canonically canonical
Robert Scott
>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
Robert Scott
>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
Robert Scott
>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
Robert Scott
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
Matthew P Wiener
>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)
Jozsef Ferincz
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
John C. Baez
>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.
John C. Baez
>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.
Robert Scott
>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
John C. Baez
>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.
Robert Scott
>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
Robert Scott
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
Angus H Rodgers
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
Angus H Rodgers
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!)