 2004 (UTC)

Dual versus opposite[edit]

Following a question I asked in Talk:Equivalence_of_categories, I'd like to suggest we move from talking of the dual of a category to the opposite of a category: while the usage dual category is recognised, opposite category generally appears to be regarded as more correct (see, eg. p.31, MacLane 1971), and is consistent with the notation we use. ---- Charles Stewart 05:41, 22 Sep 2004 (UTC)

Module (category theory)[edit]

Hi there! Does the concept of Module (category theory) make any sense to you? If so, would anybody write an article about this? This article seems to be requested on Wikipedia:Requested articles/mathematics. I suspect the person requesting this article confused something, but I could be wrong. Thanks. Oleg Alexandrov 00:16, 29 Dec 2004 (UTC)

See this: module in nLab linas (talk) 15:23, 2 September 2012 (UTC)

Non-technical explanation[edit]

I wouldn't expect non-technical readers to really understand category theory after reading this article, but from the start of the article until when one gives up reading because one is totally lost, *some* hint of what the subject of the article actually is should be obtained.

I think it would help if the Examples sections were moved up. Also, if there is a a real-world analogy that the Person on the Street can say, "Ah, categories are kind of like that." that would be very helpful.

The current introduction gives the impression that category theory is controversial among mathematicians and may be false or non-rigorous. Since the rest of the article doesn't bear that out, perhaps something like the following should be added to the end of the intro:

The use of this term does not mean that mathematicians consider category theory to be fuzzy or non-rigorous, merely that it is too complex to follow the details in casual conversation.

-- Beland 07:24, 9 Jan 2005 (UTC)

I added an analogy as you suggested to the background section and also the clarification to the intro. I think that should help.

--Carl 01:52, 22 Jan 2005 (UTC)

I agree with Beland. The second paragraph after the table of contents begins, "The most accessible example of a category is the category of sets, ...". Well, this is analogous to beginning the second lesson of an introductory course in calculus with "The most accessible example of a derivative is in the differential equation for simple harmonic motion." Set has an infinite number of objects and there is no need of jumping immediately to that. A really accessible example is category 2 suggested by Fred Linton in the Categories List, 2008-02-14. It is small enough to offer exercises for an introductory study; yet not trivial. There are other examples bigger than 2 and smaller than Set. Regards, PeterEasthope (talk) 18:06, 26 December 2013 (UTC)

Locally small categories.[edit]

The collection of morphisms from object to another does not form a set by definition. When it does form then category is called locally small as stated in book by Michael Barr and Charles Wells .— Preceding unsigned comment added by (talk) 10:31, 10 January 2005‎


Anyone brave enough to start an article on n-categories?

Try n = 2 first. Charles Matthews 17:16, 24 Mar 2005 (UTC)

Current state of the article[edit]

It needs a severe edit, no? Charles Matthews 18:50, 18 Apr 2005 (UTC)

I would say it could do with some more coherency. Much of the material that used to be in here has now been moved to its own page (e.g. category (mathematics), morphism, functor, natural transformation). The article now reads in a rather choppy fashion. In my opinion, this article should give an overview of the history and motivation for category theory with brief blurbs on the main concepts and links to the appropriate articles. -- Fropuff 19:29, 2005 Apr 18 (UTC)

I have the same comment, four years later; the first seven paragraphs are very cumbersome and leave me more confused than before I read them. I don't know enough to edit the article, but it seems we could start with a sentence of user-friendly definition, a sentence of user-friendly motivation (a hint about wide variety of uses), then a couple paragraphs with the real mathematical definition of category, morphism, and functor. Perhaps better, this page could focus on all the diverse fields of computers, math, science, philosophy, etc. in which category theory is used and how it is used. (Ezekiel, 2009 Apr 20) —Preceding unsigned comment added by (talk) 16:35, 20 April 2009 (UTC)

