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following Zoran’s suggestion I added to the beginning of the Idea-section at monad a few sentences on the general idea, leading then over to the Idea with respect to algebraic theories that used to be the only idea given there.
Also added a brief stub-subsection on monads in arbitrary 2-categories. This entry deserves a bit more atention.
I’ve reorganized monad a bit, trying to remove some of the cruft that had gathered there. Also I’ve added a new section on Street’s bicategory of monads in a given bicategory, and moved the section on the successor monad to its own page.
there is also a notion of a monad in the context of holomorphic vector bundles (or coherent sheaves) over projective spaces (or more generally, (compact) complex manifolds): A monad in this context is a complex of holomorphic vector bundles $0 \to A \to B \to C \to 0$ which is exact at A and C. If I’m not mistaken, these first appeared in
G. Horrocks, Vector bundles on the punctured spectrum of a ring, 1964, Proc. London Math. Soc. (3) 14, 689-713
and often appear in the same context as the Beilinson spectral sequence.
Is there any relation to the monoids in the category of endofunctors? Is it an example? If so, how? Is it worth mentioning this in the article on monads?
ad 3: The name is not motivated by any kind of similarity to monoids, as far as I know. Once we create the $n$Lab page for it, it should be Beilinson’s monad, what is the standard full name for it.
i was unable to find the page you mentioned, but thanks anyway.
I just checked up on Geof Horrocks in Wikipedia and that construction is known as the ADHM construction. (NB. the H is not Horroocks here but Hitchin.) This is relevance to instantons (or so it say on Wikipedia). Wikipedia also calls it the monad construction. (Perhaps disambiguation is called for. I know of another use of the term monad in non-standard analysis, and lots of others, see the wikipedia page on monad and the nLab entry).
No, Tim. ADHM construction is quite specific and later. Though it does involve Beilinson monads in some formalism. ADHM construction is from mid 1970s, see the paper under Yang-Mills instanton:
and for a noncommutative version the paper of Kapustin, Kuznetsov and Orlov which does explicitly talk about (noncommutative) Beilinson monads.
I have to say that I disagree with Wikipedia (although I’m not sure what the standard terminology is): From my point of view, the monad construction is just a part of the ADHM construction (another part being the Twistor construction which identifies instantons with certain real algebraic bundles over $\mathbb{P}^3$ and makes it possible to use monads at all)
also note that the ADHM construction was reformulated by W. Nahm in a way that does not use any monads at all (at least not explicitly).
I have started Beilinson monad.
Also created monad (disambiguation).
here is the reference to Nahm’s paper:
As David Corfield noticed in another thread, the amount of discussion of monadicity on the $n$Lab had been rather lacking.
As a first-aid means, I have
added a brief paragraph with the basic statements to monad – Properties – Relation to adjunctions
copied the same to adjoint functor – Properties – Relation to monads
cross-linked the two entries monadic adjunction and monadic functor (which didn’t know of each other and which should maybe be merged alltogether).
More deserves to be done here, eventually.
Is there something ’modal’ going on here, an adjunction suspended between two moments - the Eilenberg-Moore adjunction and the Kleisli adjunction?
In Section 1 we read:
the concept of a monad is the horizontal categorification of that of a monoid.
however horizontal categorification suggests that the h.c. of a monoid is a category, so I’m wondering how useful the above remark is.
Well, I think the preceding sentence explains what was meant by that. Also, “categorification” to my mind isn’t quite well-defined (“decategorification” fares better).
Also, categories are a special case of monads. (They are monads in Span.) But perhaps it would be better for the sentence to say “a horizontal categorification” rather than “the horizontal categorification” (and now it does).
Absolutely. If you do have the energy to improve this entry, please do.
Added:
An elementary proof of the equivalence between infinitary Lawvere theories and monads on the category of sets is given in Appendix A of
While statement is (probably) true for Cartesian closed categories, the example of the double-dual monad for vector spaces is not an example of double-dual monads for Cartesian closed categories; indeed, the Vect_k double dual monad makes use of the other monoidal product on Vect_k, namely the tensor product, which is explicitly, and very importantly here, the product which induces the appropriate tensor-hom adjunction to apply.
