Dubuc topos

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#Contents#
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## Idea

The _Dubuc topos_ is a model in which one may apply [[synthetic differential geometry]] to attain results in classical [[differential geometry]]. It is constructed as the [[topos]] of [[sheaves]] for the [[open covering]] [[Grothendieck topology|topology]] on the dual of the [[category]] of [[smooth algebra|C-infinity rings]] presented by a [[germ determined ideal]].

## Definition

+-- {: .num_defn #GermDeterminedCInfinityRing}
###### Definition
**([[germ-determined C-infinity ring]])**

A finitely generated [[C-infinity ring]] of the form $C^\infty(M)/I$ for some [[smooth manifold]] $M$ and [[ideal]] $I$ is __germ-determined__ if the [[ideal]] $I$ is a [[germ-determined ideal]], which means that for any $f\in I$, if the [[germ]] of $f$ at any point $x$ in the [[zero locus]] of $I$ in $M$ coincides with the [[germ]] of some element of $I$ at $x$, then $f$ itself belongs to $I$.

Equivalently, germ-determined [[ideals]] are precisely those [[ideals]]
that are closed under (possibly infinite) sums of [[locally finite]] families. One can also take sums with coefficients in a [[partition of unity]].

=--

+-- {: .num_defn #DubucTopos}
###### Definition
**(Dubuc topos)**

The _Dubuc topos_ is the [[sheaf topos]] on the [[site]] $G$, whose underlying [[category]] is the [[opposite category]] of the [[category]] of [[germ-determined C-infinity ring|germ-determined]] finitely generated [[C-infinity rings]] (Def. \ref{GermDeterminedCInfinityRing}).

The resulting [[opposite category]] is equipped with a [[Grothendieck topology]] as follows. A collection of morphisms $\{A_i\to B\}_{i\in I}$ in this category is a [[covering family]] if for any $i\in I$, the corresponding homomorphism of [[C-infinity rings]] $B \to A_i$ is isomorphic to a [[localization]] map at some element $b_i\in B$.

Here [[localization]] is defined using a universal property formulated in the category of [[C-infinity rings]] and need not  coincide with the usual [[localization of commutative rings]]. Furthermore, we require that applying the functor $\gamma$ to the family $\{A_i\to B\}_{i\in I}$
produces a jointly surjective family of [[continuous maps]] of [[topological spaces]].

Here the functor $\gamma$ is the [[right adjoint functor]] of the [[opposite functor]] of the inclusion of finitely generated point-determined [[C-infinity rings]] into finitely generated germ-determined [[C-infinity rings]]. Concretely, $\gamma(C^\infty(M)/I)$ for a germ-determined [[ideal]] $I$ can be described as the [[zero locus]] of $I$ in $M$,
converted into a finitely generated point-determined [[C-infinity ring]]
using its [[C-infinity ring]] of smooth functions.

Here a finitely generated [[C-infinity ring]] of the form $C^\infty(M)/I$ for some [[smooth manifold]] $M$ and [[ideal]] $I$ is __point-determined__ if the [[ideal]] $I$ is point-determined, which means that for any $f\in I$, if $f(x)=0$ for any point $x$ in the [[zero locus]] of $I$ in $M$, then $f$ itself belongs to $I$.

=--

## Some history
 {#SomeHistory}

_This section is written by [[Eduardo Dubuc]]_.

In papers on [[synthetic differential geometry|SDG]] one can find phrases of the type "We build our model after the standard manner of Moerdijk & Reyes" ([MR] "[[Models for Smooth Infinitesimal Analysis]]", Moerdijk and Reyes, Springer Verlag 1991). Now, this "standard manner" of building models was  developed by Dubuc, and not by Moerdijk-Reyes. Of course, I understand that when a monograph is available, the proper reference is that, and not the original papers. But, phrases as the one quoted above are ambiguous, and the fault is not of the authors.

I resume the history of the subject: In a series of papers and many lectures given specially in Montreal, Sydney, Oberwolfach and elsewhere, I created and started the development of the subject of models of SDG adapted  for the applications to classical [[differential geometry]].

1. _Sur les Modeles de la Geometrie Differentielle
Sinthetyque_, [[Cahiers|Cahiers de Topologie et Geometrie Differentielle]] Vol. XX-3
(1979).

1. _Schemas C-inf (amplified version of [3], with detailed proofs and many examples)_, "Prepublications de la Universite de Montreal" 80-81 edited by  G. Reyes (1980).

1. _C-inf Schemes_, American Journal of Mathematics, John Hopkins University, Vol. 103-4 (1981).

1. _Open Covers and Infinitary Operations in C-inf-rings_, Cahiers de Topologie et Geometrie Differentielle Vol. XXII-3 (1981).

1. _Integracion de campos vectoriales y Geometria Diferencial Sintetica_, Proceedings of the VII Sem. Nac. de Mat., FAMAF, Univ. Nac. de Cordoba (1984).

1. _Germ representability and Local integration of vector fields in a well adapted model of SDG_, Aarhus Univ. Math. Inst. preprint series (1985/1986),
(Journal of Pure and Applied Algebra Vol. 64, 1990).

1. _Archimedian Local C-inf-rings and Models of SDG_ (with [[Marta Bunge]]), Cahiers de Topologie et Geometrie Differentielle Vol. XXVII-3 (1986).

1. _Local concepts in SDG and germ representability_ (with Marta Bunge), in  D. Kueker et al. (ads), Mathematical Logic and Theoretical Computer Science, Lecture Notes in Pure and Applied Mathematics 106, Marcel Dekker (1987)

A) I introduced the concept, coined the name "[[well-adapted model]]", and constructed the first ones. I considered the two essential properties:

i) preservation of all [[open covers]].

ii) preservation of [[transversal maps|transversal]] [[pullbacks]].

B) I started a systematic study of [[smooth algebra|C-infinity rings]] as such. Of course, they were already there, but nothing had been done with them. I even had to state and prove such simple facts as that the algebraic quotient of a C-infinity ring by an R-algebra ideal had a canonical structure of C-infinity ring, and was then the quotient in this category.

I introduced the key notion of Germ Determined Ideal (or ideal of local nature), which was, as such, nowhere to be found in the literature, and stated and proved their basic properties. This, I think, is the most important concept in the subject. It is the basic definition to start to build upon. It is just the right concept needed. Among other things, I first proved it contains all finitely generated ideals, and defines the largest possible class of C-infinity rings consistent with the [[Nullstellensatz]]. This means that the ring can be seen as the ring of global sections of a C-infinity [[scheme]]. The notion of germ determined ideal also determines the right notion of C-infinity local ring (notice that I do not say local C-infinity ring), that I then developed. I also developed the relative C-infinity version of inverting elements universally, and proved the essential fact that the ring of C-infinity functions defined in an open set $U$ inverts universally a function which is non-zero exactly on $U$ (every Euclidean open is C-infinity  Zariski).

_When incorporating this work in their monograph Moerdijk and Reyes say (I quote): "Although this general notion of C-infinity  ring does not occur as such in classical analysis and differential geometry, the main examples do ...  Given the role of these examples of C-infinity rings in the classical literature, it is not surprising that although the statements of several of the results in this chapter seem new, most of their proofs are either known or easily derivable from known ones". This is, at the least, misleading, and  I see a clear intention to disqualify my work.  Of course, when the new concepts are introduced, the examples are already there, and the proof of the basic properties is easy. The important thing is to identify explicitly the concept, and to identify the right statements and properties, and this does not come easily. And I repeat, even if C-infinity rings may have been there, the concept of germ determined ideal was not, neither the concept of $C^\infty$-local ring, and several derived concepts and the statements of their basic essential properties neither)._

C) With this in hand, I introduced the [[topos]] $G$ of [[sheaves]] for the open covering topology on the dual of the category of C-infinity rings presented by a germ determined ideal, and proved all the basic important properties, which many times are the correct relative C-infinity versions of corresponding properties in [[algebraic geometry]]. This is the analogue in SDG of the [[Zariski topos]] of algebraic geometry (here we should notice that the Zariski topos is defined by the topology of all open covers, this is essential. Moerdijk-Reyes, misunderstanding Zariski, call Smooth-Zariski, the topos determined by the finite covers).

The topos $G$ is the best known model in order to do applications of SDG to classical differential geometry, and as such, it is the most utilized in practice. Many early workers in the subject ([[Jacques Penon|J. Penon]], O. Bruno, M. Bunge, F. Gago, [[David Yetter|Yetter]], among others) called this topos "The Dubuc Topos". Even Moerdijk and Reyes did so in some preprints, although they changed this in the published versions.

_In their monograph they say (I quote): "As far as terminology is concerned, we have tried to avoid descriptions of the type "the [[Ieke Moerdijk|Moerdijk]] envelop of the Reyes topos", in favor of more informative ones". But the true fact was that the only name that was involved was "Dubuc", since no other new name was being utilized at the time (of course, things as [[Kock-Lawvere axiom]] or [[Weil algebras]] were not aimed by this philosophy, and "Moerdijk envelop of the Reyes topos" was an invention to reduce to the ridiculous, the fact of naming things after "Dubuc"._

D)  I should mention here that I do not ignore the fact that my work is  acknowledged and referenced in the monograph, but the matter is much more subtle, and nobody can deny the evidence of the following consequence of their maneuvering:

My name, as time passes, and as young people appear, is less and less credited with a subject that I created and developed, namely, models of SDG adapted to classical differential geometry. People talk as if the well adapted models were always there, or start referring to them in a way that lead inexperienced (in the subject)  readers  to believe that these constructions are "M-R way of doing models", as I quoted at the  beginning of this note.

E) This does not do justice to my work, and does not corresponds to the true history of the subject.  As an starting point to remedy it, I request all workers that need to use the topos $G$, to refer to it as "Dubuc Topos". After all, it is a long tradition in mathematics to associate proper names to important concepts or constructions when it is justified, as I believe it is the case here.

**REMARK.**  Proposition 18 and Theorem 22 of 6. cited above establish that postulate WA2 (II, 3.2, 3.3) of 8. holds in the Dubuc Topos. However, there is a mistake (discovered by [[Michael Makkai|M. Makkai]]) in Lemma 21 of 6. which invalidates the proof. I believe that the result is true but I am not working to fix the proof now. The external statement holds, it is easy and it proved in 5. cited above.

## Related entries

* [[Cahiers topos]]

## References

* [[Eduardo Dubuc]], _Sur les modèles de la géométrie différentielle synthétique_, [[Cahiers|Cahiers de Topologie et Geometrie Differentielle]] Vol. XX-3
(1979) ([numdam:CTGDC_1979__20_3_231_0](http://www.numdam.org/item?id=CTGDC_1979__20_3_231_0))

* [[Eduardo Dubuc]], _Schemas $C^\infty$_, "Prepublications de la Universite de Montreal" 80-81 edited by  G. Reyes (1980).

