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

I'd have to agree with the comment that this page, for all its content, lacks motivation. Put it another way, such feeling as I have for the adjunction concept didn't come from reading this sort of account. It is more difficult to know exactly what to do about it. The sort of example that might help is the way implication in logic can be defined (introduced) as an adjoint. (But it depends on your background.)

Charles Matthews 17:51, 4 Nov 2003 (UTC)

Hmmm - generalised inverses - I'd have to say my understanding of the concept improved when I stopped trying to use this as an intuition.

Charles Matthews 12:08, 20 Feb 2004 (UTC)

## Plural in the title

Why the plural in the title? I realize that two functions are adjoints to each other, but the term can also be used in the singular: This functor is the adjoint of that functor. Michael Hardy 23:26, 22 Jul 2004 (UTC)

FWIW, I came here typing "adjoint functors" in the search box. For singular, I would have typed "left adjoint" or "right adjoint f." BACbKA 20:50, 7 Aug 2004 (UTC)
I agree that the title should be singular 145.97.223.187 17:32, 9 Mar 2005 (UTC)

Plural is better, really. Pair of adjoint functors is a fuller version. Left adjoint or right adjoint is OK; but 'adjoint functor' on its own is a bit like 'scissor', IMO. Charles Matthews 15:51, 16 Mar 2005 (UTC) See also stilts. Charles Matthews

It is a little bit different, in that adjoint functors have a built-in asymmetry that scissors or stilts don't have...but agreed that left adjoint and right adjoint should still just redirect here. Revolver 05:38, 14 May 2005 (UTC)
Adamek, Herrlich, and Strecker give a definition of "adjoint functor" and "co-adjoint functor", but I think these are just names for "left adjoint" and "right adjoint" (or the other way around), curiously, they don't bother to mention the relation to the usual left/right adjoint terminology. Revolver 05:35, 14 May 2005 (UTC)

## Notation

I'd like an explanation of the notation in Adjoint functors#Formal_definitions. I found the beginning of one on pages 29 and 34 of [1] (they call Mor(), Hom()), but haven't done the work yet to elaborate it to fit here. Perhaps there should be another page defining the Mor Functor? JeffreyYasskin 18:48, 3 Apr 2005 (UTC)

Not really. Mor for morphism is more correct, pedantically speaking, but I suppose Hom for homomorphism is very common. Charles Matthews 20:26, 3 Apr 2005 (UTC)

I changed from "Mor" to "hom" on the category pages just because that's the most widely used (e.g. Mac Lane, the standard reference). In an ideal world, "Mor" would have become the standard notation, but somehow that didn't happen. Both are common...once you're aware they mean the same thing, it shouldn't be confusing. Revolver 05:42, 14 May 2005 (UTC)

## Which John Conway?

John Conway is now a dab page. I assume the reference here should be to John B. Conway, whereas most of the other uses are for John Horton Conway. But I don't want to change it myself, because I'm just assuming. --Trovatore 07:09, 28 November 2005 (UTC)

