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From rrosebru@mta.ca Wed Apr 1 13:52:46 2009 -0300
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From: Thorsten Altenkirch
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Subject: categories: Where does the term monad come from?
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A question just came up at the Midland Graduate School (actually in
the functional programming lecture):
Where does the word monad come from?
I know that a monad is a monoid in the category of endofunctors, but
what is the logic monoid => monad?
Btw, I frequently encounter monads in a categories of functors which
are not endofunctors. An example are finite dimensional vectorspaces
which can be constructed via a monoid in the category of functors
FinSet -> Set, here I is the embedding and (x) can be constructed from
the left kan extension and composition.
The unit is given by the Kronecker delta and join can be constructed
from Matrix multiplication. Should one call these beasts monads as
well? Is there a good reference for this type of construction?
Cheers,
Thorsten
From rrosebru@mta.ca Wed Apr 1 13:54:55 2009 -0300
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Subject: categories: Lecturer in Computer Science, University of Leicester
Date: Wed, 01 Apr 2009 17:48:39 +0100
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Lecturer in Computer Science
Department of Computer Science
University of Leicester
Salary Grade 8: =A335,469 to =A343,622 p.a.
Available from: 1 September 2009
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The successful candidate will have a strong or promising research =20
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From rrosebru@mta.ca Thu Apr 2 09:10:58 2009 -0300
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Date: Wed, 1 Apr 2009 13:13:55 -0500 (EST)
From: Michael Barr
To: Thorsten Altenkirch ,
Subject: categories: Re: Where does the term monad come from?
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I have told this story many times, but I guess one more can't hurt. Of
course, it was originally used by Leibniz to describe the set of
infintesimals surrounding an ordinary point.
In the summer (or maybe late spring, the Oberwohlfach records will show
this) of 1966, there was a category meeting there. It was, as far as I
know, the third meeting ever devoted to categories. The first was the
first Midwest Category meeting, an invitation affair that consisted of
five people from Urbana (Jon Beck, John Gray, Alex Heller, Max Kelly, and
me), John Isbell and Fred Linton visiting Chicago that year, and a couple
people from U. Chicago, Mac Lane who was the host and arranged to pay our
expenses, Dick Swan, and maybe a couple others. The second was in La
Jolla and this was the third. The attendance consisted of practically
everyone in the world who had any interest in categories, with the notable
exception of Charles Ehresmann.
What, with one exception, most categorists call monads had by that time
been called "Standard constructions", "fundamental constructions" (in a
little-known paper by Jean-Marie Maranda pointed out to me by Peter
Huber), and, of course, "Triples". The latter was created by
Eilenberg-Moore and I once asked Sammy (who was known to agonize over good
terminology--e.g. "Exact") why. He answered that the concept seemed to be
of little importance, so he and John Moore spent no time on it! So much
for the predictive ability of a great mathematician.
At any rate, the big unanswered question of the meeting, where the
importance of the concept was becoming clear (Jon and I had proved our
Acyclic models theorem, for example, and the fact of the triplebleness of
compact Hausdorff spaces over sets, along with many mor familiar
examples), the search was on for a better name. We tried many ideas (mine
was "Standard Natural Algebraic Functor with Unit" (try the acronym). One
day at lunch or dinner I happened to be sitting next to Jean Benabou and
he turned to me and said something like "How about `monad'?" I thought
about and said it sounded pretty good to me. (Yes, I did.) So Jean
proposed it to the general audience and there was general agreement. It
suggested "monoid" of course and it is a monoid in a functor category.
The one dissenter was Jon Beck, who had invested as much into studying
them as anyone. His argument was that while "triples" was not a good
name, "monad" wasn't either and we shouldn't change the name from a poor
one to a mediocre one, but instead continue to search for a better one.
Out of solidarity with Jon (we collaborated on several papers), I
continued to use "triple". SLN 80 was (and is) known as the "Zurich
Triples Book". By 1980, Jon was no longer doing serious mathematics and I
was ready to change. Except that the book title "Toposes, Triples and
Theories" was too attactive to let go of. Try "Toposes, Monads and
Theories".
Incidentally, Peter May also claims to have invented the term. Treat that
claim with the contempt it deserves. The most charitable explanation I
have is that he heard it from Mac Lane, forgot that he had and then came
up with it later.
On Wed, 1 Apr 2009, Thorsten Altenkirch wrote:
> A question just came up at the Midland Graduate School (actually in
> the functional programming lecture):
> Where does the word monad come from?
>
> I know that a monad is a monoid in the category of endofunctors, but
> what is the logic monoid => monad?
>
> Btw, I frequently encounter monads in a categories of functors which
> are not endofunctors. An example are finite dimensional vectorspaces
> which can be constructed via a monoid in the category of functors
> FinSet -> Set, here I is the embedding and (x) can be constructed from
> the left kan extension and composition.
> The unit is given by the Kronecker delta and join can be constructed
> from Matrix multiplication. Should one call these beasts monads as
> well? Is there a good reference for this type of construction?
>
> Cheers,
> Thorsten
>
>
From rrosebru@mta.ca Thu Apr 2 09:12:55 2009 -0300
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Date: Wed, 1 Apr 2009 20:45:33 +0200 (CEST)
Subject: categories: Re: Where does the term monad come from?
From: Johannes.Huebschmann@math.univ-lille1.fr
To: "Thorsten Altenkirch" ,
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>From my recollections, the terminology monad was suggested by P. May
as a replacement for triple.
The terminology was intended to match with "operad".
At the time, S. Mac Lane has taken up that suggestion.
In his book "Categories for the working mathematician"
Mac Lane uses the terminology monad and comonad rather than triple
and cotriple.
If Peter May participates in this board I am sure he will react.
Johannes
> A question just came up at the Midland Graduate School (actually in
> the functional programming lecture):
> Where does the word monad come from?
>
> I know that a monad is a monoid in the category of endofunctors, but
> what is the logic monoid =3D> monad?
>
> Btw, I frequently encounter monads in a categories of functors which
> are not endofunctors. An example are finite dimensional vectorspaces
> which can be constructed via a monoid in the category of functors
> FinSet -> Set, here I is the embedding and (x) can be constructed from
> the left kan extension and composition.
> The unit is given by the Kronecker delta and join can be constructed
> from Matrix multiplication. Should one call these beasts monads as
> well? Is there a good reference for this type of construction?
>
> Cheers,
> Thorsten
>
>
>
From rrosebru@mta.ca Thu Apr 2 09:13:42 2009 -0300
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Date: Wed, 01 Apr 2009 20:47:12 +0100
From: Venanzio Capretta
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The philosopher Gottfried Leibniz believed that every entity in the
Universe is a separate substance that doesn't interact with others. He
called these substances "monads". All properties and events that happen
to a monad are implicit in its nature from its creation. So if an apple
falls from a tree and bounces off my head, there is actually no contact:
the apple-monad bounces by itself without the help of my head and the
Venanzio-monad feels pain without the intervention of the apple. All
monads are synchronized from creation by the wisdom of God.
This implies that every monad has an internal representation of every
entity in the universe and these representations can never influence
objects outside the monad.
The analogy with our monads should be evident!
Thorsten Altenkirch wrote:
> A question just came up at the Midland Graduate School (actually in
> the functional programming lecture):
> Where does the word monad come from?
>
> I know that a monad is a monoid in the category of endofunctors, but
> what is the logic monoid => monad?
>
> Btw, I frequently encounter monads in a categories of functors which
> are not endofunctors. An example are finite dimensional vectorspaces
> which can be constructed via a monoid in the category of functors
> FinSet -> Set, here I is the embedding and (x) can be constructed from
> the left kan extension and composition.
> The unit is given by the Kronecker delta and join can be constructed
> from Matrix multiplication. Should one call these beasts monads as
> well? Is there a good reference for this type of construction?