In any case, (assuming that the Cartesian product adjunction is the internal hom functor) all Cartesian closed categories are automatically closed symmetric monoidal categories (so automatically bi-closed monoidal categories), so the remark falls under this more expansive and contextually accurate case one way or the other.
Lillian Ryan Uhl
The comment #27 is probably referring to the last paragraph in this example.
While looking over it, I have added some more hyperlinking and adjusted some wording in this example.
Thanks for the alert. I’ll forward this to the technical team.
Thanks for highlighting. The reference to Voutas’ article was deleted in revision 93 and actually announced so in comment #26 above, but nobody reacted.
In general, though, if there is anything cite-worthy in a pdf sitting otherwise unpublished somewhere on the web, then we shoould upload it to the nLab server to preempt the almost inevitable link rot.
I have done so now for Voutas’ file (the second pdf link here)
back to #30:
Christian Sattler has kindly deleted the spurious entry “monad+”, along with a slew of similar ghost entries.
So the issue at hand should be solved. But apparently it arises due to a bug deep in the database and may reoccur over time.
The first commutative diagram is a bit confusing I think, since those diagonal arrows are unlabeled. From what I can tell they are identity 2-cells?
added pointer to
as an early reference for of the induced (co)monad of an adjunction. Since Huber attributes his notion of (co)monad (“standard construction”) to
I have added that pointer, too (the monad laws appear on p. 272 there, as part of the structure of the induced canonical resolution).
added (here) pointer to the origin of the term “monad”:
with some commentary before and after
Thanks. I have made that reference a reference (here), alongside the parallel comment by Ross Street from the same discussion thread.
While this old thread is a historically interesting read, your headlining it “Etymology” (here) seems rather optimistic: While Barr’s message mentions Leibniz’s “monads” as lead-in, it makes no attempt to make a connection.
In fact, one senses the absence of any verbalized rationalization of the terminology where Barr says:
Bénabou turned to me and said something like “How about ‘monad’?”
Several other contributors to the old thread tried to second-guess the intended connection to Leibniz’s monads — but do they succeed?
One contributor recalls from Leibniz that:
This implies that every monad has an internal representation of every entity in the universe and these representations can never influence objects outside the monad.
to immediately conclude:
The analogy with our monads should be evident!
Should it?! :-)
While this old thread is a historically interesting read, your headlining it “Etymology” (here) seems rather optimistic: While Barr’s message mentions Leibniz’s “monads” as lead-in, it makes no attempt to make a connection.
Yes, I had wondered whether “Etymology” might be a stretch… Do rename it to whatever you feel is more appropriate.
Interesting anecdote! I am grateful that SNAFU did not catch on…
What surprises me is that except for the title “Toposes, Triples, and Theories”, Barr was willing to switch over to “monad”. I can’t think of a single circumstance where Barr hasn’t stuck (stubbornly) to “triples”.
FWIW, I think “Etymology” is fine. An etymology traces how the word came to be in the language, and so I think it fits. Barr of course would deny there was a preconceived connection with Leibniz’s monads (and would probably reject a posteriori rationalizations for such a connection).
Barr’s message gives the right clue in the first line, whether intentionally or not:
Leibniz monads in the specific sense of “infinitesimal neighbourhoods” — as still used in non-standard analysis and synthetic differential geometry – are indeed monads in the mathematical sense: as discussed at infinitesimal disk bundle (here).
I’ll add a note to this effect…
Have rewritten the “Etymology” section (here):
keeping from Barr 2009 only the relevant passages
then highlighting that a few years earlier the non-standard analysts had adopted Leibniz’s “monad” to refer to “infinitesimal neighbourhood”
and finally remarking that passing to infinitesimal neighbourhoods around each point is indeed a monad (in the sense of category theory), left adjoint to the jet comonad.
The note is a reasonable (and amusing!) thing to mention, but I’d prefer having the Barr quote reproduced in full. (Simply linking to the cat-list archive is suboptimal because you have to scroll through to find it, and the jpg link is suboptimal because it’s small and hard to read.)
Barr brings up the connection with monoid, but for some reason that relevant bit got pruned out of what remains of the quote. It would be better to have the quote in full and let readers take away what they will from this. I feel like they could get a skewed idea from the nLab section as it reads now.
