Extending the n-category table
David Corfield
It has been argued that the k-tuply monoidal n-category table be extended
westward to complete three further points: (0, -2), (0, -1), (1, -1).
But how are we to understand westward movement as decategorification?
Presumably there needs to be a notion of an (-1) arrow and a (-2) arrow.
Once we have these, are there any objections to continuing northward?
You can see the delooping process going from
(0,0)-----> (-1,1)
set ---> {*}, 1 object set.
So why not deloop from
(-1,0)----->(0, -1)
? ------> empty set or {*}
What could '?' be ?
David Corfield
Squark
> It has been argued that the k-tuply monoidal n-category table be extended
> westward to complete three further points: (0, -2), (0, -1), (1, -1).
Where? Can you give any references or explain on your own?
Best regards,
Squark
------------------------------------------------------------------
Write to me using the following e-mail:
Skvark_N...@excite.exe
(just spell the particle name correctly and use "com" rather than
"exe")
John Baez
David Corfield <dn...@cam.ac.uk> wrote:
>It has been argued that the k-tuply monoidal n-category table be extended
>westward to complete three further points: (0, -2), (0, -1), (1, -1).
>But how are we to understand westward movement as decategorification?
>Presumably there needs to be a notion of an (-1) arrow and a (-2) arrow.
No; did you see my post on the category theory mailing list on
-1-categories and -2-categories? This explains what's really
going on. If you didn't see it, you should be able to find it
on the web, since all old posts to the category theory mailing
list are available there. Also try the posts by Toby Bartels
and james dolan here on s.p.r. concerning "properties", "structure",
and "stuff"; again, all old s.p.r. posts have been archived.
To whet your appetite: a 0-groupoid is a set, a -1-groupoid
is a truth value (true or false), and a -2-groupoid is a vacuous
truth value (true). It was a pleasant surprise to discover that
these curious concepts illuminate large tracts of logic and topology
that were previously obscured! When James and I write our paper
"Higher-dimensional algebra V: Feynman diagrams", I plan to put in
a lot about this, simply because it needs to be written down somewhere.
(I'll cc this to you since it's taken ages to reply.)
David Corfield
> David Corfield <dn...@cam.ac.uk> wrote:
> >But how are we to understand westward movement as decategorification?
> >Presumably there needs to be a notion of an (-1) arrow and a (-2) arrow.
> No;
...
> To whet your appetite: a 0-groupoid is a set, a -1-groupoid
> is a truth value (true or false), and a -2-groupoid is a vacuous
> truth value (true).
If the boundary of a point is the empty set, could we see a -1-groupoid as
the potential to bear a point (or label), either realised or not?
Then suspension of S^-1, takes us from a potential to bear a point to a
potential to bear a line, and then to the realisation of its extremities,
i.e., S^0.
About the march further West, does this mean you no longer see spectra as
Z-groupoids? In your
Categorification paper you describe a Z-groupoid as "some sort of gadget
with j-morphisms for all j in Z, all of which are equivalences." What were
these j-morphisms for j < -2 supposed to be?
You also said there that the theory of weak Z-categories "remains largely
terra incognita". Incognita is not the same as non-existent. So, do we fall
off the edge of the world if we head further west, or do we discover
America?
By the way, could you give us any idea as to when the long-awaited HDA5 will
appear?
David Corfield
John Baez
David Corfield <dn...@cam.ac.uk> wrote:
>John Baez wrote:
>> To whet your appetite: a 0-groupoid is a set, a -1-groupoid
>> is a truth value (true or false), and a -2-groupoid is a vacuous
>> truth value (true).
For a better explanation, I'll append some stuff that James Dolan, Toby
Bartels and myself wrote on -1-categories and -2-categories. See the
end of this post.
>If the boundary of a point is the empty set, could we see a -1-groupoid as
>the potential to bear a point (or label), either realised or not?
Yeah, I guess so...
>Then suspension of S^-1, takes us from a potential to bear a point to a
>potential to bear a line, and then to the realisation of its extremities,
>i.e., S^0.
I'm not quite sure what "and then to the realisation of its extremities"
means, but the first part seems okay - though it sounds more like the
way Aristotle might put it than the way I'd put it.
>About the march further West, does this mean you no longer see spectra as
>Z-groupoids?
Sure, Jim and I still see spectra as Z-groupoids, and by now other
people do too. -1-groupoids and -2-groupoids are just not very much
like Z-groupoids.
