Talk:Groupoid

Disambiguation
Would it be agreeable to write
 * In Wikipedia, we reserve the term "groupoid" for the first meaning, and use "magma" for the second.

Otherwise, we can never use "groupoid" anywhere without a half sentence explanation. AxelBoldt 17:47 Oct 29, 2002 (UTC)

That would be nice. But, what if someone carelessly or innocently writes groupoid when (s)he means magma ? We could keep "groupoid" as a disambiguation page as it is now, and have articles called "groupoid category" and "magma (mathematics)" (or better names). FvdP 18:10 Oct 29, 2002 (UTC)

The thing is that the two meanings are similar enough that it is not always immediately clear from the context whether the link groupoid was meant to refer to groupoid (category) or to magma. So the innocent groupoid link would have to be fixed no matter what. AxelBoldt 22:01 Oct 29, 2002 (UTC)

I agree with Axel; we can have a disambiguation block here. &mdash; Toby 02:14 Nov 3, 2002 (UTC)

Disambiguation revisited
I have the impression that "groupoid" in the algebraic sense, a set with binary operation, is still common. In that case it seems to me we ought to have a disambiguation page with links to "Magma" and "Groupoid (category theory)" (this page, renamed). I'm willing to do it as time allows, and I'm willing to be convinced it's not a good idea. Any comments? Zaslav 04:33, 18 September 2006 (UTC)


 * A disambiguation page is acceptable. It is true that occasionally people link to Groupoid when they mean Magma, but this is usually fixed pretty quickly. I disagree that the magma sense of "groupoid" is more algebraic than the Brandt groupoid sense. The redirect for Groupoid (algebra) is probably harmless, though, since I suspect nothing will link to it.
 * Besides the links for those who want to move on to one or the other meaning, a disambiguation page could briefly mention the origins of the two uses of the term. "Groupoid" in the magma sense is due to Oystein Ore sometime in the late 1930s, I believe. "Groupoids" in the categorical sense used to be called Brandt groupoids, but were not originally defined as categories with inverses, of course. I am not sure when Brandt's name was dropped, perhaps in Bourbaki? Certainly by the time Lie groupoids became all the rage in Poisson geometry, Brandt's name had vanished. "Magma" is a Bourbaki-ism. Michael Kinyon 12:23, 18 September 2006 (UTC)


 * It looks like the ambiguity problem has been solved nicely. Zaslav 23:07, 13 May 2007 (UTC)


 * I'd like to vote for a LARGE font on the initial statement saying that many people use "groupoid" to refer to magmas, though. (I feel like an outsider, so I won't do the edit myself.)  I work in Universal Algebra, and the standard term there is "groupoid" instead of "magma".  I've certainly used "groupoid" in print myself to refer to magmas, and never had a referee complain. DavidHobby 22:06, 21 July 2007 (UTC)


 * Me again. I see there weren't any comments, so I came back and made the edit. DavidHobby (talk) 03:24, 19 January 2008 (UTC)


 * Sorry, I hadn't seen your comment above. I think the large font here is really unnecessary and rather distracting. I think a standard disambiguation notice works fine. I did reword the notice to be a little more helpful. -- Fropuff (talk) 06:03, 19 January 2008 (UTC)

Fropuff-- No, the large font in NOT "really unnecessary". Look at it this way, we could have an entire disambiguation page. Your group is getting away with claiming this is the main meaning of "groupoid". I dispute that, but am prepared to compromise. DavidHobby (talk) 20:52, 14 July 2008 (UTC)


 * I don't think the bolding of the disambig notice is necessary.—3mta3 (talk) 11:45, 6 May 2009 (UTC)

O.K., and I think that since your group is getting away without a disambiguation PAGE, that having a bold notice should be a fair compromise. Look at the initial discussion--most people thought a disambiguation page was needed. Would you like to address this point? (I don't care if the notice is in bold, or whatever. It should just be more noticible than usual.) DavidHobby (talk) 03:18, 8 May 2009 (UTC)


