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Aug 30, 1993, 3:48:57â€¯PM8/30/93

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john baez writes:

>This is probably not hard but whenever I want to think about it I get an

>attack of dizziness and the desire fades.

.

.

.

>Maybe some concrete examples of 3 situations would be nice. First,

>natural transformations between 2 functors that are *unique*. Second,

>ones that aren't. And third, one's that aren't, but have some nice

>"good-as-unique" property.

i don't think you've really given us enough to go on here to tell if

you're on to something or if it's just your category-theoretic intuition

malfunctioning.

there is a general class of properties (called "coherence properties",

or at least that's what they used to be called) that natural

isomorphisms can possess in certain contexts, that have to do with

there being fewer different competing such isomorphisms than there

might otherwise have been, by virtue of certain equations holding

between potentially distinct such isomorphisms. (the "pentagonal

coherence" condition for a natural associativity is such a coherence

property, for example.) have you gotten any replies suggesting that

such "coherence properties" might be related to whatever it is that you

are trying to get at?

however, in the example that you mention of a natural isomorphism from

the identity functor of a category c to an endo-functor f:c->c

(specifically, the case where c is the category of finite-dimensional

vector spaces over a field k, and f is the "double dual" functor), there

just doesn't seem to be any room for any extra coherence properties to

be satisfied; the generic case of a natural isomorphism from an identity

functor to another endo-functor is already as coherent as it can get.

perhaps there is some extra structure lying around that could be used to

formulate some non-trivial coherence property, but this really doesn't

sound promising.

of course, one rather trivial way to give a reasonable interpretation

to the idea of "naturally natural" is to have a natural morphism

between functors of two variables. (i will use the terminology

"natural" instead of "canonical".) such a morphism is "naturally

natural", as compared to a natural morphism between functors of one

variable, which is merely "natural". i doubt this is what you're

looking for, though.

as i think matt wiener pointed out, mere non-uniqueness of natural

isomorphisms from a functor f:x->y to a functor g:x->y is so common

that it would be difficult to ever think of it as being pathological,

one classic example being the case of the identity functor of the

category of modules over a commutative ring r, which has one natural

automorphism for each invertible element of r. of course there is one

very special natural automorphism of the identity functor that stands

apart from all of the others: namely the identity automorphism.

obviously the identity natural automorphism is in some intuitive sense

more like "canonical equality" than just some plain ordinary natural

isomorphism is, but of course this idea is of no use when you are

dealing with natural automorphisms from the identity functor to some

other functor, such as the double dual functor.

-james dolan

Aug 31, 1993, 2:28:05â€¯AM8/31/93

to

i wrote:

>i don't think you've really given us enough to go on here to tell if

>you're on to something or if it's just your category-theoretic intuition

>malfunctioning.

hmm, perhaps i overlooked one very interesting and surprisingly simple

way in which the natural embedding of a module of a commutative ring

into its double dual can be seen as a "naturally natural morphism":

consider the 3-category of all 2-categories. then the assignment to [any

module of any commutative ring] of [its standard embedding into its

double dual] appears as a 3-cell in this 3-category,as follows:

first we have two 0-cells a,b in the 3-category:

1. a is the opposite of the category of all commutative rings,

elevated into a 2-category by taking as the 2-cells just the identity

2-cells of all of the 1-cells.

2. b is the 2-category of all categories.

then we have a 1-cell c:a->b :

c is the 2-functor which assigns to each commutative ring the category

of all of its modules, and which assigns to each homomorphism of

commutative rings the corresponding "restriction of scalars" functor

between the corresponding module categories. (this is a contravariant

assignment, but that agrees with the fact that we chose a to be the

_opposite_ of the usual category of commutative rings.)

then we have a pair d,e:c->c of endo-2-cells of the 1-cell c:

1. d is the identity 2-cell of the 1-cell c; thus it is the "natural

functor" assigning to each commutative ring the identity endo-functor

of the category of all of its modules.

2. e is the "natural functor" assigning to each commutative ring the

"double dual" endo-functor of the category of all of its modules.

then finally we have a 3-cell f:d->e :

f is the "natural natural morphism" assigning to each commutative

ring the natural morphism from the identity functor of its category of

modules to the double dual functor of its category of modules.

a few comments:

1. i checked a lot, but not all, of the details that need to be

checked to make sure that this all actually works.