I asked Charles, who originally wrote the text in question, and I infer from his reply that it is John H Conway that is meant. -lethe talk 10:29, 14 December 2005 (UTC)
Yup, lectures from 1976. Charles Matthews 11:09, 14 December 2005 (UTC)
By the way, I didn't find the reference to John Conway very helpful. Maybe it can go in a footnote, along with a lot of the other tangential remarks? A5 13:56, 1 April 2006 (UTC)
Aren't tangential remarks helpful, to some people, in getting to grips with something so abstract? It is quite possible to cut to the chase - the formal definition - if that's what suits. Charles Matthews 14:31, 1 April 2006 (UTC)
Sometimes, certainly, but I think it can be overwhelming to present too much information at once. Footnotes give an easy way for people to skip the extraneous stuff on a first read. Most importantly, for example for this remark it isn't obvious why John Conway is mentioned (out of other mathematicians), nor what part of his work is being referred to. From what I know about abstract algebra and John Conway, I can imagine that he would prefer a synthetic rather than analytic approach to defining things - but this only helps me understand what the comment might be talking about, I don't actually learn anything new by reading it. I see a lot of comments like this in the maths sections of Wikipedia which read like they were written by someone who was more interested in showing off his or her own knowledge, than in communicating something useful to readers. A5 15:23, 2 April 2006 (UTC)
You know, we get a lot more criticism here for stark presentation of mathematical facts, without some sort of background. And rightly so. Charles Matthews 19:23, 2 April 2006 (UTC)
I think you're missing the point, or perhaps you didn't read all of what I wrote above. I'm certainly not opposed to helpful comments which communicate useful information to the reader. These can even be tangential; and can talk about mathematicians and their work, as this one tries to. What I'm opposed to is comments that mention facts without indicating why they are true, where they came from, why they are relevant, why they are more relevant than other things, etc. It's true that such useless comments often happen to be tangential, but that's not what makes them useless. A5 21:12, 12 April 2006 (UTC)
So we blitz all that: facts without indicating why they are true, where they came from, why they are relevant, why they are more relevant than other things, etc.. And whenever anyone says 'that's not relevant', we have to cut it out. How much supporting material would we be left with? Actually, we'd be left with the kind of treatment there is on PlanetMath. Written largely for grad students, by grad students.
I have never believed that this kind of treatment is appropriate for WP. Of course we can include (mathematical) facts without indicating why they are true. Every mathematical encyclopedia does that. This is not a textbook effort. Wikibooks does that. The kind of supporting comments, in detail, is of course negotiable. But the principle that articles aren't purely logic-driven expositions is not. Charles Matthews 21:27, 12 April 2006 (UTC)
You say: "And whenever anyone says 'that's not relevant', we have to cut it out." That's not what I'm advocating. You're using strawmen again.
Personally, I would advocate deleting it, and moving it to the talk page, but not because I think it can't be turned into a useful remark - rather, because I don't know how to make it useful, and given that the author seems unwilling to make it useful himself, I'm inclined to question how important it was in the first place. A5 15:57, 21 April 2006 (UTC)

I just requested a citation for the statement about Conway's preferred formulation both because I'd like to read what he had to say about it and since it might in part resolve the above disagreement. (If it came from a private communication like a letter or conversation, oh well, cwea.)Rich 09:09, 5 August 2006 (UTC)

## Comparison to Hilbert Spaces

Is there a category where adjoint operators in Hilbert space correspond to adjoint functors? It would be nice to give a formal version of that analogy. Sorry if I missed it. A5 13:56, 1 April 2006 (UTC)

Baez defines in q-alg/9609018 2Hilb to be the 2-category of all 2-Hilbert spaces, defining a 2-Hilbert space to be a category with zero object (like the zero vector in Hilbert space), coproducts (vector sums in Hilbert space) (this is like scalar multiplication by complexes. Instead you have tensor multiplication by Hilbert spaces), cokernels (vector differences), a Hilb-module (a module category over Hilb, the category of all (finite-dim?) Hilbert spaces), and an enriched category over Hilb along with a *-morphism over the hom-sets (the inner product of two vectors should live in C). OK, that sounds like a lot of abstract nonsense, but then, Hilbert spaces have a lot of structures, and they all have to be categorified. The upshot? Any Hilbert space makes a (2-Hilbert space) category whose objects are the vectors and whose morphisms are the amplitudes between vectors, and adjoint maps between two Hilbert spaces are adjoint functors of the corresponding categories. The other way, one can say that if two morphisms in 2Hilb, that is, two functors between 2-Hilbert spaces, are adjoint functors, then they induce adjoint linear operators on each hom-set. -lethe talk + 16:39, 1 April 2006 (UTC)
It would be cool if we could put this stuff into the pedia, but it would require having an entire article devoted to 2-Hilbert spaces. Mebbe I'll do that soon, but I've got other tasks to finish first. -lethe talk + 17:16, 1 April 2006 (UTC)