 * I'm not in any group (I don't have anything to do with either article). I just think that from a style point of view, the bolding is ugly. —3mta3 (talk) 12:56, 9 May 2009 (UTC)


 * So what SHOULD be there to make it very noticable that there is another meaning of the word "groupoid"? (One which is arguably as common as the one in the article?)  Do "blinking" tags work?  : )  As an outsider, what would your opinion be of me simply putting in a disambiguation page? DavidHobby (talk) 18:19, 10 May 2009 (UTC)


 * I don't see what is wrong with the standard dablink template. It works for all other pages in this situation. I don't really know enough about it as to whether it warrants a disambig page, but I guess the main argument against it would that the other article is titled magma (algebra), and not groupoid (algebra). —3mta3 (talk) 08:14, 11 May 2009 (UTC)


 * Since you mention the other article, its first sentence is "In abstract algebra, a magma (or groupoid) is a basic kind of algebraic structure." And then the entire first paragraph goes on to explain the situation.  I believe that my kind of groupoid is listed under "magma" since this was an easy way to resolve the conflict.  I believe the term "magma" is European--I had never heard it until a few years ago.  Hence my desire for a noticable mention of my meaning of "groupoid" on this page.DavidHobby (talk) 00:55, 13 May 2009 (UTC)


 * Fair enough, but I still don't see why a standard (unbolded) hatnote isn't sufficient. —3mta3 (talk) 12:34, 13 May 2009 (UTC)

O.K., 3mta3, you may have a point. Suppose I let it go with a standard hatnote, and instead retitle this article, calling it "Groupoid (category theory)". Then "Groupoid" could be a disambiguation page, to here and to "Magma (algebra)". This seems technically correct, and it would have the effect I want. I didn't go for it in the first place because doing so seemed much more intrusive.DavidHobby (talk) 03:51, 14 May 2009 (UTC)


 * In the meantime, I have removed the bolding from the hatnote. —3mta3 (talk) 08:35, 20 May 2009 (UTC)


 * Give me a break, I'm grading final exams. So I reverted it.  I'd also like to hear from somebody else on this...  DavidHobby (talk) 12:02, 20 May 2009 (UTC)


 * I really don't see why a standard disambiguation notice would not be sufficient. —Tobias Bergemann (talk) 14:35, 20 May 2009 (UTC)


 * Well, the latest thing coming out of my conversation with 3mta3 is that I'll just put in a disambiguation page for "Groupoid", instead. Would that be O.K.?DavidHobby (talk) 17:34, 20 May 2009 (UTC)

So I'm now having the same discussion with "Anonymous Dissident". I would prefer to have that discussion HERE, rather than on my talk page. Where I come from, "Groupoid" commonly means "Magma". From my point of view, a less common meaning has taken over the term. (I can provide numerous references where "groupoid" is used to mean "magma".) I was TRYING to compromise, since the meaning I'm familiar with already has another word that also refers to it. But now I give up, and will put in a disambiguation page instead. O.K.???

(Anonymous Dissident wrote the following on my talk page: I think the argument that emphasising the hatnote on groupoid is necessary because magma (algebra) uses "magma" and "groupoid" interchangeably is valid. However, I've instead made a note of the fact in the first sentence of magma (algebra). What do you think? —Anonymous DissidentTalk 05:31, 10 October 2009 (UTC)

Here's my reply: Putting a note on the magma page is no good, unless we put in a disambiguation page. My whole problem is that THIS page is written as if the meaning of groupoid used here was the only one. Changing other pages doesn't really solve that.) DavidHobby (talk) 02:58, 11 October 2009 (UTC)

O.K., so I've been talking about making a disambiguation page for a year or so, and haven't gotten much in the way of responses. If I bother to do it, I don't want it removed later. So please, comment if you care. By the way, this page would be retitled in the process. How about "Groupoid (Category Theory)"? DavidHobby (talk) 22:54, 11 October 2009 (UTC)