Category theory is half-jokingly known as "generalized abstract nonsense".
It seems to me that leaving this statement stand without explanation could pose problems. A lot of math students, when first hearing about categories, only hear phrases such as this and hand-waving used in proofs, and many of them DO have the impression that it really is "all just a bunch of abstract nonsense". Revolver 21:35, 18 Apr 2005 (UTC)
It's a fair warning, then. We should get deeper into the debate in the article, actually: what Mac Lane believed and so on. Charles Matthews 07:00, 19 Apr 2005 (UTC)

Math markup[edit]

The article uses the Latin letter 'o' for function composition (g o f) which looks kind of dumb. The proper Unicode symbol is U+2218, or &#x2218; in HTML (g ∘ f), as anyone can easily find out. Is there any reason not to use it? The function composition article uses <small>o</small>, by the way (g o f).
Herbee 18:48, 2005 Apr 20 (UTC)

The proper symbol (U+2218) doesn't display correctly on some browsers, most notably Internet Explorer. On IE it displays as a small box. -- Fropuff 19:14, 2005 Apr 20 (UTC)

Mistake in the definition!?[edit]

Well, for any tow objects A,B in a category C, Hom_C(A,B) should be a set not a class, at least this is the normal definition. I'v not see a definition where Hom_C(A,B) is allowed to a class. But maybe it's me? The collection of all C-morphisms (the union of all the sets Hom_C, is that what is meant by hom(C)?) is a class. If we allow Hom_C(A,B) to be a class, then the collection of all C-morphisms will be a conglomerate (collection of classes). So, i think, the second line of the definition should go something like. For any to objects A,B a set Hom_C(A,B)....
— Preceding unsigned comment added by (talk) 20:27, 7 March 2006

Some authors allow a more general definition where Hom(A,B) is indeed allowed to be a class. These authors refer to categories where Hom(A,B) is always a set, as locally small categories. -- Fropuff 20:34, 7 March 2006 (UTC)

Ah i was not aware of that. But, if indeed Hom(A,B) is allowed to be large classes, how then can Hom(C) be a class? Should it not be a conglomerate? Don't we get a kind of Russels paradox if we claim Hom(C) to be a class. For instance let U be the collection of all classes, and assume U to be a class. Then consider the class A costing of all classes x \in U for which x \in x. Then A\in A iff A \not\in A. MaVincent 06:35, 8 March 2006 (UTC)

If you read the definition carefully, you'll see that's not what's going on here. One starts with a class Hom(C) that contains all morphisms in C. The class Hom(A,B) is then defined as the subclass of those morphisms with source A and target B. That is to say, Hom(C) is not a class containing other classes but rather a (disjoint) union of classes. I'm not a set-theorist so I'm a little fuzzy how exactly how classes work but think my explanation is essentially correct. Someone please correct me if I'm wrong. -- Fropuff 07:01, 8 March 2006 (UTC)


Under the header "Categories, objects, and morphisms", morphisms is the only one with its own header. It would make much more sense if each of those words had its own definition so that information could be more easily understood. Fresheneesz 01:44, 17 April 2006 (UTC)

Category of trees[edit]

Reposting from User:Lethe/list of categories:

Anyone know if there's been a category defined for trees/binary trees? I note that the p-adic numbers, as well as the real numbers when represented as strings of integers, are actually trees (viz yea olde 0.999..=1.000... debate). I also note that grassmanians (and thus supernumbers, supermanifolds and other bits of supersymmetry) can be represented as binary trees (although not uniquely/naturally). The various cantor sets are also binary trees, as is alpeh_one = powerset of aleph_zero. In a more hand-waving way, it also resembles a recursive application of a Subobject classifier (In the cae of a cantor set, "is it on the left or the right?"). Topologically, this seems like a special case of lattice (order). linas 23:47, 20 July 2006 (UTC)

Joke in the introduction[edit]

