Symplectic forms and Categories
Raymond E. Rogers
the Symplectic one and two forms on manifolds? I believe that I have
worked out the details but the picture seems very complicated.
Ray
James Gibbons
I'll take a stab at it. If we consider sym_1 and sym_2 as functors
from the category of diff. manifolds to the category of vector
spaces,and for a manifold X; sym_1(X)= all sympletic one forms on X
and sym_2(X)= all sympletic two forms on X. (Actually we are
considering cross sections of fiber bundles here). If f: X-->Y
is a map of diff. manifolds, sym_1(f)= f_1^* and sym_2(f)= f_2^*
as the pullback functions. Note what is occuring is that we have
represented the category Diff. into the category Vect. via the
sympletic functors. By the way, if we have a representation, say
F: Diff --> Vect,
we can define a "Fibered Cat." over Diff. by defining a "fiber"
over X as the vector space V_X= Hom(F(X),F(X)). Also can
define a "Gauge Field" via "parallel transport" along a "path"
f: X-->Y in Diff., by defining Del_f: V_X-->V_Y as:
[Del_f](K)= F(f) o K o F(f)^-1 : F(Y)-->F(Y).
(where K: F(X)-->F(X), F(f):F(X)-->F(Y), and "o" is composition),
thus Del_f turns endomorphisms on F(X) into endomorphisms on F(Y),
wich is just V_X and V_Y, the fibers (morphism vector spaces)
over X and Y, hence "parallel transport" is defined. Also, we can
define a functor T: [representations of Diff]-->Grps
by: T(F)= nat(F,F)= set of natural transformations F to F.
with a piontwise mult. given by: (h . g)_X = h_X o g_X.
where h and g are natural transformations. Can define
a reverse functor L: Grps-->[representations on C] , first defining C
by C= rep(G) = cat. of group representations on vector spaces.
and L(G)= F , where F:C-->Vect is just the forgetful functor
that just remembers the vector space the group was represented on.
Why all this nonsense? Because physics can be stated in terms of
category theory alone I believe. For instance, the strong
equivalence principle states that expressions make sense on all
manifolds and commute with all morphisms between those manifolds.
(John Baez, do I have this right?)
Jim Gibbons
john baez
Raymond E. Rogers <rero...@gte.net> wrote:
>Can somebody detail or point me to a Categorical style definition of
>the Symplectic one and two forms on manifolds? I believe that I have
>worked out the details but the picture seems very complicated.
Can you say what you mean by a "categorical style definition" in this
context? I'm not quite sure what you're looking for.
Btw, the usual sort of "symplectic form" on an arbitrary manifold
is a 2-form, not a 1-form. In certain cases, e.g. when the manifold
is a cotangent bundle with its usual symplectic structure, the
symplectic form is the exterior derivative of a 1-form called the
"symplectic potential". Is this what you're calling a "symplectic
1-form"? That terminology is not standard so I'm just guessing this
is what you might mean.
-----------------------------------------------------------------------
"Kill cross-platform Java by growing the polluted Java market" -
Paul Gross, head of Microsoft tools division, January 1998.
[Quoted in the Los Angeles Times, Monday November 9, 1998.]
john baez
James Gibbons <gib...@easyon.com> wrote:
>Why all this nonsense? Because physics can be stated in terms of
>category theory alone I believe.
Probably, but the goal should always be clarity, not "nonsense". :-)
>For instance, the strong
>equivalence principle states that expressions make sense on all
>manifolds and commute with all morphisms between those manifolds.
>(John Baez, do I have this right?)
Yes, though the manifolds may be required to have extra structure like
an "orientation", or extra stuff like a "spin structure". (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.) It's an
interesting question, how much extra structure should be allowed
in a "background-free" (aka "generally covariant") theory. Certainly
the structure should not be so much as to make the group of
diffeomorphisms preserving this structure finite-dimensional.
Btw, it sounds like you too know what a "symplectic 1-form" is -
at least you show no hesitancy in using the term. What is it??
James Gibbons
>
> Btw, it sounds like you too know what a "symplectic 1-form" is -
> at least you show no hesitancy in using the term. What is it??
Isn't it a canonical one form given by the inverse boundry operator
acting on a symplectic form? (with extra conditions).
Jim Gibbons
Toby Bartels
>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
to...@ugcs.caltech.edu
Raymond E. Rogers
Sorry I didn't get back sooner; my news server has dropped the last two
days of this group. Typically a symplectic two form is defined as
closed so there is a corresponding one form (locally). This is the one
form I was referring to. I am directly interested in cotangent bundles
but I decided to leave the condition out and see what a more general
question would produce.
Categorical (in the cotangent bundle case): I was referring to just
elementary mappings between the the various manifolds involved in the
cotangent bundle. A simple case seems to be T*M -> M -> T*(M) -> T(T*M)
; but when the actual parts are written for the symplectic one form it
looks like a mathematical pun with entities wildly jumping between
levels. Is this really the nature of the items and is there an elegant
expression? It all seemed simple till I started thinking about it! BTW:
I am only superficially familiar with Category theory; i.e. Geroch's
book.
Ray
***
Raymond E. Rogers
Thank for the detailed reply! It looks like what I had in mind. One
other item: What book would you recomend (Baez's ?) so I could read
replies like yours without going home and drawing pages of diagrams and
doing cross references for some hours? I have read (more or less)
Geroch's book but it didn't prepare me for this. Either that or I
should reread more carefully!
Ray
James Gibbons
Dear Ray,
Any good book on category theory should prepare you for
the articles (papers) in the book "Differential Geometric Methods
in Theoretical Physics", (Proceedings of 19th International confer-
ence in Rapallo,Italy , June 1990, edited by C. Bartocci, U. Bruzzo,
and R. Cianci, published by Springer-Verlag 1991. In it they detail
among other things some categorical applications to physics, where
I got my ideas. Also, another book: "Sheaves in Geometry and Logic,
(a first intoduction to topos theory)" by Saunders Mac Lane and
Leke Moerdijk, Springer-Verlag 1992, was also useful to me. A useful
hint: if we consider the algebra of all real valued functions
defined on a differential manifold and call it "A", and if we
consider a vector bundle over this manifold and consider the vector
space of all it's sections, this vector space of sections is really
a projective A-module! This is how we can move freely between
Algebra and Geometry in the Vector bundle case. Sheaves is just
when we consider the functions defined only on an open set of the
manifold. Since these open sets form a lattice, what we really have
is a functor S: [cat.lattice of open sets]-->[cat. of algebras],
where the sheaf S sends an object (open set) to the algebra of
functions defined on it (algebra). Using sheaves, (and it's
generalization to locales) we can construct a link between topology
and category theory. By the way, a topos is a special kind of
category, one that mimicks the category of sets. Since logic and
set theory are intertwined, using this more "general category"
(topos) we can construct all kinds of neat mathematical stuff.
Coming back to physics, since it appears we need topology
to do physics, this link might prove useful.
Jim Gibbons
john baez
Toby Bartels <to...@ugcs.caltech.edu> 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".
Toby Bartels
>Toby Bartels <to...@ugcs.caltech.edu> 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
to...@ugcs.caltech.edu
james dolan
-james dolan <jdo...@math.ucr.edu> 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.
Toby Bartels
>Toby Bartels <to...@ugcs.caltech.edu> 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 <jdo...@math.ucr.edu> 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
to...@ugcs.caltech.edu
james dolan
-james dolan <jdo...@math.ucr.edu> wrote:
-
->Toby Bartels <to...@ugcs.caltech.edu> 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.)