Property, Structure and Stuff
John Baez, Toby Bartels, James Dolan and David Corfield
In the Spring 2004 session of
the Quantum Gravity
Seminar we talked a bit about
properties, structure and stuff.
The idea in a nutshell is simple: mathematical gadgets are
defined by specifying some stuff equipped with
structure satisfying some properties.
For example, a group is a set equipped with some operations
satisfying some equations.
But James Dolan realized something very important:
these concepts can be made completely precise using category theory!
You can read how in week 1 of the
Spring 2004 seminar notes.
Toby Bartels
played a crucial role in clarifying the details, so you should
also read his
pedagogical paper about properties, structure and stuff.
Unfortunately, these writeups only scratch
the surface of a wonderful and profound subject, which has
deep links to homotopy theory.
Someday I plan to explain it all in my
book on
higher-dimensional algebra. But for now you can get the gist
by reading the following discussion, which took place on the newsgroup
sci.physics.research.
John Baez wrote parenthetically:
>I will
>leave it to James Dolan to explain the technical distinction between
>"extra properties", "extra structure", and "extra stuff" - there is
>a nice category-theoretic way of making this precise.
Ooh, let me guess!
Given a functor U: C -> D, interpret U as a forgetful functor.
Then C is D with extra *structure* if U is surjective on the objects
and, given a pair of objects, injective on the morphisms between them;
and C is D with extra *properties* if U is injective on the morphisms
(meaning injective on the objects and on the morphisms between a given pair);
Otherwise, I guess C is just D with extra *stuff*
if, given a pair of objects, U is injective on the morphisms between them.
For example, the forgetful functor Groups -> Sets
shows that groups are sets with extra structure,
while the forgetful functor Abelian Groups -> Groups
shows that Abelian groups are groups with extra properties.
Or you can turn around and use the free functor Sets -> Groups
and say that sets are groups with extra properties
(to wit, the property of being free).
OTOH, the Abelianization functor Groups -> Abelian groups
is surjective on the objects (and on the morphisms for that matter),
but groups are not Abelian groups with extra structure,
because the functor isn't injective on the morphisms between a given pair.
-- Toby
From: John Baez
Subject: Just Categories now
Date: 1998/11/17
In article <72b3pd$...@gap.cco.caltech.edu>,
Toby Bartels wrote:
>Given a functor U: C -> D, interpret U as a forgetful functor.
Of course, part of the point of this puzzle is that the term
"forgetful" is usually given no precise definition, so here we
are seeking a precise definition. Usually people don't bother
to define "forgetful functor" very precisely - like pornography,
you're just supposed to know a forgetful functor when you see it.
>Then C is D with extra *structure* if U is surjective on the objects
>and, given a pair of objects, injective on the morphisms between them [...]
I'm a little unhappy with this for two reasons: one nitpicky and one
more serious.
The nitpicky reason is that it's bad to care if a functor U: C -> D
is surjective on objects. If you think you want this, all you *really*
should want is that U be "essentially surjective". This means that
every object of D is, not necessarily equal, but *isomorphic* to an
object of the form U(x) for some object x of C.
In general, the interesting properties of functors must be preserved
by natural isomorphisms. If you have two naturally isomorphic functors
and one is essentially surjective, so is the other. This is not true
of "surjective on objects".
However, the more serious reason I'm unhappy is that sometimes C-objects
are D-objects with extra structure but *not every D-object can be made
into a C-object*. In this case U: C -> D is not essentially surjective.
For example, consider the forgetful functor U: Vect -> Set where Vect
is the set of real vector spaces. There's no way to equip a set with
2 elements with the structure of a real vector space! Thus U is not
essentially surjective.
Anyway, I think we should say that C-objects are D-objects with
extra structure if your second criterion holds: given any pair of
objects x,y in C,
U: hom(x,y) -> hom(Ux,Uy)
is injective. By the way, a functor with this property is called
"faithful".
>[...] and C is D with extra *properties* if U is injective on the morphisms
>(meaning injective on the objects and on the morphisms between a given pair);
Hmm, again I'm unhappy for the same sort of nitpicky reason. Again,
it's bad to care if U is injective on objects, because this property
is not preserved by natural isomorphisms. I believe the politically
correct substitute for this property is called "reflecting isomorphisms":
we say a functor U: C -> D "reflects isomorphisms" if U(f) being an
isomorphism in D implies that f is an isomorphism in C. In particular,
nonisomorphic objects in C can't get sent to isomorphic objects in D
by a functor that reflects isomorphisms.
It seems that whenever C-objects are D-objects with extra properties,
the forgetful functor U: C -> D reflects isomorphisms. For example,
if two groups are isomorphic, and they happen to be abelian,
they are isomorphic in the category of abelian groups.
But I'd give a slightly different criterion for when C-objects are
D-objects with extra properties. I'd say this happens when U: C -> D
is "fully faithful". This means that given any pair of objects
x,y in C,
U: hom(x,y) -> hom(Ux,Uy)
is 1-1 and onto. (If this map is always injective, we say it's
"faithful". If it's always surjective, we say it's "full". If
both, we say it's "full and faithful", or "fully faithful" for
short.)
Note that a fully faithful functor always reflects isomorphisms -
this is a fun little exercise - so my criterion is stronger than
yours, at least modulo political correctness, which forbids me
from saying that a functor is "injective on objects".
Also note that the way I'm setting things up, "extra properties"
is a special case of "extra structure".
>Otherwise, I guess C is just D with extra *stuff*
>if, given a pair of objects, U is injective on the morphisms between them.
Hmm, I wouldn't demand that. An example of "extra stuff" would
be the functor U: Vect^2 -> Vect that takes a pair of vector spaces,
or a pair of linear map, and discards the second one. In other words,
I'd say a pair of vector spaces is a vector space "with extra stuff",
namely another vector space. The functor U: Vect^2 -> Vect doesn't
have the property you demand - i.e., it's not faithful.
Now I forget if there is *any* property we should demand of U: C -> D
when D-objects are supposed to be C-objects with "extra stuff"!
Maybe James Dolan will remind me what he told me about this case.
>For example, the forgetful functor Groups -> Sets
>shows that groups are sets with extra structure,
>while the forgetful functor Abelian Groups -> Groups
>shows that Abelian groups are groups with extra properties.
Let's check these examples: yes, U: Groups -> Sets is
faithful, while U: Abelian Groups -> Sets is fully faithful.
>Or you can turn around and use the free functor Sets -> Groups
>and say that sets are groups with extra properties
>(to wit, the property of being free).
Hmm, F: Sets -> Groups is faithful, but not full. Thus I'd
say that a set can be viewed as a group with extra *structure*:
namely, the property of being free together with the *structure*
of a specific set of generators. The point is that not all
homomorphisms between free groups come from functions between
their set of generators.
Fun stuff, eh? But I'm afraid it's drifting rather far afield
from physics, except insofar as every mathematical physicist
should spend a little time thinking about "properties" vs
"structure".
John Baez wrote:
>Toby Bartels wrote:
>>Given a functor U: C -> D, interpret U as a forgetful functor.
>Of course, part of the point of this puzzle is that the term
>"forgetful" is usually given no precise definition, so here we
>are seeking a precise definition. Usually people don't bother
>to define "forgetful functor" very precisely - like pornography,
>you're just supposed to know a forgetful functor when you see it.
Yeah, that's kind of the point of these definitions;
it's reasonable to regard U as a forgetful functor
if that allows you to think of C as D with something extra.
>>Then C is D with extra *structure* if U is surjective on the objects
>>and, given a pair of objects, injective on the morphisms between them.
>However, the more serious reason I'm unhappy is that sometimes C-objects
>are D-objects with extra structure but *not every D-object can be made
>into a C-object*. In this case U: C -> D is not essentially surjective.
Point taken.
>Note that a fully faithful functor always reflects isomorphisms -
>this is a fun little exercise.
Yeah, that was fun.
>Also note that the way I'm setting things up, "extra properties"
>is a special case of "extra structure".
I know you've been saying before that the two concepts are related,
and I thought I found the difference between them,
that properties was all about adding extra requirements,
while structure was about doing everything except extra requirements.
But I see that was wrong, because we don't hesitate to add extra requirements,
even to the old structure, when we add new structure.
Thus, a ring is not just an Abelian group with a new monoid structure;
we add new requirements to the Abelian group,
to wit, that the new monoid should distribute over it.
Similarly, we can start with a group, add a new group structure,
and then require that the two group structures be compatible in that
they commute with each other and have the same identity --
but now all we've really done is require that the original group be Abelian!
So extra properties really are a case of extra structure.
>Now I forget if there is *any* property we should demand of U: C -> D
>when D-objects are supposed to be C-objects with "extra stuff"!
>Maybe James Dolan will remind me what he told me about this case.
Well, I just chose the generalization of the other concepts I'd come up with.