2. if you are worried that my usage of constructions like "the

3-category of all 2-categories" sounds dangerously close to

problematic constructions such as "the set of all sets", don't worry

about it. we category-theorists are a very peace-loving people with

no desire to cause any foundational crisis that might cause the

edifice of mathematics to crash down upon us all, and that gives us a

special license to commit all sorts of set-theoretic mayhem that

would be off-limits to the less pure of heart. (seriously, there are

always ways around this sort of difficulty.)

3. i called the 2-cells d,e in the 3-category of all 2-categories

"natural functors", and i called the 3-cell f in this 3-category a

"natural natural morphism", but i just made that terminology up on

the spur of the moment, partially in response to john baez's original

question about "canonically canonical morphisms". i think that this

terminology is somewhat appropriate in this particular situation,

though it is far from systematic. i don't know if there is any

standard systematic terminology for "k-cell in the [n+1]-category of

all n-categories" other than "k-cell in the [n+1]-category of all

n-categories", though i have sometimes played around with inventing

such a systematic terminology.

-james dolan

Aug 31, 1993, 9:06:41â€¯AM8/31/93

to

i wrote:

>consider the 3-category of all 2-categories. then the assignment to [any

>module of any commutative ring] of [its standard embedding into its

>double dual] appears as a 3-cell in this 3-category,as follows:

hmm, i'm going to have to withdraw this claim for the moment (though

there may be some way to salvage the situation). the problem is that

upon closer examination i haven't yet found any way to see "double

dual" as a "natural functor", since it doesn't "commute" with

"restriction of scalars".

-james dolan

Aug 31, 1993, 9:38:11â€¯AM8/31/93

to

i wrote:

well, here is one very simple quick fix, obtained by lowering ambitions

somewhat: simply restrict consideration to only the invertible ring

homomorphisms. in the special case where the ring homomorphism is

invertible, "double dual" _does_ commute with "restriction of scalars".

-james dolan

Aug 31, 1993, 8:55:48â€¯AM8/31/93

to

In article <25ur1l$5...@max.physics.sunysb.edu>, rscott@ic (Robert Scott) writes:

>hmm, perhaps i overlooked one very interesting and surprisingly simple

>way in which the natural embedding of a module of a commutative ring

>into its double dual can be seen as a "naturally natural morphism":

>hmm, perhaps i overlooked one very interesting and surprisingly simple

>way in which the natural embedding of a module of a commutative ring

>into its double dual can be seen as a "naturally natural morphism":

You haven't overlooked anything. John was asking for "canonically

canonical", not "naturally natural".

>[...]

>then finally we have a 3-cell f:d->e :

> f is the "natural natural morphism" assigning to each commutative

> ring the natural morphism from the identity functor of its category of

> modules to the double dual functor of its category of modules.

Note that -f is also a natural natural morphism.

--

-Matthew P Wiener (wee...@sagi.wistar.upenn.edu)

Aug 31, 1993, 6:05:42â€¯PM8/31/93

to

If you feel like to continue this discussion with some practical

use, please contact me. I'm a chemist (do not underestimate the

problem!) and I have a - for me - big problem with the canonical

labelling.

Regards,

Jozsef

--

Jozsef Ferincz INTERNET : fer...@chemie.fu-berlin.de

Freie Universitaet Berlin UUCP : fer...@fub.uucp

Institut fuer Organische Chemie Tel.: (+49 30) 838-2677, 838-5363

Takustr. 3, D-14195 Berlin Fax : (+49 30) 838-5163, 838-4248

Aug 31, 1993, 11:33:58â€¯PM8/31/93

to

In article <25tlj9$p...@max.physics.sunysb.edu> rsc...@ic.sunysb.edu (Robert Scott) writes:

>john baez writes:

>

>>This is probably not hard but whenever I want to think about it I get an

>>attack of dizziness and the desire fades.

>.

>.

>.

>>Maybe some concrete examples of 3 situations would be nice. First,

>>natural transformations between 2 functors that are *unique*. Second,

>>ones that aren't. And third, one's that aren't, but have some nice

>>"good-as-unique" property.

>john baez writes:

>

>>This is probably not hard but whenever I want to think about it I get an

>>attack of dizziness and the desire fades.