By the way, Mac Lane’s Categories for the Working Mathematician also has historical notes on the matter (at least the first edition did; I’d have to check the second).
I have now uploaded (and linked in the entry) the (truncated) message by Barr09 as a txt-file: ncatlab.org/nlab/files/Barr-HistoryOfMonadTerminology.txt
Okay, fair enough. I can see why you wouldn’t want to have the paragraph about Peter May left in (when I said “full quote”, I was going by varkor’s version).
I see that 10 years ago at monad (disambiguation) (revision 7) I had argued that category-theoretic monads can roughly be thought of as “atoms” (and hence of vaguely Leibnizian kind) in the sense that they are “lax points” in $Cat$, namely lax 2-functors $\ast \to Cat$. Maybe worth adding to the present discussion.
My two cents: if this is supposed to be part of the etymology discussion, then I think it sounds really far-fetched. (A Procrustean bed: stretching to try to make a point.) I’m not sure it would be particularly worth adding in any case.
have added the following paragraph to “Etymology”:
But it is striking that Bénabou 1967, Def. 5.4.1 defines a monad to be a lax 2-functor from the terminal category to the 2-category of categories (and more generally to whatever given ambient 2-category) and then proceeds to unwind the equivalence of this definition to the traditional one
$Monads(Cat) \;\;\; \simeq \;\;\; \Big\{ \ast \xrightarrow{\; lax \;} Cat \Big\} \,.$In this sense, monads are “points” in a 2-category theoretic sense, which may square well with Leibniz’s notion of monads as a kind of atom.
Re #54:
Incidentally, this suggestion was mentioned originally by Jonas Frey (Dec. 2013, here), who in turn had hear it elsewhere.
It seems a rather plausible candidate for Benabou’s actual reasoning, given that Benabou defines monads as lax 2-functors out of the terminal category.
Huh, interesting what Jonas said. Anyway, I think the addition looks good.
Of course the use of the word “monad” in mathematics is much older than Leibniz and goes back at least to Euclid, in whose books on number theory the noun “μονάς” (stem μοναδ-), usually translated into English as “unit”, appears throughout, e.g. in the definition (Elements Book VII Def 2) of number as a multitude composed out of units: ἀριθμὸς δὲ τὸ ἐκ μονάδων συγκείμενον πλῆθος.
Thanks, that’s a great point.
I have accordingly adjusted the wording here and added some relevant pointers at monad (disambiguation) (see there).
Incidentally, I am wondering if something was lost in translation for “Definition 1” in Euclid’s Book VII:
His “Definition 2” is clear enough:
Definition 2: A number is a multitude composed of monads.
From this alone one gathers that “monad” must be something that is not a multitude. It must be that “Definition 1” means to say just that, but in its usual English translation it is is not actually quite intelligible:
Definition 1: A monad is that by virtue of which each of the things that exist is called one.
Now, Gordon, Kusraev & Kutateladze 2002, §2.2.7; quote commentary on this definition by Sextus Empiricus:
A whole as such is indivisible and a monad, since it is a monad, is not divisible. Or, if it splits into many pieces it becomes a union of many monads rather than a [simple] monad.
Since this is again quite clear, I am wondering now if Euclid’s Definition 1 transports a different meaning in the original ancient Greek than comes across in its English translation.
I am wondering now…
On this point, Kutateladze (p. 2 here) seems to offer the following reading:
We begin counting with making “each of the things one.”
If that’s the right reading, then Def 1 & 2 from Euclid’s Book VII could jointly be understood as saying:
Counting is establishing a bijection with a finite set, and a monads are (the identification with) the corresponding elements.
This would also clarify the relation to Euclid’s notion of “point” from Book I
a point is that which has no parts
Namely, in counting sheep we do not claim that sheeps are points and yet we call, for the purpose of counting, each sheep “one” (one sheep here, other one there). That must be the sense in which Def.1 in Book VII speaks of “things” being “one”.
So I gather Euclid meant to distinguish between what we would call the singleton set $\ast$ and its maps, such as $\ast \to \mathbb{R}^3$ (a geometric point) or $\ast \to SetOfSheep$ (a sheep regarded as “one”).
That’s pretty clever of Euclid!