This preprint is an amplified version of the following article, with detailed proofs and many examples:

* [[Eduardo Dubuc]], _$C^\infty$ Schemes_, American Journal of Mathematics, John Hopkins University, Vol. 103-4 (1981) ([jstor:2374046](http://www.jstor.org/stable/2374046))

* [[Eduardo Dubuc]], _Open Covers and Infinitary Operations in C-inf-rings_, [[Cahiers|Cahiers de Topologie et Geometrie Differentielle]] Vol. XXII-3 (1981) ([numdam:CTGDC_1981__22_3_287_0](http://www.numdam.org/item?id=CTGDC_1981__22_3_287_0))

* [[Eduardo Dubuc]], _Integracion de campos vectoriales y Geometria Diferencial Sintetica_, Proceedings of the VII Sem. Nac. de Mat., FAMAF, Univ. Nac. de Cordoba (1984). [journal volume](http://inmabb.criba.edu.ar/revuma/revuma.php?p=toc/vol35), [pdf](http://inmabb.criba.edu.ar/revuma/pdf/v35/p151-162.pdf)

* [[Eduardo Dubuc]], _Germ representability and Local integration of vector fields in a well adapted model of SDG_, Aarhus Univ. Math. Inst. preprint series (1985/1986), published in Journal of Pure and Applied Algebra Vol. 64, (1990) (doi:10.1016/0022-4049(90)90152-8)

* [[Marta Bunge]], [[Eduardo Dubuc]], _Archimedian Local C-inf-rings and Models of SDG_, [[Cahiers|Cahiers de Topologie et Geometrie Differentielle]] Vol. XXVII-3 (1986) ([numdam:CTGDC_1986__27_3_3_0](http://www.numdam.org/item?id=CTGDC_1986__27_3_3_0))

* [[Marta Bunge]], [[Eduardo Dubuc]], _Local concepts in SDG and germ representability_, in  D. Kueker et al. (ads), Mathematical Logic and Theoretical Computer Science, Lecture Notes in Pure and Applied Mathematics 106, Marcel Dekker (1987)

Penon's thesis has an appendix on the Dubuc topos:

* [[Jacques Penon]], _De l'infinitésimal au local_ (Thèse de Doctorat d'État). [[Diagrammes]], S13 (1985), p. 1-191 ([numdam:DIA_1985__S13__1_0](http://www.numdam.org/item?id=DIA_1985__S13__1_0))


Dudley E. Littlewood

[[!redirects Littlewood]]
[[!redirects D. E. Littlewood]]
Dudley Earnest Littlewood (1903 -  1979) was a British mathematician whose main research was in group represntation theory. From 1948 until his retirement in 1970, he was professor of Pure Mathematics at the  then University College of North Wales, Bangor.

*  [wikipedia entry](https://en.wikipedia.org/wiki/Dudley_E._Littlewood)

* [St. Andrews History entry](https://mathshistory.st-andrews.ac.uk/Biographies/Littlewood_Dudley/)

* [Math Genealogy page](https://www.mathgenealogy.org/id.php?id=79948)

##Related entries

* [[plethysm]]

* [[combinatorial representation theory]]

* [[Littlewood-Richardson coefficients]]

category:people

[[!redirects D. E. Littlewood]]


Duff interview at M-Theory-Mathematics 2020

* [[Mike Duff]]:

  \linebreak

  **Perspectives on M-Theory**

  \linebreak

  interview/opening remarks at

  _[M-Theory and Mathematics 2020](/nlab/show/M-Theory+and+Mathematics#2020)_

  January 2020, NYU Abu Dhabi

  \linebreak

  questions by [[Hisham Sati|H. Sati]]

  \linebreak

  view recording:

  * first min: [YT](https://youtu.be/hCCmYwwKmhc)

  * full: [YT](https://youtu.be/aHj_Thu8d54)

  * [pre-recording](https://nyu.zoom.us/rec/share/zJBpF4Ch-39JWdLy62vjQvQiGaq6aaa8hCQX_qILzRpfMN17dpCEJSXld0pzPxFC?startTime=1579698234000)

on the early history of and the perspective of [[M-theory]].

\linebreak

#Contents
* table of contents
{:toc}

## Transcript

**Q**: You had codiscovered, early in the 1980s, some core ingredients of [[M-theory]], like [[M2-brane|membranes]]. Later on, in 1995, this came to be known as "M-theory". How do you reflect on that?

**Duff**:

(00:43) Reflecting on M-theory is quite a big challenge.

(00:49) The story of M-theory began with the story of eleven dimensions.

(00:54) It was 1978, I think, when [[Werner Nahm]] pointed out $[$[Nahm 78](supersymmetry#Nahm78)$]$ that [[supersymmetry]] places an upper limit on the dimension of spacetime, which is 11.

(01:05) And so in the early 80s, my colleagues and I looked to [[11-dimensional supergravity]] as a candidate for a unified theory of the fundamental forces.

(01:16) That involves [[KK-compactification|compactifying]] the 11 dimensions down to 4.

(01:23)  That had its problems. The theories we looked at were not [[phenomenology|phenomenologically]] very promising.

(01:31) Whether you have extra dimensions or supersymmetry, the problem still remains that Einstein's gravity is incompatible with quantum mechanics, in the sense that the theory is not [[renormalization|renormalizable]] in the  sense of [[perturbative QFT|conventional quantum field theory]].

(01:51) When [[string theory]] came along, in the 1984 string revolution, 11 dimensions got pushed to the sidelines, and we were told that 11 dimensions was barking up the wrong tree.

(02:08) But some of us thought, even then, that there is something not quite right: Why do [[superstrings]] live in 10 spacetime dimensions, if supersymmetry allows 11?

{#0224} (02:24) The next major contribution was when Bergshoeff, Sezgin and Townsend discovered $[$[Bergshoeff-Sezgin-Townsend 87](M2-brane#BergshoeffSezginTownsend87)$]$ the [[M2-brane|11-dimensional supermembrane]].

(02:35) For me that was the starting point of what I would call [[M-theory]].

(02:42) Because it said: Fine, strings are fundamental and live in 10 dimensions, but you have these membranes, which seem equally fundamental, living in 11 dimensions.

{#0255} (02:55) My colleagues and I -- Stelle, Howe, Inami -- were able to show $[$[Duff-Howe-Inami-Stelle 87](double+dimensional+reduction#DuffHoweInamiStelle87)$]$
that if you [[wrapped brane|wrap]] this [[M2-brane|11-dimensional membrane around a circle, it looks like]] a 10-dimensional string, in fact it looks like the [[type IIA string theory|type IIA string]].

(03:14) So for us, that is proof, if proof is needed, that membranes in 11-dimensions were part of the big picture, including strings.

(03:26) However, that view was not a popular one, I have to say.

(03:31) The overriding view at that time was that spacetime had 10 dimensions, it was occupied by strings, which were the theory of everything, and all these other objects, these branes, were theories of nothing.

(03:46) That was the conventional view.

(03:50) So in the early 90s, there were sort of two communities, if you like: The string people were doing their thing in 10 dimensions; the membrane people were doing theirs in 11; and it wasn't clear if we were on the same page, or what.

(04:06) And then, as you know, [[Edward Witten]] made this startling speech at the University of Southern California
$[$[Witten 95](M-theory#Witten95Talk)$]$, where he pointed out that the five consistent string theories and 11-dimensional supergravity were not, as we previously thought, six rival Theories of Everything: They were six different corners in the deeper and more profound theory that he called "[[M-theory]]".

{#0438} (04:38) Now, given that we have been arguing in favour of membranes, the fact that the theory got called "M" was something of a Pyrrhic victory.

(04:50) It was saying: "Well, maybe membranes, we're not completely sure." $[$see [Hořava-Witten 95, p. 2](M-theory#NonCommittal)$]$

(04:56) For me, anyway, it was clear that branes were just as important as strings.

{#0505} (05:05) [[Joseph Conlon]], writing in a recent book on string theory, says that when he saw our paper $[$[Duff-Howe-Inami-Stelle 87](double+dimensional+reduction#DuffHoweInamiStelle87)$]$ about wrapping the brane around the 11th dimension, from the late 80s, he was shocked, because the history of the theory that he had been brought up with would not allow such a thing until 1995.

{#0531} (05:31) So, M-theory had a strange history.

{#0538} (05:38) I could summarize my research in the early 1980s as arguing for spacetime dimensions greater than 4, and for worldvolume dimensions greater than two in the late 80s, and that struggle was by far the harder of the two.

**Q**: Similarly in the late 80s, also the regularized quantization of the super-membrane led to the [[BFSS matrix model|matrix model]], which later on was re-discovered as [[D0-brane]] quantum mechanics, and then hailed as a contender for a definition of M-theory. How do you reflect on this curious M-theory conceivement?

**Duff**:

{#0630} (06:30) It was quite usual for discoveries that were made in the 80s to re-appear in the 90s.

(06:39) I don't want to diminish the importance of the matrix model. It was very important. They built on earlier work -- for what we would now call D0-branes -- of the late 80s.

(06:59) But the [[BFSS matrix model|matrix model]] itself was not all of M-theory; it was a corner of M-theory, and it told us certain interesting things, but there were interesting things about M-theory that it didn't tell us.

{#0713} (07:13) I think we are still looking, in fact, for what M-theory really is.

(07:19) We have a patchwork picture. We understand various corners. But the overarching big picture of [M-theory is still waiting to be discovered](M-theory#TheOpenProblem), in my view.

**Q**: In your famous review "M-Theory (The theory formerly known as strings)" $[$[Duff 96](#Duff96)$]$, in the concluding section, you wrote: "The overriding problem in super-unification, in the coming years, will be to take the Mystery out of M-theory, while keeping the Magic and the Membranes." What do you think is the status of this "overriding problem" today?

**Duff**:

(08:08) It's still there, of course.

(08:12) M-theory in 1995 was very promising, and it's taught us a lot about the fundamental interactions; but the final theory is still not with us.

(08:24) My view, and I consistently held this view since I first started in [[quantum gravity]] is that we have to take a long-term stance.

(08:37) There'll be no overnight miracles; we have to keep beavering away, chipping away, and hopefully, one day, we'll find the answers.

(08:48) I am optimistic that there is an M-theory without nagging mystery, but I wouldn't like to put a time-scale on how long it's going totake us to find it.

(09:01) My argument would be for patience.; this is what we need right now. Of course that's not popular with the journalists, or for quick gratification.

(09:16) But if we look at the discussions that had the greatest impacts, recently: the Higgs boson took 50 years between its prediction and its discovery; the cosmological constant, if indeed that's what dark energy is, would be a hundred years between the prediction and the discovery; similarly gravitational waves.

(09:42) There is no reason to think that a Theory of Everything is going to come along any quicker than these other discoveries.

(09:51) We just have to keep hoping for the best.

**Q**: In the late 90s you wrote in Scientific American $[$[Duff 98](#Duff98)$]$ and also in your book on M-theory
$[$[Duff 99](#Duff99)$]$: "Future historians may judge the late 20th century as a time when theorists were like children playing on the sea shore, diverting themselves with the smoother pebbles or prettier shells of superstrings, while the great ocean of M-theory lay undiscovered before them." How do you look at his prediction 20 years into the 21st century?

**Duff**:

(10:30) I hope you recognize that most of these words were of Isaac Newton, I just made a few substitutions in the right places.

(10:40) I still hold the view that perturbative 10-dimensional strings, as they were pursued in the late 80s, will be seen to be a small corner of the final theory.

(10:57) That's not a view that's necessarily shared by others: Ed Witten, in fact, wrote to me, after I wrote that, to argue that he didn't share that view.

{#1110} (11:10) There is a certain faction that believed that strings are more fundamental than branes, because [string theory admits a perturbation expansion](string+theory+FAQ#WhatIsStringTheory).

(11:23) But in my view, that's not the criterion for what is fundamental and what isn't.

{#1129} (11:29) God does not do [[perturbation theory]], perturbation theory is what we do because we don't know any better.

(11:38) So the fact that branes do not admit a perturbative treatment, as strings do, is not, in my view, a reason for thinking that branes are less fundamental than strings -- especially since strings are just a limiting case, as we see, of branes.

(11:58) I don't see how you can maintain that one object which is not fundamental has a limiting case which is.

(12:08) So we have to treat them democratically.

**Q**: In your interview by Farmelo, last year
$[$[Farmelo interview Duff](https://grahamfarmelo.com/the-universe-speaks-in-numbers-interview-14/)$]$, you said: "The problem we face is that we have a patchwork understanding of M-theory, like a quilt: We understand this corner and that corner, but what's lacking is the overarching big picture. So directly or indirectly, my research hopes to explain what M-theory really is. We don't know what it is." Do you have a hunch what form the answer might eventually take?

**Duff**:

(12:44) No, actually I don't know.

(12:46) I wouldn't like to predict what the ultimate picture of M-theory will be; I imagine it will be something quite different from what we can imagine now.

(12:58) Going back to how the M-theory has developed:

(13:05) One curious feature of the big story is the [[AdS/CFT correspondence]]. Because when [[Juan Maldacena|Maldacena]] wrote his paper
$[$[Maldacena 97](AdS-CFT#Maldacena97a)$]$, he had three examples: $AdS_4 \times S^7$, $AdS_5 \times S^5$, $AdS_7 \times S^4$. Two of those were 11-dimensional M-theory, compactified, and the other one was Type IIB.