>
> Cheers,
> Thorsten
>
>
From rrosebru@mta.ca Thu Apr 2 09:14:50 2009 -0300
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Date: Wed, 01 Apr 2009 23:19:50 +0200
From: burroni@math.jussieu.fr
To: Thorsten Altenkirch ,
Subject: categories: Re: Where does the term monad come from?
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Cher Thorsten,
toutes mes excuses pour ce message en fran=E7ais.
Le terme "monade" a =E9t=E9 employ=E9 par Benabou (LNM Springer no 47, si je=
=20
ne me trompe) et dans un sens abstrait : pseudofoncteur 1 --> B de la =20
bicat=E9gorie finale 1 vers une bicat=E9gorie arbitraire B. Par la suite =20
il a =E9t=E9 convenu de le r=E9sever au cas particulier o=F9 B=3DCat (en =20
remplacement du terme "triple").
A mon avis, le terme est remarquable car il combine ceux de "monoides" =20
et de "monades", concept utilis=E9 par Leibnitz, mais qui, ind=E9pendement =
=20
de l'usage fait par ce philosophe, signifie : unit=E9 simple, =20
ind=E9composable. Cette simplicit=E9, cette ind=E9composabilit=E9 est celle =
de =20
la bicat=E9gorie 1.
Aujourd'hui, on appelle monoide, les monades au sens g=E9n=E9ral de =20
Benabou. (Personnellement, je ne trouve cela imparfait car un vrai =20
monoide est une structure beaucoup plus riche : exemple x |--> x^2 n'a =20
pas de sens en general.)
amiti=E9s,
Albert
Thorsten Altenkirch a =E9crit=A0:
> A question just came up at the Midland Graduate School (actually in
> the functional programming lecture):
> Where does the word monad come from?
>
> I know that a monad is a monoid in the category of endofunctors, but
> what is the logic monoid =3D> monad?
>
> Btw, I frequently encounter monads in a categories of functors which
> are not endofunctors. An example are finite dimensional vectorspaces
> which can be constructed via a monoid in the category of functors
> FinSet -> Set, here I is the embedding and (x) can be constructed from
> the left kan extension and composition.
> The unit is given by the Kronecker delta and join can be constructed
> from Matrix multiplication. Should one call these beasts monads as
> well? Is there a good reference for this type of construction?
>
> Cheers,
> Thorsten
From rrosebru@mta.ca Fri Apr 3 10:00:00 2009 -0300
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Date: Thu, 2 Apr 2009 07:51:22 -0500 (CDT)
From: Peter May
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Michael, where on earth did that piece of contemptible writing
come from. I never claimed to invent the term monad. I did
invent the term operad, as a portmanteau of operation and monad.
And I convinced MacLane to change from the silly term `triple'
to `monad' in Categories for the working mathematician. He is
not here to corroborate, but look at his note on terminology,
page 138 of the second edition: ``The frequent but unfortunate
use of the word `triple' in this sense has achieved a maximum
of needless confusion, what with the conflict with ordered
triple, plus the use of associated terms such as ``triple
derived functor''for functors which are not three times
derived from anything in the world. Hence the term monad.''
Michael, shame on you!
Peter May
From rrosebru@mta.ca Fri Apr 3 10:00:01 2009 -0300
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Date: Thu, 02 Apr 2009 09:31:00 -0400
From: jim stasheff
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Whereas my recollection (from those dear dim days beyond recall when I
was present on a weekly basis for ND about that time)
was that the terminology went from Mac Lane to May with
operad to match monad
as I recall, Mac Lane liked monad because of the philosophical connection
Leibniz as philosopher not as mathematician?
* Monad (Greek philosophy) a term used by ancient philosophers
Pythagoras, Parmenides, Xenophanes, Plato, Aristotle, and Plotinus as a
term for God or the first being, or the totality of all being.
* Monism, the concept of "one essence" in the metaphysical and
theological theory
* Monad (Gnosticism), the most primal aspect of God in Gnosticism
****** Monadology, a book of philosophy by Gottfried Leibniz in
which monads are a basic unit of perceptual reality
* Monadologia Physica by Immanuel Kant
* The Cup or Monad, a text in the Corpus Hermetica
from the Wiki
Johannes.Huebschmann@math.univ-lille1.fr wrote:
> >From my recollections, the terminology monad was suggested by P. May
> as a replacement for triple.
> The terminology was intended to match with "operad".
> At the time, S. Mac Lane has taken up that suggestion.
> In his book "Categories for the working mathematician"
> Mac Lane uses the terminology monad and comonad rather than triple
> and cotriple.
>
> If Peter May participates in this board I am sure he will react.
>
> Johannes
>
>> A question just came up at the Midland Graduate School (actually in
>> the functional programming lecture):
>> Where does the word monad come from?
>>
>> I know that a monad is a monoid in the category of endofunctors, but
>> what is the logic monoid => monad?
>>
>> Btw, I frequently encounter monads in a categories of functors which
>> are not endofunctors. An example are finite dimensional vectorspaces
>> which can be constructed via a monoid in the category of functors
>> FinSet -> Set, here I is the embedding and (x) can be constructed from
>> the left kan extension and composition.
>> The unit is given by the Kronecker delta and join can be constructed
>> from Matrix multiplication. Should one call these beasts monads as
>> well? Is there a good reference for this type of construction?
>>
>> Cheers,
>> Thorsten
>>
>>
>>
>>
>
>
>
>
>
From rrosebru@mta.ca Fri Apr 3 10:00:20 2009 -0300
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for categories-list@mta.ca; Fri, 03 Apr 2009 10:00:16 -0300
Date: Fri, 03 Apr 2009 15:28:26 +1100
Subject: categories: Re: Where does the term monad come from?
From: Steve Lack
To: Thorsten Altenkirch ,
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Dear All,
As usual, there have been plenty of people with comments about history.
There was also a second part to the question:
>
> Btw, I frequently encounter monads in a categories of functors which
> are not endofunctors. An example are finite dimensional vectorspaces
> which can be constructed via a monoid in the category of functors
> FinSet -> Set, here I is the embedding and (x) can be constructed from
> the left kan extension and composition.
> The unit is given by the Kronecker delta and join can be constructed
> from Matrix multiplication. Should one call these beasts monads as
> well? Is there a good reference for this type of construction?
The category of functors from FinSet to Set is equivalent to the category
of endofunctors of Set which preserve filtered colimits: such endofunctors
are usually called finitary. Thus a monoid in [FinSet,Set] with respect to
this tensor product is the same thing as a monad on Set whose endofunctor
part is finitary: this is called a finitary monad.
These finitary monads on Set are equivalent to Lawvere theories and so in
turn to (finitary, single-sorted) varieties.
Finitary monads can also be considered on other base categories than Set,
especially on locally finitely presentable ones.
It is true that vector spaces are the algebras for a finitary monad on Set.
There is no need to restrict to finite-dimensional vector spaces; in fact it
is not true that there is a monad on Set whose algebras are the
finite-dimensional vector spaces.
Steve.
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Date: Fri, 03 Apr 2009 15:33:13 +1100
Subject: categories: Re: Where does the term monad come from?
From: Steve Lack
To: , Thorsten Altenkirch ,
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Dear All,
Just another quick comment about monads:
On 2/04/09 8:19 AM, "burroni@math.jussieu.fr"
wrote:
> Cher Thorsten,
>=20
> toutes mes excuses pour ce message en fran=E7ais.
>=20
> Le terme "monade" a =E9t=E9 employ=E9 par Benabou (LNM Springer no 47, si je
> ne me trompe) et dans un sens abstrait : pseudofoncteur 1 --> B de la
> bicat=E9gorie finale 1 vers une bicat=E9gorie arbitraire B. Par la suite
> il a =E9t=E9 convenu de le r=E9sever au cas particulier o=F9 B=3DCat (en
> remplacement du terme "triple").