If someone can think of something useful to require of U,
then we can call that "extra stuff" if we like,
but otherwise we can just let that be perfectly general.
(There is one concept left: the case where U is faithful but not full.
But we can't call that "extra stuff",
since stuff should be more general than structure.)
>Fun stuff, eh? But I'm afraid it's drifting rather far afield
>from physics, except insofar as every mathematical physicist
>should spend a little time thinking about "properties" vs
>"structure".
If that's good enough for you, it's good enough for me.
s.p.r is more fun than s.math.r anyway -- and I say that as a mathematician.
(Just like you, I guess -- and yet you're even a moderator here.)
-- Toby
toby bartels writes:
->I will leave it to James Dolan to explain the technical
->distinction between "extra properties", "extra structure",
->and "extra stuff" - there is a nice category-theoretic way
->of making this precise.
-
-Ooh, let me guess!
-
-Given a functor U: C -> D, interpret U as a forgetful functor.
-Then C is D with extra *structure* if U is surjective on the
-objects and, given a pair of objects, injective on the
-morphisms between them; and C is D with extra *properties* if
-U is injective on the morphisms (meaning injective on the
-objects and on the morphisms between a given pair); Otherwise,
-I guess C is just D with extra *stuff* if, given a pair of
-objects, U is injective on the morphisms between them.
here's my classification:
given groupoids c,d and a functor u:c->d, the objects of c can
be thought of via the forgetful functor u as objects of d with
an extra _property_ iff u is full and faithful, as objects of d
with extra _structure_ iff u is faithful, and as objects of d
with extra _stuff_ regardless.
(some category-theoretic jargon:
1. a "groupoid" is a category where all the morphisms are
invertible. it may very well be interesting to generalize the
subject matter of this discussion to the case where c and d are
not necessarily groupoids, but to keep things simple for now i
won't do that in this post.
2. a functor u:c->d is "full" iff for any pair c1,c2 of
objects in c, the map from the hom-set hom(c1,c2) to the
hom-set hom(u(c1),u(c2)) given by u is surjective.
3. a functor u:c->d is "faithful" iff for any pair c1,c2 of
objects in c, the map from the hom-set hom(c1,c2) to the
hom-set hom(u(c1),u(c2)) given by u is injective.)
one reason i don't (as i think toby was suggesting) require the
forgetful functor u to be surjective on (isomorphism classes
of) objects in order for the objects of c to qualify as objects
of d with extra "structure" is as follows:
consider for example the case where c is the category of rings,
d is the category of groups, and u is the functor assigning to
each ring its underlying additive group. clearly the objects
of c are objects of d with extra "structure" in the intuitive
sense that i'm trying to capture; we can say that "a ring is
defined to be a group (henceforward referred to as "the
underlying additive group of the ring") equipped with an extra
multiplication operation on it satisfying certain equational
laws...", and although this may sound like the equational laws
only constrain the ring structure on the additive group, they
in fact also implicitly constrain the additive group itself:
it's easy to show that even if you don't explicitly require the
additive group of a ring to be commutative, it will
automatically be forced to be commutative by the other clauses
in the usual definition of "ring" (left and right
distributivity plus multiplicative unit laws, in combination
with the group axioms for addition, should do it, i think).
thus this example is supposed to demonstrate the fact that as
soon as you generally allow yourself to invent a new "type of
structure that an object of d can be equipped with" by starting
with an arbitrary existing such type of structure and
constraining the structures to satisfy some property, it's
awkward to prevent an arbitrary "property that can be
predicated of an object of d" from being considered as a "type
of structure that an object of d can be equipped with" by being
looked at as a constraint on [the degenerate "type of structure
that an object of d can be equipped with" given by "no extra
structure at all"]. thus you should probably broaden your
concept of "type of structure that an object of d can be
equipped with" to include "property that can be predicated of
an object of d" as a special case.
for similar reasons you should probably broaden your concept of
"type of stuff that an object of d can be equipped with" to
include "type of structure that an object of d can be equipped
with" as a special case, if it isn't that broad already.
-For example, the forgetful functor Groups -> Sets shows that
-groups are sets with extra structure, while the forgetful
-functor Abelian Groups -> Groups shows that Abelian groups are
-groups with extra properties.
i agree with those examples (at least if i interpret them in
accordance with my self-imposed restriction to consider only
the case where all of the morphisms in c and d are invertible).
-Or you can turn around and use
-the free functor Sets -> Groups and say that sets are groups
-with extra properties (to wit, the property of being free).
i disagree with that example, for reasons that hopefully are
clear from my explanations above. thus i would _not_ say that
a set is a group with the extra property of being free; rather
i'd say that a set is a group with the extra _structure_ of
being equipped with a favored "basis" of mutually free mutual
generators.
-OTOH, the Abelianization functor Groups -> Abelian groups is
-surjective on the objects (and on the morphisms for that
-matter), but groups are not Abelian groups with extra
-structure, because the functor isn't injective on the
-morphisms between a given pair.
i think i agree with this, but it sounds like you're using
my rules here rather than the rules i thought you tried to
spell out in your post.
another example of an object equipped with extra "stuff" would
be a set equipped with another _set_; that is, take c to be
the category of ordered pairs of sets, d to be the category of
sets, and u to be the "projection" functor assigning to an
ordered pair (x,y) of sets its first coordinate x. i hope
this example helps to show why i consider the terminology
"stuff" reasonably descriptive of the intuition involved.
another example (maybe or maybe not causing some additional
(?) number of people to see this post as having some relevance
to physics) of an object equipped with extra "stuff" rather
than merely with extra "structure" is a manifold equipped with
an unfortunately so-called "spin structure". the point is
that if we define the concept of "morphism between spin
manifolds" in what seems to me to be the most advantageous
way, then taking c to be the category of spin manifolds, d the
category of manifolds, and u the hopefully obvious forgetful
functor assigning to a spin manifold its underlying ordinary
manifold, u is not faithful.
thus a "spin structure" is not merely "structure"; it's
"stuff". so what is this extra "stuff"?? you can think of it
as "spin frames" if you want to. (a "spin frame" is what a
spin manifold has two of where an ordinary manifold has only
one.) or you can think of it as "spinors"; morphisms between
spin manifolds have an extra discrete degree of freedom to
flip the sign of spinors even after their action on ordinary
manifold points has been completely nailed down.
a deeper understanding of how the classification offered here
arises involves the relationship between groupoid theory and
homotopy theory, as follows:
for any integer n greater than or equal to -1, a space x is
defined to be of "homotopy dimension n" iff for any integer
j strictly greater than n, every continuous map from the
j-dimensional sphere s^j to x is homotopic to a constant map.
using this terminology, every space of homotopy dimension n
is also of homotopy dimension m for any integer m greater than
n. a crucial fact is that the world of spaces of homotopy
dimension 1 is secretly isomorphic in a very strong way to the
world of groupoids; there's an amazingly perfect "dictionary"
linking concepts from the world of spaces of homotopy
dimension 1 to their secret equivalents in the world of
groupoids. the groupoid corresponding to a space x of
homotopy dimension 1 is called the "fundamental groupoid" of
x, and the space of homotopy dimension 1 corresponding to a
groupoid g is called the "classifying space" of g.
inside the world of spaces of homotopy dimension 1 are of
course the sub-world of spaces of homotopy dimension 0, and
the sub-sub-world of spaces of homotopy dimension -1. the
secret equivalent inside the world of groupoids of the
sub-world of spaces of homotopy dimension 0 is the sub-world
of so-called "discrete groupoids", and the secret equivalent
of the sub-sub-world of spaces of homotopy dimension -1 is
the sub-sub-world of just those special discrete groupoids
which have either just one object and one morphism, or no
objects and morphisms at all.
the discrete groupoids are also known as "sets", or
(exploiting the [homotopy dimension 1]/groupoids dictionary)
"groupoids of homotopy dimension 0". the special discrete
groupoids corresponding to the spaces of homotopy dimension
-1 are called "truth values", or "groupoids of homotopy
dimension -1". the groupoid with just one object and one
morphism is called "true" (aka "the terminal groupoid" aka
"yes" aka "in") while the empty groupoid is called "false"
(aka "the initial groupoid" aka "no" aka "out").
given a pair c,d of groupoids and a functor u:c->d and an
object d1 in d, we can construct a new groupoid called "the
homotopy fiber of u over d1". roughly, the homotopy fiber
of u over d1 is the groupoid of "objects of c equipped with
designated isomorphisms from their images under u to d1";
the morphisms in the homotopy fiber are required to preserve
the designated isomorphisms. as you might guess from the
name "homotopy fiber", the groupoid-theoretic concept of
"homotopy fiber" has a very direct equivalent in homotopy
theory.