(13:34) But if you look at the papers that have been written since Maldacena's seminal contribution, the vast majority of them have been on the $AdS_5 \times S^5$-story.

(13:48) There hasn't, for whatever reason, been the same progress in M-theory, as a result of that.

{#1358} (13:58) The [[AdS/CFT correspondence]] in a way diverted attention away from the goal of finding a unified theory of all the fundamental forces, starting from 11 dimensions.

(14:13) It's been tremendously successful in its own right, Maldacena's paper $[$[Maldacena 97](AdS-CFT#Maldacena97a)$]$ is the most highly cited paper in history.

{#1422} (14:22) But it has not, strangely enough, contributed to how we unify the strong, weak and electromagnetic forces with gravity. At least I don't think it has.

(14:35) That's a different problem.

(14:38) I would have been happier, I think, looking back, if we were further down that road, since 1995, than we are.

(14:48) But I am still optimistic that we are on the right track.

**Q**: So you believe we ought to get back to the original big question about: what's the nature of M-theory.

**Duff**:

(15:03) That's my view, yes.

(15:05) But of course, in order to do that, you have to have some good ideas. We need more good ideas.

**Q**: In that same interview with Farmelo, you said: "In a certain sense, and this is not a popular statement, I think it's premature to be asking: 'What are the empirical consequences?', because it's not yet in a mature enough state, where we can sensibly make falsifiable predictions." Would you like to expand on that?

**Duff**:

(15:39) Yes, it was harking back to the remarks I made a few minutes ago, about the timeline between the germ of a theoretical idea and its ultimate empirical confirmation.

(15:53) Another good example would be quantum entanglement: The problem was spotted in the 1930s, by Einstein, Rosen and Podolsky, but it wasn't until the 1960s that John Bell came up with a falsifiable prediction, and it was another 20 years before Alain Aspect did the experiment.

(16:19) My view is that Theories of Everything are going to similarly require a large timeline between the original idea and the confirmation.

{#1636} (16:36) That's why I think [M-theory is not yet in a mature enough stage](M-theory#TheOpenProblem) for us to make falsifiable predictions.

{#1644} (16:44) We don't understand the theory sufficiently well yet to do so.

**Q**: Should we try to focus on understanding it first, and then it would naturally give us predictions?

**Duff**:

(17:00) That's what I would recommend.

(17:03) But of course progress comes in directions that you often don't expect.

(17:11) So I don't want to predict that progress will come from a particular direction.

**Q**: A similar point was made recently by [[Tom Banks]], in a critique of the 15 year old discussion of the [[landscape of string theory vacua|landscape]] of [[de Sitter spacetime|de Sitter]] [[string theory vacuum|vacua]]
$[$[Banks 19](swampland#Banks19)$]$. Do you think it's possible that an actual formulation of M-theory will show of the putative landscape of string theory is not actually there?

**Duff**:

(17:59) Superficially, I would say No, because M-theory has even more vacua than string theory has.

(18:07) The landscape problem is not going away.

(18:13) What's often not appreciated, though, is that if you follow the critics in string theory and M-theory and stop doing them today, and start tomorrow with a blank sheet of paper and a sharp pencil, and say: "Where do we go from here?", the landscape problem would still be there!

(18:36) The problem of why you pick one physical vacuum from an infinity of mathematically possible vacua is a problem that is going to be there whether you do string theory/M-theory or not.

(18:55) I am agnostic about the [[multiverse]]. I don't know whether we live in one universe or many. But there, again, we have to keep beavering away, and hopefully the truth will out.

**Q**: Do you envision any role for mathematics? If you are able to map M-theory to mathematics, would you think mathematics would have something to say?

**Duff**:

(19:25) It certainly does, because in the absence of empirical data, what do we have to guide us? We have consistency as our criterion.

(19:37) Whatever our theories are, they must be, first of all, mathematically consistent -- no self-contradictions -- and they must be consistent with what we already know to be true.

(19:51) This is, in a way, a straightjacket, that constrains our speculations and our projections where the theory is going to go.

(20:03) And its a very severe straightjacket: We can't just dream up any old thing and expect it to be mathematically consistent, or consistent with our knowledge.

(20:17) Mathematical consistency has a vital part to play.

{#2021} (20:21) As you and your colleagues are aware, sophisticated mathematics is becoming more and more important in understanding M-theory; and it is going to require the kind of mathematical [approaches that you are doing](https://ncatlab.org/schreiber/show/Hypothesis+H).

(20:37) You'll sort it all out, I am sure.

**Q**: How do you envision the future of activity in M-theory should look like?

**Duff**:

(20:54) There again I don't want to make any rash predictions.

(21:00) My view of research is a Darwinian one: You find the brightest people, you give them the money, you let them get on with it and they will go wherever they will go. I don't think one should try and engineer, from above,  the directions that research should take. Let the imagination of the best people -- let their imagination run them, and we'll see where it takes us.

\linebreak

\linebreak

## Further reading

A commented collection of the early articles on the [[super-membrane]] and what came to be called [[M-theory]]:

* {#Duff99} [[Michael Duff]],

  _[[The World in Eleven Dimensions]]_

  IoP (1999)

Further early exposition of [[M-theory]]:

* {#Duff96} [[Michael Duff]], _M-Theory (the Theory Formerly Known as Strings)_,  Int. J. Mod. Phys. A11 (1996) 5623-5642 ([arXiv:hep-th/9608117](http://arxiv.org/abs/hep-th/9608117))

* {#Duff98} [[Michael Duff]], _The Theory Formerly Known as Strings_, Scientific American 1998 ([pdf](https://www.nikhef.nl/pub/services/biblio/bib_KR/sciam14395569.pdf), [[DuffFormerlyStrings98.pdf:file]])

* {#Duff98b} [[Michael Duff]], _A Layman's Guide to M-theory_, 	Abdus Salam Memorial Meeting, Trieste, Italy, 19 - 22 Nov 1997, pp.184-213 ([arXiv:hep-th/9805177](https://arxiv.org/abs/hep-th/9805177), [cds:355721](https://cds.cern.ch/record/355721))

***

\linebreak

\linebreak

category: reference


Duflo isomorphism

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* table of contents
{:toc}

## Idea

A form of [[PBW theorem]] says that the [[symmetric algebra]] and the [[universal enveloping algebra]] of a [[Lie algebra]] $g$ are [[isomorphism|isomorphic]] as [[vector spaces]] (in fact as [[Lie algebra module|$\mathfrak{g}$-modules]], cf. eg. [Dixmier 1974, §2.4.10](#Dixmier74)).

However this is not an isomorphism [[associative algebras]], but rather an isomorphism of [[filtered object|filtered]] [[coalgebras]]. One can compose the PBW isomorphism with an additional automorphism to get an isomorphism of vector spaces which restricts to isomorphism of __algebras__ when restricted to the subalgebras of $g$-invariant functions.

The original proof in [Duflo 1977](#Duflo77) is rather case-by-case, using the structure theory of [[Lie algebras]]. Kontsevich in 1998 gave a new proof which generalizes to some geometric situations in deformation quantization and which refines to a fact at the [[derived geometry|derived]] level (expressed in terms of an isomorphism at the level of [[Hochschild cohomology]]). At the level of derived geometry this expresses the property of certain derived exponential map constructed using [[Hochschild-Kostant-Rosenberg map]] precomposed by a square root of the Todd class in appropriate setup. This has prompted a series of articles by Markarian, Caldararu, Chen and others to explain the appearance of Todd class (or related [[Atiyah class]]). Refinements and analogues include [[Kashiwara-Vergne conjecture]].

## Related concepts

* [[PBW theorem]]

* [[Wheels theorem]]

## References

Original article:

* {#Duflo77} [[Michel Duflo]], *Opérateurs différentiels bi-invariants sur un groupe de Lie*, Ann. Sci. École Norm. Sup. (4) 10 (1977), 265–288  ([numdam:ASENS_1977_4_10_2_265_0](http://www.numdam.org/item/ASENS_1977_4_10_2_265_0/), [MR56:3188](http://www.ams.org/mathscinet-getitem?mr=56:3188))

* [[M. Kontsevich]], _Deformation quantization of Poisson manifolds_, Lett. Math. Phys. __66__ (2003), no. 3, 157–216; _Operads and motives in deformation quantization_, Lett. Math. Phys. __48__ (1999), no. 1, 35–72.

Review:

* [[Damien Calaque]], [[Carlo A. Rossi]], _Lectures on Duflo isomorphisms in Lie algebra and complex geometry_, European Math. Soc. (2011) [[doi:10.4171/096](https://doi.org/10.4171/096), [pdf](https://imag.umontpellier.fr/~calaque/lecturenotes-LecturesDufloETH)]

See also:

* Wikipedia, *[Duflo isomorphism](https://en.wikipedia.org/wiki/Duflo_isomorphism)*

* [[Adrien Brochier]], _A Duflo star-product for Poisson groups_([arxiv/1604.08450](http://arxiv.org/abs/1604.08450))

[[!redirects Duflo map]]


Dugger's theorem

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#Contents#
* table of contents
{:toc}

## Idea

Dugger's theorem identifies [[combinatorial model categories]] as the [[model category]]-presentations of [[locally presentable (infinity,1)-categories]].

## Statement

+-- {: .num_theorem }
###### Theorem
**(Dugger's theorem)**

Every [[combinatorial model category]] $C$ is [[Quillen equivalence|Quillen equivalent]] to a left [[Bousfield localization of model categories|Bousfield localization]] $L_S SPSh(K)_{proj}$ of the global projective [[model structure on simplicial presheaves]] $SPSh(K)_{proj}$ on a [[small category]] $K$

$$
  L_S SPSh(K)_{proj} \stackrel{\simeq_{Quillen}}{\to}
  C
  \,.
$$

=--

This is [Dugger (2001), theorem 1.1](#Dugger01) building on results in [Dugger (2001b)](#Dugger01b).

+-- {: .proof}
###### Proof

The proof proceeds (the way Dugger presents it, at least) in roughly three steps:

1. Use that $[C^{op}, sSet_{Quillen}]_{proj}$ is in some precise sense the _homotopy-_ [[free cocompletion]] of $C$. This means that every functor $\gamma : C \to B$ from a plain category $C$ to a model category $B$ factors in an essentially unique way through the [[Yoneda embedding]] $j : C \to [C^{op},sSet]$ by a [[Quillen adjunction]]

   $$
     (\hat \gamma \dashv R) :
     B
     \stackrel{\overset{\hat \gamma}{\leftarrow}}
      {\underset{R}{\to}}
     [C^{op}, sSet_{Quillen}]_{proj}
     \,.
   $$

   The detailed definitions and detailed proof of this are discussed at [[(∞,1)-category of (∞,1)-presheaves]].

1. For a given combinatorial model category $B$, choose $C := B_\lambda^{cof}$ the full [[subcategory]] on a [[small set]] (guaranteed to exist since $B$ is [[locally presentable category|locally presentable]]) of cofibrant $\lambda$-[[compact object]]s, for some [[regular cardinal]] $\lambda$, and show that the induced Quillen adjunction

   $$
     B \stackrel{\overset{\hat i}{\leftarrow}}{\underset{R}{\hookrightarrow}}
     [(B_\lambda^{cof})^{op}, sSet]_{proj}
   $$

   induced by the above statement from the inclusion $i : B_\lambda^{cof} \hookrightarrow B$ exhibits $B$ as a homotopy-[[reflective subcategory]] in that the [[derived adjunction counit]] $ \hat i \circ Q \circ R \stackrel{\simeq}{\to} Id$ ($Q$ some cofibrant replacement functor) is a [[natural transformation|natural]] [[weak equivalence]] on fibrant objects (recall from [[adjoint functor]] the characterization of adjoints to full and faithful functors).

1. Define $S$ to be the set of morphisms in $[(C_\lambda^{cof})^{op}, sSet]$ that the left [[derived functor]] $\hat i \circ Q$ of $\hat i$ (here $Q$ is some cofibrant replacement functor) sends to weak equivalences in $B$. Then form the left [[Bousfield localization of model categories|Bousfield localization]] $L_S [(C_\lambda^{cof})^{op},sSet]_{proj}$ with respect to this set of morphisms and prove that this is [[Quillen equivalence|Quillen equivalent]] to $B$.

