Just Categories now
james dolan
->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 tha
-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.
john baez
Toby Bartels <to...@ugcs.caltech.edu> wrote:
>james dolan <jdo...@math.ucr.edu> 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.
john baez
>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?