we can now re-state the definitions of "property",
"structure", and "stuff" in terms of homotopy dimension of
homotopy fibers, as follows:
given groupoids c,d and a functor u:c->d, the objects of c can
be thought of via the forgetful functor u as objects of d with
an extra _property_ iff the homotopy fibers of u are all of
homotopy dimension -1, as objects of d with extra _structure_
iff the homotopy fibers of u are all of homotopy dimension 0,
and, and as objects of d with extra _stuff_ iff the homotopy
fibers of u are all of homotopy dimension 1.
hopefully this makes the intuition behind the concepts a bit
clearer. a "property" is something which, if you possess it
at all, then you have no choice in _how_ to possess it, you
just do. a "structure" is something which if you possess it
then possessing it involves picking a particular structure in
a way analogous to picking an element of a set. "stuff" is
something which if you possess it then possessing it amounts
to picking some particular stuff in a way analogous to picking
an object of a groupoid.
of course as with most concepts of groupoid theory, the
concepts discussed here should be generalized to the case of
"higher-dimensional groupoid theory" which corresponds to
the homotopy theory of spaces with arbitrary homotopy
dimension in the same way that ordinary groupoid theory
corresponds to the homotopy theory of spaces of homotopy
dimension 1. thus the stunted progression property,
structure, stuff becomes a genuine open-ended progression:
property, structure, stuff, eka-stuff, eka-eka-stuff, ... .
thus given arbitrary spaces c,d and a continuous map u:c->d,
we should say that "the objects of the fundamental
infinity-groupoid of c can be thought of via the forgetful
infinity-functor induced by u as objects of the fundamental
infinity-groupoid of d equipped with extra eka^n-stuff" iff
all of the homotopy fibers of u are of homotopy-dimension
n+1.
Toby Bartels wrote:
>james dolan wrote:
>>A "groupoid" is a category where all the morphisms are
>>invertible. it may very well be interesting to generalize the
>>subject matter of this discussion to the case where c and d are
>>not necessarily groupoids, but to keep things simple for now i
>>won't do that in this post.
>You seem to agree with John Baez's classification,
>but he doesn't feel the need to limit to groupoids;
>perhaps a word on how you think that complicates things?
>Or is it just that groupoids are needed for the deep homotopy connection?
He'd darn well BETTER agree with it, because I learned everything
I said from him!
In all the examples I know, James' definition of "structure"
and "properties" works nicely for categories as well as just
groupoids. And certainly it's nice to have *some* definition
of this sort for categories, not just groupoids. So my hunch is
that he restricted attention to groupoids so that he could
Effortlessly ascend the dimensional ladder to n-groupoids,
using the conjectured equivalence between n-groupoids and
homotopy n-types (which for now can be taken as a definition
of n-groupoids if one likes).
But he's back in Riverside now so I should ask him.
Once upon a time, I wrote:
>Hmm, again I'm unhappy for the same sort of nitpicky reason. Again,
>it's bad to care if U is injective on objects, because this property
>is not preserved by natural isomorphisms. I believe the politically
>correct substitute for this property is called "reflecting isomorphisms":
>we say a functor U: C -> D "reflects isomorphisms" if U(f) being an
>isomorphism in D implies that f is an isomorphism in C. In particular,
>nonisomorphic objects in C can't get sent to isomorphic objects in D
>by a functor that reflects isomorphisms.
Jim Dolan kindly pointed out that the last sentence is in error.
For example, if D is a category with lots of isomorphisms, and C is
the category with the same objects and only identity morphisms,
there's an obvious functor U: C -> D. This reflects isomorphisms
but maps nonisomorphic objects in C to isomorphic ones in D.
However, if U: C -> D reflects isomorphisms and is also full, it can't
map nonisomorphic objects to isomorphic ones. In the context of my
remark, this fact is all we really need. Recall that we defined objects
of C to be objects of D "with extra properties" if U: C -> D was full
and faithful. This implies that U reflects isomorphisms. So it also
implies that U can't send nonisomorphic objects to isomorphic ones.
And that's reassuring, because we expect that forgetting extra properties
can't make nonisomorphic objects isomorphic --- though forgetting extra
*structure* can.
But enough of this --- back to physics! Has anyone read Wilczek's paper
"Beyond the Standard Model: This Time for Real"? What do you think?
It argues that the recent neutrino oscillation results support a
supersymmetric SU(5) or SO(10) grand unified theory. Do people really
believe this?
toby bartels writes:
-james dolan wrote:
-
->given groupoids c,d and a functor u:c->d, the objects of c can
->be thought of via the forgetful functor u as objects of d with
->an extra _property_ iff u is full and faithful, as objects of d
->with extra _structure_ iff u is faithful, and as objects of d
->with extra _stuff_ regardless.
-
->A "groupoid" is a category where all the morphisms are
->invertible. it may very well be interesting to generalize the
->subject matter of this discussion to the case where c and d are
->not necessarily groupoids, but to keep things simple for now i
->won't do that in this post.
-
-You seem to agree with John Baez's classification,
-but he doesn't feel the need to limit to groupoids;
-perhaps a word on how you think that complicates things?
it complicates things in the obvious way: a single concept in groupoid
theory (for example the concept of "faithful functor between
groupoids") may bifurcate into non-equivalent concepts in category
theory (for example the concepts of "faithful functor between
categories" and "functor between categories which is faithful on
isomorphisms"); the necessity of worrying about the distinctions
between such non-equivalent concepts is eliminated by discussing only
the groupoid case. but presumably you're also asking why it is that
in this tradeoff between simplicity and generality i chose simplicity,
so i'll try to say something about that too.
-Or is it just that groupoids are needed for the deep homotopy connection?
that's part of my motivation by now, but i think my original
motivation had less to do with the "dictionary" that relates groupoid
theory to a special part of homotopy theory than with a different but
in its own way equally powerful "dictionary" relating groupoid theory
to a special kind of predicate logic. in the world of predicate logic
there's an obvious sense in which adding extra "properties" to the
models of a theory means adding new axioms to the theory, adding extra
"structure" to the models means adding new predicate symbols (possibly
supplemented by new axioms) to the theory, and adding extra "stuff" to
the models means adding new "types" (possibly supplemented by new
predicate symbols and axioms) to the theory. this
property/structure/stuff distinction in predicate logic matches
perfectly the property/structure/stuff distinction in groupoid theory
if groupoids are interpreted as a certain sort of logical theories in
a certain way.
the more i think about this the more it seems that there should be
some nice big picture that links together the predicate logic aspects
of the situation with the homotopy theory aspects of the situation,
but if so it's a bit too big for me to fully grasp yet so i won't try
to say any more about it at the moment.
i will say though that if someone would show how to generalize the
correspondence between groupoids and logical theories of a certain
sort to a correspondence between categories and logical theories of
some more general sort, then i might be willing to agree that there is
some obvious way of extending the property/structure/stuff
classification of groupoid theory to apply to category theory as well.
i have a vague suspicion that in fact this has already been done and
that the logical theories corresponding to categories differ from the
logical theories corresponding to groupoids more or less precisely in
being "intuitionistic" rather than "classical", but i'm not at all
clear on the details of how this works if it's even correct.
->a deeper understanding of how the classification offered here
->arises involves the relationship between groupoid theory and
->homotopy theory, as follows:
-
->for any integer n greater than or equal to -1, a space x is
->defined to be of "homotopy dimension n" iff for any integer
->j strictly greater than n, every continuous map from the
->j-dimensional sphere s^j to x is homotopic to a constant map.
-
-You can even generalize this to n = -2, noting that s^{-1} is the empty set.
yes, very much so, though i don't think i thought about this until
afterwards.
-Of course, no map from s^{-1} to any space can ever be homotopic to a
-constant, yet there is always some map from s^{-1} to any space (the
-empty map), so no space has homotopy dimension -2, which must be why
-nobody talks about it.
hmm. first of all, i think i should revise my definition of homotopy
dimension to eliminate the idea of "homotopic to a constant map",
because people seem to disagree on the meaning of "constant map" when
the domain is empty. (some people think that constantness of maps is
the property of factoring through the one-point set, others think it's
the _structure_ of being equipped with a specific factorization
through the one-point set, and toby apparently thinks it's the
property of having the one-point set as image.)
here's the revised version:
for any integer n greater than or equal to -2, a space x is defined to
be of "homotopy dimension n" iff for every continuous map m from the
[n+1]-dimensional sphere s^[n+1] to x, the space of extensions of m to
the [n+2]-dimensional disk d^[n+2] is contractible.
(here the sphere s^[-1] is defined to be empty, the disk d^[j+1] is
defined to be the mapping cylinder of the map s^j->1, and the sphere
s^[j+1] is defined to be the pushout d^[j+1] +_[s^j] d^[j+1].