Some people may reserve monad for the case B=3DCat, but not all. After Benabo=
u
demonstrated the incredible importance of this idea in various B, the theor=
y
of monads in 2-categories/bicategories has been widely developed, starting
(I believe) with Ross Street's "Formal theory of monads".
Steve.
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From: Ross Street
Subject: categories: RE: Monads
Date: Sat, 4 Apr 2009 14:31:39 +1100
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I hope I can add some jigsaw pieces towards the history of the term
"monad" in category theory without offending anyone.
1) It is clearly a fact that the term "monad" is used in Benabou's
paper SLNM 47 (1967). He recognized that it is a morphism of
bicategories from the terminal category 1.
2) I have a clear memory that Mac Lane told me (perhaps at Chicago
while I was a postdoc at Champaign-Urbana 1968-69) that Benabou
courteously asked him (possibly by airmail, maybe by phone call,
maybe at a conference) whether Mac Lane would mind whether he used
the term "bicategory" in the sense we now use it. Mac Lane had used
"bicategory" to mean a category with two distinguished classes of
morphisms: roughly speaking, what we now call a category with a
factorization system. Mac Lane told Benabou he did not mind. So
Benabou used it in SLNM 47.
3) Less clearly I remember Mac Lane said Benabou also suggested the
term "monad" for use in SLNM 47.
4) It is again my clear memory that, in his lecture marathon at the
Summer School on Category Theory at Bowdoin College (Maine,
mid-1969), Mac Lane expressed strong dislike for the term "triple"
but had not really settled on a term. Mac Lane actually used the term
"triad" in his lectures at Bowdoin.
5) At CT08, Lawvere told me Eilenberg suggested the term "monad".
Best wishes,
Ross
From rrosebru@mta.ca Sun Apr 5 18:01:25 2009 -0300
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Date: Sat, 04 Apr 2009 11:45:47 -0400
From: jim stasheff
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Ross Street wrote:
>
> I hope I can add some jigsaw pieces towards the history of the term
> "monad" in category theory without offending anyone.
>
>
...
> Best wishes,
> Ross
>
>
Some version of this and other responses whould be added to the Wiki
I'm not up to the job.
jim
From rrosebru@mta.ca Mon Apr 6 13:38:38 2009 -0300
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Date: Mon, 6 Apr 2009 06:52:25 +0200 (MEST)
From: Patrik Eklund
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"Operads" are like sets of operations.
A monad is an extension of a functor. If the functor is the term functor,
then the operations of the signature lies inside the functor, and the
"operations" eta and mu are identities, or at least something very
isomorphic to identities.
In the filter functor eta is point filters and mu is Kowalsky's
diagonalization.
In my view there is no logic monoid => monad, and I cannot see the
full idea behind using "operads", so help me Mona.
Patrik
On Thu, 2 Apr 2009, jim stasheff wrote:
> Whereas my recollection (from those dear dim days beyond recall when I
> was present on a weekly basis for ND about that time)
> was that the terminology went from Mac Lane to May with
> operad to match monad
>
> as I recall, Mac Lane liked monad because of the philosophical connection
> Leibniz as philosopher not as mathematician?
>
> * Monad (Greek philosophy) a term used by ancient philosophers
> Pythagoras, Parmenides, Xenophanes, Plato, Aristotle, and Plotinus as a
> term for God or the first being, or the totality of all being.
> * Monism, the concept of "one essence" in the metaphysical and
> theological theory
> * Monad (Gnosticism), the most primal aspect of God in Gnosticism
> ****** Monadology, a book of philosophy by Gottfried Leibniz in
> which monads are a basic unit of perceptual reality
> * Monadologia Physica by Immanuel Kant
> * The Cup or Monad, a text in the Corpus Hermetica
> from the Wiki
>
>
From rrosebru@mta.ca Mon Apr 6 13:44:16 2009 -0300
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Date: Mon, 6 Apr 2009 06:33:17 +0200 (MEST)
From: Patrik Eklund
To: categories@mta.ca
Subject: categories: RE: Monads
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[Note from moderator: this thread has strayed; although this post is
allowed, comments closer to categories (not Kant's) are preferred.]
Reference to Leibniz is nice, and so is going back even more in history.=20
Going forward into modern history leads to problems of who actually cause=
d=20
what. Probably because we then tend to mix history and politics.
Anyway, also having googled, I found this about Leibniz:
=A7. 1. Die Monaden (Das Worte Monade oder Monas) wovon wir allhier reden=
=20
werden / sind nichts anders als einfache Substanzen / woraus die zusammen=
=20
gesetzten Dinge oder composita bestehen. Unter dem Wort / einfach /=20
verstehet man dasjenige / welches keine Teile hat.
"sind nichts anders als einfache Substanzen"
"is nothing but simple substances"
They are, but it is not a mathematical statement.
"woraus die zusammen gesetzten Dinge oder composita bestehen"
"using which you put them together or compose(!) them together"
Now he is cooking. Monad compositions are important. Leibniz and Beck=20
working together, I like it. This is closer to mathematics.
"verstehet man dasjenige / welches keine Teile hat"
"is to be understood as something which doesn't have subparts"
I am sure there are non-trivial monads which are not composed (in Beck's=20
sense) by other non-trivial monads. But more interestingely, composed=20
monads are indeed monads, and even worse (from leibniz point of=20
view) submonads do exist, like the filter monad being submonad to the=20
ultrafilter monad (with the astonishing fact, yes, I know, I am repreatin=
g=20
myself, that their respective algebras are Scott lattices and compact=20
Hausdorff spaces).
So, basically I like Leibniz, even if he was wrong at this point. History=
=20
is not easy. We say "Rome was destroyed" and we frequently say by the=20
goths. Saying that leads us to ask "how could it be destroyed". Seldom do=
=20
we hear "how could it stay alive so long".
Best,
Patrik
PS "Monas" seems mostly to be used for a sailing boat, the "Kiel", and=20
"the Mona" is Louvre in Paris.
From rrosebru@mta.ca Fri Apr 3 20:46:01 2009 -0300
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From: Marta Bunge
To: ,
Subject: categories: RE: Monads
Date: Fri, 3 Apr 2009 09:56:46 -0400
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Dear all=2CSomething to corroborate MacLane's abhorrence of the word "tripl=
e" is=2C in my view=2C his refusal to communicate my first paper (Marta Bun=
ge=2C Relative Functor Categories and Categories of Algebras=2C J.of Algebr=
a 11 (1969) 64-101) unless I changed the word "triple" in it for that of "m=
onad". In order to show my independence (!)=2C yet wishing to have the pape=
r published=2C I changed "triple" back to "standard construction". This he =
accepted without objections. Nowadays I use monads like everybody else. I h=
ave no idea which=2C among the many possible reasons suggested in categorie=
s=2C was MacLane's reason for insisting on "monads"=2C whether philosophica=
l or mathematical. However=2C his acceptance of my use of "standard constru=
ction" suggests that his dislike of "triple" was stronger than his preferen=
ce for "monad". Cordial regards=2CMarta Bunge
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Date: Fri, 03 Apr 2009 15:55:16 +0200
From: burroni@math.jussieu.fr
To: jim stasheff , categories@mta.ca
Subject: categories: Re: Where does the term monad come from?
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> * The Cup or Monad, a text in the Corpus Hermetica
> from the Wiki
Je trouverais normal que le mod=C3=A9rateur ne permette pas cette =20
intervention. C'est en partie =C3=A0 titre de plaisanterie et en compl=C3=A9=
ment =20
=C3=A0 une information de Jim que je la fait.
Pour des sources mystiques sur la monade :
http://www.esotericarchives.com/dee/monade.htm
on y trouve des th=C3=A9or=C3=A8mes in=C3=A9dits en th=C3=A9orie des cat=C3=
=A9gories.
Albert
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Date: Fri, 03 Apr 2009 10:07:56 -0400
From: jim stasheff
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Peter May wrote:
> I never claimed to invent the term monad. I did
> invent the term operad, as a portmanteau of operation and monad.