>.

>.

>.

>>Maybe some concrete examples of 3 situations would be nice. First,

>>natural transformations between 2 functors that are *unique*. Second,

>>ones that aren't. And third, one's that aren't, but have some nice

>>"good-as-unique" property.

>i don't think you've really given us enough to go on here to tell if

>you're on to something or if it's just your category-theoretic intuition

>malfunctioning.

Well, since I possess no such intuition, it can't be malfunctioning.

:-)

>there is a general class of properties (called "coherence properties",

>or at least that's what they used to be called) that natural

>isomorphisms can possess in certain contexts, that have to do with

>there being fewer different competing such isomorphisms than there

>might otherwise have been, by virtue of certain equations holding

>between potentially distinct such isomorphisms. (the "pentagonal

>coherence" condition for a natural associativity is such a coherence

>property, for example.) have you gotten any replies suggesting that

>such "coherence properties" might be related to whatever it is that you

>are trying to get at?

I believe Matt Wiener's reply referred to coherence conditions and other

nice things in "Categories for the Working Mathematician." Certainly I

know a bit about how these conditions play a role in braided monoidal

categories, and how in braided monoidal 2-categories the commutativity

of the pentagon diagram is weakened to the existence of a 2-isomorphism

between the two different composites of morphisms occuring in that

diagram, with this 2-morphism itself satisfying a coherence condition.

(I found Kapranov and Voevodsky's paper again after you left, by the

way.)

>however, in the example that you mention of a natural isomorphism from

>the identity functor of a category c to an endo-functor f:c->c

>(specifically, the case where c is the category of finite-dimensional

>vector spaces over a field k, and f is the "double dual" functor), there

>just doesn't seem to be any room for any extra coherence properties to

>be satisfied; the generic case of a natural isomorphism from an identity

>functor to another endo-functor is already as coherent as it can get.

>perhaps there is some extra structure lying around that could be used to

>formulate some non-trivial coherence property, but this really doesn't

>sound promising.

>

>of course, one rather trivial way to give a reasonable interpretation

>to the idea of "naturally natural" is to have a natural morphism

>between functors of two variables. (i will use the terminology

>"natural" instead of "canonical".) such a morphism is "naturally

>natural", as compared to a natural morphism between functors of one

>variable, which is merely "natural". i doubt this is what you're

>looking for, though.

No. Of course I'm just vaguely fishing around here, but that doesn't

sound like the kind of fish I'm dreaming of. Maybe there's no such

fish. Maybe however this might work. The natural transformation

between the identity functor and the double dual functor doesn't just

exist in the context of the category Vec, it works in all sorts of

categories. (Maybe compact categories is the right class; I always

forget the name for monoidal categores that have a nice notion of dual,

and I forget the nuances of right and left duals.) Given a category C

with the necessary structure to play this game, let me write T(C) for

the natural transformation from the identity to the double dual.

One might ask for some coherence condition expressing the fact that all

these T(C)'s fit together nicely. That is, if we have categories C and

C', and a functor from C to C', T(C) and T(C') are related in some nice

way I'm too tired to conceive of. Whereas some of the other natural

transformations between the identity functor and the double dual that

make sense for Vec might not make sense at all for other Cs, because

those Cs might never have heard of "multiplying by a scalar".

Let me attempt to appeal to your prejudices, nudging you in the ribs and

saying that "OBVIOUSLY" the standard natural transformation between the

identity and double dual in Vec is "more natural" (in the nontechnical

sense) than all the other ones. If one believes this, one should be

able to formalize it.

Aug 31, 1993, 11:45:01â€¯PM8/31/93

to

In article <25ur1l$5...@max.physics.sunysb.edu> rsc...@ic.sunysb.edu (Robert Scott) writes:

>consider the 3-category of all 2-categories.

> 2. if you are worried that my usage of constructions like "the

> 3-category of all 2-categories" sounds dangerously close to

> problematic constructions such as "the set of all sets", don't worry

> about it. we category-theorists are a very peace-loving people with

> no desire to cause any foundational crisis that might cause the

> edifice of mathematics to crash down upon us all, and that gives us a

> special license to commit all sorts of set-theoretic mayhem that

> would be off-limits to the less pure of heart.