"contractible" means equivalent to the 1-point space.)
hopefully with this revised definition it's still true that being of
homotopy dimension n implies being of homotopy dimension n+1. the
spaces of homotopy dimension -2 are the contractible spaces, and the
spaces of homotopy dimension n for higher n are hopefully just as
before.
the spaces of homotopy dimension n taken from a sufficiently
"convenient" category s of spaces form a cartesian closed category
s_n, and the spaces of homotopy dimension n+1 in s are the spaces
equivalent to classifying spaces of groupoids enriched over s_n.
the class of continuous maps with all homotopy fibers of homotopy
dimension -2 is the class of all homotopy equivalences. in the world
of groupoids this corresponds to the class of all functors that are
"invertible up to natural isomorphism". thus eka^[-3]-stuff is
_vacuous_ properties; that is, given groupoids c,d and a functor
f:c->d with f invertible up to natural isomorphism, objects of c can
be thought of as objects of d equipped with a _vacuous_ property. (as
throughout this discussion, we are interested in everything only "up
to natural isomorphism" or "up to homotopy" in groupoid theory or in
homotopy theory theory respectively.)
notice that the class of all maps with all homotopy fibers of homotopy
dimension n is closed under composition because the homotopy fibers of
a composite map fg are themselves the total spaces of fibrations with
base spaces which are homotopy fibers of g and fibers which are
homotopy fibers of f, and because the class of spaces of homotopy
dimension n is closed under the process of forming a new space as the
total space of a fibration with its base and all its fibers in the
class.
finally, if there's anything such as "spaces of homotopy dimension
-3", i don't want to hear about it.
james dolan wrote:
>Toby Bartels wrote:
>>Or is it just that groupoids are needed for the deep homotopy connection?
>that's part of my motivation by now, but i think my original
>motivation had less to do with the "dictionary" that relates groupoid
>theory to a special part of homotopy theory than with a different but
>in its own way equally powerful "dictionary" relating groupoid theory
>to a special kind of predicate logic. in the world of predicate logic
>there's an obvious sense in which adding extra "properties" to the
>models of a theory means adding new axioms to the theory, adding extra
>"structure" to the models means adding new predicate symbols (possibly
>supplemented by new axioms) to the theory, and adding extra "stuff" to
>the models means adding new "types" (possibly supplemented by new
>predicate symbols and axioms) to the theory. this
>property/structure/stuff distinction in predicate logic matches
>perfectly the property/structure/stuff distinction in groupoid theory
>if groupoids are interpreted as a certain sort of logical theories in
>a certain way.
OK, I tried to think about this, but I don't really know where to start.
Give me a clue: what famous groupoid corresponds to
what I've been taught to regard as the basic predicate calculus:
ordinary logic with forall, forsome, and equality?
>>james dolan wrote:
>>>for any integer n greater than or equal to -1, a space x is
>>>defined to be of "homotopy dimension n" iff for any integer
>>>j strictly greater than n, every continuous map from the
>>>j-dimensional sphere s^j to x is homotopic to a constant map.
>>You can even generalize this to n = -2, noting that s^{-1} is the empty set.
>>Of course, no map from s^{-1} to any space can ever be homotopic to a
>>constant, yet there is always some map from s^{-1} to any space (the
>>empty map), so no space has homotopy dimension -2, which must be why
>>nobody talks about it.
>hmm. first of all, i think i should revise my definition of homotopy
>dimension to eliminate the idea of "homotopic to a constant map",
>because people seem to disagree on the meaning of "constant map" when
>the domain is empty. (some people think that constantness of maps is
>the property of factoring through the one-point set, others think it's
>the _structure_ of being equipped with a specific factorization
>through the one-point set, and toby apparently thinks it's the
>property of having the one-point set as image.)
Another example of my remaining uncategorical thinking, clearly.
I remember thinking that some fool might argue
that S^{-1} can't be empty (and is indeed nonexistent),
because S^{n+1} is the suspension of S^n
(or S^{n+1} = S(S^n), which is obvious when you look at it),
whereas S^0 = {0,1} is not the suspension of {}.
But to argue S({}) != {0,1} is to make the same uncategorical mistake
of worrying more about the image of a map than what it factors through.
I caught the mistake that time, because I didn't want to make it,
but this time it was easier just to say there was no dim -2.
>for any integer n greater than or equal to -2, a space x is defined to
>be of "homotopy dimension n" iff for every continuous map m from the
>[n+1]-dimensional sphere s^[n+1] to x, the space of extensions of m to
>the [n+2]-dimensional disk d^[n+2] is contractible.
What topology are you putting on this space of functions?
>finally, if there's anything such as "spaces of homotopy dimension
>-3", i don't want to hear about it.
That would require S^{-2}.
I've been thinking about it, and I don't think that exists.
For no definition of S^n that I can think of does S^{-2} make sense.
(And I can be quite sure that {} isn't the suspension of anything.)
For example, you say
>the disk d^[j+1] is
>defined to be the mapping cylinder of the map s^j->1, and the sphere
>s^[j+1] is defined to be the pushout d^[j+1] +_[s^j] d^[j+1].
No matter what space we take for S^{-2},
applying this definition to get D^{-2} and then S^{-1}
will never yield that S^{-1} is the empty set.
Therefore, S^{-2} doesn't exist.
(Similarly, B^{-1} doesn't exist.)
Therefore, homotopy dimension -3 is a meaningless concept.
Until, of course, someone comes up with
another way to assign meaning to it ....
-- Toby
toby bartels wrote:
-james dolan wrote:
-
->Toby Bartels wrote:
-
->>Or is it just that groupoids are needed for the deep homotopy
->>connection?
-
->that's part of my motivation by now, but i think my original
->motivation had less to do with the "dictionary" that relates groupoid
->theory to a special part of homotopy theory than with a different but
->in its own way equally powerful "dictionary" relating groupoid theory
->to a special kind of predicate logic. in the world of predicate logic
->there's an obvious sense in which adding extra "properties" to the
->models of a theory means adding new axioms to the theory, adding extra
->"structure" to the models means adding new predicate symbols (possibly
->supplemented by new axioms) to the theory, and adding extra "stuff" to
->the models means adding new "types" (possibly supplemented by new
->predicate symbols and axioms) to the theory. this
->property/structure/stuff distinction in predicate logic matches
->perfectly the property/structure/stuff distinction in groupoid theory
->if groupoids are interpreted as a certain sort of logical theories in
->a certain way.
-
-OK, I tried to think about this, but I don't really know where to
-start. Give me a clue: what famous groupoid corresponds to what I've
-been taught to regard as the basic predicate calculus: ordinary logic
-with forall, forsome, and equality?
the correspondence is between individual groupoids and individual
_theories_ of a particular form of predicate logic. the particular
form of predicate logic involved is pretty much just "the basic" form,
with the allowed syntactic constructions including:
1. the usual finitary boolean connectives obeying the usual finitary
boolean equational laws
2. the universal quantifier "for all" (and therefore also the
existential quantifier "for some" via the equivalence between "for
some x, p(x)" and "not (for all x, (not p(x)))")
3. the built-in binary predicate "equality" with it's usual built-in
reflexivity, symmetry, transitivity, and substitutability properties
plus one more construction going beyond what's ordinarily considered
"the basic":
4. the restriction in #1 above against the _infinitary_ boolean
connectives (such as n-fold conjunction for an arbitrary infinite
cardinality n) is lifted.
given a theory t expressed in this kind of logic, we obtain the
groupoid of models of t. when all the i's are dotted and the t's
crossed in the right way, this process of passing from the theory t to
the groupoid of models of t becomes a "bi-equivalence from the
bi-category of theories to the bi-category of groupoids".
for example, let t be the theory presented by giving no predicate
symbols, plus the one axiom "there are exactly seven things". (of
course this axiom can be expressed using the allowed syntatic
constructions.) the groupoid of models of t is the groupoid of
seven-element sets. this groupoid has just one isomorphism class
because the theory t is "categorical" (in a sense of the word
"categorical" having not much relationship to category theory!).
that's not a complete exposition of the situation, rather just a clue
of the sort i hope you wanted. i will mention further though that to
develop the full correspondence between theories and groupoids, the
theories should be allowed to be "multi-typed". if only
"single-typed" theories are considered then the most straightforward
correspondence is not with "abstract" groupoids but rather with
"concrete" groupoids, a "concrete groupoid" being a groupoid equipped
with a faithful functor to the groupoid of sets. it might be a good
idea to develop the correspondence between single-typed theories and
concrete groupoids before developing the full correspondence between
multi-typed theories and abstract groupoids. one of the basic lemmas
you should try to understand is as follows:
let x be a set. let c be the collection of all pairs (s,p) with s a
(possibly infinite) set and p an s-ary relation on x. let d be the
hyper-collection of all sub-collections of c that are closed under all
of the operations on relations alluded to in #1-#4 above. then d is
in canonical bijection with the set of subgroups of the group of
permutations of x (taking "permutation" to mean "auto-bijection").