Carrying this program through requires the following intermediate results.

First recall from the discussion at [[(∞,1)-category of (∞,1)-presheaves]] that to produce the Quilen adjunction $(\hat i \dashv R)$ from $i$, we are to choose a [[cofibrant resolution]] functor

$$
  I : C \to [\Delta,B]
$$

of $i : C= B_\lambda^{cof} \to B$.

The [[adjunct]] of this is a functor $\tilde I : C \times \Delta \to B$. For each object $b \in B$ write $(C \times \Delta \downarrow b)$ for the [[slice category]] induced by this functor.

**Lemma** (Dugger, prop. 4.2)

For every fibrant object $b \in B$ we have that the [[homotopy colimit]] $hocolim (C \times \Delta \downarrow b) \to B)$
is weakly equivalent to $\hat i \circ Q\circ R (b)$.

**Corollary** (Dugger, cor. 4.4) The induced Quillen adjunction

$$
  B \stackrel{\leftarrow}{\to}
  [C^{op}, sSet]
$$

is a homotopy-reflective embedding precisely if the canonical morphisms

$$
  hocolim (C \times \Delta \downarrow b) \to b
$$

are weak equivalences for every fibrant object $b \in B$.

...

=--

Notice that the theorem just mentions plain combinatorial model categories, not [[simplicial model category|simplicial model categories]]. But of course by basic facts of [[enriched category theory]] $Funct(C^{op}, SSet)$ is an [[SSet]]-[[enriched category]] and its projective [[global model structure on functors]] $Func(C^{op}, SSet)_{proj}$ is compatibly a [[simplicial model category]], as are all its [[Bousfield localization of model categories|Bousfield localizations]]. (See [[model structure on simplicial presheaves]] for more details.) Therefore an immediate but very useful corollary of the above statement is

+-- {: .num_corollary }
###### Corollary

Every combinatorial model category is [[Quillen equivalence|Quillen equivalent]] to one which is

* a [[simplicial model category]]

* a left [[proper model category]].

=--

## Related concepts

[[!include locally presentable categories - table]]

## References

Dugger's theorem is due to

* {#Dugger01} [[Daniel Dugger]], _[[Combinatorial model categories have presentations]]_, Adv. Math. **164** (2001) 1 177-201 [[arXiv:math/0007068](http://arxiv.org/abs/math/0007068), [doi:10.1006/aima.2001.2015](https://doi.org/10.1006/aima.2001.2015)]

based on results in

* {#Dugger01b} [[Dan Dugger]], _[[Universal homotopy theories]]_, Advances in Mathematics **164** (2001)
144-176 [[arXiv:math/0007070](https://arxiv.org/abs/math/0007070)]

The interpretation in terms of [[locally presentable (infinity,1)-categories]] is due to

* {#Lurie} [[Jacob Lurie]], _[[Higher Topos Theory]]_


Duiliu-Emanuel Diaconescu

__Duiliu-Emanuel Diaconescu__ is a mathematical physicist ([[superstring theory]], [[algebraic geometry]]).

* page at [Rutgers](http://www.physics.rutgers.edu/people/pips/Diaconescu.html)

* _D-branes and categories_, lecture at Kavli [ITP Santa Barbara](http://online.itp.ucsb.edu/online/mtheory_c01/diaconescu)

* SLAC spires [list of papers](http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=ea+Diaconescu,+Duiliu+Emanuel)

## Selected writings

On the [[McKay correspondence]]:

* [[Duiliu-Emanuel Diaconescu]], Mauro Porta, Francesco Sala, _McKay correspondence, cohomological Hall algebras and categorification_, Represent. Theory **27** (2023) 933-972 [[arXiv:2004.13685](https://arxiv.org/abs/2004.13685), [doi:10.1090/ert/649](https://doi.org/10.1090/ert/649)]

## Related entries

* [[Diaconescu-Moore-Witten anomaly]]

* [[wall crossing]], [[BPS state]], [[Donaldson-Thomas invariant]]

category: people


Duistermaat-Guillemin trace formula

## Idea

## Related concepts

* [[zeta function of an elliptic differential operator]]

* [[Selberg trace formula]]

## References

* {#DuistermaatGuillemin} [[Hans Duistermaat]], [[Victor Guillemin]], _The Spectrum of Positive Elliptic Operators and Periodic Bicharacteristics_,Inventiones mathematicae (1975) Volume: 29, page 39-80 ([EuDML](https://eudml.org/doc/142329))


Duncan Haldane

* [Wikipedia entry](https://en.wikipedia.org/wiki/Duncan_Haldane)

## Selected writings

Introducing the "Haldane phase of the 1d spin chain", the first example of a [[symmetry protected topological phase]]:

* {#Haldane83} [[Duncan Haldane]], *Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the $O(3)$ nonlinear sigma model*, Physics Letters A **93** 9 (1983) 464-468 $[$doi:10.1016/0375-9601(83)90631-X $]$

Introducing the [[Haldane model]]:

* [[Duncan Haldane]], *Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the "Parity Anomaly"*, Phys. Rev. Lett. **61** (1988) 2015 $[$[doi:10.1103/PhysRevLett.61.2015](https://doi.org/10.1103/PhysRevLett.61.2015)$]$

On [[photonic crystals]] and [[topological photonics]]:

* [[Duncan Haldane]], S. Raghu, *Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry*, Phys. Rev. Lett. **100** (2008) 013904 [[doi:10.1103/PhysRevLett.100.013904](https://doi.org/10.1103/PhysRevLett.100.013904)]

category: people

[[!redirects Frederick Duncan Michael Haldane]]
[[!redirects F. Duncan M. Haldane]]
[[!redirects F. Duncan Haldane]]
[[!redirects F. D. M. Haldane]]


Dunkl operator

Let $G$ be a [[Coxeter group]] with a reduced [root system](http://en.wikipedia.org/wiki/Root_system) $R$. A multiplicity function $k$ on $R$ satifies by definition property $k(\lambda)=k(\mu)$ iff the corresponding reflections $s_\lambda$ and $s_\mu$ are conjugate each to another. A __Dunkl operator__ is defined on smooth functions in $\mathbb{R}^N$ by the formula
$$
T_i f(x) = (\partial_i f)(x) - \Sigma_{\lambda\in R^+} k(\lambda)\frac{f(x)-f(s_\lambda x)}{\langle x,\lambda\rangle}
\lambda_i
$$

There are also many variants and generalizations of this definition to various setups.
Dunkl operators appear in the theory of [[Calogero model]] and of the Cherednik (= [[double affine Hecke algebra|double affine Hecke]]) algebras.

* C. Dunkl, _Differential-difference operators associated to reflection groups_, Trans. AMS __311__ (1989), 167--183.
* Ivan Cherednik, _Introduction to double Hecke algebras_, [arXiv/math/0404307](http://arxiv.org/abs/math/0404307)
* [[Pavel Etingof]], _Lectures on Calogero-Moser systems_, [pdf](http://www-math.mit.edu/~etingof/zlecnew.pdf)
* P. Etingof, X. Ma, _On elliptic Dunkl operators_, [arXiv/0706.2152](http://arxiv.org/abs/0706.2152)
* Yu. Berest, P. Etingof, V. Ginzburg, _Finite-dimensional representations of rational Cherednik algebras_, Int. Math. Res. Not. 2003, no. 19, 1053--1088.
* C. Dunkl, E. Opdam, _Dunkl operators for complex reflection groups_, Proc. London Math. Soc. __86__ (2003), 70--108.
* A.N. Sergeev, A.P. Veselov, _Dunkl operators at infinity and Calogero-Moser systems_, [arxiv/1311.0853](https://arxiv.org/abs/1311.0853)

There is also a braided/$q$-deformation:

* Yuri Bazlov, [[Arkady Berenstein]], _Noncommutative Dunkl operators and braided Cherednik algebras_, Selecta Math. (N.S.) __14__ (2009), no. 3-4, 325--372, [pdf](https://pages.uoregon.edu/arkadiy/doubles.pdf), [MR2010k:16044](https://mathscinet.ams.org/mathscinet-getitem?mr=2511188) [doi](https://doi.org/10.1007/s00029-009-0525-x)

Dunkl operators are named after [Charles Dunkl](https://math.virginia.edu/people/cfd5z).


duoid

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\tableofcontents

## Idea

A *double monoid* [[Aguiar & Mahajan (2010)](#AguiarMahajan10) or *duoid* [[Batanin & Markl (2012)](#BataninMarkl12)] is an [[object]] of a [[duoidal category]] equipped with two compatible [[monoid]] [[structures]].

## Definitions

### With a binary operation and an element

Naively, a **duoid** is a [[set]] $D$ equipped with a [[pair]] of [[monoid]] [[structures]] $(\otimes, I)$ and $(\odot, J)$ such that:

\[
  (v \otimes x) \odot (y \otimes z)
  \;=\;
  (v \odot y) \otimes (x \odot z)
  \,.
\]

In this form, a duoid can be viewed as a [[strict double category]] with a single [[object]], one [[horizontal morphism]], and one [[vertical morphism]].

However, due to the [[Eckmann–Hilton argument]], this is equivalent to a [[commutative monoid]]. In fact, this is true for a duoid in any [[braided monoidal category]] (viewed as a [[duoidal category]] in which both tensor products coincide).

### In a duoidal category

See [[duoidal category]].

## References

Duoids were introduced as "double monoids" in the context of [[duoidal categories]] in :

* {#AguiarMahajan10} [[Marcelo Aguiar]], Swapneel Arvind Mahajan, Definition 6.28 of: _Monoidal functors, species and Hopf algebras_ **29** Providence, RI: American Mathematical Society (2010) [[pdf](https://pi.math.cornell.edu/~maguiar/a.pdf), [ams:crmm-29](https://bookstore.ams.org/crmm-29)]

The name "duoid", following a suggestion by [[Ross Street]], is due to:

* {#BataninMarkl12} [[Michael Batanin]], [[Martin Markl]], *Centers and homotopy centers in enriched monoidal categories*, Advances in Mathematics **230** 4-6 (2012) 1811-1858 [[doi:10.1016/j.aim.2012.04.011](https://doi.org/10.1016/j.aim.2012.04.011)]

.

[[!redirects duoids]]


duoidal category

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# Duoidal categories

* table of contents
{: toc}

## Idea

A *duoidal category* is a [[category]] with two [[monoidal category|monoidal structures]] which [[interchange law|interchange]] laxly.

## Definition

### Duoidal categories

A **duoidal category**, or **2-monoidal category**, is a [[pseudomonoid]] in the [[2-category]] $MonCat_l$ of [[monoidal categories]] and [[lax monoidal functors]].  Thus it is a monoidal category, say $(C,\diamond,I)$, together with an additional monoidal structure $(C,\star,J)$ such that $\star:C\times C\to C$ and $J:1\to C$ are [[lax monoidal functors]] with respect to $(\diamond,I)$ and the [[coherence]] axioms of $(C,\diamond,I)$ are [[monoidal natural transformations]] with respect to $(\diamond,I)$.  The laxity of $\star$ consists of [[natural transformations]]
$$ (A\star B) \diamond (C\star D) \to (A\diamond C) \star (B\diamond D) $$
and
$$ I\to I\star I $$
while the laxity of $J$ consists of transformations
$$ J\diamond J \to J $$
and
$$ I \to J. $$
There are then various axioms, which in particular say that $J$ is a $\diamond$-[[monoid]] and $I$ is a $\star$-[[comonoid]].

It is equivalent to ask that $(C,\diamond,I)$ is a pseudomonoid structure on $(C,\star,J)$ in the 2-category $MonCat_c$ of monoidal categories and *colax* monoidal functors.

The map $I\to J$ is actually determined by the rest of the structure; it is the composite
$$I \xrightarrow{\cong} (J\star I)\diamond (I\star J) \to (J\diamond I) \star (I\diamond J) \xrightarrow{\cong} J $$
as well as the dual composite.  If this map is an isomorphism, the duoidal category is called **normal**.