As far as language is concerned, Heath’s English translation that you quote is as close as you can get to a word-for-word translation from the Greek:
Μονάς ἐστιν καθ’ ἣν ἕκαστον τῶν ὄντων ἓν λέγεται.
The only room for movement is in how you translate the preposition κατά (elided here to καθ’), which Heath translates as “by virtue of” but which has a very wide range of possible meanings; the basic sense in this context is something like “according to”. But beyond this, I find the original Greek definition no clearer than the English translation.
Since this definition of μονάς is nowhere explicitly used in Euclid (just like the enigmatic definitions in Book I of point and straight line), it’s hard to tell from the Elements alone what this definition is supposed to mean. I suspect that the proper context for understanding this definition is less Mathematics and more Philosophy; indeed the wording of this definition is very reminiscent of some of the definitions in the Platonic work “Definitions”, such as this definition (among several) of virtue:
Ἀρετὴ … ἕξις καθ’ ἣν τὸ ἔχον ἀγαθὸν λέγεται.
which translates literally to something like “virtue (is) a habit according to which the thing having (it) is called good”.
As ever, the actual meaning of the mathematical notion of μονάς is best shown by how it is employed in the theorems and proofs of the Elements themselves.
Thanks!
the actual meaning of the mathematical notion of μονάς is best shown by how it is employed in the theorems and proofs of the Elements themselves.
Sounds good. Might you know of a good example of Euclid using μονάς later on?
Yes, I have more to say on this, but textual examples will have to wait for another day, as it’s past my bed time now. In briefest summary, the word μονάς plays two main roles, one when it is used in the singular, in which it denotes a notion more or less equivalent to our natural number 1, considered on an equal footing with the other natural numbers, and the other when it is used in the plural, in which case μονάδες refer to the constituent elements of ἀριθμοί (numbers), in a manner which reminds me of Lawvere’s way of talking about finite sets as “bags of dots”.
Sounds great. No rush, but I am looking forward to hearing more from you on this.
I’d want to re-typeset the definition of the 2-category of monads (here) since the current choice of display of the commuting diagrams actively hides what is going. Maybe I find the time to do so later.
For the moment I have added an Example (here) spelling out the simple but important case of a transformation of monads on a fixed category.
added pointer to
who gives an early statement of morphimsm of monads (2 years before Street)
In fact, Pumplün on p.334 already raises and resolves the issue of whether or not to have the natural transformations run along or reverse to the monad homomorphism.
Prefixed the special case of monad transformations over a fixed category (here) with pointer to Barr & Wells 1983 §6.1 and to Moggi 1989 Def. 4.0.11.
who gives an early statement of morphimsm of monads (2 years before Street)
I’m unable to check right now, because my internet connection is too poor to load a PDF, but I believe Coppey also defined both notions of morphisms of monads in the 1970 paper “Morphismes et comorphismes de cotriples” (available on gallica, e.g. as linked to from CRAS).
It might be worth mentioning that Street’s convention of the 2-cells in a monad morphism is the opposite to the usual notion of monad morphism for monads on a fixed category (i.e. monoid morphisms in an endofunctor category). (Personally, I feel that Street’s choice of direction for the 2-cells is not the most natural choice, this being one of the reasons.)
Coppey also defined both notions of morphisms of monads
Thanks – have added it (now Coppey 1970).
It might be worth mentioning
Absolutely – yesterday I had added a brief remark on this (here) with pointer to page 334 in Pumplün 1970 who already nicely sorts out what’s happening (as noticed in #67 above):
$Mnd$ defined with the reverse $\lambda$ as by Street 1972 is actually equivalent to $CoMnd$ with the natural handedness of morphisms. (Street 1970 almost makes that observation on his pp 158)
But I gather Street’s reverse convention is convenient when discussing the functor of categories of modales (monad modules) induced by a monad morphism, since that’s covariant in Street’s (and Coppey’s) convention but contravariant in Pumplün and Barr & Well’s convention. I suppose.
Mnd defined with the reverse $\lambda$ as by Street 1972 is actually equivalent to CoMnd with the natural handedness of morphisms.
You are saying that for any 2-category K, there is a 2-functor Mnd(K) → CoMnd(K^co)^co (using Street’s convention), which is a 2-equivalence? What is its action on objects? (Sorry, I should try to figure it out from Pumplün myself, but I read German very slowly.)