I think the joke really should not be in the introduction. As someone pointed out above, it gives the impression that Catgory theory is somehow considered a joke to mathematicians (as if its numerology or something.) I'm not qualified to say if its a fruitful line of questioning as I'm still struggling with set-theory, but I don't think its apporpriate to give such an impression to an intrested laymen especially when there are apparently many mathematicians who take category theory seriously. Or atleast some. Even if it was a fringe theory it still would not be appropriate to have such a joke in the intro. As it does seem pretty clear to me to be innapropriate I'm going to take the liberty of removing it. If someone feels the joke is worth mentioning, please mention it as an aside at the bottom of the article (and source the joke if possible. I think Douglas Hoffstadter may have started that joke or made it popular; I think I remember reading it in one of his sci-am Metamagical Themas columns, so try to source it if you put it back.). Brentt 12:23, 16 August 2006 (UTC)

The 'joke' goes back long before Hofstadter. Please don't just remove it. If you want to qualify the intro, go ahead. Charles Matthews 12:26, 16 August 2006 (UTC)
I agree with Brentt, the "joke" should not be in the intro, especially the part mentioning something about non-sense absolutely irrelevant. Please remove. Tamokk 20:45, 3 September 2006 (UTC)

P.S. I remember reading in "Algebra" by S. Lang, that the term is due to Steenrod. So he refered to the homological algebra done in categories (himself being one of the creators of this theory), rather than the category theory itself, which by then was in a very early stage of development, and was seen by mathematicians as no more then a mere language. Latter the term has been not always neutral, used in different context and with different attitudes (sometimes unsypathetic). Tamokk 20:45, 3 September 2006 (UTC)

Article rating B+[edit]

This is a good article for a technical subject. It may be slightly more technical than is necessary in the beginning parts; the intro (the background section) at least should be readable by an undergrad.

  • The second para of the lead is surprising to me; I know the phrase "abstract nonsense" but it is a stretch to say that "nonsense" refers to commutativity of diagrams. If this is what the original statement meant, I think a citation would be useful to back it up.
  • The historical notes section needs several inline citations, especially for the "it has been claimed" part. Well-known and accepted facts don't need inline cites in my opinion, but direct quotes or attributions of opinion do. The fourth para of that section also needs citations, since the fact that one book was received better than another is not a well-known mathematical fact.
  • The section on Categories, Objects, and Morphisms is very terse; one or two introductory sentences would make it more readable.
  • Here are a few questions that a naive reader might ask. What are contemporary trends in category theory? How is it related to computer science?

CMummert 14:52, 25 October 2006 (UTC)

I certainly agree with the rating the writing in the introduction does a good job of explaining the concept with a minimum of jargon. I wonder if there is any way it could be made visually more appealing? --Salix alba (talk) 15:24, 25 October 2006 (UTC)
The thing about diagram chasing strikes me as possibly original research. I think it need sourcing, at least. And I'll add more about diagrams. Charles Matthews 15:45, 25 October 2006 (UTC)
I think that a section of commutative diagrams in this article, maybe in summary style, would be nice, because they are the picture that most people get when they hear the phrase category theory. The article on commutative diagrams is very short right now, and could use attention. CMummert 15:55, 25 October 2006 (UTC)

I agree that this is a pretty good intro. I have tried to achieve an understanding of category theory a number of times, and this is the article that has brought me closest. --Doradus 02:41, 5 November 2007 (UTC)

I'm working on wikiversity learning project Introduction to Category Theory, maybe it helps you understand, if you're not scared of math... Tlepp 19:10, 5 November 2007 (UTC)

Portal:Category theory[edit]

I added {{Template:Portal}}. The portal is still a "stub", any help is welcome. I added {{Template:Portal|Category theory}} at the pages of the main subjects of category theory too. Cenarium (talk) 17:25, 14 February 2008 (UTC)

A question on one-to-many relationships[edit]

How do you represent one-to-many relationships in category theory? Skylarkmichelle (talk) 9 Aug 2008 —Preceding undated comment was added at 14:12, 9 August 2008 (UTC)

This page is intended for discussing improvements to the article on Category theory, and not for discussing issues related to category theory. A more suitable place for this question is Wikipedia:Reference desk/Mathematics. By the way, as posed the question has no proper answer; compare the question: "How do you represent one-to-many relationships in logic?".  --Lambiam 16:25, 15 August 2008 (UTC)