 * Given that you've been talking about a disambiguation page for a year or so and no one has objected to it in all this time, you should probably go ahead and make the disambiguation page. In the meantime, since everyone here (except for you) have objected to bolded hatnote, I'm going to unbold it. Garyzx (talk) 02:00, 12 October 2009 (UTC)


 * Fair enough, it's a deal. So "Groupoid (Category Theory)" is a good title?  DavidHobby (talk) 03:19, 12 October 2009 (UTC)


 * Yeah, I think so.Garyzx (talk) 04:08, 12 October 2009 (UTC)

Incidentally, I replaced dablink with otheruses4 since the latter is simpler and produces exactly the same wording. Also, there should not be any links other than the disambiguating ones. See WP:HAT. Hairy Dude (talk) 03:17, 7 February 2010 (UTC)

Removed
I removed the sentence that said that the collapse from groupoids to collections of groups doesn't lose information; in fact it does. If you view a groupoid as a category, and you only apply category theoretical concepts, then indeed equivalent categories are pretty much identical for all practical purposes. However, if you view a groupoid as an algebraic structure, a group with partially defined group law, then you want to ask many algebraic questions, and you need to distinguish between non-isomorphic equivalent groupoids. I could for instance ask "how many elements of order 2 does this groupoid have?". AxelBoldt 20:33 Nov 4, 2002 (UTC)

Yes, just as I might ask of a group "how many elements of this group are real numbers?". If the group is the group of real numbers under addition, then the answer is "all 2&#1488;0 of them", whereas if the group is the group of purely imaginary numbers under addition, then the answer is "only one of them, the identity". The answer is irrelevant from certain points of view (the groups are isomorphic), but relevant from others (they're different subgroups of the additive group of the field of complex numbers). &mdash; Toby 09:35 Nov 17, 2002 (UTC)

From one who doesn't actually know any category theory, this is a nice point to clarify, since I think I almost "get" it :). But it still seems to me like in both your examples, the collapse into groups doesn't so much lose information as add a layer of obfuscating information - can't you always create the same groupoid from algebraic structures which differ only by arbitrary choice of x when making G(x)? (as opposed to trying to figure out which specific algebraic structure was used to make a particular groupoid) Chas zzz brown

I don't understand the question, specifically "can't you always create the same groupoid from algebraic structures which differ only by arbitrary choice of x when making G(x)". You "create" a groupoid by taking a bunch of groups and connecting some of them by a bunch of isomorphisms. The labeling of the groups as G(x) is indeed arbitrary. But which and how many isomorphisms you take to connect those groups matters.

Toby, I just thought of a better example: the fundamental groupoid of the reals is equivalent, but not isomorphic, to the fundamental groupoid of two separate copies of the reals. AxelBoldt 16:17 Nov 18, 2002 (UTC)

These are definitely not equivalent. If they are according to the definitions in Wikipedia, then something's wrong (or ambiguous) with them somewhere.If x and y aren't (path-)connected, then G(x) and G(y) can't be identified. -- Toby 06:24 Feb 2, 2003 (UTC)

Here's my thinking: the category * which has only one object and one morphism (the identity) is equivalent to the category * * which has two objects and two morphisms (both identities). The proof of this equivalence should carry over to the proof of the equivalence of the two fundamental groupoids discussed above. AxelBoldt 23:30 Feb 2, 2003 (UTC)

I suppose that such a proof would carry over, but there is no such proof (or had better not be!), since these categories are not equivalent. If you throw in two more morphisms (to * *), one going from each object to the other (with composition defined the only way possible), then you get a category equivalent to * -- and this is analogous to throwing in a path from a point in one copy of the real line to a point in the other. (IOW, * and *&harr;* are equivalent, but * * is different.) -- Toby 00:05 Feb 3, 2003 (UTC)

Yes, you're right. AxelBoldt 18:36 Feb 4, 2003 (UTC)

Algebroid
I said "algebroid" instead of "algebraoid", since I've only ever seen the former. So I don't know if it's a mistake on your part, or if it appears both ways. Feel free to change it back if the latter! -- Toby Bartels 02:27 6 Jun 2003 (UTC)

Inaccuracy
This paragraph is not really accurate:


 * Note that the isomorphism described above is not unique, and there is no natural choice. Choosing such an isomorphism for a connected groupoid essentially amounts to picking one object x0, a group isomorphism h from G(x0) to G, and for each x other than x0 a morphism in G from x0 to x.