(in the above lemma, among the operations that should count as
"alluded to" is the operation of replacing an s-ary relation by the
obvious corresponding t-ary relation given a bijection from s to t,
even though this operation was perhaps _not_ very explicitly alluded
to.)
Here's a much later post of mine, on the category theory mailing list:
Frank Atanassow emailed me saying that he's "dying to know" the
answer to my puzzle about (-1)-categories and (-2)-categories.
That's good! So far the silence has been deafening, and I can't
tell if it indicates bewilderment, lack of interest, or disgust
with an imprecisely posed question.
I guess I'll give away the answer.
As I said, the trick is to figure out what (-1)-groupoids and
(-2)-groupoids are, and then cross our fingers and hope that
the answer is the same for (-1)-categories and (-2)-categories.
We start with the basic principle that people use when trying
to cook up definitions of "weak n-groupoid": weak n-groupoids should
be the same as nice topological spaces of homotopy dimension n.
Here "nice" means something like a CW complex, and "the same" must
be taken in a suitably n-categorical/homotopical spirit. But
what does "homotopy dimension n" mean?
Well, the usual definition is that a space has homotopy dimension n
if all its homotopy groups above the nth are trivial. But if we
carefully unpack this, we'll see it's an interesting condition even
when n is -1 or -2.
Here goes:
A topological space X has homotopy dimension n if given k > n,
any continuous map from the k-sphere to X extends to a continuous
map from the (k+1)-disc to X.
So: X has homotopy dimension 0 if its arc-components have
vanishing homotopy groups. If X is nice, this means it's a
disjoint union of contractible spaces, so it's the same as a *set*.
That's good: 0-groupoids should be sets.
(See? I'm using "the same" in a suitably homotopical spirit!
A disjoint union of nice contractible spaces is homotopy equivalent
to a set with the discrete topology.)
Next: X has homotopy dimension -1 if it has homotopy dimension
0 and also any continuous map from the 0-sphere to X extends to
a continuous map from the 1-disc to X. In other words, X is either
empty, or it consists of a single arc-component with vanishing
homotopy groups. If X is nice, this means it's the same as a
*set with cardinality 0 or 1*. So we declare that a (-1)-groupoid
is a set with cardinality 0 or 1.
Next: X has homotopy dimension -2 if it has homotopy dimension
-1 and also any continuous map from the (-1)-sphere to X extends
to a continuous map from the 0-disc to X. The 0-disc is a single
point, and its boundary the (-1)-sphere is the empty set. So X
must be a *nonempty* space with homotopy dimension -1. So X
consists of a single arc-component with vanishing homotopy groups.
If X is nice, this means it's the same as a *set with cardinality
1*. So we declare that a (-2)-groupoid is a set with cardinality 1.
Crossing our fingers, we therefore guess:
A (-1)-category is a set with cardinality 0 or 1.
A (-2)-category is a set with cardinality 1.
This may seem silly, but it's not! There is a nice relation
between all this business and the notion of "n-stuff". But I'm
getting worn out, so instead of explaining that, I'll just quote
some articles that James Dolan and Toby Bartels wrote on
sci.physics.research when they were first figuring out this "n-stuff" stuff.
Frank also asked if the error in our definition of n-categories
appears in HDA3. Yes! We will correct it in HDA5, which will
be about Feynman diagrams and "n-stuff". As I said, it's easy
to fix: just one number is wrong.
John Baez
And here's another by Toby on sci.physics.research:
John Baez wrote in part:
>This may seem silly, but it's not! There is a nice relation
>between all this business and the notion of "n-stuff". But I'm
>getting worn out, so instead of explaining that, I'll just quote
>some articles that James Dolan and Toby Bartels wrote on
>sci.physics.research when they were first figuring out this "n-stuff" stuff.
Ah, the sins of my wayward youth.
>From: james dolan
>Subject: Re: Just Categories now
>Date: 16 Nov 1998 00:00:00 GMT
>Toby Bartels wrote:
>>Given a functor U: C -> D, interpret U as a forgetful functor.
>>Then C is D with extra *structure* if U is surjective on the
>>objects and, given a pair of objects, injective on the
>>morphisms between them; and C is D with extra *properties* if
>>U is injective on the morphisms (meaning injective on the
>>objects and on the morphisms between a given pair); Otherwise,
>>I guess C is just D with extra *stuff* if, given a pair of
>>objects, U is injective on the morphisms between them.
>given groupoids c,d and a functor u:c->d, the objects of c can
>be thought of via the forgetful functor u as objects of d with
>an extra _property_ iff u is full and faithful, as objects of d
>with extra _structure_ iff u is faithful, and as objects of d
>with extra _stuff_ regardless.
Well, Jim is right,
but it turns out that my definition for "structure"
wasn't as bad as we originally thought.
It's wrong -- acording to it, groups aren't sets with extra structure,
simply because the empy set cannot be made into a group --
but it *is* essentially the definition of being *only* extra structure.
First note the theorem that a functor between categories
is an equivalence iff it's full, faithful, and essentially surjective
(that is surjective, not on objects, but on isomorphism classes of objects).
This is analogous to the theorem in set theory that
a function is a bijection iff it's injective and surjective,
as you'll see below.
Now, a forgetful functor is *only* extra stuff iff
it is both essentially surjective and full.
A forgetful functor is *only* extra structure iff
it is both essentially surjective and faithful.
And of course it is (only) extra property iff
it is both faithful and full.
So groups are indeed sets with extra structure,
but they aren't *only* extra structure --
they *also* have the property of being nonempty.
Just as a function between sets can be factored
in a unique way (up to bijection of sets)
into a surjection followed by an injection,
so a functor between categories can be factored
in a unique way (up to equivalence of categories)
into a functor that is only extra stuff,
followed by one that is only extra structure,
followed by one that is extra property.
Also note that while only stuff, only structure, and property
form a complete trio (so long as we stick to 1categories),
stuff, structure, and property is an incomplete list -- downwards.
We can have an arbitrary functor -- that's stuff.
Then we can require that the functor be faithful -- that's structure.
Then we also can require that the functor be full -- that's property.
But there is one more requirement to add of course,
that it be essentially surjective -- that's an equivalence.
So it really goes: equivalence, property, structure, stuff.
>given groupoids c,d and a functor u:c->d, the objects of c can
>be thought of via the forgetful functor u as objects of d with
>an extra _property_ iff the homotopy fibers of u are all of
>homotopy dimension -1, as objects of d with extra _structure_
>iff the homotopy fibers of u are all of homotopy dimension 0,
>and, and as objects of d with extra _stuff_ iff the homotopy
>fibers of u are all of homotopy dimension 1.
To continue with the dimension that Jim forgot at first (-2),
U is an equivalence of categories iff
its homotopy fibres all have dimension -2.
>hopefully this makes the intuition behind the concepts a bit
>clearer. a "property" is something which, if you possess it
>at all, then you have no choice in _how_ to possess it, you
>just do. a "structure" is something which if you possess it
>then possessing it involves picking a particular structure in
>a way analogous to picking an element of a set. "stuff" is
>something which if you possess it then possessing it amounts
>to picking some particular stuff in a way analogous to picking
>an object of a groupoid.
So if U: C -> D is an equivalence of categories,
then an object of D just *is* an object of C
and (once U has been specified) that's all that there is to say about it.
Similarly, there just *is* a -2category,
and that's all that there is to say about it.
But if U: C -> D is full and faithful (extra property),
then given an object of D, it either is or is not an object of C.
The answer to the question is a truth value, a -1category.
Then if U: C -> D is faithful (extra structure),
then given an object of D,
it may be given the structure of being an object of C
in many ways, or one way, or none.
The answer to the question is a set, a 0category.
More recently, the philosophers
David Corfield and
Jean-Pierre Marquis
began writing a paper that mentions the above ideas.
In the process, David Corfield revived the above discussion in
mail:
Hi, one teeny problem from my exchange with Marquis:
"Now, a forgetful functor is *only* extra stuff iff
it is both essentially surjective and full.
A forgetful functor is *only* extra structure iff
it is both essentially surjective and faithful.
And of course it is (only) extra property iff
it is both faithful and full.
So groups are indeed sets with extra structure,
but they aren't *only* extra structure --
they *also* have the property of being nonempty.
Just as a function between sets can be factored
in a unique way (up to bijection of sets)
into a surjection followed by an injection,
so a functor between categories can be factored
in a unique way (up to equivalence of categories)
into a functor that is only extra stuff,
followed by one that is only extra structure,
followed by one that is extra property". (1)
Actually, what about the slippery nature of the identity of a monoid? It can
be included as a piece of structure or as a property. At the level of
groupoids it doesn't matter which. Similarly here,
"Similarly, we can start with a group, add a new group structure,
and then require that the two group structures be compatible in that
they commute with each other and have the same identity --
but now all we've really done is require that the original group be Abelian!