But I gather Street’s reverse convention is convenient when discussing the functor of categories of modales (monad modules)
Yes, my assumption has always been that Street chose his convention to avoid taking opposites when considering Eilenberg–Moore objects (which is essentially the motivating construction in The formal theory of monads). On the other hand, this introduces an opposite when considering Kleisli objects, which could be argued to be more fundamental (for instance, a V-enriched monad always admits a Kleisli V-category, but only admits an EM V-category when V has enough limits).
You are saying that for any 2-category $K$
Ah, no, just for $K = Cat$, where the equivalence is given by conjugating the monad data with the contravariant equivalence of any category to its opposite.
Oh, I see! That makes more sense :)
Street chose his convention to avoid taking opposites when considering Eilenberg–Moore objects
Just to remark, for what it’s worth, that this seems a somewhat weird motivation, given that the action of monad morphisms on EM-categories is an internal instance of extension of scalars of modules along monoid homomorphisms. There is no reason for this not being contravariant (unless one wants to be very fancy and regard monads as a generalized kind of affine schemes…).
At the beginning of the section “Etymology” (here) I have worked in Bénabou’s footnote on why he chose to say “monad” – thanks to Varkor kindly pointing this out in another thread.
Also added a line on “polyads”.
66 Urs, I believe that analysis of functoriality of EM construction for Cat has been known before 1970, the point of Street’s paper is to do it abstractly in any 2-category with certain limits. In fact, anybody studying Eilenberg-Moore 1965 paper in much detail would probably work out this. People in coring theory were also forced to discover this notion independently in 2000s, and then learned a posteriori the relation with Street’s 2-category. (See the paper by Brzezinski et al. where several 2-categories of corings were studied).
73 “You are saying that for any 2-category K, there is a 2-functor Mnd(K) → CoMnd(K^co)^co (using Street’s convention), which is a 2-equivalence”
This is definitely widely used and often taken as a definition of CoMnd 2-category (at least I remember from discussions that Gabi Bohm often used it).
The earliest reference that I am aware of so far is Frei 1969. If there are earlier ones, let’s add them.
(If we could trust that authors would have cited precursors with the same observation, it would be easier…)
I’m travelling at the moment, so it’s not convenient for me to double check right now, but I believe functoriality of the EM construction (and Kleisli construction) is due to Maranda’s 1966 “On fundamental constructions and adjoint functors”. This paper is also the one where the universal property of the Kleisli category as the initial resolution of the monad is first proven. In the 1968 “Sur les propriétés universelles des foncteurs adjoints”, Maranda then proves a 2-categorical universal property of the Kleisli and EM constructions. However, I don’t believe it follows immediately from the results in these papers that monad morphisms are (dually) in bijection with functors between the EM categories: as far as I’m aware, the Frei paper is the earliest reference for this fact. The corresponding statement for Kleisli doesn’t explicitly appear in the literature for some time, but it is implicit in Linton’s early papers (e.g. the 1969 “An outline of functorial semantics”).
Thanks for this. I have added pointer to
(also at monad morphism and at Kleisli category).
However, while this article defines monad morphisms (in the “right” direction! :-) and discusses them in the context of initiality of the Kleisli category, I haven’t spotted a declaration of the induced functor between categories of modales.
(I admit that I haven’t read line-by line, but such a functor would need to involve the symbol “$\overline{\mathcal{C}'}$” in Maranda’s notation, and that seems to never appear.)
have now also added pointer to:
As before, I haven’t yet spotted a definition of the induced functor on EM-categories in there.
(But it’s strainful reading, and not due to the French. If I missed it and you show me it’s somewhere in there, I won’t be surprised.)
Actually, could you maybe say what you mean by:
The corresponding statement for Kleisli
?
The functor induced from a monad morphism $\phi$ preserves free modales (up to iso) iff $\phi$ is an isomorphism, no?
Re. 83:
However, while this article defines monad morphisms (in the “right” direction! :-) and discusses them in the context of initiality of the Kleisli category, I haven’t spotted a declaration of the induced functor between categories of modales.