While such a choice is sufficient to constuct the isomorphism, it's not necessary. In particular, the general case won't pick out any particular object x0. -- Toby Bartels 03:13 6 Jun 2003 (UTC)

Types of morphisms
I have added comments about the various types of morphisms of groupoids, and also the example of the classification of groupoids with one endomorphism. I am not sure if this has or can be done! Recall that the classification of vector spaces is easy; there is just the dimension. The classification of vector spaces with one endomorphisms involves theorems on normal forms of square matrices, and so is very interesting. I am told the classification of vector spaces with two endomorphisms is hard, and with three has not been done!

I also mention that there are recent applications of groupoids to combinatorics, shown by papers on the arXiv in 2005; and that the objects of a groupoid give a spatial component to group theory. -- Ronnie Brown 22:25 27 April, 2006 GMT

Cleanup, accuracy
I'm sort of just skiming this article, but I get the sense that it contains numerous minor fibs and inaccuracies, especially during the categorical discussions. Things like the "general linear groupoid" set me off as well; last I looked, that thing was a full-fledged group (matrix multiplication is always defined), and so it makes a terrible example of a groupoid. The style sometimes goes conversational; sometimes deep and shallow topics get mixed together. The category discussion in "examples" needs wikilinks, we have articles now for most of these concepts. This article needs cleanup. linas 03:27, 10 April 2007 (UTC)

I removed this:
 * Covariance in special relativity An example of this phenomenon that is well known in physics is covariance in special relativity. Working with a single group corresponds to picking a specific frame of reference, and you can do all of physics in this fashion. But it's more natural to describe physics in a way that makes no mention of any particular frame of reference, and this corresponds to using the entire groupoid.

I can't begin to imagine what this paragraph is trying to say. linas 03:44, 10 April 2007 (UTC)


 * linas said:
 * Things like the "general linear groupoid" set me off as well; last I looked, that thing was a full-fledged group (matrix multiplication is always'' defined), and so it makes a terrible example of a group.


 * I'm assuming you meant "...terrible example of a groupoid" rather than "...terrible example of a group", but matrix multiplication is certainly not always defined. For example, you can't multiply an n &times; n matrix by an m &times; m matrix unless n = m.  As such, the set of all invertible matrices over a given field cannot be a group. shonk 01:25, 8 June 2007 (UTC)


 * Oh, yes, right. OK. Now that is a good example. The quoted paragraph should have said "working with a group requires picking a specific spin representation..." and then it would have made sense.  linas 24 Aug 2013

Categorical -> Category Theory
I changed the sec head 'Categorical definition' to 'Category theory definition' to avoid the possible (albeit unlikely) chance of confusion with the term 'categorical' as it is meant in model theory. 'Category theoretical' seemed too awkward. Of course an argument could be made that the possibility of confusion is so remote as to make my edit unnecessary; Zero sharp 14:54, 21 June 2007 (UTC)

Problem in the algebraic definition ?
I don't see how one can derive from the given axioms that if a*b and b*c are defined, then (a*b)*c is. I think one should add this axiom, or explain why it is a consequence of the others.

Also, it is not clear what the "uniquely" in the identity axiom means. 64.81.240.241 (talk) 10:16, 19 November 2007 (UTC) Benoît

You're right, I'll fix these. —Toby Bartels (talk) 18:48, 16 October 2008 (UTC)

Is there any particular reason why the axioms mentioned here differ so much from the four axioms Brandt wrote in his article in 1926? The mentioned axiom sets for sure aren't equivalent and at first sight the three axioms mentioned only correspond to Brandt's second and third axiom. The entire subtleties of left and right unit seem to be lost this way. HSNie 00:18, 20 October 2009 (UTC) —Preceding unsigned comment added by HSNie (talk • contribs)