### Duoidal functors

There are three natural kinds of functors between duoidal categories, which are compatibly lax or colax with respect to $\star$ and $\diamond$.  The possible choices of lax and colax seem like there should be four, but in fact only three of them work; in the fourth case the compatibility condition would involve wrong-way composites due to the non-invertibility of the duoidal interchange constraints.

### Virtual duoidal categories

A **virtual duoidal category** is a pseudomonoid in the 2-category of [[multicategories]] (called **monoidal multicategories** in Appendix B of [López Franco & Vasilakopoulou](#FrancoVasilakopoulou)).  That is, it is a multicategory with a monoidal structure $(C,\star,J)$: the $\diamond$-monoidal structure is replaced by the multicategory structure.  Since every monoidal category can be regarded as a [[representable multicategory]], and multicategory functors correspond exactly to lax monoidal functors, duoidal categories can be identified with virtual duoidal categories in which the multicategory structure is representable.

In particular, note that $J$ in a virtual duoidal category is a functor of multicategories $1\to C$, i.e. precisely a [[monoid]] in the multicategory $C$.  As part of this monoid structure it has a unit map $()\to J$; we say that the virtual duoidal category $C$ is **normal** if this map exhibits $J$ as a unit object for the multicategory structure.

Nearly all the definitions given below for duoidal categories (except for coduoids and colax/colax functors) make sense just as well for virtual ones.

### Produoidal categories

A [[produoidal category]] is a pair of pseudomonoids that interchange laxly (a *duoidale*) in the monoidal bicategory of [[profunctor|profunctors]]. Produoidal categories are the profunctorial analogue of duoidal categories, much in the same way that [[promonoidal category|promonoidal categories]] are the profunctorial analogue of monoidal categories.

## Examples

* Any [[braided monoidal category]] can be made into a (normal) duoidal category in which the two monoidal structures agree.  The interchange transformation is built out of the braiding.  Indeed, by the Eckmann-Hilton argument, a duoidal category whose above structure transformations are invertible necessarily arises in this way (up to equivalence).

* Similarly but somewhat more generally, an 2-monoidal category, as a special case of [[iterated monoidal category]] (in the sense of Balteanu, Fiedorowicz, Schwänzl, and Vogt), is a duoidal category.  More precisely, 2-monoidal categories are the duoidal categories such that $\star$ and $J$ are *normal* lax monoidal functors, meaning concretely that the morphisms $I\to I\star I$ and $I\to J$ are isomorphisms.

* If $(C,\diamond,I)$ is a monoidal category with [[finite products]], then it becomes duoidal with $\star$ the [[cartesian monoidal category|cartesian monoidal structure]].  Dually, if $(C,\star,J)$ is a monoidal category with finite coproducts, then it becomes duoidal with $\diamond$ the cocartesian monoidal structure.

* If $A$ and $B$ are monoidal categories, with $A$ small and $B$ cocomplete and closed, then the functor category $[A,B]$ is duoidal with $\star$ the pointwise tensor product, $(X\star Y)(a) = X(a) \otimes Y(a)$, and $\diamond$ the [[Day convolution]] tensor product.

* In particular, if $A$ is a monoid (in Set) and $B$ a cocomplete closed monoidal category, then the category $B^A$ of "$A$-graded objects of $B$" is duoidal with $\star$ the pointwise (or "Hadamard") product and $\diamond$ the "Cauchy" product, $(X\diamond Y)_c = \sum_{a+b=c} X_a \otimes Y_b$.  If $B$ is [[pointed category|pointed]], then the interchange law in this case is naturally a [[split mono]], and its canonical retraction gives $B^A$ a different duoidal structure with the roles of $\star$ and $\diamond$ switched.

* Similarly, the category $[core(FinSet),Vect]$ of vector [[species]] has both a "Hadamard product" $(p\diamond q)(I) = p(I) \otimes q(I)$. and a "Cauchy product" $(p\star q)(I) = \bigoplus_{I = S\sqcup T} p(S) \otimes q(T)$, and is duoidal in two ways.

* The category $FF(C)$ of [[functorial factorizations]] on a category $C$ is a duoidal category.  The $\star$ product of functorial factorizations $F$ and $F'$ is obtained by $F$-factoring a morphism and then $F'$-factoring the first factor, while the $\diamond$ product is obtained by $F$-factoring a morphism and then $F'$-factoring the *second* factor.

* The category $C=[V,V]$ of endofunctors of a closed symmetric monoidal category $V$ is *virtually* duoidal, with $\star$ being functor composition the multicategory structure being "the one that would be represented by Day convolution if it existed".  That is, a morphism $(F,G) \to H$ is a natural transformation $F\otimes G \to H\circ \otimes$ between functors $V\times V \to V$, and so on.  This multicategory structure is not generally representable, since the convolution product requires colimits of the size of the domain category, but $V$ can't in general be expected to have $V$-sized colimits.  However, if $V$ is monoidally [[locally presentable category|locally presentable]], then we can restrict to the subcategory $End_\kappa(V)$ of $\kappa$-accessible endofunctors, and in this case there is no problem, since $End_\kappa(V) \simeq [V_\kappa,V]$  where $V_\kappa$ is the essentially small subcategory of $\kappa$-presentable objects.  If $V=Set$ then this (virtual) duoidal structure is normal with $I=J$ being the identity functor, but for general $V$ it need not be normal.

* As special case of the preceding when $V=Set$ and $\kappa=\omega$, consider the functor category $[FinSet,Set]$, which is equivalent to the category $[Set,Set]_f$ of [[finitary endofunctors]] of $Set$.  Since the composite of finitary endofunctors is finitary, we have an induced composition monoidal structure $\star$.  We can define $\diamond$ by convolution with the monoidal structure of $FinSet$, making $[FinSet,Set]$ duoidal.

* More generally, instead of $End_\kappa(V) = [V_\kappa,V]$ we can consider any category $[V',V]$ where $V'$ is a small monoidal subcategory of $V$ as long as the category of functors $V\to V$ that are left Kan extended from $V'$ is closed under composition.  For instance, if $A$ is a monoidal category and $V=[A^{op},Set]$ with $V'=A$, then $[V',V] \simeq Prof(A,A)$ is the category of endo-profunctors of $A$, which is therefore duoidal.  More generally, the category of endo-proarrows of any monoidal object in a [[proarrow equipment]] is duoidal by a similar construction: $\star$ is proarrow composition and $\diamond$ is convolution with the monoidal structure on both sides.

* We can do something similar for [[generalized operads]] with respect to any [[monoidal monad|monoidal]] [[pseudomonad]] on [[Prof]].  For instance, with the pseudomonad for symmetric monoidal categories, we obtain a duoidal structure on the category of [[collections]].

## Structures in duoidal categories

### Bimonoids, bicomonoids, and duoids

In a duoidal category $C$, the category $Mon_\diamond(C)$ of [[monoid]] objects with respect to $\diamond$ is again a monoidal category under the $\star$ product.  Specifically, if $A$ and $B$ are $\diamond$-monoids, then the multiplication of $A\star B$ is
$$ (A\star B) \diamond (A\star B) \to (A\diamond A) \star (B\diamond B) \to A\star B.$$
We define a **[[bimonoid]]** in $C$ to be a [[comonoid]] in the monoidal category $(\Mon_\diamond(C),\star)$.  An equivalent definition is obtained by considering $\diamond$-monoids in the category of $\star$-comonoids.

Of course, if $C$ is [[braided monoidal category|braided]], this reduces to the usual definition of [[bimonoid]] in a braided monoidal category.

A nontrivially duoidal example is that a bimonoid with respect to the duoidal structure on functorial factorizations described above is precisely an [[algebraic weak factorization system]].

Similarly, a **[[duoid]]** is defined to be a $\star$-monoid in the monoidal category $(\Mon_\diamond(C),\star)$ of $\diamond$-monoids, while a *coduoid* is a $\diamond$-comonoid in the category of $\star$-comonoids.  Duoids, bimonoids, and coduoids in $C$ are equivalently duoidal functors $1\to C$ of the three possible kinds (see above).

### Rings and near-rings

We can define [[near-rig|(near-)]][[rigs]] and near-[[rings]] in a duoidal category, where the additive and multiplicative monoidal structures are expressed with respect to $\star$ and $\diamond$ respectively.  Since the distributive law requires duplication of variables, we need the additive monoidal structure to actually be a [[bimonoid]] (or a [[Hopf monoid]] in the case of a ring), and hence that $\star$ is (on its own) braided.

More precisely, let $C$ be a duoidal category such that the tensor $\star$ is braided (and probably some compatibility holds between the braiding and the duoidal structure).  A **near-rig** in $C$ is of an object $R$ together with

* A [[bimonoid]] structure on $R$ with respect to the braided monoidal category $(C,\star,J)$, written as addition $add:R\star R \to R$ and "coaddition" $\Delta:R\to R\star R$;

* A [[monoid]] structure on $R$ with respect to $(C,\diamond,I))$, written as multiplication $mult : R \diamond R \to R$;

* The following diagram commutes, expressing right distributivity:
  $$ \array{
    (R\star R) \diamond R & \xrightarrow{add\diamond id} &
    R\diamond R & \xrightarrow{mult} &
    R\\
    ^{id \diamond \Delta}\downarrow & & & &
    \uparrow^{add}\\
    (R\star R) \diamond (R\star R) &\to &
    (R\diamond R) \star (R\diamond R) & \xrightarrow{mult \star mult} &
    R\star R
  }
  $$

* The following diagram commutes, expressing right absorption:
  $$ \array{
    R & \xrightarrow{\epsilon} &
    J & \xrightarrow{0} &
    R\\
    ^\cong\downarrow & & & & \uparrow^{mult} \\
    I\diamond R & \to &
    J \diamond R & \xrightarrow{0\diamond id} &
    R\diamond R
  }
  $$

It is a **near-ring** if the additive bimonoid structure is a [[Hopf monoid]], and a **rig** or **ring** if the left distributive law and left absorption also hold.

Of course, this all makes sense in particular in a braided monoidal category regarded as duoidal.  In particular, in a [[cartesian monoidal category]], it reduces to the obvious internal definition of (near-)rigs and (near-)rings (since every object of a cartesian monoidal category is a comonoid in a unique way).

One can show that for near-rings, cocommutativity is automatic, in which case the Hopf monoid $(R,add,\Delta)$ can equivalently be regarded as a Hopf monoid in the *cartesian* monoidal category $CComon_\star(C)$ of cocommutative comonoids in $(C, \star, J)$.  In this case, the above definitions coincide with the definitions of (near-)rings in the [[colax-distributive rig category]] induced by $\diamond$ and the cartesian structure of $CComon_\star(C)$. However, for general (near)-rigs, the definitions do not coincide, because cocommutativity is not automatic for near-rigs.

### Strong and commutative monoids

Note that the constraints $J\diamond J\to J$ and $I\to J$ make $J$ into a $\diamond$-monoid, and indeed also a bimonoid (since it is certainly a $\star$-comonoid).  Also, just as the category of $\diamond$-monoids is $\star$-monoidal, the category of left, right, or bi-$\diamond$-modules over any bimonoid $A$ is $\star$-monoidal, with actions such as
$$(M\star N) \diamond A \to (M\star N) \diamond (A\star A) \to (M\diamond A) \star (N\diamond A) \to M\star N.$$
If $C$ admits reflexive coequalizers preserved in each variable by $\diamond$, then the category of $(A,\diamond)$-bimodules is in fact duoidal, with $\diamond$ the usual tensor product of bimodules.  (In the absence of such coequalizers, this category is virtually duoidal.)

We define a **bistrong $\star$-monoid** in a duoidal category to be a $\star$-monoid in the monoidal category of $J$-$\diamond$-bimodules.  In the duoidal categories of endofunctors considered above, this reproduces the usual notion of "bistrong functor" (i.e. functor with a compatible left and right [[tensorial strength]]).  Note that in a *normal* duoidal category, every object is uniquely a $J$-bimodule, so every $\star$-monoid is bistrong; and conversely, the duoidal category of $(J,\diamond)$-bimodules is (when it exists) always normal.