> Peter May
>
>
which I am happy to confirm
I think I was there at the time
or at least nearby
jim
Rainer,
Is this covered in your memoir of those miravulaous years?
jim
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From: Michael Barr
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Subject: categories: Re: Monads
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Sorry if this offended you, but I heard from several places that you
claimed the invention of the term. You will note that one other
respondent credited it to you, so there must have been a meme to that
effect. If you never made that claim then I truly apologize. Perhaps
people confused "monad" with "operad", which I do believe you invented.
Michael
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Subject: categories: Re: Where does the term monad come from?
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Patrik Eklund wrote:
In my view there is no logic monoid => monad...
It's pretty much been said, but I'll say it again:
We can generalize the concept of monoid from Set to any monoidal category
and then to any bicategory. A monoid in Cat is then a monad.
Indeed, most people seem to call a "monoid" in a bicategory a "monad".
Best,
jb
From rrosebru@mta.ca Tue Apr 7 08:25:21 2009 -0300
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Date: Mon, 6 Apr 2009 23:06:37 -0300 (ADT)
From: RJ Wood
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John Baez wrote:
It's pretty much been said, but I'll say it again:
We can generalize the concept of monoid from Set to any monoidal category
and then to any bicategory. A monoid in Cat is then a monad.
Indeed, most people seem to call a "monoid" in a bicategory a "monad".
Best,
jb
John, given the didactic nature of this thread, I think we should be
more precise about what you mean by `a "monoid" in a bicategory'. For
a bicategory B and an object X therein, B(X,X) (together with composition,
1_X, and the inherited constraints of B) i s a monoidal category and a
monad in B is an object X in B together with a monoid in B(X,X).
Rj
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From: Vaughan Pratt
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Patrik Eklund wrote:
> "Operads" are like sets of operations.
>
> A monad is an extension of a functor. If the functor is the term functor,
> then the operations of the signature lies inside the functor, and the
> "operations" eta and mu are identities, or at least something very
> isomorphic to identities.
>
> In the filter functor eta is point filters and mu is Kowalsky's
> diagonalization.
>
> In my view there is no logic monoid => monad, and I cannot see the
> full idea behind using "operads", so help me Mona.
In one paragraph, a monad can be understood as a set T of operations
graded by their arity X, that is, a variable set T(X) where the set X is
the parameter of variation. This set is made a monoid by the operation
of substitution interpreted as the multiplication of the monoid.
Substitution is associative (terms of height three can be built top-down
or bottom-up) and has a two-sided identity interpretable ambiguously as
the identity operation of T (when applied to the top of a term) and
substitution of variables for themselves (when applied to the bottom of
a term).
The back-to-back talks of Walter Taylor on his general theory of
varieties and Fred Linton on monads at the Universal Algebra and
Category Theory (UACT) meeting held at MSRI in 1993 were like ships
passing in the night. No one thought to mention, either before, during,
or after the talks, that they were describing essentially the same thing
(or if they did both George McNulty and I missed it), with the result
that many of the algebraists at the meeting just assumed that these were
unrelated talks.
Monads can be explained in terms of their associated Kleisli category,
or their Eilenberg-Moore category, or as the composition UF of any
set-valued functor U with its left adjoint. After Fred's talk I had
lunch with George and tried out the third of these on him. However we
got bogged down in the definition of adjunction.
In hindsight I think the quickest way to explain a monad to an
algebraist is to do so directly in terms of T, \mu, and \eta, without
the distraction of the additional machinery of the three above methods.
It would go something like the following, which of the above three is
closest to the Kleisli category approach. I'll ignore the inconsistent
monads, those axiomatized by x=y, for which T(X) = 1 for nonempty X and
T(0) <= 1.
A monad specifies the language and equations of an equational theory.
The functor T specifies the language by providing for each set X the set
of operations (more properly polynomials or abstract terms) of arity X,
e.g. T(2) is the set of binary operations of the theory. The
multiplication \mu_X: T(T(X)) --> T(X) specifies the theory by mapping
terms of height at most two to operations (identified with terms of
height at most one). Terms s and t of height two identified by \mu,
e.g. x(y+z) = xy+xz in the case of ring theory, constitute the axioms s
= t of the theory determined by \mu.
Hardware types and visual thinkers can picture T(X) as a black box
containing all operations of arity X. X can be thought of as a row (or
any other layout, I like the unit interval [0,1] of reals for picturing
an uncountable set as a row) of input sockets on one side and T(X) as a
row of output plugs on the opposite side, one per operation. The unit
\eta_X: X --> T(X) at X, necessarily an injection, ensures that the
operations include the variables, qua (formal) projections. Thinking of
T(X) as consisting of terms, define the height of each variable to be
zero and that of the remaining operations of T(X) as one.
Since T can take any set of variables it can take in particular T(X),
whence there is also a box with input set T(X) and output set T(T(X)).
The boxes containing T(X) and T(T(X)) can be plugged together to form a
single black box with set X of inputs and set T(T(X)) of outputs.
However this latter set must now be interpreted as consisting of
entities of arity X instead of arity T(X). Viewed syntactically (taking
into account the separate contents of the two boxes and how they attach)
we can consider the outputs of T(T(X)) as terms of height two in the
variables in X, or rather at most two since the unit of the monad embeds
X in T(X) and T(X) in T(T(X)).
One can then ask whether any of these terms realize some operation not
among those of T(X). The function \mu_X: T(T(X)) --> T(X) accomplishes
three things.
(i) It interprets every term of height up to two as an operation of
T(X), a form of abstract evaluation that hides the two-level term structure.
(ii) In so doing it answers the above question in the negative: no new
operations, all terms of height up to two realize operations already
present in T(X). In this sense T as a graded set of operations is
closed under substitution.
(iii) As noted above it axiomatizes the equational theory associated
with the monad with all equations of that theory involving terms of
height at most two, namely all equations s = t such that \mu_X maps s
and t to the same operation of T(X).
\mu can be extended to evaluate terms of height h inductively. If
\mu_h: T^h --> T evaluates terms of height up to h then the vertical
composite \mu T(\mu_h): T^{h+1} --> T^2 --> T evaluates terms of height
up to h+1, starting from \mu_2 = \mu. (So \mu_h = \mu T(\mu)
T^2(\mu)...T^{h-2}(\mu).) This is the categorical counterpart of using
equational logic to inductively build up the height of equations in the
theory one level at a time via \mu, starting from the equations
constituting the kernel of \mu.
(Although height is always finite there is no such restriction on arity
and hence on width of a term, which can be any set. Even for a finitary
monad an operation can take uncountably many arguments, e.g. for the
monad for Vct_R as the variety of vector spaces over the reals, each
operation in T(T(2)) takes as many parameters as there are linear
combinations ax+by, namely uncountably many, though it depends on only
finitely many of them because Vct_R is a finitary variety in the sense
Steve Lack referred to on Friday.)
If the boxes really do consist of operations then the two pluggings
required to form a chain T(X), T(T(X)), T(T(T(X))) of three black boxes
should have the same operational effect regardless of the order in which
they're performed. The associativity axiom for a monad enforces this
property of black boxes containing operations. The above inductive
definition of \mu_h was bottom-up (T^{h+1} = TT^h) but there is an
equivalent top-down one (T^h T) producing the same \mu_h.
The Eilenberg-Moore category of a monad is the variety of algebras it
axiomatizes, modulo details of the treatment of the associated
signature. In general the signature will be a proper class but in many
cases encountered in practice one can pick out of this class a (small)
set sufficient for a basis of operations, e.g. +, -, and the scalar
multiplications for Vct_k, or NAND and a constant for Boolean algebra.
The variety of sigma-algebras is not finitary although it can still be
furnished with a small signature, but this is not possible for the
varieties of complete semilattices and of complete atomic Boolean
algebras, which old-school algebraists would not consider varieties for
that reason.