The first equality in the first proof, which is of the fact that inversion is an involution, is not correct. Can it be fixed? Qweyui (talk) 03:17, 25 July 2012 (UTC)

I think the basic identity needed -- a-1 * a = a * a-1 -- is quickly derived as: a-1 * a = (a-1 * a) * a * a-1 = a-1 * a * (a * a-1) = a * a-1 (by axioms 3 and 1). However, I think this will require an additional axiom, such as: for all a, a * a exists. Perhaps a note to this extent should be in the main article? Also, in the proof of the second property, doesn't the first equality hold the hidden assumption that (a * b)-1 * a is defined? Or can this existence be derived (if so maybe the derivation should be shown)? --Tboul (talk) 22:07, 23 December 2012 (UTC)

There's still a slight problem with the algebraic definition, the proof of $$(a^{-1})^{-1} = a$$ claims that the step $$(a^{-1})^{-1} = (a^{-1})^{-1}a^{-1}a$$ follows from axiom 3, by which I presume they mean $$abb^{-1} = a$$, but it doesn't because plugging in $$a \mapsto (a^{-1})^{-1}$$ and $$b \mapsto a^{-1}$$ gives $$(a^{-1})^{-1} = (a^{-1})^{-1}a^{-1}(a^{-1})^{-1}$$. Instead we need the axiom to be $$ab^{-1}b = a$$, then the proof goes through as stated. Once $$(a^{-1})^{-1} = a$$ is established plugging $$b \mapsto b^{-1}$$ into the new axiom recovers the old one and all the standard $$aa^{-1} = a^{-1}a$$ stuff follows. I'm gonna make the change, just leaving this note to explain why. 2601:144:C004:1590:44EE:DDE5:D556:DE (talk) 22:26, 13 January 2018 (UTC)

Both of the previous comments are incorrect, as in fact it is not true that $$a * a^{-1} = a^{-1} * a$$. Considering the category theoretic definition should make this clear: if $$a \in G(X, Y)$$, then $$a^{-1} * a \in G(X, X)$$ but $$a * a^{-1} \in G(Y, Y)$$. A correct proof that $$(a^{-1})^{-1} = a$$ is as follows (leaving out details): $$a = a * a^{-1} * (a^{-1})^{-1} = (a^{-1})^{-1} * a^{-1} * a * a^{-1} * (a^{-1})^{-1} = (a^{-1})^{-1} * a^{-1} * (a^{-1})^{-1} = (a^{-1})^{-1}$$ In the second-to-last step, we have replaced the middle $$a^{-1} * a * a^{-1}$$ with just $$a$$. I will make this edit now, but feel free to revert if I'm wrong. Conleyw (talk) 05:38, 11 May 2018 (UTC)

Requested move

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section. 

The result of the move request was not done. Skomorokh, barbarian  11:09, 27 October 2009 (UTC)

Groupoid → Groupoid (category theory) — see "Disambiguation revisited" in talk DavidHobby (talk) 04:34, 17 October 2009 (UTC)

Survey

 * Feel free to state your position on the renaming proposal by beginning a new line in this section with  or  , then sign your comment with  . Since polling is not a substitute for discussion, please explain your reasons, taking into account Wikipedia's naming conventions.



Discussion

 * Any additional comments:


 * That discussion is 3 years old??? There's no way it can be used to support a requested move. Please state your reasoning. 76.66.194.183 (talk) 01:00, 19 October 2009 (UTC)


 * The start of it was 3 years old. But there's more recent stuff further down.  Basically, I want a disambiguation page.  "Groupoid" is also used for a set with a single binary operation.  I dispute that the category-theoretic meaning of the word used on this page is the primary one.  DavidHobby (talk) 03:26, 19 October 2009 (UTC)
 * We do not create disambiguation pages for titles covering only two pages; a hatnote sufficies. Skomorokh,  barbarian  11:09, 27 October 2009 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