So extra properties really are a case of extra structure." (2)
Aargh, I'm losing it. (2) sounds like this could be factored
Ab - > Ab -> Groups or Ab-> Group -> Group
with the first Ab-> groups forgetting only property, and the second forgetting
only structure. But the first paragraph of (1) sounds like the latter
factorisation is the right one.
Is this a difference between the august authors of (1) and (2), or am I being
dim?
Best, David
Subject: Re: Teeny bit of help needed
From: John Baez
To: David Corfield
Cc: James Dolan, Toby Bartels
Date: Wed, 9 Feb 2005 22:13:28 -0800 (PST)
Hi -
> Hi,
> one teeny problem from my exchange with Marquis:
>
> "Now, a forgetful functor is *only* extra stuff iff
> it is both essentially surjective and full.
> A forgetful functor is *only* extra structure iff
> it is both essentially surjective and faithful.
> And of course it is (only) extra property iff
> it is both faithful and full.
> So groups are indeed sets with extra structure,
> but they aren't *only* extra structure --
> they *also* have the property of being nonempty.
>
> Just as a function between sets can be factored
> in a unique way (up to bijection of sets)
> into a surjection followed by an injection,
> so a functor between categories can be factored
> in a unique way (up to equivalence of categories)
> into a functor that is only extra stuff,
> followed by one that is only extra structure,
> followed by one that is extra property." (1)
>
> Actually, what about the slippery nature of the identity of a monoid? It can
> be included as a piece of structure or as a property. At the level of
> groupoids it doesn't matter which.
I'm not sure how the above quote is relevant to *this*.
If we work with the groupoid of monoids, there's no trouble.
Otherwise:
There are two different categories of monoids: one with identity-preserving
morphisms, one with identity-nonpreserving morphisms.
We can take either of these categories - call it C - and use
the factorization theory described in (1) to factor the forgetful
functor
C -> Set
into three functors, the first of which forgets only stuff,
the second of which forgets only structure, and the third of
which forgets only properties. It should be fun to work out the details.
> Similarly here,
>
> "Similarly, we can start with a group, add a new group structure,
> and then require that the two group structures be compatible in that
> they commute with each other and have the same identity --
> but now all we've really done is require that the original group be Abelian!
> So extra properties really are a case of extra structure." (2)
>
> Aargh, I'm losing it.
You aren't saying what reference (2) is, but I suspect it was
written before (1). If so, it's crucial to know that the factorization
in (1) was discovered fairly late in the game (by Toby and Jim).
Before it was discovered, we always used "stuff" to mean "stuff, including
properties and structure", "structure" to mean "structure, including
properties", and "properties" to mean "properties".
After it was discovered, we were able to isolate the three concepts
more sharply and talk about, e.g., functors that only forget stuff
but not properties.
Toby's exposition in terms of polynomials explains the difference
in attitude by an analogy. We can use "quadratic polynomial" to
include "linear polynomial" and "constant" as special cases, or not.
Both attitudes have their advantages.
We're now in the position to take either attitude when it comes
to stuff, structure and properties.
> (2) sounds like this could be factored
> Ab - > Ab -> Groups or Ab-> Group -> Group
> with the first Ab-> groups forgetting only property, and the second
> forgetting only structure. But the first paragraph of (1) sounds
> like the latter factorisation is the right one.
>
> Is this a difference between the august authors of (1) and (2), or am I being
> dim?
If we want to factor the forgetful functor Ab -> Grp using the
method described in 1, we are forced to do
Ab -> Ab which forgets only stuff (and in fact forgets no stuff)
and then
Ab -> Ab which forgets only structure (and in fact forgets no structure)
and then
Ab -> Grp which forgets only properties
I hope I understood your question; if not, maybe one of the other
guys will.
Best,
jb
From: Toby Bartels
Subject: Re: Teeny bit of help needed
To: David Corfield
Cc: John Baez, James Dolan
Date: Wed, 9 Feb 2005 23:06:31 -0800
John Baez wrote in part:
>David Corfield quoted:
>>"Now, a forgetful functor is *only* extra stuff iff
>>it is both essentially surjective and full.
>>A forgetful functor is *only* extra structure iff
>>it is both essentially surjective and faithful.
>>And of course it is (only) extra property iff
>>it is both faithful and full.
>>So groups are indeed sets with extra structure,
>>but they aren't *only* extra structure --
>>they *also* have the property of being nonempty.
>>Just as a function between sets can be factored
>>in a unique way (up to bijection of sets)
>>into a surjection followed by an injection,
>>so a functor between categories can be factored
>>in a unique way (up to equivalence of categories)
>>into a functor that is only extra stuff,
>>followed by one that is only extra structure,
>>followed by one that is extra property."
This quotation was posted by me to s.p.r on 2002 April 18:
http://www.lns.cornell.edu/spr/2002-04/msg0041086.html.
I'll note that I prefer the term "purely" to "only" now,
since I think that "only" has more ambiguous connotations.
(It can sometimes mean "at most" instead of "purely".)
>>"Similarly, we can start with a group, add a new group structure,
>>and then require that the two group structures be compatible in that
>>they commute with each other and have the same identity --
>>but now all we've really done is require that the original group be Abelian!
>>So extra properties really are a case of extra structure." (2)
This quotation was written by me in s.p.r on 1998 Nov 29:
http://www.lns.cornell.edu/spr/1998-11/msg0013559.html.
I referred to the posts from 1998, in the posts from 2002,
as "the sins of my wayward youth" for some good reasons;
nevertheless, I still stand by the quotation above.
>You aren't saying what reference (2) is, but I suspect it was
>written before (1).
Well caught! (2) is more than 3 years earlier than (1), in fact.
So while (2) is perfectly correct, it's not the full story;
and (2) doesn't know how careful to be with the terminology.
>Before it was discovered, we always used "stuff" to mean "stuff, including
>properties and structure", "structure" to mean "structure, including
>properties", and "properties" to mean "properties".
When I'm being careful, I call this "forgetting at most stuff",
"forgetting at most structure", and "forgetting only property".
There is also "forgetting nothing" (which John calls "vacuous property").
>After it was discovered, we were able to isolate the three concepts
>more sharply and talk about, e.g., functors that only forget stuff
>but not properties.
I call these "forgetting purely stuff", "forgetting purely structure",
and "forgetting only property" (again!). There is now NO next concept.
>Toby's exposition in terms of polynomials explains the difference
>in attitude by an analogy. We can use "quadratic polynomial" to
>include "linear polynomial" and "constant" as special cases, or not.
>Both attitudes have their advantages.
Actually, the ~usual~ meaning of "quadratic polynomial"
(which doesn't include the special cases but does allows inhomogeneity)
doesn't match EITHER of these attitudes. This is because
the usual meaning of "quadratic polynomial" includes the clause
"a is distinct from zero", whose analogue isn't useful for functors.
(That in turn is because when an element of a field is distinct from 0,
there's a special thing that you can do with it -- division --
that makes the concept more significant than the mere negation
of being equal to zero.)
So in the analogy, "forgetting purely stuff" is analogous
to "homogeneous quadratic polynomial", while "forgetting at most stuff"
is analogous to "at-most-quadratic polynomial" (a nonstandard term
that I use for ~any~ polynomial that's quadratic, linear, OR constant).
This analogy is further exposited in A Pedagogical Analogy,
at http://math.ucr.edu/home/baez/qg-spring2004/polynomials.pdf.
>>Is this a difference between the august authors of (1) and (2)
The august author of (2) is a younger me than the august author of (1),
so (1) should be trusted over (2) in case of disagreement.
But the quotations that you gave above are both correct!
-- Toby
From: David Corfield
Subject: Teeny bit of help needed
To: Toby Bartels
Cc: John Baez, James Dolan
Date: Thu, 10 Feb 2005 10:07:06 +0100
Thanks everyone for the help. The factorisation idea really gets to the root
of the matter.
By the way, anyone had any further profound thoughts about the top left corner
of the periodic table? You may remember I wanted to line things up so that a
row told you the starting dimension for any potential homotopy (staying with
n-groupoids), and the column specified in how many consecutive dimensions
homotopy is allowable:
m consecutive possible dimensions
1 2 3 4
-1 truth value set groupoid 2-groupoid
potentially
starts at 0 n.e. set n.e. groupoid n.e. 2-groupoid n.e. 3-groupoid
dim n
1 group groupal groupoid groupal 2-grouopid groupal 3-groupoid
where n.e. is non-empty, or perhaps pointed would do.
Killing off the puny scope for a truth value to have homotopy by asking for a
pointed or n.e. truth value sends you south west into the abyss to the left.
Are there different rows of homotopic nothingness? I.e., something has no
possible dimensions of homotopy starting from dimension n.
Then, of course, there's Toby's point - what's the big deal in having
dimensions consecutive.