Morphisms, maps and arrows[edit]

I've added 'arrow' and 'map' as alternatives to 'morphism'. All three terms are in common use. Three authoritative sources: 'arrow' is the favoured term in Mac Lane's Categories for the Working Mathematician, 'map' is the favoured term in Lawvere and Schanuel's Conceptual Mathematics, and 'morphism' is the favoured term in Grothendieck's work. I see that a previous attempt to include 'map' was reverted, but that's a mistake. If you don't believe me, go look in the archives of the categories mailing list: (talk) 03:55, 22 February 2009 (UTC)

I have removed the sentence "The influence of commutative diagrams has been such that "arrow" and "morphism" are now synonymous." The statement that "arrow" and "morphism" are now synonymous is redundant, since that follows from the definition. And to say "now" is silly since it has been used since the 1970's. And I know of no published or unpublished justification for the statement that "arrow" came into use because of the influence of commutative diagrams. SixWingedSeraph (talk) 21:42, 10 April 2009 (UTC)

External Wiki links that may need deletion[edit]

The following links were deleted based on ELNO, specifically "one should generally avoid" "Links to open wikis, except those with a substantial history of stability and a substantial number of editors...."

  • nLab a wiki project on mathematics and physics with emphasis on role of category and higher category theory; see nLab
  • Joyal's CatLab a wiki project on foundations of categorical mathematics with theorems and proofs

I propose we determine if they indeed do not have "a substantial history of stability and a substantial number of editors" before deleting them. Note that the first has its own Wikipedia page. Hga (talk) 14:08, 11 September 2010 (UTC)

So, Were is the Category?[edit]

What link does Category in mathmatics has with the concept of categorization, kantic categories, or Aristotelian categories? Or, from whence the name here comes from? (talk) 09:54, 9 November 2011 (UTC)

Why should it link to any of them? It is just a good name for a theory which categorizes various structures in mathematics as in the very first dictionary I came across on the web ' A general class of ideas, terms, or things that mark divisions or coordinations within a conceptual scheme'. It is more descriptive than a lot of terms in maths which have been taken from normal language like group theory for instance! Dmcq (talk) 10:14, 9 November 2011 (UTC)

There is no other link than a "syntactic" one, which is to say, via its name. However, Eilenberg and MacLane are quite open with the fact that they "purloined the terms of the philosophers" whilst choosing the terms that they would associate with the objects they set out to describe in their initial description of categories. Thus, as they say, the term "functor" comes from the logical positivist Carnap "category" from Kant, and so on. — Preceding unsigned comment added by (talk) 00:18, 25 July 2013 (UTC)

Is category theory of high importance in computer science[edit]

I disputed marking category theory as of high importance in computer science and was reverted with a pointer to Stanford philosophy where they say it is. I can see it being seized on by some philosophy people, but exactly what has it brought to computer science? My feeling is that it sounds like the new math being taught in schools as far as computer science is concerned. Is there any good example of something it does actually help with rather than just being jargon by people puffing up their importance to put promising undergraduates off the whole idea of computer science? Dmcq (talk) 16:19, 10 March 2012 (UTC)

Apparently my exposure to Haskell has led me to overestimate the importance of category theory in computer science. (One of the characteristics of Haskell is the use of monads to represent side effects, and many introductions to Haskell and functional programming in general will mention category theory in passing. See also and
There have been several international conferences on ‘Category Theory and Computer Science’,[1] and I think every computer science department offers some course or another titled ‘Category theory for computer scientists’ or similar, at least at German universities. However, I don't see category theory proper used much in theoretical computer science other than providing the vocabulary for type theory. See for example Crole's book Categories for Types, or the paper ‘Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire’ and our own articles Anamorphism and Catamorphism.
Cartesian closed categories are a natural setting for models of lambda calculus, see e.g. Barendregt's book on lambda calculus or Barr's and Wells's book Category Theory for Computing Science.
Anyway, I have changed the importance back to "low". Sorry for the confusion. — Tobias Bergemann (talk) 20:17, 10 March 2012 (UTC)
Well what you said plus it being pretty standard fare in computer departments sounds like a mid so I'll go halfway up again ) Dmcq (talk) 22:28, 10 March 2012 (UTC)
The role of category in Computer Science is very important: even for those who do not directly apply it in their own work, it constitutes an extra tool in the "thinking toolkit" (cultural background). A large amount of work was done starting already decades ago in the IFIP WG 2.1 (Algorithmic Languages and Calculi) by Lambert Meertens, Richard Bird, Oege De Moor and many others. Some of this work has resulted in one of the most enjoyable books on the subject, which seems missing in the reference list:
Richard Bird and Oege De Moor, Algebra of Programming. Pearson (1997) ISBN 0-13-507245-X
An added benefit (which some readers will appreciate) is that proofs are presented in the calculational style, which makes them very streamlined. Boute (talk) 20:58, 29 July 2012 (UTC)