Thanks. I've looked at the article. I don't really understand it. I wish it were written in terms of categories instead of 2-categories, but I assume the extra complexity has a purpose. A5 17:16, 3 April 2006 (UTC)

The fact that the category of 2-Hilbert spaces is actually a 2-category doesn't seem to be too important for just understand when adjoint functors are adjoint maps. I think he does that as part of his n-category quantization program. Anyway, I think I will try to distill some of the easier parts of the paper and make an article here. I'll keep you posted. -lethe talk + 20:43, 3 April 2006 (UTC)

I have previously encountered a defintion for adjoint morphisms in an arbitrary category, of which the definition of adjoint functors is a special case for say the category of small categories. There does not seem to be an article on these types of morphisms, and while I don't know if they merit their own article, perhaps a mention here would be appropriate? Marc Harper 19:10, 3 August 2006 (UTC)

I don't see why not. Melchoir 03:49, 25 September 2006 (UTC)
In retrospect, I think a discussion of adjoint 2-morphisms may move a bit to much in the direction of 2-category theory to be useful here. A mention back to this article as a special case seems immediate should someone decide it is an article worth writing. Marc Harper 06:04, 28 February 2007 (UTC)

## From monoids and groups to rings

For the examples listed under "From monoids and groups to rings", surely one needs to restrict to the integral monoid ring and the integral group ring constructions?

And is there a reference for these adjunctions? Melchoir 03:45, 25 September 2006 (UTC)

Exercise in MacLane-Birkhoff? Charles Matthews 08:12, 25 September 2006 (UTC)
That would sure work, but I don't have the book... is it in there? Melchoir 15:55, 25 September 2006 (UTC)
It was, some time in the 1970s. It's reference madness, though, when a simple exercise has a reference demanded for it. Charles Matthews 16:00, 25 September 2006 (UTC)
I am challenging the factual accuracy of the item as it appears on this article. Is it reference madness to demand a citation for disputed material? Melchoir 16:13, 25 September 2006 (UTC)

Let's walk through this then. The monoid ring construction gives a functor from monoids to rings. This functor is left adjoint to the functor that associates to a given ring its underlying multiplicative monoid.

Hommonoid(M,R)= Homring(Z[M],R).

Yup. You tell me where m in M goes, either way, equally good. You are correct that it is the integral version of monoid ring/group ring; you can get a group algebra version with appropriate algebras instead of rings. It is pretty much a tautology when you see it written down. Charles Matthews 16:21, 25 September 2006 (UTC)

If that italicized bit is a quote from MacLane-Birkhoff, then the article definitely needs a citation so that the reader can, in theory, go back to the source and determine what assumptions the authors make. In this case, they seem to be using a convention that if the ring is unspecified, it is understood to be Z by default. I don't think that's standard, and if we're going to assume the convention here, then it has to be noted at least in Monoid ring and Group ring, with citations there too. Melchoir 16:42, 25 September 2006 (UTC)

Well no, the italicised bit is a quote from the page, here. This is getting too pedantic for me. I'll make it integral group ring and monoid ring. Charles Matthews 16:49, 25 September 2006 (UTC)

## Primes put on variables are mispaced?

Is the Oct 5 2006 contribution (done in the Adjoint functors#Formal definitions) correct? It has made a rearrangement of primes on the X, Y objects in case of the f morphism. But I think this rearrangement is mistaken.

In the Adjoint functors#Formal definitions, and important notion is that we regard set of morphisms between between two objects as itself an object in $\mathcal{SET}$.

$X, Y, X^\prime, Y^\prime \in \mathbf{Ob}(\mathcal C)$
$f, g \in \mathbf{Ar}(\mathcal C)$
$f \in \mathrm{Hom}_{\mathcal C}(X^\prime,\;X)$
$g \in \mathrm{Hom}_{\mathcal C}(Y,\;Y^\prime)$

Let us note the strange (crossed) arrangement of primes on variables: in case of f it is reversed (compared to the case of g).