The Kleisli category is the full subcategory of the variety consisting
of its free algebras. The latter is intimately linked to the above
intuitions, and amounts to an equivalent way of seeing that T is closed
under substitution by formulating substitution as the composition of the
Kleisli category.
The essential point of departure from the usual notion of a monoid as a
fixed set is that for the type T^2 --> T of multiplication, T^2 is
defined via composition instead of cartesian product. Monads are
therefore simply monoids adapted in this way to accommodate their
variability.
Monads need not be sets. Just as a ring can be defined as a monoid
object in Ab, with the domain of multiplication being formed via
cartesian product in Ab, so can a monad be a monoid object in the
category of endofunctors of Ab, with the domain of multiplication being
formed by functor composition in Ab^Ab. Although not all categories
have finite products, T^2 is defined for every category C and
endofunctor T: C --> C.
Vaughan Pratt
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Date: Tue, 07 Apr 2009 08:50:29 +0100
From: Tim Porter
To: "categories@mta.ca"
Subject: categories: L'Aquila category theorists?
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I like many other categorists have very fond memories of the welcome
given us on visits to l'Aquila. Can I request that as soon as news
comes through, of either sort, about our friends there, that the news
be shared with this list.
Tim Porter
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Date: Tue, 07 Apr 2009 12:40:53 +0100
From: Maria Manuel Clementino
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[Note from moderator: several posts with essentially the same message
follow; the details mentioned vary slightly so all will be forwarded.
Thanks to all who responded. It is a relief to know that the news is not
worse.]
Tim Porter wrote:
> I like many other categorists have very fond memories of the welcome
> given us on visits to l'Aquila. Can I request that as soon as news
> comes through, of either sort, about our friends there, that the news
> be shared with this list.
>
> Tim Porter
I received a message from Anna Tozzi saying that Eraldo Giuli and her
were fine.
Maria Manuel Clementino
From rrosebru@mta.ca Tue Apr 7 19:55:57 2009 -0300
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From: luciano stramaccia
Subject: categories: Re: L'Aquila category theorists?
Date: Tue, 7 Apr 2009 14:27:30 +0200
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Dear Tim,
I heard from Eraldo Giuli and Anna Tozzi yesterday morning soon after
the quake.
They all are safe (and their families). My be Eraldo's hause was
slightly damaged, not Anna's.
Anyway, they plan to move to their hauses along the coast where the
quake had no effect.
Luciano
Il giorno 07/apr/09, alle ore 09:50, Tim Porter ha scritto:
I like many other categorists have very fond memories of the welcome
given us on visits to l'Aquila. Can I request that as soon as news
comes through, of either sort, about our friends there, that the news
be shared with this list.
Tim Porter
From rrosebru@mta.ca Tue Apr 7 19:56:44 2009 -0300
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Date: Tue, 07 Apr 2009 09:57:32 -0400
From: Walter Tholen
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Subject: categories: Re: L'Aquila category theorists?
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Tim -
I have been in telephone contact with Eraldo Giuli. His and Anna Tozzi's
families are fine. They are currently staying with their families away
from the town of L'Aquila, in order to avoid the aftershocks. They do,
however, report considerable damage to some of their properties in L'Aquila.
Regards,
Walter.
Tim Porter wrote:
> I like many other categorists have very fond memories of the welcome
> given us on visits to l'Aquila. Can I request that as soon as news
> comes through, of either sort, about our friends there, that the news
> be shared with this list.
>
> Tim Porter
From rrosebru@mta.ca Tue Apr 7 19:57:54 2009 -0300
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Date: Tue, 07 Apr 2009 18:33:38 +0200
To: t.porter@bangor.ac.uk
From: aurelio carboni
Subject: categories: Re: L'Aquila category theorists?
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Dear Tim,
I just succeeded in getting in touch by phone with Eraldo,
after trying for quite a long time. Eraldo, his family and all the other
colleagues are fine. Eraldo lost a family flat in the center of the town,
but nobody was injured. He will not be able to be contacted by e-mail
for quite a long time, and is now staying in his house at the beach
nearby L'Aquila. Eraldo asked me to inform the colleagues and friends.
Best wishes,
Aurelio.
>At 09.50 07/04/2009, you wrote:
>>I like many other categorists have very fond memories of the welcome
>>given us on visits to l'Aquila. Can I request that as soon as news
>>comes through, of either sort, about our friends there, that the news
>>be shared with this list.
>>
>>Tim Porter
>>
From rrosebru@mta.ca Tue Apr 7 20:00:29 2009 -0300
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Date: Tue, 07 Apr 2009 11:10:41 -0400
From: jim stasheff
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Subject: categories: Re: Where does the term monad come from?
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Vaughan Pratt wrote:
> Patrik Eklund wrote:
>> "Operads" are like sets of operations.
>>
>> A monad is an extension of a functor. If the functor is the term
>> functor,
...
> endofunctor T: C --> C.
>
> Vaughan Pratt
>
Someone should up date the Wiki
jim
From rrosebru@mta.ca Tue Apr 7 20:01:13 2009 -0300
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To: Categories list
Subject: categories: virtual special session
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Last weekend there was a special session on Homotopical Algebra with
Applications to Mathematical Physics
at eh AMS meeting in Raleigh. I was unable to attend in person but Tom
Lada set up via Skype and opportunity for me to participate. I strongly
encourage this technology for virtual attendance at such meetings or
informal equivalents.
Slides from the session are becoming available at
http://www4.ncsu.edu/~lada/NCSU%20special%20session.htm
jim
From rrosebru@mta.ca Tue Apr 7 20:02:13 2009 -0300
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for categories-list@mta.ca; Tue, 07 Apr 2009 20:02:05 -0300
Date: Tue, 7 Apr 2009 18:05:22 +0200 (CEST)
Subject: categories: a workshop on commutativity of diagrams, Toulouse
From: soloviev@irit.fr
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CAM-CAD
Workshop on Computer Algebra Methods and Commutativity of Algebraic Diag=
rams
IRIT, Toulouse (France)
October 2009
*** FIRST CALL FOR PARTICIPATION ***
*** AND ***
*** PAPERS ***
Categorical diagrams have multiple applications
in mathematics (algebra, topology) and computer science
(models and metamodels, rewriting systems, higher order
languages). Diagrams (understood less strictly) can be found
in physics, chemistry and other scientific domains.
One meets many similar problems in
computer-assisted treatment of diagrams in all these
domains concerning algorithms, graphic interfaces,
interaction with systems of computer algebra and
other software. In spite of importance of diagrammatic
methods, they are relatively little developped
and underrepresented in the world of computer-assisted
reasoning.
This workshop aims to bring together researchers working on
these subjects, to assess the current state of the art and
identify open problems and future research directions.
The main topics may be listed (non-exhaustively) as follows:
algorithms that may be used in computer-assisted treatment of diagrams,
treatment of diagrams in existing computer algebra systems,
formal developments related to diagrams and
category theory in proof-assistants,
user interfaces and graphics for categorical diagrams.
There will be space for talks presenting original work,
work in progress, applications, survey of previous works.
We plan to provide sufficient time for discussions. Details on
paper submission will be given in a further announcement.
Papers presented at the workshop will be published on the
web site of the workshop and may be selected for submission,
in complete and revised form, to a special issue of an
international journal, in case their number and quality
justify it.