Best, David
From: Toby Bartels
Subject: Re: Teeny bit of help needed
To: David Corfield
Cc: John Baez, James Dolan
Date: Thu, 10 Feb 2005 01:24:42 -0800
>By the way, anyone had any further profound thoughts about the top left corner
>of the periodic table? You may remember I wanted to line things up so that a
>row told you the starting dimension for any potential homotopy (staying with
>n-groupoids), and the column specified in how many consecutive dimensions
>homotopy is allowable.
> m consecutive possible dimensions
> 1 2 3 4
>potentially -1 truth value set gpoid 2-gpoid
>starts at 0 n.e. set n.e. gpoid n.e. 2-gpoid ....
>dim n 1 gp gpal gpoid ....
> ...
>where n.e. is non-empty, or perhaps pointed would do.
I think that this table is correct, as well as complete
(given that you want the dimensions to be consecutive).
And I'm leaning towards interpreting "n.e." here as pointed.
>Killing off the puny scope for a truth value to have homotopy by asking for a
>pointed or n.e. truth value sends you south west into the abyss to the left.
>Are there different rows of homotopic nothingness? I.e., something has no
>possible dimensions of homotopy starting from dimension n.
I don't think that there are different rows of the m = 0 column,
which is why the table properly doesn't need that column.
This weirdness at m = 0 has nothing to do with groupoids particularly;
it's just the weirdness that you get way say such a phrase as
"0 consecutive integers starting with n". Which is why I said:
>Then, of course, there's Toby's point - what's the big deal in having
>dimensions consecutive.
Right!
I ~have~ thought of some more things, actually,
but only about extending from groupoids to categories.
-- Toby
From: David Corfield
To: Toby Bartels
Subject: Re: Teeny bit of help needed
Cc: John Baez, James Dolan
Date: Thu, 10 Feb 2005 11:01:09 +0100
On Thursday 10 February 2005 10:24, Toby Bartels wrote:
> >By the way, anyone had any further profound thoughts about the top left
> > corner of the periodic table? You may remember I wanted to line things up
> > so that a row told you the starting dimension for any potential homotopy
> > (staying with n-groupoids), and the column specified in how many
> > consecutive dimensions homotopy is allowable.
> >
> > m consecutive possible dimensions
> > 1 2 3 4
> >potentially -1 truth value set gpoid 2-gpoid
> >starts at 0 n.e. set n.e. gpoid n.e. 2-gpoid ....
> >dim n 1 gp gpal gpoid ....
> > ...
> >
> >where n.e. is non-empty, or perhaps pointed would do.
> I think that this table is correct, as well as complete
> (given that you want the dimensions to be consecutive).
> And I'm leaning towards interpreting "n.e." here as pointed.
In the part of the factorisation of a functor which is forgetting purely
structure the fibre is a n.e. set rather than a pointed set. Does this not
count against the pointed interpretation?
One decategorifaction down in the factorisation of a function between sets
into an epi followed by a mono, the fibre above an element of the image is
n.e. rather than pointed.
David
From: Toby Bartels
Subject: Re: Teeny bit of help needed
To: David Corfield
Cc: John Baez, James Dolan
Date: Thu, 10 Feb 2005 02:18:48 -0800
David Corfield wrote:
>Toby Bartels wrote:
>>David Corfield wrote:
>>> m consecutive possible dimensions
>>> 1 2 3 4
>>>potentially -1 truth value set gpoid 2-gpoid
>>>starts at 0 n.e. set n.e. gpoid n.e. 2-gpoid ....
>>>dim n 1 gp gpal gpoid ....
>>> ...
>>And I'm leaning towards interpreting "n.e." here as pointed.
>In the part of the factorisation of a functor which is forgetting purely
>structure the fibre is a n.e. set rather than a pointed set. Does this not
>count against the pointed interpretation?
Hey, that's a good point!
>One decategorifaction down in the factorisation of a function between sets
>into an epi followed by a mono, the fibre above an element of the image is
>n.e. rather than pointed.
I never noticed it before, but homotopy fibres under factorisation fragments
are always in the m = 1 column, just as you're describing here.
That's pretty neat!
-- Toby
From: David Corfield
Subject: Re: Teeny bit of help needed
To: Toby Bartels
Cc: John Baez, James Dolan
Date: Thu, 10 Feb 2005 12:33:36 +0100
On Thursday 10 February 2005 11:18, Toby Bartels wrote:
> David Corfield wrote:
> >Toby Bartels wrote:
> >>David Corfield wrote:
> >>> m consecutive possible dimensions
> >>> 1 2 3 4
> >>>potentially -1 truth value set gpoid 2-gpoid
> >>>starts at 0 n.e. set n.e. gpoid n.e. 2-gpoid ....
> >>>dim n 1 gp gpal gpoid ....
> >>And I'm leaning towards interpreting "n.e." here as pointed.
> >In the part of the factorisation of a functor which is forgetting purely
> >structure the fibre is a n.e. set rather than a pointed set. Does this not
> >count against the pointed interpretation?
> Hey, that's a good point!
> >One decategorifaction down in the factorisation of a function between sets
> >into an epi followed by a mono, the fibre above an element of the image is
> >n.e. rather than pointed.
> I never noticed it before, but homotopy fibres under factorisation
> fragments are always in the m = 1 column, just as you're describing here.
> That's pretty neat!
Does that tally with the forgetting purely stuff functor projecting from
Set X Set to the first component? Isn't the fibre above any set the groupoid
of sets?
David
From: Toby Bartels
Subject: Re: Teeny bit of help needed
To: David Corfield
Date: Thu, 10 Feb 2005 04:04:03 -0800
Cc: John Baez, James Dolan
David Corfield wrote:
>Toby Bartels wrote:
>>I never noticed it before, but homotopy fibres under factorisation
>>fragments are always in the m = 1 column, just as you're describing here.
>>That's pretty neat!
>Does that tally with the forgetting purely stuff functor projecting from
>Set X Set to the first component? Isn't the fibre above any set the groupoid
>of sets?
That functor doesn't forget purely stuff, because it's not full.
(However, it is essentially surjective.) It factors as follows:
Set x Set -> Set x [Set] -> Set -> Set
Here, [C] is the set of (isomorphism classes of) objects of C,
as a discrete category. The fibre over (X,Y) in Set x [Set]
under the left map is the group Y! of autobijections of Y.
The fibre over X in Set under the middle map is the nonempty set [Set].
(And the fibre over X in Set under the right map is the truth value 1.)
It is by combining the nonempty set [Set] and the group Y!
(actually a family of groups indexed by an element Y of [Set])
that you get the nonempty groupoid Set, which is the fibre
of X under the composition of the left two maps (which is the whole map,
since the right map is an equivalence).
So conjecture: If you compose m maps in a row from a factorisation,
then the fibre will lie in column m. The row n is given by
how far to the right your composition begins (n = -1 on the far right).
Illustration: The first-component forgetful functor from Mon x Mon -> Set,
where Mon is the groupoid of monoids. This functor factorises as follows:
Mon x Mon -F-> Mon x [Mon] -G-> NonEmptySet -H-> Set
The fibre of (M,N) under F is Aut(N), the group of automorphisms of N.
The fibre of nonempty X under G is MonStruct(X) x [Mon], where MonStruct(X)
is the set of monoid structures on X; note that this a nonempty set.
And the fibre of arbitrary X under H is the truth value {X is nonempty};
this may be empty (specifically, whenever X is empty).
Thus the m = 0 column is produced (for n = 1, n = 0, and n = -1).
But also! The fibre of nonempty X under FG is MonStruct(X) x Mon,
the product of a nonempty set and a nonempty groupoid,
which is itself a nonempty groupoid. That's m = 1, n = 0.
And the fibre of arbitrary X under GH is MonStruct(X) x [Mon],
a set that may be empty (and is iff X is empty): m = 1, n = -1.
Finally, the fibre of arbitrary X under FGH is Mon(X),
the groupoid of monoids whose underlying set is (of cardinality) X.
This is a groupoid that may be empty: m = 2, n = -1.
After working through that example,
I now officially believe my conjecture!
(I'll even prove the conjecture for m = 0, the degenerate case.
The composition of 0 functors in a factorisation is the identity,
and every fibre under the identity is trivial. The m = 0 column
consists only of the trivial w-groupoid, as well.)
-- Toby
Here's a little discussion between David Corfield and John
Baez, which relates the above ideas to path integrals and
"idempotent analysis", also known as
"tropical mathematics". The idea of this subject
is to view minimization problems as sums or integrals in
which addition is replaced by the operator "min",
which picks out the smaller of two numbers. Thus, a
sum
a + b + c
turns into
a min b min c.
Why is this a good idea? Well, the nonnegative
real numbers form a "rig" under ordinary addition
and multiplication - where a "rig" is a ring without
negatives (additive inverses). The real numbers together
with positive infinity, R ∪ {+∞}, also form a rig under
minimization and addition, since addition distributes over minimization:
a + (b min c) = (a + b) min (a + c).