Such foundational issues are the least of my worries at present; I'm

much more worried about how my brain overheatead and crashed when I

contemplated your construction. I'll try again after a good night's

rest.

Aug 31, 1993, 11:34:30â€¯PM8/31/93

to

matt wiener writes:

>You haven't overlooked anything. John was asking for "canonically

>canonical", not "naturally natural".

...

>Note that -f is also a natural natural morphism.

yes, to the same extent that f itself is. (that is, modulo the blunder

that i committed and then hopefully repaired.)

are you assuming that john actually knows what john wants?

there's not really a whole big difference between "canonicalness" and

"naturalness". canonicalness is the special case of naturalness where

the naturalness is with respect to the invertible morphisms only.

(notice that in fact that's how i managed to get f to be naturally

natural: by settling for the special case where "naturally natural"

really means just "canonically natural"; that is, the case of invertible

ring homomorphisms.) or to put it the other way round, naturalness is

a "constructive" generalization of canonicalness, in roughly the same

sense that, for example, topos theory is a constructive generalization

of classical logic.

i don't know whether we've succeeded in communicating any actual ideas

to anybody with this thread, but i'll bet we've succeeded in scaring

john.

-james dolan

Sep 1, 1993, 10:41:01â€¯AM9/1/93

to

In article <261586$j...@max.physics.sunysb.edu> rsc...@ic.sunysb.edu (Robert Scott) writes:

>matt wiener writes:

>

>>You haven't overlooked anything. John was asking for "canonically

>>canonical", not "naturally natural".

>matt wiener writes:

>

>>You haven't overlooked anything. John was asking for "canonically

>>canonical", not "naturally natural".

>are you assuming that john actually knows what john wants?

I'm afraid he *is* making that fatal blunder. If I knew what I wanted I

would probably already have it.

>

>there's not really a whole big difference between "canonicalness" and

>"naturalness". canonicalness is the special case of naturalness where

>the naturalness is with respect to the invertible morphisms only.

>(notice that in fact that's how i managed to get f to be naturally

>natural: by settling for the special case where "naturally natural"

>really means just "canonically natural"; that is, the case of invertible

>ring homomorphisms.) or to put it the other way round, naturalness is

>a "constructive" generalization of canonicalness, in roughly the same

>sense that, for example, topos theory is a constructive generalization

>of classical logic.

>

>i don't know whether we've succeeded in communicating any actual ideas

>to anybody with this thread, but i'll bet we've succeeded in scaring

>john.

Watching category theorists quarrel is in itself a most edifying

spectacle, but I may have obtained a few ideas as well. For one, I had

never known there was a technical distinction between "natural" and

"canonical." Working mathematicians seem to use these terms pretty

much interchangeably. I'm not sure I get the precise scope of these

terms even now. Are these terms that live at the level of functors,

natural (canonical?) transformations, or n-morphisms in general. Feel

free to go tell me to read CFTWM.

I also found Jim Dolan's construction quite charming, and I think I may

be able, with a half-hour of work sometime, to really understand the

notion of the unit of an adjunct.

I'm certainly no more scared than I ever was. In fact, most of my

professed fear of category theory is really just a way to keep from

feeling guilty that I haven't put in enough time studying it.

Sep 3, 1993, 3:59:31â€¯AM9/3/93

to

john baez writes:

>Watching category theorists quarrel is in itself a most edifying

>spectacle, but I may have obtained a few ideas as well. For one, I had

>never known there was a technical distinction between "natural" and

>"canonical." Working mathematicians seem to use these terms pretty

>much interchangeably. I'm not sure I get the precise scope of these

actually i don't know whether anybody else besides me likes to think of

"canonical" as meaning "natural with respect to _invertible_

morphisms". matt wiener seems to agree with me that "canonical" and

"natural" are not synonymous, though matt's position may be just that

the difference between them is that "canonical" is a colloquialism of

rather flexible meaning whereas "natural" has a precise technical

meaning- or at least it does nowadays; don't forget that it too was

just a colloquialism at one time, and that it is still arguable whether

maclane's formalization of the concept really captured the essence of

the colloquial use.

but let me give a few examples to try to show the difference between

"canonical" and "natural" in my sense, and why i think my usage of

"canonical" is appropriate terminology.