Important dates:
Submission deadline:
send short abstract (title) by 30 June 2009 by e-mail
to soloviev@irit.fr
send either a full paper or an extended abstract by 30 September 2009
Workshop: Friday 16 and Saturday 17 October 2009
Organising/program committee:
P. Damphousse (Universite de Tours)
Y. Lafont (IML, Universite Aix-Marseille 2)
R. Matthes (IRIT, Universite Paul Sabatier, Toulouse)
S. Soloviev (IRIT, Universite Paul Sabatier, Toulouse)
Local organisation:
S. Soloviev
R. Matthes
A. El Khoury
Contact:
Sergei Soloviev
IRIT
University Toulouse-3
118, route de Narbonne,
31062 Toulouse
France
E-mail: soloviev@irit.fr
Tel: (+33) 5 61 55 62 55
Fax: (+33) 5 61 55 62 58
From rrosebru@mta.ca Tue Apr 7 20:03:32 2009 -0300
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Date: Tue, 7 Apr 2009 12:50:21 -0400
Subject: categories: Re: Where does the term monad come from?
From: Zinovy Diskin
To: Steve Lack , categories@mta.ca
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On Fri, Apr 3, 2009 at 12:28 AM, Steve Lack wrote:
>
> Finitary monads can also be considered on other base categories than Set,
> especially on locally finitely presentable ones.
>
> It is true that vector spaces are the algebras for a finitary monad on Set.
> There is no need to restrict to finite-dimensional vector spaces; in fact it
> is not true that there is a monad on Set whose algebras are the
> finite-dimensional vector spaces.
>
there is something similar in algebraic logic. The class of locally
finite cylindric/polyadic algebras is not a variety and the forgetful
functor to Set is not monadic (l.f. means that all relations are of
finite arities). In categorical logic (hyperdoctrines), these algebras
are considered in many-sorted signatures, in fact, as algebras over
graphs, and their theory becomes equational (= the corresponding
forgetful functor to Graph is monadic). Probably, it's a general
phenomenon wrt specifying finitary objects: by indexing them with
finite sets (contexts, supports,arities), we get equational theories
over graph-like structures.
In a wider (and partly speculative) setting, the shift from classical
algebraic to categorical logic is a shift from simple signatures and
complex theories to complex signatures and simple theories. In a
sense, this is what category theory does wherever it applies to
classical problems: it greatly simplifies the logic (and the internal
structure), but pays for this by a complex vocabulary (the external
structure, interface). A typical example is classical vs. categorical
set theories.
Thus, a categorical model is a device with a structurally complex
interface and simple internal logic. An average user prefers, of
course, simple-looking interfaces of classical theories (and
eventually has to pay for this choice but it happens later on...). So,
for marketing categorical models, it's important to provide good
manuals for their complicated interfaces -- what Vaughan just did for
monads.
Zinovy
From rrosebru@mta.ca Wed Apr 8 19:47:06 2009 -0300
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for categories-list@mta.ca; Wed, 08 Apr 2009 19:44:28 -0300
Date: Wed, 08 Apr 2009 10:44:18 +0200
From: James McKinna
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To: categories@mta.ca
Subject: categories: iBourbaki in constructive type theory? [Fwd: 1 Postdoc and 1 PhD vacancy in the MathWiki project]
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Dear categories-subscribers,
Following the recent discussion of iBourbaki and related ideas on this
list, you (or your students) may be interested in the following jobs
being advertised in our Foundations group at Nijmegen:
1 Postdoc and 1 PhD vacancy in the MathWiki project
Please contact Herman Geuvers directly if interested.
James McKinna
====================================================
1 POSTDOC and 1 PHD POSITION in the MathWiki project
at Radboud University Nijmegen (NL)
http://www.fnds.cs.ru.nl/fndswiki/Vacancies
The Institute for Computing and Information Science of the Radboud
University Nijmegen (NL) is looking for 2 researchers to work on the NWO
project "MathWiki a Web-based Collaborative Authoring Environment for
Formal Proofs".
The vacancies are:
- a POSTDOC for the period of 3 years
vacancy number: 62-16-09
- a PHD POSITION for the period of 4 years
vacancy number: 62-17-09
AIMS OF THE PROJECT
===================
The aim of the MathWiki project is to open up to a wider community the
rich collections of knowledge stored in the repositories of proof
assistants.
To this end we will build a web-based collaborative authoring
environment for formal mathematics, the MathWiki system. This system
will provide interactive web access through a standardized interface to
a number of proof assistants. The MathWiki system will also be a
platform for the development of formal proofs within those proof
assistants and it will provide high level access (through Wikipedia-like
web pages) to their repositories of formalised mathematics. These
repositories will reside on the server.
In the project we will study and further develop Wiki technology and
semantic web technology, all in the context of proof assistant
repositories of formalized mathematics. The project thus brings together
the open nature of Wiki authoring with expertise in Proof Assistants and
Semantic Web technologies to build a new Wiki for mathematics,
supporting content creation, search and retrieval.
>From the perspective of the ordinary user of mathematics, MathWiki will
be important because it will provide high-level mathematical content on
the web in a much more coherent and precise way than is available at
present.
>From the proof assistant user perspective, MathWiki will be important
because it will provide an advanced environment for the collaborative
authoring of verified mathematics, mediated simply by a web interface.
The MathWiki system will be based on our existing experience with proof
assistant technology on the web, the "ProofWeb" systems, see
http:://prover.cs.ru.nl
-----------------------------------------------------------------
Requirements for the PhD student position:
- A master's (or equivalent) degree in Computer Science, Mathematics or
a related field, with a strong interest in proof assistants and/or
semantic web technology (preferably both)
- Commitment and a cooperative attitude.
- Very good written and oral English skills.
Requirements for the Postdoc position:
- A PhD in Computer Science, Mathematics, or a related field with
expertise in proof assistants and/or semantic web technology (preferably
both).
- A strong publication record.
- Commitment and a cooperative attitude.
- Very good written and oral English skills.
-----------------------------------------------------------------
Conditions of employment:
The PhD students will be employed for a period of 4 years (40 hrs/week).
The Postdocs will be employed for a period of 3 years (40 hrs/week).
Supervision for the projects will be done by Prof. Dr. Herman Geuvers
and Dr. F. Wiedijk
Postdoc and PhD student will be appointed by the Radboud University
Nijmegen. Both positions shall start before October 1 2009, but
preferably earlier.
The salary for the PhD position starts at 2042 Euro per month,
increasing to 2612 Euro per month in the fourth year.
The maximum salary for the Postdoc is 3755 Euro per month (salary scale
10).
-----------------------------------------------------------------
Information:
For more information, see http://www.fnds.cs.ru.nl/fndswiki/Vacancies.
For inquiries about the project and its positions, please contact the
project leader Prof. Dr. Herman Geuvers (H.Geuvers@cs.ru.nl, +31
243652603). Interested candidates can ask the project leader for the
complete project application text.
-----------------------------------------------------------------
Application:
Deadline for application is May 1, 2009.
Send an application letter with CV and 3 references, mentioning the
vacancy number by e-mail to
RU Nijmegen, FNWI, P&O
mrs. D. Reinders
Postbus 9010
6500 GL Nijmegen
Netherlands
e-mail: pz@science.ru.nl
telephone: +31 243652764
From rrosebru@mta.ca Fri Apr 10 09:46:17 2009 -0300
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for categories-list@mta.ca; Fri, 10 Apr 2009 09:43:28 -0300
Date: Thu, 09 Apr 2009 18:19:47 -0500 (CDT)
From: zackluo@j4.com
Subject: categories: Monads over a subcategory
To: categories
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There is yet another way to see logic monoid => monad.
Definition. Let N be a full subcategory of a category G. A monad (or clone) over N is a pair (K, T) where K is a category with Ob K = Ob N and T is a functor T: K -> G such that for any A, B, C in N
(i) K(A, B) = G(A, TB).
(ii) f(Tg) = fg for any f in K(A, B) and g in K(B, C) (the order of composition is from left to right).
A monad over a category G (in the usual sense) is a monad over the subcategory N = G of G, with K as the Kleisli category and T the right adjoint of the adjunction. A monoid is simply a monad over a singleton (as a subcategory of the category of sets).
Examples: Let G = Set be the category of sets.