The part I was referring to was this line in Theorem 2:
such that if is an adjoint morphism defining the fundamental construction $(T, U, k', u) \colon \mathscr C(S', p', k')$ and if $m \colon (S, p, k) \to (S', p', k')$, then there exists a unique functor […]
While Maranda doesn’t actually show this assignment of a functor from a monad morphism is functorial, he gives its action on morphisms.
Re. 85: There are two relevant functors here: one $EM \colon Mnd(C)^\circ \to CAT/C$ which is the one currently described in the article as Example 3.3 (though I notice this functor is not explicitly mentioned); and one $Kl \colon Mnd(C) \to C/CAT$ sending each monad to its Kleisli category (which is currently referenced in Example 3.2, but again not explicitly written down). It is the full faithfulness of this latter functor that I meant when I wrote “the corresponding statement for Kleisli” (rather than a restriction of the former to free algebras).
The part I was referring to was this line in Theorem 2:
Thanks, I see now.
This functor $Z$ goes $\mathcal{C}' \to \underline{\mathcal{C}}$. So I guess to get the desired functor on EM-categories $\underline{(-)}$ we need to assume that $U$ is monadic.(?)
There are two relevant functors here:
Right, thanks.
I think I am out of energy with editing on this point now, but this would be good to clarify in the entry.
This functor $Z$ goes $\mathcal{C}' \to \underline{\mathcal{C}}$. So I guess to get the desired functor on EM-categories $\underline{(-)}$ we need to assume that $U$ is monadic.(?)
Yes, that’s right. The presentation of the result in Maranda is certainly more awkward than one would hope, though I think the essence of the idea is contained there.
I think I am out of energy with editing on this point now, but this would be good to clarify in the entry.
I will try to find some time soon to clarify these aspects.
re #87:
So what’s a reference discussing both of these functors, clearly?
For Eilenberg–Moore, Theorem 3 of Frei’s Some remarks on triples is the result that the functor from the category of monads on a category $C$, to the category of monadic slices over $C$, is an equivalence. (It is immediate that this equivalently states that the functor from the category of monads on $C$ to the category of slices over $C$ is fully faithful.) For Kleisli, Lemma 10.2 of Linton’s An outline of functorial semantics is the result that the functor from the category of monads on $C$, to the category of opmonadic coslices under $C$, is an equivalence. (It is immediate that this equivalently states that the functor from the category of monads on $C$ to the category of coslices under $C$ is fully faithful.)
If you would really like a reference that makes these two statements entirely explicit, they appear as Corollary 6.40 and Corollary 6.49 of Arkor–McDermott’s The formal theory of relative monads (in a formal context).
Thanks! Okay, I am finally taking a closer look at Arkor & McDermott…
There is currently further discussion on the history of the terminology “monad” on the Categories-mailing list, some of which might be worth pointing to.
Now a couple of contributors mean to attribute the term to private conversation by Eilenberg. Concretely, Jirí Adámek writes (categories@mq.edu.au
, dated Nov 13, 2023, 2:02 AM):
Bill Lawvere once told me that ’monad’ had been the idea of Eilenberg. Later I asked him by email about the details and he answered the following:
I do not remember in which year it was. (Maybe 1968, judging from vague allusions in Springer Lecture Notes 80.) In any case it was in the common room of the old castle at Oberwolfach when Sammy came out from behind the piano and announced the change. His informal speech emphasized that the word would inflect well: ’monadic’ etc. He also explicitly said that nobody would ever confuse it with Leibnitzian monads.
(Of course 1968 $\gt$ 1967, but it may still be worth recording.)
[edit:
Further discussion on the list seems to have clarified the issue, with people jointly sorting out their memories of the past. Michael Barr (Nov 13, 2023, 7:30 PM) recalls that this all happened at one and the same 1966 meeting at Oberwolfach:
I am more than willing to believe that it was Benabou sitting next to me who proposed monad. It is entirely possible that Sammy came down and pronounced it “official”. And it was certainly in the old castle.
]
Just a lazy question: Can anyone give me a hint on how to access the CatList archives online? I gather the latest messages are no longer available on the former archive but now require some extra software?
At some point I got the impression I would be able to access the latest archives via my GoogleGroups account, but if that is so I didn’t manage to figure it out in the time I spent with the issue.
added pointer to:
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