I cannot resist observing that the statement "We do not create disambiguation pages for titles covering only two pages" is false in general; as the linked policy page states, when neither topic is primary a disambiguation page is considered the correct thing. It is debated whether such is the case here, though I see little evidence that those who feel, as I do, that the topic covered by this article is the primary sense of "groupoid" are willing to insist upon this. Archelon (talk) 03:09, 28 May 2015 (UTC)
 * Hi. Still watching the article, still want a disambiguation page.  But it seems I never get around to doing it, because it seems like a project. DavidHobby (talk) 10:46, 28 May 2015 (UTC)

Groupoids vs. groups
Towards the end of the article it mentions that in writing a groupoid as a product $G \times X$ where $G$ is a group and $X$ is a set ($X$ a groupoid with one morphism between any two objects) some information is lost. Is there a way to be more quantitative -- perhaps an example where this is relevant?

A related question, would it be possible to mention the relationship between the classifying space of a groupoid and that of its maximal subgroup?

Rybu (talk) 13:24, 5 October 2010 (UTC)

Not a generalisation of categories
The lead section said:
 * In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group and of category in several equivalent ways.

and then went on to define a groupoid as a type of category. A "generalisation of the notion of category" would entail that a groupoid is not necessarily a category, which is incorrect. (To compare the group-theoretic definition, a group is a groupoid whose operation happens to be total, but a groupoid whose operation is properly partial is not a group.) So I removed the phrase "and of category". Hairy Dude (talk) 23:26, 13 October 2010 (UTC)

A small point about "operation" versus "function"
Near the beginning of the article, one of the four descriptions reads, "Group with a partial function replacing the binary operation."

Since "function" so often refers to a function of one variable, shouldn't that read something like "Group with a partial binary operation replacing the binary operation," or possibly "Group with a partial function of two variables replacing the binary operation"?

See, for example, http://mathworld.wolfram.com/Groupoid.html, where "partially defined binary operation" is used instead of "function".

Composition?
"For elements x and y of X, let the set of morphisms from x to y be G. Composition of morphisms is the group operation of G."

Those morphisms will have no composition (unless x=y)!

77.125.6.136 (talk) 07:55, 18 August 2012 (UTC)

Brandt
"Heinrich Brandt introduced groupoids implicitly via Brandt semigroups in 1926" is most probably false. See Clifford, Preston "The Algebraic Theory of Semigroups" vol. 1. — Preceding unsigned comment added by 178.36.13.246 (talk) 03:20, 5 April 2013 (UTC)

As far as I can tell, he indeed more or less straightaway defines groupoids: H. Brandt,̈Uber eine Verallgemeinerung des Gruppenbegriffes, Math.Ann. 96, 360 (1926). It's also somewhat surprising this article isn't mentioned in the wikipedia article. HSNie 09:21, 31 May 2013 (UTC)

Exists symbol
It looks weird using the $\exists$ symbol for an expression. It's fine saying $a^{-1} * a$ exists, although maybe is defined is better, but using the symbol that way gets strange. — Preceding unsigned comment added by 31.52.247.252 (talk) 17:45, 7 April 2013 (UTC)

Category of groupoids
Could someone please explain what are the natural equivalences of morphisms. Could someone please explain what is meant by conjugacies of morphisms. If Grpd is the category of groupoids, and G and H are two groupoids, what is Grpd(G, H)? Howard McCay (talk) 20:54, 4 January 2014 (UTC)

Examples
It would help if the first example, "Linear algebra" were easier to understand. I think I can follow the first two sentences: they are saying that union of the groups GL(n,k), where k is a specified field and n ranges over all positive integers, forms a groupoid. The third sentence uses the undefined term G0, making it hard to understand. Similarly, the fourth sentence uses the undefined term G(m,n). Maproom (talk) 09:24, 14 May 2014 (UTC)

Oriented graph?
Sounds to me like it would an unoriented one since every arrow has a pair going in the opposite direction. Sure one can trivially orient an unoriented graph by replacing each edge with two arcs going in opposite directions, but that doesn't seem to be the essence of an unoriented graph, does it? Maybe I'm missing something here... JMP EAX (talk) 01:54, 24 August 2014 (UTC)