for example, let's consider the question of what are all of the

"canonical maps from a set to itself". one obvious map from any set s

to itself is the identity map of s. there is however one other

slightly peculiar canonical map (let's call it c(s)) from s to itself,

which is defined as follows: if the cardinality of s is two, then c(s)

maps each element of s to the other element; otherwise c(s) maps every

element to itself.

while the assignment "s |-> identity map of s" is clearly both

canonical and natural, the assignment "s |-> c(s)" is only canonical.

if you carefully analyze why it fails to be natural, you'll see that it

has a lot to do with the intuitively "non-constructive" nature of its

definition. that is, from a "constructive" viewpoint, conclusions that

you reach about the cardinality of a set are only tentative, and

therefore you can't be sure which clause of the definition of c(s) is

the one that you should use.

(the reason that conclusions about the cardinality of a set are

tentative from a constructive point of view is that constructivists see

a set as something that is not handed to you once and for all, but

rather as something about which you may learn more as time goes on.

sometimes you learn that things that you used to think of as different

are really the same (the morning-star is really the evening-star, clark

kent is really superman), while other times you learn about the

existence of things that you didn't know existed (uranus, neptune,

pluto, bizarro). thus new discoveries can "change" the cardinality of

a set, and thus cardinality is a very tentative property.)

it's sometimes said that the "natural" maps are just "the ones that you

can think of (without taking advantage of any special knowledge about

the object whose points are being mapped)". but this is really more

accurately a description of the "canonical" maps, not the "natural"

maps. i, for example, have absolutely no trouble thinking of the map

c(s). in order to be unable to think of any but the natural maps, you

would have to not only not have any special knowledge about the object

whose points are being mapped, but also be a mental cripple like errett

bishop.

consider another example: consider the category of finite totally

ordered sets. this is a monoidal category under the operation of

"concatenation". that is, the concatenation of a pair of finite

totally ordered sets is a new finite totally ordered set, and this

concatenation operation has all of the nice properties (functoriality,

natural associativity, existence of a unit object, and so forth) needed

to form a monoidal category. now consider this question: is this

monoidal category "commutative"?

well, from one point of view, the answer would seem to be yes. namely,

it is certainly true that given finite totally ordered sets x and y,

the concatenation x+y is isomorphic to the concatenation y+x.

furthermore, the isomorphism between x+y and y+x is _absolutely_

unique. thus, what could possibly be more canonical than that? and

indeed, the commutativity isomorphisms x+y = y+x is entirely canonical-

but it is _not_ natural, as you can check for yourself. this is a very

good example to think about if you want to explore the relationship

between the precise technical concept of naturality as it exists today,

and whatever intuitive concepts that may have motivated and preceded

it.

[to be continued, perhaps]

-james dolan

Sep 4, 1993, 12:55:19â€¯PM9/4/93

to

i may have gone a bit overboard in my last post taking the intuitive

approach to understanding canonicalness and naturalness without

sufficiently spelling out some of the technical details, so i will try

to remedy that somewhat in this post.

approach to understanding canonicalness and naturalness without

sufficiently spelling out some of the technical details, so i will try

to remedy that somewhat in this post.

(i'm not sure what's the best level of technical preparedness to assume

of whatever audience there may be here, but more or less arbitrarily i

will assume as a prerequisite the knowledge of what a category is, and

what a functor is (namely, a "homomorphism of categories", more or

less), but not much idea of what you can do with such things. i guess

that as usual the best reference book to fall back on for this stuff is

"categories for the working mathematician" by maclane.)

a family of morphisms

{n(x) | x is an object of c}

in a category d, indexed by the objects of another category c, is said

to form a "natural transformation from the functor f:c->d to the

functor g:c->d" iff:

1. for each object x in c, n(x) is a morphism from f(x) to g(x),

and:

2. for each morphism m:x->y in c, the following "naturality diagram"

commutes:

n(x)

f(x) -------> g(x)

| |

f(m) | | g(m)

\|/ \|/

f(y) -------> g(y)

n(y) .