If the monoidal structure $\diamond$ is [[braided monoidal category|braided]] (compatibly with $\star$), then a one-sided module automatically acquires a module structure on the other side.  In this case, we define a **strong $\star$-monoid** to be a bistrong one whose $J$-$\diamond$-bimodule structure is obtained in this way.

Now let $S$ and $T$ be arbitrary objects and $M$ a (bi)strong $\star$-monoid.  We say that two morphisms $\sigma:S\to M$ and $\tau:T\to M$ be morphisms **commute** if the following diagram commutes:
$$\array{
  T\diamond S & \xrightarrow{\cong} &
  (T\star J) \diamond (J \star S) & \to &
  (T\diamond J) \star (J\diamond S) \\
  ^\cong\downarrow  &&&& \downarrow^{\tau \star \sigma}\\
  (J\star T) \diamond (S \star J) &&&& M\star M\\
  \downarrow &&&& \downarrow^{mult}\\
  (J\diamond S) \star (T\diamond J) & \xrightarrow{\sigma \star \tau} &
  M\star M & \xrightarrow{mult} & M.
}
$$
We say that $M$ is **commutative** if $id_M$ and $id_M$ commute.

In a braided monoidal category regarded as a normal duoidal one, this reduces to the usual notion of [[commutative monoid]] object.  On the other hand, in the endofunctor-like examples above, it results in the notion of [[commutative monad]] or [[commutative theory]].

## Relation to intercategories

A duoidal category is the same as an [[intercategory]] with one object and only identity horizontal arrows, vertical arrows, horizontal cells, and vertical cells.  The objects of the duoidal category are the basic cells of the intercategory, and its morphisms are the cubes.  Like duoidal categories, intercategories support three (but not four) different kinds of morphism.

## Related concepts

* [[distributivity for monoidal structures]]
* [[intercategory]]

## References

* [[Marcelo Aguiar]] and [[Swapneel Mahajan]], *Monoidal Functors, Species and Hopf Algebras*, [pdf](http://www.math.iitb.ac.in/~swapneel/aguiar.pdf).  Here the notion is called a "2-monoidal category".

* [[Richard Garner]], _Understanding the small object argument_, [arXiv](http://arxiv.org/abs/0712.0724).  Here the notion is called a "two-fold monoidal category", although [[iterated monoidal category|that term]] is also used for the case when the two units coincide.

* [[Michael Batanin]] and [[Martin Markl]], _Centers and homotopy centers in enriched monoidal categories._ Advances in Mathematics 230 , 4-6 (2012), 1811–1858.  Here apparently the term "duoidal category" was introduced.

* [[Richard Garner]], [[Ignacio López Franco]], _Commutativity_, [arXiv](http://arxiv.org/abs/1507.08710).  See also _Commutativity and tensor products of theories, monads, and operads_, talk at CT2013, [slides](https://www.dpmms.cam.ac.uk/~ill20/ct2013-talk.pdf)

* Zoran Petrić and [[Todd Trimble]] 2014, _Symmetric bimonoidal intermuting categories and $\omega \times \omega$ reduced bar constructions_, Applied Categorical Structures 22(3): 467-499, [arXiv:0906.2954](http://arxiv.org/abs/0906.2954)

* {#FrancoVasilakopoulou} [[Ignacio López Franco]] and [[Christina Vasilakopoulou]], _Duoidal categories, measuring comonoids and enrichment_, [arXiv:2005.01340](https://arxiv.org/abs/2005.01340) (2020).

[[!redirects duoidal category]]
[[!redirects duoidal categories]]
[[!redirects 2-monoidal category]]
[[!redirects 2-monoidal categories]]

[[!redirects virtual duoidal category]]
[[!redirects virtual duoidal categories]]


Duong Phong

* [webpage](http://www.math.columbia.edu/~phong/)

## Selected writings

On [[2-loop]] [[string scattering amplitudes]] of the [[superstring]]:

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings I, Main Formulas_, Phys. Lett. B529:241-255, 2002 ([arXiv:hep-th/0110247](http://arxiv.org/abs/hep-th/0110247))

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings II, The Chiral Measure on Moduli Space_, Nucl. Phys. B636:3-60, 2002 ([arXiv:hep-th/0110283](http://arxiv.org/abs/hep-th/0110283))

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings III, Slice Independence and Absence of Ambiguities_, Nucl. Phys. B636:61-79, 2002 ([arXiv:hep-th/0111016](http://arxiv.org/abs/hep-th/0111016))

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings IV, The Cosmological Constant and Modular Forms_, Nucl. Phys. B639:129-181, 2002 ([arXiv:hep-th/0111040](http://arxiv.org/abs/hep-th/0111040))

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings V: Gauge Slice Independence of the N-Point Function_, Nucl. Phys. B715:91-119, 2005 ([arXiv:hep-th/0501196](http://arxiv.org/abs/hep-th/0501196))

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings VI: Non-Renormalization Theorems and the 4-Point Function_,  	Nucl. Phys. B715:3-90, 2005 ([arXiv:hep-th/0501197](http://arxiv.org/abs/hep-th/0501197))

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings VII, Cohomology of Chiral Amplitudes_, Nucl. Phys. B804:421-506, 2008 ([arXiv:0711.4314](http://arxiv.org/abs/0711.4314))

## Related entries

* [[string scattering amplitude]]

category: people


duploid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Duploid
* table of contents
{:toc}

## Idea

A **duploid** is a [[category]]-like [[structure]] where [[objects]] have a specific [[polarity in type theory|polarity]] and [[composition]] is not necessarily [[associative]].

Duploids are models of [[programming languages]] with [[side effects]] and explicitly polarized [[positive type|positive]] and [[negative type|negative]] [[types]], accomodating both [[call-by-value]] and [[call-by-name]] programming paradigms. Duploids axiomatize the algebraic structure of effectful morphisms, and to a certain extent "pure" morphisms (those with no side effects) can be characterized by equational properties as thunkable and linear morphisms. Contrast with [[call-by-push-value]] which directly provides a syntax for the pure morphisms (as values and stacks).

A duploid can be constructed from an [[adjunction]] and the category of duploids is [[reflective subcategory|coreflective]] in the category of adjunctions, equivalent to the [[subcategory]] of adjunctions satisfying an equalizing requirement. The intuition is that a duploid is what is left of an adjunction if we only consider (co)Kleisli morphisms and duploids can be identified with adjunctions whose categories can be recovered from the Kleisli morphisms.

## Definition

A **pre-duploid** $\mathcal{D}$ consists of

1. a set $|\mathcal{D}|$ of objects with a polarity map $\varpi : |\mathcal{D}| \to \{+,-\}$. If $\varpi(A) = +$, we say $A$ is positive and otherwise negative.
2. for every pair $A,B \in |\mathcal{D}|$ a set of morphisms $\mathcal{D}(A,B)$.
3. for every compatible pair of morphisms $f \in \mathcal{D}(A,B), g\in \mathcal{D}(B,C)$ a composite $g \odot f \in \mathcal{D}(A,C)$.
4. for every object $A$ an morphism $id_A \in \mathcal{D}(A,A)$ that is the identity with respect to composition.
5. For every $f \in \mathcal{D}(A,B), g \in \mathcal{D}(B,C), h \in \mathcal{D}(C,D)$, an associative law $h \odot (g \odot f) = (h \odot g) \odot f$ when $B$ is negative or $C$ is positive.

Note that when restricted to positive objects or negative objects, composition and identities form a category, written $\mathcal{P}$ and $\mathcal{N}$. When the polarity is known we write positive objects as $P,Q,R$ and negative objects as $L,M,N$. We call composition where the middle object is positive "positive composition" or "by-value composition" and notate it as $g \bullet f$ and when the middle object is negative we call it "negative composition" or "by-name composition" and notate it as $g \circ f$.
Then the associativity laws can be restated as:

1. $\bullet\bullet$: $f \bullet (g \bullet h)  = (f \bullet g) \bullet h$
2. $\circ\circ$: $f \circ (g \circ h)  = (f \circ g) \circ h$
3. $\bullet\circ$: $f \bullet (g \circ h)  = (f \bullet g) \circ h$

and we can see that the only obstacle to associativity is that $f \circ (g \bullet h)$ is not necessarily equal to $(f \circ g) \bullet h$.

A **duploid** is a pre-duploid $\mathcal{D}$ plus two **polarity shifts**
$\Downarrow : |\mathcal{N}| \to |\mathcal{P}| and \Uparrow : |\mathcal{P}| \to |\mathcal{N}|$, and for each $P \in |\mathcal{P}|, N \in |\mathcal{N}|$, morphisms:

1. $delay_P : P \to \Uparrow P$
2. $force_P : \Uparrow P \to P$
3. $wrap_N : N \to \Downarrow N$
4. $unwrap_N : \Downarrow N \to N$

such that

1. $\forall f : A \to P, force_P \circ (delay_P \bullet f) = f$
2. $\forall f : N \to A, (f \circ unwrap_N) \bullet wrap_N = f$
3. $delay_P \bullet force_P = id_{\Uparrow P}$
4. $wrap_N \circ unwrap_N = id_{\Downarrow N}$

In light of the definition of linear and thunkable morphisms, these identities are equivalent to saying

1. $wrap_N$ is thunkable.
2. $wrap_N, unwrap_N$ is an isomorphism.
3. $force_P$ is linear.
4. $force_P, thunk_P$ is an isomorphism.

## Linear and Thunkable Morphisms

Just as thunkable morphisms can be defined in a [[thunk-force category]] and linear morphisms can be defined in a category with a [[runnable monad]], they can be defined in a duploid in  a similar way using $delay_P$ in place of the thunk and $unwrap_N$ in place of the run morphism.
However, thunkable and linear morphisms can also be defined in a _pre-duploid_, giving a simple characterization just in terms of associativity of composition.

A morphism $f$ is **thunkable** if for all compatible $g,h$,
$$h \odot (g \odot f) = (h \odot g) \odot f.$$
and a morphism $f$ is **linear** if for all compatible $g,h$,
$$f \odot (g \odot h) = (f \odot g) \odot h.$$

Observe that all $f : P \to B$ are trivially linear and $g : A \to N$ are trivially thunkable. Further thunkable and linear morphisms form (non-full) subcategories $\mathcal{D}_t, \mathcal{D}_l$.

To understand these concepts, consider the non-trivial situation where $f : A \to P$ is thunkable:
$$h \circ (g \bullet f) = (h \circ g) \bullet f.$$
on the left side, since we have a by-name composition, $h$ is "evaluated first", whereas on the right side we have a by-value composition, so $f$ is evaluated first.
Thus these two morphisms being equal implies in many semantics $f$ cannot perform any [[side-effect]]. For instance if $f$ prints to the screen and so does $h$, then each composite will print in a different order. Furthermore, if $f$ can manipulate its [[continuation]] explicitly, it has to treat it linearly otherwise $h$ may be evaluated twice or not at all.

In the dual, if $f : N \to B$ and $f$ is linear:
$$f \circ (g \bullet h) = (f \circ g) \bullet h$$
holds and similarly $f$ is evaluated first in the left morphism and $h$ is evaluated first in the right. Then as above it cannot perform any effects and has to treat its *input* linearly, as duplicating or dropping $h$ would make the equation fail to hold.

## Relation to Adjunctions

Briefly, the category of duploids and duploid functors is a [[reflective subcategory]] of the category of adjunctions and morphisms of adjunctions.