Now we build an interesting thing in the $\mathcal{SET}$ category out of all these above $\mathcal{C}$-things:

$\mathrm{Hom}_{\mathcal C}(X,\;Y) \in \mathbf{Ob}(\mathcal{SET})$
$\mathrm{Hom}_{\mathcal C}(X^\prime,\;Y^\prime) \in \mathbf{Ob}(\mathcal{SET})$

We have just constructed objects in $\mathcal{SET}$ category, built of the above $\mathcal{C}$-things. But what about the corresponding morhisms (i.e. ordinary functions) in $\mathcal{SET}$? How shall we build them in an appropriate way out of the above $\mathcal{C}$-things?

$\mathrm{hom}(f,\;g) : \mathrm{Hom}_{\mathcal C}(X,\;Y) \to_{\mathcal{SET}} \mathrm{Hom}_{\mathcal C}(X^\prime\;Y^\prime)$

defined as

$\forall a \in \mathrm{Hom}_{\mathcal C}(X,\;Y)\longmapsto g \cdot a \cdot f \in \mathrm{Hom}_{\mathcal C}(X^\prime\;Y^\prime)$

Let us turn back again to the typing of f and g

$f \in \mathrm{Hom}_{\mathcal C}(X^\prime,\;Y)$
$g \in \mathrm{Hom}_{\mathcal C}(X,\;Y^\prime)$

Now the strange, “crossed” arrangement of primes does not look so strange any more — the construction has showed the main point.

I think the 5 Oct2006 contribution (that has made the strange “crossed” arrangement of primes “straight”) was benevolent and (in a sense) logical, but erranous.

Physis 05:36, 16 October 2006 (UTC)

Thank You for having corrected it. Now, the above argumentation of mine has lost its original purpose. By the way, is my above argumentation correct at all? I do not really understand much to category theory, The above things are just my conjecture based upon esthetical consideration — I have not proven it. I suppose the answer lies in the article Hom functor

Physis 02:56, 17 October 2006 (UTC)

Yes, your argument is essentially correct. Thanks for pointing out the mistaken edit. In fancy language one can say that the Hom functor Hom(-,-) is contravariant in the first argument and covariant in the second. Hence the funny arrangement of the primes. -- Fropuff 03:14, 17 October 2006 (UTC)
Thank You very much for Your quick answer. Physis 14:03, 17 October 2006 (UTC)

## Missing hypothesis in statement of adjoint functor theorem?

[Mistaken objection (based on incorrect interpretation of "continuous" on zero-ary limits) retracted. 8 June 2007]

Kevin.watkins 18:23, 15 May 2007 (UTC)

## "Level beyond the everyday usage of most mathematicians"

I removed from the opening paragraph the sentence "Like much of category theory, the general notion of adjoint functors arises at an abstract level beyond the everyday usage of most mathematicians", for several reasons.

First, it's not NPOV; it contains the assertion that "much of category theory... arises at [a]... level beyond the everyday usage of most mathematicians". This is highly debatable. Many of the people who understand category theory best believe that it arises at the most basic level of mathematics. See for instance Lawvere and Schanuel's book Conceptual Mathematics (Cambridge University Press, 1997).

Second, the "beyond" is slightly pejorative and unnecessarily intimidatory to the reader. Saying "beyond" just makes it sound harder. And what does it mean to say that category theory is "beyond", say, number theory? Most category theorists regard number theory as in some respect a much harder subject than their own; in that sense, number theory is "beyond" category theory. It's pointless to try to make comparisons like this.