The relation to graphs is not discussed further in the article after being mentioned in the lead... JMP EAX (talk) 02:00, 24 August 2014 (UTC)

I found an explanation of how a directed graph induces a free groupoid on nLab but I don't fully understand it (I think it's the "formal inverses" bit). I've added the link as a reference and hopefully someone else knows about it. http://ncatlab.org/nlab/show/free+groupoid.Grabigail (talk) 17:01, 25 September 2014 (UTC)


 * I guess this may be the connection:
 * In a group, every operation can be reversed because every element has an inverse; and in an unoriented graph likewise, any edge can be traversed in either direction. In a groupoid, not all elements have inverses; and in an oriented graph, some edges are one-way only. Maproom (talk) 17:19, 25 September 2014 (UTC)


 * All of this is a red herring, and I will remove the reference to oriented graphs. The point of the first part of the introduction is to state that there are several equivalent ways of defining groupoids; however, the concept of a directed graph is neither equivalent to that of a groupoid, nor do the cited references claim that it is. Rather, one reference claims that every groupoid is a directed graph (which is certainly true, but the converse isn't); another reference claims that one can generate a free groupoid from a graph (which is again true; however, not all groupoids are free). What is in fact true is that there is a forgetful functor from groupoids to graphs, and this functor has a right adjoint. However, the pair of adjoint functors does not form an equivalence. In any case, these facts are kind of irrelevant and don't belong in the introduction or even in the article. Selinger (talk) 21:02, 19 October 2014 (UTC)

Reverted move
So this edit has moved the article to Groupoid (category theory) (a rather ungainly title in my opinion, but let us leave that aside for the moment) and left Groupoid a redirect to this article (i.e. ). This is of course not a proper state for things to be left in. But I don't feel like writing a disambiguation page to occupy the Groupoid space and fixing all the broken links that would result; that would have been the responsibility of the editor who performed the move, anyway. So I'm moving the page back. This is not because I insist that the category-theoretic meaning of "groupoid" is the primary one (although I admit that is my own preference). Archelon (talk) 03:37, 28 May 2015 (UTC)

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Equivalence
In the section on comparing the definitions it's stated that the algebraic and category theoretic definitions are equivalent and two constructions are given, one to produce an algebraic groupoid from a category theoretic groupoid, and one to produce a category theoretic groupoid from an algebraic groupoid. These two constructions are clearly not inverse to each other. Starting with an algebraic groupoid G, constructing a category theoretic one, and then from that constructing another algebraic groupoid G' one can get that G and G' have different cardinalities, so may not be isomorphic. In what sense then are we saying that the two definitions are equivalent? 2601:144:C004:1590:44EE:DDE5:D556:DE (talk) 22:51, 13 January 2018 (UTC)

Algebraic definition of groupoid - mistake in identity?
I'm not an expert on groupoids (hence I'm not making an edit just yet), but in "identity", it is said that "If $$a*b$$ is defined, then $$a*b^{-1}*b = a$$, and $$a^{-1} * a* b = b$$."

Should this perhaps be "... then $$a*b*b^{-1} = a$$, and ..." ? — Preceding unsigned comment added by 2A02:2C40:2E0:0:590B:B35A:9621:172F (talk) 07:57, 26 March 2018 (UTC)

"General linear groupoid" listed at Redirects for discussion
An editor has asked for a discussion to address the redirect General linear groupoid. Please participate in the redirect discussion if you wish to do so. –Deacon Vorbis (carbon &bull; videos) 19:08, 25 December 2019 (UTC)

Confusion in category definition
The category definition ends with this: "and the two arrows $$G_1 \to G_0$$ represent the source and the target." Can that be right? I don't think it is meaningful. $$G_1$$ is the set of morphisms and $$G_0$$ is the set of objects; there is no mapping from the one to the other. Neither is a source nor a target. Should this be removed? Zaslav (talk) 05:03, 16 January 2024 (UTC)