### A Duploid from an Adjunction

First, we construct a duploid from an adjunction, which we think of as a [[forgetful functor]] that only remembers the Kleisli morphisms of the adjunction. To keep the proof unbiased, we start with a [[profunctor]] $O : C_- ⇸ C_+$ that the adjunction [[representable functor|represents]]. This gives us a notion of "oblique-" or [[heteromorphism|hetero-]]morphism from a positive object $P \in C_+$ to a negative object $N \in C_-$ as $O(P,N)$ and can be used to define the Kleisli and coKleisli categories. In programming applications these are the "possibly effectful" morphisms. Then let $F \dashv G$ be adjoint functors representing $O$, i.e. there is a natural equivalence:
$$ C_-(F P, N) \cong O(P,N) \cong C_+(P, G N).$$
The pre-duploid $\mathcal{D}$ has as objects $|\mathcal{D}| = |C_+| + |C_-|$ with $\varpi(P) = +, \varpi(N) = -$.
The morphisms are defined as:
$$\mathcal{D}(A,B) = O(A^+,B^-)$$

where
$$P^+ = P$$
$$N^+ = G N$$
$$P^- = F P$$
$$N^- = N$$

and identities are given by the identities/unit and counit of the adjunction:

$$\id_P : O(P, F P) \equiv C_-(F P,F P)$$
$$\id_N : O(G N, N) \equiv C_+(G N,G N)$$

To define composition of $f \in \mathcal{D}(A,B), g \in \mathcal{D}(B,C)$, we inspect $B$. If $B = P$ is positive, composition is interpreted as composition in $C_-$, or equivalently in the [[Kleisli category]] of $GF$, using
$$f \in O(A^+,F P) \cong C_-(F A^+, F P)$$
$$g \in O(P, C^-) \cong C_-(F P, C^-)$$
and similarly if $B = N$ is negative, composition is interpreted as composition in $C_+$, or equivalently in the [[Kleisli category]] of $FG$, using
$$f \in O(A^+,N) \cong C_+(A^+, G N)$$
$$g \in O(G N, C^-) \cong C_+(G N, G C^-)$$

This definition makes it clear that the identities are identity and the $\bullet\bullet$ and $\circ\circ$ laws hold from identity and associativity of $C_+,C_-$.

### Adjunction from a Duploid

From a duploid, we can construct the "cofree" adjunction: i.e., a right adjoint to the forgetful functor just defined. Intuitively, we want to recover the homomorphisms just from the heteromorphisms. The maximal choice is to consider all thunkable morphisms between positive types and linear morphisms between negative types to be homomorphisms.

Furthermore, this functor is fully faithful, making the category of duploids a [[reflective subcategory]] of adjunctions: they can be identified with exactly those adjunctions in which the unit and counit are mono and epi respectively and in which all thunkable heteromorphisms are the image under $F$ of some homomorphism and vice-versa all linear heteromorphisms are in the image of $G$.
This is equivalent to saying that the unit $\eta$ is the [[equalizer]] of $GF\eta$ and $\eta GF$ and dually that the counit $\epsilon$ is the [[coequalizer]] of $FG\epsilon$ and $\epsilon FG$

## Related Concepts

* [[polarity in type theory]]
* [[thunk-force category]]
* [[runnable monad]]

## References

Duploids were introduced in chapter II of

* {#Mthesis} Guillaume Munch-Maccagoni, _Syntax and Models of a Non-Associative Composition of Programs and Proofs_, PhD thesis University of Paris Diderot, 2013 [link](https://hal.inria.fr/tel-00918642)


Durov's generalized scheme

[[!redirects generalized scheme after Durov]]
[[Nikolai Durov]] constructed [[spectrum (geometry)|spectra]] of commutative [[finitary monads]] (cf. [[commutative algebraic theory]]) which he calls __generalized__ [[commutative ring]]s; using a class of categorically defined [[localizations]] (which for some reason he calls pseudolocalizations), he glues such spectra to obtain __generalized [[schemes]]__ which are a class of [[generalized ringed space]]s.

There are many other [[generalized scheme|generalizations of schemes]] in the literature.

* [[Nikolai Durov]], _A new approach to Arakelov geometry_, [arXiv:0704.2030](https://arxiv.org/abs/0704.2030)
* Stella Anevski, _Algebraic K-theory of generalized schemes_, PhD thesis, 2013 [pdf](http://web.math.ku.dk/noter/filer/phd13sa.pdf)

category: algebraic geometry


Dusa McDuff

**Dusa McDuff** is a English mathematician, who worked in the USA. She has proved profound results in operator algebras and in [[differential geometry]]. In recent years she has concentrated on [[symplectic geometry|symplectic]], [[contact geometry]] and [[symplectic topology]].

* [homepage]; a [biography](http://www.agnesscott.edu/Lriddle/women/mcduff.htm); some [autobiographical notes](http://www.math.sunysb.edu/~tony/visualization/dusa/dusabio.html)

## Selected writings

On the [[group completion theorem]]:

* {#McDuffSegal76} [[Dusa McDuff]], [[Graeme Segal]], _Homology fibrations and the “group-completion” theorem_, Inventiones mathematicae, October 1976, Volume 31, Issue 3, pp 279–284 ([doi:10.1007/BF01403148](https://doi.org/10.1007/BF01403148))

On [[configuration spaces of points]]:

* {#McDuff75} [[Dusa McDuff]], _Configuration spaces of positive and negative particles_, Topology Volume 14, Issue 1, March 1975, Pages 91-107 (doi:10.1016/0040-9383(75)90038-5)

On [[symplectic topology]]:

* [[Dusa McDuff]], [[Dietmar Salamon]], _Introduction to symplectic topology_, 2 ed. Oxford Mathematical Monographs 1998. x+486 pp.

* [[Dusa McDuff]], [[Dietmar Salamon]], _J-holomorphic curves
and quantum cohomology_, AMS, University Lecture Series __6__, 1994 ([revised pdf](http://www.barnard.edu/math/mcduff/survey/6manjl26.pdf) [1995 pdf](http://barnard.edu/sites/default/files/inline/jholsm_0.pdf))

* [[Dusa McDuff]], D.A. Salamon, _J-holomorphic curves and symplectic topology_, AMS Colloquium Publications __52__, 2004.

category: people


Dusko Pavlovic > history

see at *[[Duško Pavlović]]*


Dustin Clausen

* [website](https://www.hcm.uni-bonn.de/people/profile/dustin-clausen/)

### Related entries

* [[condensed set]]

category: people


Dustin Ross

* [webpage](http://online.sfsu.edu/rossd)

## Selected writings

On the [[Gromov-Witten/Donaldson-Thomas correspondence]]:

* [[Dustin Ross]], _On GW/DT and Ruan's Conjecture in All Genus for Calabi-Yau 3-Orbifolds_, [arXiv:1409.7015](https://arxiv.org/abs/1409.7015v3)

category: people


Dutch Categories and Types Seminar

\tableofcontents

## Idea

The **Dutch Categories and Types Seminar** is a seminar on [[category theory]] and [[type theory]] in the Netherlands.

## References

* [website](https://dutchcats.github.io/)

[[!redirects DutchCATS]]


Duško Pavlović

* [webpage](http://dusko.org/)

* [webpage](http://www.cs.ox.ac.uk/people/dusko.pavlovic/)

## Selected writings

On [[categorical semantics]] of [[constructive logic]] and [[intuitionistic type theory]]:

* {#Pavlovic1990} [[Duško Pavlović]], *Predicates and Fibrations*, PhD thesis, Utrecht (1990) [[pdf](https://dusko.org/wp-content/uploads/2020/03/1990-proefschrift-dusko.pdf), [[Pavlovic-PredicatesAndFibrations.pdf:file]]]

On [[monadic descent]] and the [[Beck-Chevalley condition]]:

* {#Pavlović91} [[Duško Pavlović]], *Categorical interpolation: Descent and the Beck-Chevalley condition without direct images*, in: *Category Theory*, Lecture Notes in Mathematics **1488** (1991) [[doi:10.1007/BFb0084229](https://doi.org/10.1007/BFb0084229), [pdf](http://dusko.org/wp-content/uploads/2020/03/1990-Como-BCDE.pdf), [[Pavlovic-CategoricalInterpolation.pdf:file]]]

and its role in [[categorical logic]]/[[proof theory]]:

* [[Duško Pavlović]], *Maps II: Chasing Diagrams in Categorical Proof Theory*, Logic Journal of the IGPL, **4** 2 (1996) 159–194 [[doi:10.1093/jigpal/4.2.159](https://doi.org/10.1093/jigpal/4.2.159), [pdf](http://dusko.org/wp-content/uploads/2020/03/00-95-IGPL-mapsII.pdf)]

On [[quantum measurement]] formulated in [[finite quantum mechanics in terms of dagger-compact categories ]] in terms of [[Frobenius algebras]] and the [[quantum reader monad]]:

* [[Bob Coecke]], [[Duško Pavlović]], *Quantum measurements without sums*, in [[Louis Kauffman]], [[Samuel Lomonaco]] (eds.), *Mathematics of Quantum Computation and Quantum Technology*, Taylor & Francis (2008) 559-596 [[arXiv:quant-ph/0608035](https://arxiv.org/abs/quant-ph/0608035), [doi:10.1201/9781584889007](https://doi.org/10.1201/9781584889007)]

* [[Bob Coecke]], [[Eric Paquette]], [[Dusko Pavlovic]], *Classical and quantum structures* (2008) [[pdf](http://www.comlab.ox.ac.uk/files/627/RR-08-02.pdf), [[Coecke-Paquette-Pavlovic-CQS.pdf:file]]]

* [[Bob Coecke]], [[Eric Oliver Paquette]], [[Duško Pavlović]], *Classical and quantum structuralism* [[arXiv:0904.1997](https://arxiv.org/abs/0904.1997)]

* [[Bob Coecke]], [[Duško Pavlović]], [[Jamie Vicary]],  *A new description of orthogonal bases*, Mathematical Structures in Computer Science **23** 3 (2012) 555- 567 [[arXiv:0810.0812](https://arxiv.org/abs/0810.0812), [doi:10.1017/S0960129512000047](https://doi.org/10.1017/S0960129512000047)]

On [[Functorial Semantics of Algebraic Theories]] generalized to [[relational theories]]:

* [[Filippo Bonchi]], [[Dusko Pavlovic]], [[Pawel Sobocinski]], _Functorial Semantics for Relational Theories_, ([arXiv:1711.08699](https://arxiv.org/abs/1711.08699))

## Related entries

* [[quantum information theory via dagger-compact categories]]

  * [[classical structure]]

category: people

[[!redirects Dusko Pavlovic]]


Dwyer map

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### Context
#### Category theory
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#### Model category theory
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[[!include model category theory - contents]]
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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
* automatic table of contents goes here
{:toc}

## Idea

A **Dwyer map** is a type of [[cofibration]] in the [[Cat|category of small categories]] designed for the purpose of constructing [[homotopy colimit|homotopy invariant colimits]] with respect to weak homotopy equivalences, i.e. weak equivalences created from [[weak homotopy equivalence|weak homotopy equivalences]] of [[simplicial set|simplicial sets]] by the [[nerve]] functor. It plays an essential role in the construction of the [[Thomason model structure]]. Its definition is inspired by [[neighborhood retract|neighborhood deformation retracts]].

A suitable generalization serves a similar purpose in the homotopy theory of [[relative category|relative categories]], see [Barwick, Kan](#BarwickKan).

## Definition

A functor of small categories $i \colon C \to D$ is a **Dwyer map** if it is a [[sieve]] and factors as a composite of $f \colon C \to C'$ and $j \colon C' \to D$ such that

* $f$ admits a deformation retraction, i.e. a functor $r \colon C' \to C$ such that $r f = \id_C$ toghether with a natural transformation $h \colon f r \to \id_{C'}$ such that $h f = \id_f$,
* $j$ is a [[cosieve]].

In the original definition $r$ was assumed to be a right adjoint of $f$. The definition above is due to Cisinski who called this more general notion a **pseudo-Dwyer map**. Cisinski's definition is now considered more useful and hence usually called by the simpler name "Dwyer map".

## Properties

+-- {: .num_prop }
###### Proposition

Dwyer maps are closed under:

* coproducts,
* pushouts along arbitrary functors,
* [[transfinite composition]],
* retracts.

Moreover, the [[nerve]] functor preserves pushouts along Dwyer maps and transfinite composites of Dwyer maps up to [[weak homotopy equivalence]].
=--

The proofs can be found in [Thomason, Proposition 4.3, Lemma 4.7](#Thomason) and [Cisinski, Lemme 4](#Cisinski).