Third, it's somewhat in conflict with the later demonstration that adjoint functors arise throughout mathematics. Strictly speaking it's not a logical conflict, because you could maintain that while the general notion is "beyond" everyday usage, many particular instances are encountered every day. But for a sentence in the opening paragraph, it's misleading. 158.109.1.23 (talk) 16:45, 27 February 2008 (UTC)

In its present state, the article itself seems far beyond the working knowledge of most mathematicians; e.g. the examples are all written for an audience that is familiar with category theory. If adjoint functors are everywhere in mathematics it should be possible, and desirable, to write an article that explains the notion to the layman. This article doesn't. Rp (talk) 15:35, 29 January 2009 (UTC)

I somewhat agree with the author of the sentence. Although I understand the beauty and power of the adjunction examples, any real usefulness outside category theory (besides a neat language) escapes me. Could someone add more concrete examples on the use of adjunction in general mathematics? Some links to the articles? Any examples of theorems proved with it? There must be any. --Anton (talk) 19:37, 28 April 2009 (UTC)

I agree with the remover. That phrase should be removed because it does not bring in a new knowledge. It does not help to learn about adjunctions. Should we in every article about any abstract notion write: "This is not for a layman. We warned you. Be afraid. Boo-boo-boo."? Any human opening this page can do some logical inference. "Category theory is not for a layman, this is about category theory, this is not for a layman." Even if you want that phrase back, if you want to analyze the use of category theory from the economical or philosophical or social viewpoint, do it in a proper and complete way. Like counting the word "adjunction" in articles, doing interviews, doing statistics, etc. Not throwing bold judgments out of nowhere. --Beroal (talk) 22:16, 10 February 2011 (UTC)
I agree with the removal. I do think that category theory is quite more abstract than many other branches of mathematics, even if it can be used at the foundational level. The point is that the statement does not belong in that form to Wikipedia, unless specifically sourced. Benjamin Pierce's primer to category theory does say that especially adjoint functors are a quite complex concept. To answer one of the points made: adjuncts are useful and do occur, but it is often significantly complex often to explain that they are there, to understand what this means, and to make use of this knowledge. --Blaisorblade (talk) 02:19, 3 July 2011 (UTC)

Shouldn't this page redirect to "Adjoint functor" (rather than having Adjoint functor redirect here)? I thought that the convention was that entries should be in the singular and not the plural. Gandalf (talk) 15:13, 26 November 2008 (UTC)

I think the reason for this redirection is probably that adjoint functors occur in pairs. Calling the page "adjoint functor" would be like having a page called "equivalent metric". True, perhaps "adjunction of functors" would be a more accurate title for an article describing the relationship of adjunction, but perhaps that would also make the article more intimidating. Thoughts? Wikimorphism (talk) 07:17, 3 December 2008 (UTC)

## question on equivalence of definitions

In the section titled "Counit-unit adjunction induces hom-set adjunction", it says to define $\Psi_{Y,X}(g) = \varepsilon_X\circ F(g)$. However, my problem is in well-definedness of this action. To be be well-defined, this should be $\varepsilon_{GFX}$ or something like this (I'm a little confused by the notation in this article, which is at odds with what I've been learning the last few weeks). In other words, how do we know that 1X to GFX to GY is unique, that is, there is no X' such that 1X' to GFX'=GFX to GY by the same natural transformation in the first mapping? —Preceding unsigned comment added by 129.107.240.1 (talk) 00:28, 11 March 2009 (UTC)

This is standard notation for natural transformations... $\varepsilon$ is a natural transformation from FG to 1C, meaning for each X we have a map $\varepsilon_X$ from FGX to X. (Also note that you wrote the composition of functors backwards: FGX is not defined!). Wikimorphism (talk) 20:16, 28 April 2009 (UTC)

## Formulaic solutions to optimization problems

I think that that interesting section needs references to further reading. It is not a usual topic in a category theory textbook. --Beroal (talk) 22:41, 26 May 2009 (UTC)

## Naturality

Could someone please create a page redirecting from naturality to this article (or a more appropriate one)? Thanks, that is all. 70.250.184.138 (talk) 05:27, 12 January 2011 (UTC)