1. A clone over a singleton is (equivalent to) a monoid.
2. A clone over a finite set is a unitary Menger algebra.
3. A clone over a countably infinite set is simply called a clone.
4. A clone over the subcategory of finite sets is a clone in the classical sense (or a Lawvere theory), which corresponds to a locally finitary clone in the sense of 3 above.
5. A clone over a one-object category is a Kleisli algebra in the sense of E. Manes.
For a monad the left algebras (Eilenberg-Moore algebras) represent the smantics (model) and right algebras represent the syntax (logic) of the monad. The theory of right algebras, which is missing from the classical approach to monads, may be applied to study mathematical logic, lambda calculus and recursion theory effectively.
From: Clones and Genoids (http://www.algebraic.net/cag/)
Zhaohua Luo
From rrosebru@mta.ca Sun Apr 12 10:35:25 2009 -0300
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Message-Id:
From: Thorsten Altenkirch
To: Steve Lack ,
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Subject: categories: Re: Where does the term monad come from?
Date: Sat, 11 Apr 2009 16:43:13 +0100
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Hi Steve,
thank you for addressing the other part of my question.
> There was also a second part to the question:
>
>>
>> Btw, I frequently encounter monads in a categories of functors which
>> are not endofunctors. An example are finite dimensional vectorspaces
>> which can be constructed via a monoid in the category of functors
>> FinSet -> Set, here I is the embedding and (x) can be constructed
>> from
>> the left kan extension and composition.
>> The unit is given by the Kronecker delta and join can be constructed
>> from Matrix multiplication. Should one call these beasts monads as
>> well? Is there a good reference for this type of construction?
>
> The category of functors from FinSet to Set is equivalent to the
> category
> of endofunctors of Set which preserve filtered colimits: such
> endofunctors
> are usually called finitary. Thus a monoid in [FinSet,Set] with
> respect to
> this tensor product is the same thing as a monad on Set whose
> endofunctor
> part is finitary: this is called a finitary monad.
>
> These finitary monads on Set are equivalent to Lawvere theories and
> so in
> turn to (finitary, single-sorted) varieties.
>
> Finitary monads can also be considered on other base categories than
> Set,
> especially on locally finitely presentable ones.
>
> It is true that vector spaces are the algebras for a finitary monad
> on Set.
> There is no need to restrict to finite-dimensional vector spaces; in
> fact it
> is not true that there is a monad on Set whose algebras are the
> finite-dimensional vector spaces.
I am not sure I completely understand your comments. I guess it may be
helpful to be more precise:
F : FinSet -> Set
F A = Real -> A
together with:
>
eta_A : A -> F A
eta a = \ b . if a=b then 1 else 0
(>>=) : F A -> (A -> F B) -> F B
v >>= f = \ b. \Sigma a:A.(v a)*(f a b)
My notation is inspired by functional programming and naturally as a
Computer Scientist I am interested in the constructive content of
theorems. This construction only works if the input is decidable
(needed for eta) and if we can define Sigma (this certainly works if A
is finite).
I can see how to lift F to a functor on Sets by using a Kan extension
(left ?). In my terminology it may be something like
F' : Set -> Set
F' X = Sigma A:FinSet. A -> X x F A
I suspect my eta and >>= give then rise to a monad on Set? However, I
don't see how to do this if the vector spaces are not finite.
Btw, I only used this as an example. My question was rather wether
people have studied monoids in categories of functors which are not
endofunctors. I believe this notion is useful in functional
programming and Type Theory as a natural generalisation of the notion
of a monad.
Cheers,
Thorsten
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Date: Sun, 12 Apr 2009 11:30:45 +1000
Subject: categories: Re: Where does the term monad come from?
From: Steve Lack
To: Thorsten Altenkirch ,
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Dear Thorsten,
I'm not familiar with the notation that you are using, although I can
guess what is meant in some cases
>
> I am not sure I completely understand your comments. I guess it may be
> helpful to be more precise:
>
> F : FinSet -> Set
> F A = Real -> A
I assume you mean A->Real. It's true that the monad for vector spaces sends
a finite set A to R^A, which can be seen as the set of functions from A to
R.
For a general set A (not necessarily finite) FA is the set of functions from
A to R of finite support. Equivalently, FA is the set of formal finite
linear combinations of elements of A.
> I suspect my eta and >>= give then rise to a monad on Set? However, I
> don't see how to do this if the vector spaces are not finite.
Yes, this gives a monad on Set whose algebras are vector spaces, not
necessarily finite dimensional. I'm not sure what it is you claim to be
doing when you "do this". In any case there is a monad on Set whose
algebras are vector spaces; there is not a monad on Set whose algebras are
finite dimensional vector spaces. You can see this last statement by noting
that the category of algebras for a monad on Set is always cocomplete.
>
> Btw, I only used this as an example. My question was rather wether
> people have studied monoids in categories of functors which are not
> endofunctors. I believe this notion is useful in functional
> programming and Type Theory as a natural generalisation of the notion
> of a monad.
>
Yes, monoids in categories of functors are useful concepts. Of course to
define a monoid you need a monoidal structure on the ambient category. There
may be many possibilities, and for some of them the corresponding notion of
monoid looks more like a monad than for others. For some monoidal structures
one should really think of the monoids as not generalizations of monads, but
special cases of monads. Your example of finitary monads is a good example.
So are operads. There are more examples in the paper "notions of Lawvere
theory" available from my home page or as arXiv:0810.2578.
Regards,
Steve Lack.
From rrosebru@mta.ca Mon Apr 13 21:38:17 2009 -0300
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Date: Mon, 13 Apr 2009 11:35:07 -0400
From: "Fred E.J. Linton"
To:
Subject: categories: Monoids in functor categories [was: Re: Where does the term monad come from?]
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On Sun, 12 Apr 2009 09:38:18 AM EDT, Thorsten Altenkirch =
, writing to Steve Lack =
and , asked, inter alia: =
> > ... [snip] ...
> =
> ... My question was rather wether
> people have studied monoids in categories of functors which are not
> endofunctors. ... [snip again] ...
> =
Yes, they have. Two quick examples of such categories, =
and what monoids in them boil down to:
1) simplicial sets (cf. Gabriel-Zisman or D.M. Kan for
just how this is a functor category) -- here monoids =
are "simplicial monoids." Of particular interest, =
of course: simplicial groups.
2) modules over a fixed (commutative, say) ring R (a
functor category of the form [R, Ab] from the one-object
additive category R to the category Ab of abelian groups,
consisting of course only of the additive functors) --
here monoids boil down to R-algebras (what in the older
van der Waerden terminology were called hypercomplex
systems over R).
In each instance, of course, one must be careful to
specify correctly just which "product" bifunctor on
the functor category is to be used in the definition
of "monoid" -- in case (2) it's not the usual =
(cartesian) product but, rather, a suitable tensor =
product one that one wants to be using.
I'll let others provide other illustrative examples.
Cheers, -- Fred
From rrosebru@mta.ca Wed Apr 15 09:25:59 2009 -0300
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Date: Wed, 15 Apr 2009 11:35:40 +0200
From: Jaap van Oosten
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To: categories@mta.ca
Subject: categories: PhD vacancy at Nijmegen
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I forward the message below from Klaas Landsman.
==============================================================
I have a vacancy for a PhD student on the project "Topos theory,
noncommutative geometry, and quantum logic", starting at a negotiable
date in 2009.
A detailed description of the project is available at
>. The position is for
four years
(with an initial one-year appointment subject to extension to four years
on satisfactory performance), and is financed by NWO. PhD positions in
the Netherlands carry a decent salary (going up from about 2000 euro per
month gross initially to about 3000 euro in the final year), and the
local group of PhD students in mathematics and mathematical physics is
exceptionally nice. Moreover, the mathematical physics group at Nijmegen
is embedded in the GQT-cluster (see >),
officially called "The Fellowship of Geometry and Quantum Theory."