+-- {: .num_prop }
###### Proposition

Let $c$ denote the left adjoint of the nerve functor and let $Sd$ denote the [[subdivision|barycentric subdivision]] functor. The composite $c Sd^2$ carries injective simplicial maps to Dwyer maps.
=--

This is [Thomason, Proposition 4.2](#Thomason).

So among the Dwyer maps are all cofibrations in the Thomason model structure. But not all Dwyer maps are Thomason cofibrations -- for example, the unique functor $\emptyset \to C$ is always a Dwyer map, but not every category is Thomason-cofibrant.

## References

*  [[Robert Thomason|R. W. Thomason]], _Cat as a closed model category_,
Cahiers Topologie Géom. Différentielle **21**, no. 3 (1980), pp. 305--324. MR0591388 (82b:18005) numdam scan {#Thomason}
* [[Denis-Charles Cisinski]], _Les morphisme de Dwyer ne sont pas stables par rétractes_, Cahiers Topologie et Géom. Différentielle Catégoriques, **40** no. 3 (1999), pp. 227--231. ([Numdam](http://www.numdam.org/item?id=CTGDC_1999__40_3_227_0)) {#Cisinski}
* [[Clark Barwick]], [[Daniel Kan]], _Relative categories: Another model for the homotopy theory of homotopy theories_. Indagationes Mathematicae 23 (2012) pp. 42–68. {#BarwickKan}


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#### Enriched category theory
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[[!include enriched category theory contents]]
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# Contents #
* table of contents
{:toc}

## Idea

The *Dwyer-Kan fundamental simplicial groupoid* (often "loop groupoid") [[Dwyer & Kan (1984), §3.1](#DwyerKan84)] of a [[simplicial set]] $K$ is the [[simplicial groupoid]] whose [[objects]] are the [[vertices]] of $K$ and whose [[hom object|hom]]-[[simplicial set]] between a [[pair]] of such picks up the composable sequences of higher dimensional simplices, where the zeroth [[vertex]] is thought of as the [[domain]] [[object]] and the first [[vertex]] as the [[codomain]].

So the "loop"-terminology is a misnomer: It really refers to a [[simplicial homotopy theory|simplicial]] version of a [[path groupoid]]-construction which may be thought of as modelling a [[fundamental infinity-groupoid|fundamental $\infty$-groupoid]] as a [[simplicial groupoid]].

But applied to the special case of a [[reduced simplicial set]] this construction reduces to the *[[Kan loop group]]* construction, which is properly a model for a [[loop space]]-construction.

The Dwyer-Kan groupoid construction is a [[left Quillen functor|left]] [[Quillen equivalence]] ([[right adjoint]] to a groupoidal [[simplicial classifying space]]-construction) from the [[classical model structure on simplicial sets]] to the [[model structure on simplicial groupoids]] [[Dwyer & Kan (1984), Thm. 3.3](#DwyerKan84)],
and as such exhibits the [[homotopy theory]] of [[simplicial groupoids]] as an [[equivalence of (infinity,1)-categories|equivalent]] incarnation of [[classical homotopy theory]].

## Definitions
 {#Definition}

There are different choices of how to define a DK-type loop groupoid functor, depending, for instance, on which [[pair]] of [[vertices]] in the [[n-simplex|$n$-simplex]] of a simpilcial set are to be regarded as the [[source]]- and [[target]]-[[object]] of the corresponding [[morphism]] in the loop groupoid. In the original definition [Dwyer & Kan (1984), §3.1](#DwyerKan84) took the direction of the morphism to point from the $1$st to the 0th vertex, $1 \to 0$. But the modern convention in analogous constructions for [[Segal spaces]] etc. is to instead take $0 \to 1$.

Moreover, apparently the original definition in [Dwyer & Kan (1984), §3.1](#DwyerKan84) had a mistake in its definition of $d_0$ (according to [Ehlers (1991), p. 10](#Ehlers91)). Both issues are said to be fixed in [Ehlers (1991), pp. 10](#Ehlers91).

Other definitions have been given, too (all necessarily isomorphic to each after their [[left adjoint|left adjointness]] to the [[simplicial classifying space]]-construction has been established, by [uniqueness of adjoint](adjoint+functor#UniquenessOfAdjoints)).

In any case, the DK loop groupoid construction is a [[functor]]

$$
  \mathcal{G}
    \colon
  sSet
  \longrightarrow
  sGrpd
  \,,
$$

which takes a [[simplicial set]] $S$ to a Dwyer-Kan [[simplicial groupoid]] (i.e. [[sSet]]-[[enriched groupoid]]) $\mathcal{G}(S)$.

### Original definition, fixed
 {#OriginalDefinition}

The following is a slight modification of the original [Dwyer & Kan (1984), §3.1](#DwyerKan84), given by [Ehlers (1991), pp. 10](#Ehlers91):

For $S \in sSet$ the simplicial groupoid $\mathcal{G}(S)$ is defined as follows:

* [[objects]] are the [[vertices]] of $S$:

  $Obj\big(\mathcal{G}(K)\big) \,\coloneqq\, K_0$;

* [[morphisms]] in degree $n \in \mathbb{N}$ form the [[groupoid]] which is the [[quotient object|quotient]]

  $$
    \array{
      S_{n+1}
        &\overset{\phantom{----}}{\rightrightarrows}
      & S_0
      \\
      \mathllap{{}^{q}}\Big\downarrow
      &&
      \Big\downarrow\mathrlap{{}^=}
      \\
      \mathcal{G}(S)_n
        &\overset{\phantom{----}}{\rightrightarrows}&
      Obj\big(\mathcal{G}(S)\big)
    }
  $$

  of the [[free groupoid]] on the [[graph]]

  $$
    s,t
     \,\colon\,
    S_{n+1}
      \overset{\phantom{--}}{\rightrightarrows}
    S_0,
    \,,
  $$

  * with [[domain]]-function $s \coloneqq d_1 \circ d_2 \cdot d_3 \cdots d_{n+1}$,

    and [[codomain]]-function $t \coloneqq d_0 \circ d_2 \cdot d_3 \cdots d_{n+1}$

  by the [[relations]]

  $q(s_0 \sigma_n) \,\sim\, id_{d_1 \cdots d_n \sigma_n}$, for all $\sigma_n \in S_n$.

* The [[face and degeneracy maps]] are given on generators by

  * $d_0\big(q(\sigma)\big) \,\coloneqq\, q\big(d_1(\sigma_n)\big) \cdot q\big( d_0(\sigma_n)\big)^{-1}$

  * $d_i\big(q(\sigma)\big) \,\coloneqq\,q\big(d_{i+1}(\sigma)\big)$ for $i \gt 0$

  * $s_i\big(q(\sigma)\big) \,\coloneqq\,q\big(s_{i+1}(\sigma)\big)$ for $i \geq 0$.

### By Goerss & Jardine
 {#DefinitionByGoerssJardine}

Here is the definition given in [Goerss & Jardine (1999) p. 322, (2009) p. 302](#GoerssJardine09):

## Properties

### General

\begin{remark}
The fundamental simplicial groupoid functor has a [[right adjoint]], $\overline{W}$ [[Dwyer & Kan (1984) §3.2](#DwyerKan84)] also called the *[[simplicial classifying space]] functor*.
\end{remark}

### Relation to rigidification of quasi-categories

Let
\[
  \label{LocalizationOfsSetCategoriesToSSetGroupoids}
  L
    \,\colon\,
  sSet\text{-}Cat
  \longrightarrow
  sSet\text{-}Grp
\]
denote the functor from [[sSet-enriched categories]] to [[sSet]]-[[enriched groupoids]] (Dwyer-Kan [[simplicial groupoids]]) which is degreewise given by the [[localization]] operation [[left adjoint]] to the [[full subcategory]] inclusion [[Grpd]] $\hookrightarrow$ [[Cat]].

Let
\[
  \label{RigidificationOfQuasiCategories}
  \mathfrak{C}
  \,\colon\,
  sSet
    \longrightarrow
  sSet\text{-}Cat
\]
denote the operation of [[rigidification of quasi-categories]].

\begin{proposition}\label{ComparisonWithRigidification}
  For $S \,\in\, sSet$ there is a [[natural transformation]]
$$
  L \circ \mathfrak{C}(S)
  \longrightarrow
  \mathcal{G}(S)
$$
which is a [[Dwyer-Kan equivalence]]
(from the localization (eq:LocalizationOfsSetCategoriesToSSetGroupoids) of the rigidification (eq:RigidificationOfQuasiCategories) to the Dwyer-Kan fundamental groupoid (as [above](#Definition)).
\end{proposition}
[[Minichiello, Rivera & Zeinalian (2023), Thm. 1.1 (Cor. 4.2)](#MinichielloRiveraZeinalian23)]

## References

The original reference for the functor $G$ is Section 9 in

* [[Daniel M. Kan]], _A combinatorial definition of homotopy groups_, Annals of Mathematics 67:2 (1958), 282–312.  [doi:10.2307/1970006](https://doi.org/10.2307/1970006).

Later, Dwyer and Kan constructed the right adjoint $\bar W$ to $G$ in

* {#DwyerKan84} [[William Dwyer]], [[Daniel Kan]], §3.1 in *Homotopy theory and simplicial groupoids*, Indagationes Mathematicae (Proceedings) **87** 4 (1984) 379-385 [doi:10.1016/1385-7258(84)90038-6]

{#ReferencesBeware} Beware that the definition in [Dwyer & Kan (1984) §3.1](#DwyerKan84) apparently contains a mistake (in its  formula for $d_0$). It may be that this problem was early on known and communicated by [[Andre Joyal|Joyal]] and [[Myles Tierney|Tierney]], but in print the correction is given in:

* {#Ehlers91} [[Philip Ehlers]], pp. 10 in: _Simplicial groupoids as models for homotopy type_, Master's thesis (1991) [[pdf](https://ncatlab.org/nlab/files/Ehlers-MSc-thesis.pdf)]

See also:

* {#JoyalTierney00} [[André Joyal]], [[Myles Tierney]], *On the theory of path groupoids*, Journal of Pure and Applied Algebra **149** 1 (2000) 69-100 [doi:10.1016/S0022-4049(98)00164-9]

* {#GoerssJardine09} [[Paul Goerss]], [[J. F. Jardine]], Section V.7 of: _[[Simplicial homotopy theory]]_, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) [[doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4), [webpage](http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/)]

Relating the Dwyer-Kan loop groupoid-construction to the [[rigidification of quasi-categories]]:

* {#MinichielloRiveraZeinalian23} [[Emilio Minichiello]], [[Manuel Rivera]], [[Mahmoud Zeinalian]], Def. 3.1 in: *Categorical models for path spaces*, Advances in Mathematics **415** (2023) 108898 [[arXiv:2201.03046](https://arxiv.org/abs/2201.03046), [doi:10.1016/j.aim.2023.108898](https://doi.org/10.1016/j.aim.2023.108898)]

[[!redirects Dwyer-Kan fundamental simplicial groupoids]]

[[!redirects Dwyer-Kan loop groupoid]]
[[!redirects Dwyer-Kan loop groupoid]]
[[!redirects Dwyer–Kan loop groupoid]]
[[!redirects Dwyer--Kan loop groupoid]]
[[!redirects Dwyer-Kan loop groupoids]]
[[!redirects Dwyer–Kan loop groupoids]]
[[!redirects Dwyer--Kan loop groupoids]]

[[!redirects Dwyer-Kan loop group]]
[[!redirects Dwyer-Kan loop groups]]

[[!redirects Joyal-Tierney loop groupoid]]
[[!redirects Joyal-Tierney loop groupoids]]

[[!redirects Dwyer-Kan path groupoid]]
[[!redirects Dwyer-Kan path groupoids]]
[[!redirects Dwyer--Kan path groupoid]]
[[!redirects Dwyer--Kan path groupoids]]

[[!redirects Dwyer-Kan simplicial path groupoid]]
[[!redirects Dwyer-Kan simplicial path groupoids]]
[[!redirects Dwyer--Kan simplicial path groupoid]]
[[!redirects Dwyer--Kan simplicial path groupoids]]


Dwyer-Kan localisation > history

< [[Dwyer-Kan localisation]]