Created, pointing to Natural transformation [2].--Blaisorblade (talk) 22:58, 2 July 2011 (UTC)

## Syntax semantics galois connection as adjoint due to Lawvere, not Hyland?

Martin Hyland's assumed comment is marked for citation needed. I'm not an expert, but I suspect that the adjoint-point-of-view for this is usually cited to originate from Lawvere. Page 15 in his ADJOINTNESS IN FOUNDATIONS for example. Article Categorical Logic mentions Lawvere and Tierney. Eivuokko (talk) 20:39, 3 June 2011 (UTC)

A Pdf I found time ago on this connection also cites Lawvere (at page 4):
http://www.logicmatters.net/resources/pdfs/Galois.pdf
The same (mis)attribution is also present in Galois_connection#Syntax_and_semantics. I am not an expert either - Looking up "adjointness in foundations" on Google Scholar finds the referenced source, but I have serious trouble in reading it. Still, your comment sounds plausible. --Blaisorblade (talk) 22:29, 2 July 2011 (UTC)
As a further comment, the first article I can ever find from Hyland is from 1976 (using either Google Scholar or his own homepage). The article "adjointness in foundations" from William Lawvere is instead from 1976. That seems enough to me. I'm probably fixing both articles. --Blaisorblade (talk) 22:37, 2 July 2011 (UTC)

## Definitions via universal morphisms - unclearness

I've read and worked out from here the definitions via universal morphisms. The text says that "F is a left adjoint... if ... there exists a terminal morphism." That seems a complete definition, even if quite terse - but I actually managed to demangle it. However, the text continue giving another statement, which seems to be an alternative definition or an explanation. However, it is clear that the hypothesis of the first definition implies the hypothesis of the second, but not viceversa - at least not obviously. The same thing (dualized) applies to the definition of right adjoint functor. Could someone clarify the exact relation between these definitions? --Blaisorblade (talk) 02:39, 3 July 2011 (UTC)

## Definition via hom-set adjunction only valid for locally small categories

It says "In order to interpret Φ as a natural isomorphism, one must recognize homC(F–, –) and homD(–, G–) as functors. In fact, they are both bifunctors from Dop × C to Set (the category of sets). For details, see the article on hom functors." But this is only true for locally small categories since it requires homC(FX, Y) and homD(X, GX) to be sets. We should therefore remark that this definition only works for locally small categories. --TurionTzukosson (talk) 17:59, 28 October 2011 (UTC)

## Just thanking the contributors

This entry is amazingly good. Thank you! Would be cool to have other categorical stuff on the same level.

Vlad Patryshev (talk) —Preceding undated comment added 00:24, 18 July 2012 (UTC)

## Unclarity about equivalence of definitions

In the section on the definition via hom-set adjunction the article says

"This definition is a logical compromise in that it is somewhat more difficult to satisfy than the universal morphism definitions, and has fewer immediate implications than the counit-unit definition. It is useful because of its obvious symmetry, and as a stepping-stone between the other definitions."

"somewhat more difficult to satisfy" suggests that the three definitions treated in the article (via hom-set adjunction, via counit/unit adjunction, and via universal morphisms) might not be equivalent, whereas elsewhere in the article it is stated that they are. Is what is meant something like that "it is (often? usually?) somewhat more difficult to prove that it is satisfied"? "Fewer immediate implications" also suggests nonequivalence, but I guess the qualifier "immediate" can save equivalence... (perhaps a reminder of equivalence here would be worthwhile, though, if in fact it holds). Calling the definition via hom-set adjunction a "logical compromise", in the context of discussing how difficult it is to satisfy and how many immediate implications it has, also suggests that it might be intermediate in logical strength between the other two... was what was meant "a reasonable compromise"?

I would prefer someone more expert than I to attend to this, but I do think it needs clarification...

MorphismOfDoom (talk) 11:15, 29 July 2014 (UTC)