Definition of hom(C) is confusing[edit]

Currently, hom(C) is defined as "A class hom(C), whose elements are called morphisms or maps or arrows. Each morphism f has a unique source object a and target object b." This can be interpreted as "f" only exists for "a". That is, that if each object were a dot, then "f" is one particular arrow between dots. (And, in fact, this is how I interpreted it!) What we want to say is that that "f" is a like a function. It maps some subset of objects to other objects and, if a given object "a" is mapped, it always maps to only one other object "b". Right? — Preceding unsigned comment added by Mdnahas (talkcontribs) 14:38, 13 September 2012 (UTC)

I'm not sure what is confusing, since you got the correct essence in your interpretation of what the article says. "f" is like a function in the sense that morphisms in the category Set literally are functions. A function prescribes a rule for going from one set to one set – is an arrow from one object to one object, since sets are the objects in Set. What that rule is doesn't matter from a Category theory perspective, since that's going on "inside" the arrow.

For the latter part of your comment, if a given object "a" is mapped to an object "b" (f: a—>b), there is no restriction on another morphism mapping "a" to a different object "c" (g: a—>c). Indeed, if there were such a restriction, commutative diagrams would be impossible! More concretely, a given set can be the domain (source object) for several functions, each with different codomains (different target objects). (talk) 08:20, 11 March 2014 (UTC)

Graphical schematic, in lead, of a cat[edit]

   Without exaggerating the seriousness of the situation, i think it would be worth some attention to the "f o g" typography in the lead section's graphic. It appears that the composition symbol is misrepresented by use of a lower-case roman letter O rather than the proper uniform circle. IMO it is thereby too large, and improperly close to the base line, to reliably evoke the minimal-diameter, "floating", hollow dot that denotes relational composition. A proper fix may take higher-mathematics exposure in addition to routine typographical/graphics expertise!
--Jerzyt 21:40, 8 May 2015 (UTC)

Historical notes and other items[edit]

The section on Historical notes is unsourced and rambling commentary at times, e.g. how students are getting introduced to it! It seems apologetic, and remorseful. No need. Article is 50% in a state of mess really. The page of Category is shorter and better. I would delete a lot of the rambling here, and be better and shorter over all. Applications of category theory really need expansion. SaundersLane (talk) 01:24, 2 November 2015 (UTC)

Why are none of the categories in a different script?[edit]

In most of the books I've seen and lectures, categories and functors are usually written in some type of cursive script. Why do none of the wikipedia articles do this? It would avoid a lot of confusion (in my opinion) when distinguishing between categories and their objects. — Preceding unsigned comment added by Csasba2 (talkcontribs) 23:37, 29 November 2015 (UTC)

Are we talking about the names of specific categories (like Set, Grp, Top...) or about variables that take on categories as values (as in "A functor F from a category C to a category D is a mapping that...")? AFAIK, Wikipedia articles consistently use boldface for the names of specific categories. Wikipedia:Manual of Style/Mathematics apparently is silent about this.
For what it's worth, the following books too use boldface to mark categories:
So I think it's fair to say that this convention is not uncommon.
Tobias Bergemann (talk) 11:11, 17 December 2015 (UTC)