now, i would like to convince you that the above definition is not just

a random profusion of arrows flying every which way, but rather

actually captures a very interesting intuitive notion of what it means

for a morphism n(x) (depending on an object parameter x) to be

"natural", as for example when we say that the standard isomorphism

from a finite-dimensional vector space x to its "double dual" x** is

"natural", or that, even though x is also always isomorphic to its

_single_ dual x*, there is no "natural" isomorphism from x to x*. i

will try to do this by working through in some detail the example that

i dealt with on an intuitive level in my last post, of "the natural

maps from a set to itself".

so, what i mean by a "natural map from a set to itself" is the special

case of a family of morphisms as described above, where the categories

c and d are both taken to be the category of sets, and the functors f

and g are both taken to be the identity functor. so in this case, the

morphism n(x) needs to be a map from the set x to itself, and the

naturality diagram for a map m:x->y is like this:

n(x)

x -------> x

| |

m | | m

\|/ \|/

y -------> y

n(y) .

now to try to see what is going on here, i want you to consider what

the above naturality diagram says in the special case where m happens

to be an invertible map. in that case, we can think of the bijection

m:x->y as being "a coordinate system for the object y in terms of the

model object x", and the naturality diagram as saying that you can

compute n(y) equally well either directly, or else via the coordinate

system (taking the alternate route around the diagram). in other

words, naturality with respect to invertible maps is really just the

same thing as "coordinate-invariance".

thus for example consider the case where for each set x, n(x) is the

map that takes each point of x to itself, _unless_ the cardinality of x

is exactly two, in which case n(x) is the map that reverses the two

points. (this is the same map that i called "c(x)" in my last post.)

then indeed, the prescription for n(x) is completely invariant under

change of coordinates, and thus n(x) is what i call a "canonical"

morphism, meaning that it is natural with respect to invertible

morphisms m:x->y.

however, n(x) is _not_ a natural morphism, because the naturality

diagram fails for some _non_-invertible morphisms m:x->y. for example,

let m be the inclusion of a two-element subset x={p,q} into a 3-element

set y={p,q,r}. then, looking at the naturality diagram for m, we can

try to evaluate the map n(y) at the point p of y either directly,

obtaining the value p, or else indirectly via the "partial coordinate

system" m, obtaining the value q (because the model object of the

partial coordinate system has cardinality two, causing the anomalous

clause of the definition of n(x) to kick in). thus, the partial

coordinate system m is fatally incomplete; it hasn't yet heard the news

that the element r has been discovered (bringing the cardinality of y

up to 3), and because of this it miscalculates the value of n(y) at p.

thus, to summarize: the concept of "canonicalness" is really

essentially just the familiar, essentially group-theoretical concept of

"coordinate-invariance" (with respect to _global_ coordinate systems).

the concept of "naturalness", which is the central concept of category

theory in the same way that coordinate-invariance is the central

concept of group theory, is the generalization in which

coordinate-invariance may be demanded not only with respect to complete

or "global" coordinate systems, but also with respect to some

collection of partial or "local" coordinate systems, namely all of the

morphisms in some category being considered. such local coordinate

systems may in general be neither injective nor surjective. the

property of being natural (or "invariant with respect to local

coordinate systems") is stronger than the property of being merely

canonical (or "invariant with respect to global coordinate systems"),

as illustrated by the example of the canonical but un-natural map n(x)

discussed above, but it is also very interestingly more "delicate", in

the sense that many different categories can share the same collection

of invertible morphisms.

[tbc,p]

-james dolan

Sep 4, 1993, 2:34:57â€¯PM9/4/93

to

In <26ah9n$r...@max.physics.sunysb.edu>

rsc...@ic.sunysb.edu (Robert Scott) writes:

rsc...@ic.sunysb.edu (Robert Scott) writes:

>[tbc,p]

[o,pd,bmg!]

>-james dolan

--

Gus Rodgers, Dept. of Computer Science, Queen Mary & Westfield College,

Mile End Road, London, England +44 71 975 5241 arod...@dcs.qmw.ac.uk

Sep 4, 1993, 2:45:43â€¯PM9/4/93

to

In <266th3$f...@max.physics.sunysb.edu>

rsc...@ic.sunysb.edu (Robert Scott) writes:

rsc...@ic.sunysb.edu (Robert Scott) writes:

>[...] a mental cripple like errett bishop. [...]

I think you mean "differently abled". :) :)

(I *knew* there had to be at least one real use for PC-speak!)

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