The reason we include the number +∞ is that it's
the identity for minimization:
a min +∞ = a
Idempotent analysis is just analysis where the rig
R+ = ([0,∞), + , 0, ×, 1)
has been replaced by the rig
Rmin = (R ∪ {+∞}, min, +∞, +, 0)
It's called "idempotent" because minimization is
idempotent, meaning:
a min a = a
But what's so great about all this? Well, in classical statics
we figure out what a system does by taking all possibilities and
finding the one with least
action, and this is nicely done using the rig Rmin.
In classical mechanics we
figure out what a system does by taking all possibilities and
finding the one with least action
and this is also nicely done using the rig Rmin.
But in statistical mechanics we figure out what a system
does by taking all possibilities and summing over them, weighted
by probabilities.
And, there's a marvelous big story about how classical
statics is a limiting case of statistical mechanics when the
temperature approaches zero, based on a way of "deforming"
the rig Rmin into the rig R+.
The basic idea goes back to Boltzmann. He discovered that
at temperature T, the probability of a system in thermal equilibrium
being found in a specific state whose energy is E is proportional to
exp(-E/T)
where I'm working in units where "Boltzmann's constant"
is 1. So, we have a way of converting from energies to relative
probabilities which depends on temperature. Energies add, but
relative probabilities multiply. The exponential in the above
formula converts addition to multiplication. So, it defines a
homomorphism from the monoid
(R ∪ {+∞}, +, 0)
to the monoid
([0, ∞), × 1)
Since it's one-to-one and onto, we can use it to pull back
the entire rig structure on
R+ = ([0,∞), + , 0, ×, 1)
to the set R ∪ {+∞}, giving us a rig we'll call RT.
And, this rig depends on the temperature T.
The "multiplication"
in RT is always just +, and the multiplicative identity is
always just 0, but the addition depends on T.
I know this is incredibly confusing, so let me try to say it with
a teeny bit less math jargon.
The point is that multiplication of probabilities
always corresponds to addition of energies,
But, addition of probabilities
corresponds to some operation on energies which
depends on the temperature! So, we get a rig of
energies RT which depends on the temperature T.
And, finally, the punchline: as the temperature approaches absolute
zero, the operations in the rig RT approach those in
Rmin. You can just check this.
So, using a bit of standard math jargon,
we say the rig Rmin has a "one-parameter
deformation", namely the rigs RT where our parameter
is T, which approach Rmin as T → 0. Note also that
all the rigs RT for T > 0 are isomorphic to R+.
What does all this mean physically? It takes a while to unravel
all these rigs, but the upshot is familiar: as it gets really
cold, statistical mechanics a la Boltzmann reduces to the simple
business of minimizing energy. All states except the least-energy
one(s) get assigned relative probability zero, since
exp(-E/T) → 0
as T → 0, unless E = 0.
Anyway, in the following conversation David starts by alluding to this
business. Then we shoot off in the direction of letting the rig
in question be the rig of truth values, with "or"
as + and "and" as ×. Then we realize that
truth values are the same as -1-groupoids, and try to generalize
everything to n-groupoids!
All this might be more fun if you start by reading some
articles on sci.physics.research about
matrix mechanics with an arbitrary commutative rig replacing the
complex numbers. It would be even more fun if I wrote a more
detailed introduction to "rigs in physics", but alas....
From: David Corfield
Subject: Re: very quick question
To: John Baez
Date: Sun, 15 Jan 2006 13:01:30 +0100
Now for a problem for future students if you carry on ideas from your
Fall 2003 quantum gravity seminar....
In this article:
G. L. Litvinov
The Maslov dequantization, idempotent and tropical mathematics: A brief introduction
Litvinov points out that corresponding to the Fourier transform for
C, there is the Legendre transform for the rig
Rmax = (R ∪ -∞, max, +, -∞, 0)
What is the corresponding construction for the rig of truth values?
Best, David
From: John Baez
Subject: Re: very quick question
To: David Corfield
Date: Sun, 15 Jan 2006 16:17:09 -080
You wrote:
>Now for a problem for future students if you carry on ideas from your
>Fall 2003 quantum gravity seminar....
>
>In this article:
>
>G. L. Litvinov
>The Maslov dequantization, idempotent and tropical mathematics: A brief introduction
>
>Litvinov points out that corresponding to the Fourier transform for
>C, there is the Legendre transform for the rig
>
>Rmax = (R ∪ -∞, max, +, -∞, 0)
I think the Legendre transform more directly generalizes the Laplace
transform.
In fact, as Jim Dolan explained but I never got around to retelling in
the Fall '03 seminar, the Legendre transform ("finding the minimum of
energy") is the temperature -> 0 limit of the Laplace transform
("summing over states weighted by exp(-E/kT)").
In other words, classical statics, where we minimize energy, is the
temperature -> 0 limit of statistical mechanics.
(Litvinov is maximizing instead of minimizing, but that's no big deal.)
And, the ultimate reason this works is that the rig
Rmin = (R ∪ {+∞}, min, +∞, +, 0)
has a one-parameter deformation, where the deforming parameter is temperature.
When we let this parameter become complex we get quantum mechanics and
Fourier transforms....
> What is the
> corresponding construction for the rig of truth values?
Probably something like "finding the possible outcomes". Finding what's
possible to do is a simplified version of "finding the least energetic
thing to do", which is in turn a simplified version of "doing
everything, but doing something of energy E with probability
proportional to exp(-E/kT)".
All this needs to be explained very clearly to the world, so everyone
will realize how cool it is!
Best,
jb
From: David Corfield
Subject: Re: very quick question
To: John Baez
Date: Mon, 16 Jan 2006 13:04:24 +0100
> When we let this parameter become complex we get quantum mechanics and
> Fourier transforms....
and it seems that the Mellin transform is not far away:
"the theory of the Laplace-, Fourier-, Mellin-, and Z-transforms are at
bottom the same subject."
http://en.wikipedia.org/wiki/Laplace_transform
Were you hinting this in week 217?
>> What is the
>> corresponding construction for the rig of truth values?
> Probably something like "finding the possible outcomes". Finding what's
> possible to do is a simplified version of "finding the least energetic
> thing to do", which is in turn a simplified version of "doing
> everything, but doing something of energy E with probability
> proportional to exp(-E/kT)".
So maps X -> {T,F} are subsets of X, and you find out whether it's possible to
get from x to y within X. Sounds a lot like homotopy theory to me. But then
the equivalence relationship of path connectedness is reflecting that it's a
groupoid enriched over {T,F}, or was that impoverished?
Best, David
Subject: Re: very quick question
From: John Baez
To: David Corfield
Cc: James Dolan
Date: Mon, 16 Jan 2006 09:00:47 -0800 (PST)
Hi -
>> When we let this parameter become complex we get quantum mechanics and
>> Fourier transforms....
> and it seems that the Mellin transform is not far away:
>
> " the theory of the Laplace-, Fourier-, Mellin-, and Z-transforms are at
> bottom the same subject."
>
> http://en.wikipedia.org/wiki/Laplace_transform
>
> Were you hinting this in week 217?
I wasn't thinking about that, but it's good for people to know that
all the transforms listed here are related by simple changes of variable,
so you shouldn't feel ignorant of one if you know about another.
>>> What is the
>>> corresponding construction for the rig of truth values?
>> Probably something like "finding the possible outcomes". Finding what's
>> possible to do is a simplified version of "finding the least energetic
>> thing to do", which is in turn a simplified version of "doing
>> everything, but doing something of energy E with probability
>> proportional to exp(-E/kT)".
> So maps X -> {T,F} are subsets of X, and you find out whether it's possible
> to get from x to y within X.
Right: and this is a kind of "path integral".
We can compute the "least action for a path from x to y" as an integral
in a rig where addition is minimization.
We can compute the "amplitude for a path from x to y" as an integral
in a rig where addition is addition.
We can compute the "possbility of a path from x to y" as an integral
in the rig of truth values, where addition is "or".
> Sounds a lot like homotopy theory to me.
Yes, I guess any topological space gives a boolean-valued 2-variable
function "can you get from x to y along a path?"
> But then
> the equivalence relationship of path connectedness is reflecting that it's a
> groupoid enriched over {T,F}, or was that impoverished?
Right!
The really interesting 2-variable function associated to a
topological space X is "the space of paths from x to y".
If X is a homotopy n-type, this function will take values in homotopy
(n-1)-types. If we think of a homotopy n-type as an n-groupoid this
function is just hom(x,y)!
But, we can decategorify hom(x,y) down to a homotopy -1-type, aka a
truth value, which is "true" if hom(x,y) is nonempty and "false" if
it's empty.
Best,
jb
© 2006 John Baez, Toby Bartels, James Dolan, David Corfield
baez@math.removethis.ucr.andthis.edu