Relevant to this project, among its other members
are Ieke Moerdijk and Gunther Cornelissen.
Applications are welcomed by May 15, 2009, using snail mail (accompanied
by a very brief email to
to announce your application) to Prof.dr. N.P. Landsman, Radboud
Universiteit Nijmegen, Faculty of Science,
IMAPP, Heyendaalseweg 135, 6525 AJ NIJMEGEN, THE NETHERLANDS. Please
include a letter of motivation, a copy or draft
of of your M.Sc. Thesis, a list of courses (including marks), and the
names and email addresses of two or three senior
academics who could provide information about yourself. Please do not
ask them to send references yourself.
Female mathematicians or theoretical physicists (with a strong
background in mathematics) are especially encouraged to apply.
From rrosebru@mta.ca Wed Apr 15 09:25:59 2009 -0300
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for categories-list@mta.ca; Wed, 15 Apr 2009 09:22:57 -0300
Date: Tue, 14 Apr 2009 18:53:37 -0400 (EDT)
From: larry moss
To: larry moss
Subject: categories: Workshop on Quantum Logic Insipred by Quantum Computation
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There will be an informal workshop on
Quantum Logic Insipred by Quantum Computation
at Indiana University, May 11-12.
Our aim is to organize a small workshop that would bring together people
who are developing new areas of logic coming from quantum computation, and
also people who are interested in related projects coming from areas of
philosophical logic, mathematics, and theoretical computer science.
Information on speakers may be found at www.indiana.edu/~iulg/qliqc.
There is no formal registration and all are welcome, but we would
appreciate knowing ahead if you plan to come.
From rrosebru@mta.ca Wed Apr 15 14:53:59 2009 -0300
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for categories-list@mta.ca; Wed, 15 Apr 2009 14:52:19 -0300
From: Hasse Riemann
To:
Subject: categories: Smooth and proper functors
Date: Wed, 15 Apr 2009 13:45:06 +0000
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=20
Hi category gurus and categorists
=20
I have many questions about category theory but i start with one.
=20
1>
What are smooth functors and proper functors=2C originating in pursuing sta=
cks?
Both nontechnically and technicaly.
=20
I know they are dual to each other and that they are characterized by cohom=
ological properties
inspired by the proper or smooth base change theorem in algebraic geometry=
=2C but what is the relation?
(I don't know the statement of the theorems)
=20
Finally=2C what are smooth and proper functors good for?
Are smooth and proper functors fibrations and cofibrations or Grothendieck =
fibrations and
Grothendieck op-fibrations in some model categories or derivators?
=20
The only thing i could find about smooth and proper functors on internet is=
the last entrance in
http://golem.ph.utexas.edu/category/2008/01/geometric_representation_theor_=
18.html
=20
Best regards
Rafael Borowiecki
From rrosebru@mta.ca Thu Apr 16 11:11:10 2009 -0300
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From: =?UTF-8?Q?Jonathan_CHICHE_=E9=BD=90=E6=AD=A3=E8=88=AA?=
Subject: categories: Re: Smooth and proper functors
Date: Thu, 16 Apr 2009 15:46:53 +0200
To: Hasse Riemann ,
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Hi,
The following paper is very clear, I'm currently learning the basics =20
of the subject with it: http://people.math.jussieu.fr/~maltsin/ps/=20
asphbl.ps. It's written in French. Another member of this mailing-=20
list has asked me to translate it in English, I may be able to send =20
you a rough translation in a few weeks.
Best,
Jonathan
Le 15 avr. 09 =E0 15:45, Hasse Riemann a =E9crit :
> Hi category gurus and categorists
>
>
>
> I have many questions about category theory but i start with one.
>
>
>
> 1>
>
> What are smooth functors and proper functors, originating in =20
> pursuing stacks?
>
> Both nontechnically and technicaly.
>
>
>
> I know they are dual to each other and that they are characterized =20
> by cohomological properties
>
> inspired by the proper or smooth base change theorem in algebraic =20
> geometry, but what is the relation?
>
> (I don't know the statement of the theorems)
>
>
>
> Finally, what are smooth and proper functors good for?
>
> Are smooth and proper functors fibrations and cofibrations or =20
> Grothendieck fibrations and
>
> Grothendieck op-fibrations in some model categories or derivators?
>
>
>
> The only thing i could find about smooth and proper functors on =20
> internet is the last entrance in
> http://golem.ph.utexas.edu/category/2008/01/=20
> geometric_representation_theor_18.html
>
>
>
> Best regards
>
> Rafael Borowiecki
From rrosebru@mta.ca Thu Apr 16 11:11:11 2009 -0300
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for categories-list@mta.ca; Thu, 16 Apr 2009 11:08:24 -0300
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Date: Wed, 15 Apr 2009 19:44:05 +0100
Subject: categories: Re: Smooth and proper functors
From: Andreas Holmstrom
To: Hasse Riemann , categories@mta.ca
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Hi Rafael,
I don't know much about this, but I listened to an excellent talk of
Maltsiniotis a few months ago at IHES and posted the scanned notes in
a blog post here:
http://homotopical.wordpress.com/2009/01/26/maltsinotis-grothendieck-and-homotopical-algebra/
These notes (on page 11-12) contain at least the definition of proper
and smooth functors, and the duality statement, so maybe they can be
of some limited use. Hopefully other people on this list can provide
some more substantial information.
Best regards,
Andreas Holmstrom
2009/4/15 Hasse Riemann :
>
>
>
> Hi category gurus and categorists
>
>
>
> I have many questions about category theory but i start with one.
>
>
>
> 1>
>
> What are smooth functors and proper functors, originating in pursuing stacks?
>
> Both nontechnically and technicaly.
>
>
>
> I know they are dual to each other and that they are characterized by cohomological properties
>
> inspired by the proper or smooth base change theorem in algebraic geometry, but what is the relation?
>
> (I don't know the statement of the theorems)
>
>
>
> Finally, what are smooth and proper functors good for?
>
> Are smooth and proper functors fibrations and cofibrations or Grothendieck fibrations and
>
> Grothendieck op-fibrations in some model categories or derivators?
>
>
>
> The only thing i could find about smooth and proper functors on internet is the last entrance in
> http://golem.ph.utexas.edu/category/2008/01/geometric_representation_theor_18.html
>
>
>
> Best regards
>
> Rafael Borowiecki
>
>
>
From rrosebru@mta.ca Thu Apr 16 11:11:23 2009 -0300
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for categories-list@mta.ca; Thu, 16 Apr 2009 11:11:19 -0300
Date: Wed, 15 Apr 2009 19:35:34 +0000 (GMT)
From: Pierre Cardascia
Subject: categories: A theorem from Herrlich and Strecker
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Dear Cat=E9goristes,=0A=0AI'm working on an introduction for categorical lo=
gic, and I try to avoid using the notion of limit before introducing the no=
tion of functor in my work (because limit means limit of functors. Squarely=
, we can introduce the limit before, but we don't understand why the limit =
is called limit, and limit of what ??).=0ABut I have to introduce the notio=
n of categories finitely complete. SO I think about this theorem :=0A=3D=3D=
=3D> If C has a terminal object, and a pullback for each pair of arrows wit=
h common codomains, then C is finitively complete.=0AI found that without a=
ny proof in Goldblatt. Rob Goldblatt just said : "you can find it into such=
book from Herrlich and Strecker"... Does somebody has the proof ? Can I us=
e this theorem to define complety closed categories instead of working with=
limits ? Or does somebody have any way to define complety closed categorie=
s without any reference to functors ?=0A=0AThanks !=0A=0APierre CARDASCIA=
=0A=0A=0A
From rrosebru@mta.ca Thu Apr 16 20:30:55 2009 -0300
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Date: Thu, 16 Apr 2009 16:22:15 +0100 (BST)
From: Jocelyn Paine
To: categories@mta.ca
Subject: categories: Graphical Category Theory Demonstrations
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