Re: Where does the term monad come from?

Re: Where does the term monad come from?

[John Baez]

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

Re: Where does the term monad come from?

[Steve Lack]

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.

Re: Where does the term monad come from?

[Thorsten Altenkirch]

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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Re: Where does the term monad come from?

[Zinovy Diskin]

On Fri, Apr 3, 2009 at 12:28 AM, Steve Lack <s.lack@uws.edu.au> 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

Re: Where does the term monad come from?

[jim stasheff]

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,

<snip>...

> endofunctor T: C --> C.
>
> Vaughan Pratt
>
Someone should up date the Wiki

jim

Re: Where does the term monad come from?

[Vaughan Pratt]

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

Re: Where does the term monad come from?

[RJ Wood]

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

Re: Where does the term monad come from?

[Patrik Eklund]

"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
>
>

Re: Where does the term monad come from?

[burroni]

>    * The Cup or Monad, a text in the Corpus Hermetica
> from the Wiki

Je trouverais normal que le modérateur ne permette pas cette  
intervention. C'est en partie à titre de plaisanterie et en complément  
à 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 théorèmes inédits en théorie des catégories.


Albert

Re: Where does the term monad come from?

[Steve Lack]

Dear All,

Just another quick comment about monads:


On 2/04/09 8:19 AM, "burroni@math.jussieu.fr" <burroni@math.jussieu.fr>
wrote:

> Cher Thorsten,
> 
> toutes mes excuses pour ce message en français.
> 
> Le terme "monade" a été employé par Benabou (LNM Springer no 47, si je
> ne me trompe) et dans un sens abstrait : pseudofoncteur 1 --> B de la
> bicatégorie finale 1 vers une bicatégorie arbitraire B. Par la suite
> il a été convenu de le résever au cas particulier où B=Cat (en
> remplacement du terme "triple").

Some people may reserve monad for the case B=Cat, but not all. After Benabou
demonstrated the incredible importance of this idea in various B, the theory
of monads in 2-categories/bicategories has been widely developed, starting
(I believe) with Ross Street's "Formal theory of monads".

Steve.

Re: Where does the term monad come from?

[Steve Lack]

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.

Re: Where does the term monad come from?

[jim stasheff]

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
>>
>>
>>
>>
>
>
>
>
>

Re: Where does the term monad come from?

[burroni]

Cher Thorsten,

toutes mes excuses pour ce message en français.

Le terme "monade" a été employé par Benabou (LNM Springer no 47, si je  
ne me trompe) et dans un sens abstrait : pseudofoncteur 1 --> B de la  
bicatégorie finale 1 vers une bicatégorie arbitraire B. Par la suite  
il a été convenu de le résever au cas particulier où B=Cat (en  
remplacement du terme "triple").

A mon avis, le terme est remarquable car il combine ceux de "monoides"  
et de "monades", concept utilisé par Leibnitz, mais qui, indépendement  
de l'usage fait par ce philosophe, signifie : unité simple,  
indécomposable. Cette simplicité, cette indécomposabilité est celle de  
la bicatégorie 1.

Aujourd'hui, on appelle monoide, les monades au sens général de  
Benabou. (Personnellement, je ne trouve cela imparfait car un vrai  
monoide est une structure beaucoup plus riche : exemple x |--> x^2 n'a  
pas de sens en general.)

amitiés,
Albert


Thorsten Altenkirch <txa@Cs.Nott.AC.UK> a écrit :

> 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

Re: Where does the term monad come from?

[Venanzio Capretta]

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
>
>

Re: Where does the term monad come from?

[Johannes.Huebschmann]

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
>
>
>

Re: Where does the term monad come from?

[Michael Barr]

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
>
>

Where does the term monad come from?

[Thorsten Altenkirch]

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