Loosely speaking, a Z-groupoid is a gadget that has weakly invertible
j-morphisms for j any integer. An n-groupoid is a gadget is a gadget
that has weakly invertible j-morphisms for 1 <= j <= n. A -1-groupoid
or -2-groupoid is what we get when we specialize the latter concept to n
= -1 or -2. Clearly that makes no sense! - and yet, by appropriate
finagling, some sense can be made out of it.
See? A Z-groupoid is a big complicated generalization of a groupoid.
A -1 or -2-groupoid is a peculiar and pathetically simple degenerate
case of a groupoid.
>In your
>Categorification paper you describe a Z-groupoid as "some sort of gadget
>with j-morphisms for all j in Z, all of which are equivalences." What were
>these j-morphisms for j < -2 supposed to be?
A quasi-historical answer is probably best. In topology people are
interested in sequences of spaces X_0, X_1, X_2, ... where the (n+1)st
homotopy group of X_{i+1} is the same as the nth homotopy group of
X_i. For example, suppose X_i is the loop space of X_{i+1}.
We call this sort of setup a spectrum, and it's very natural to define
the nth homotopy group of the spectrum to be the limit as n -> infinity
of the (n+i)th homotopy group of X_i. This makes sense whenever n
is any integer, not just any natural number.
To extend the relation between spaces and weak omega-groupoids to
include spectra, we thus need weak Z-groupoids.
>You also said there that the theory of weak Z-categories "remains largely
>terra incognita". Incognita is not the same as non-existent. So, do we fall
>off the edge of the world if we head further west, or do we discover
>America?
I'm in California, so if I keep heading further west I won't discover
America - I'll loop around and discover the mysterious East. :-)
But yes, there are some examples of weak Z-categories staring us in the
face: the weak Z-groupoids (or spectra) and the stable omega-categories,
like omega-Cat itself, or various versions of omega-Cob, which have
cobordisms between cobordisms between cobordisms.... as their
j-morphisms. All the examples I know are of one or the other of these
two types, but that's probably just my own personal ignorance.
One can probably concoct other weak Z-categories from sequences of
omega-categories just as topologists concoct spectra from sequences of
spaces, as described above. And they will probably be interesting!
For what it's worth, the gadgets lying in the intersection of [weak
Z-groupoids] and [stable omega-categories] could be called "stable
omega-groupoids", but people usually call them "connective spectra" -
spectra without any homotopy groups pi_n for n less than zero.
>By the way, could you give us any idea as to when the long-awaited HDA5 will
>appear?
Ugh! My goal is to write it this Spring, but I haven't made any
progress at all so far.
.........................................................................
From the category theory mailing list at
http://north.ecc.edu/alsani/ct99-00(8-12)/msg00304.html
Subject: categories: (-1)-categories and (-2)-categories
From: ba...@newmath.ucr.edu
Date: Thu, 14 Dec 2000 13:07:32 -0800 (PST)
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
.........................................................................
From: "james dolan" <jdo...@math.ucr.edu>
Subject: Re: Just Categories now
Date: 16 Nov 1998 00:00:00 GMT
Message-ID: <72pusp$3o5$1...@pravda.ucr.edu>
Organization: Department of Mathematics, University of California, Riverside
Newsgroups: sci.physics.research
X-Newsposter: Pnews 4.0-test50 (13 Dec 96)
toby bartels writes:
-john baez wrote:
->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.
From: james dolan <jdo...@math.ucr.edu>
Subject: Re: Just Categories now
Date: 05 Jan 1999 00:00:00 GMT
Message-ID: <76rulr$h7d$1...@pravda.ucr.edu>
References: <726r22$qr$1...@news-1.news.gte.net> <3647A0...@easyon.com> <72agsd$ook$1...@pravda.ucr.edu> <72b3pd$7...@gap.cco.caltech.edu>
Organization: Department of Mathematics, University of California, Riverside
Newsgroups: sci.physics.research
X-Newsposter: Pnews 4.0-test50 (13 Dec 96)
toby bartels writes:
-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.
From: jdo...@galaxy.ucr.edu (james dolan)
Subject: Re: Just Categories now
Date: 15 Jan 1999 00:00:00 GMT
Message-ID: <77k9fr$va6$1...@pravda.ucr.edu>
References: <3647A0...@easyon.com> <72b3pd$7...@gap.cco.caltech.edu> <76rulr$h7d$1...@pravda.ucr.edu> <77h2qd$o...@gap.cco.caltech.edu>
Organization: none
Newsgroups: sci.physics.research
toby bartels wrote:
-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.)
Toby Bartels
>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" <jdo...@math.ucr.edu>
>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.
Then if U: C -> D is any functor whatsoever (extra stuff),
then given an object of D, how many ways can it be given the stuff of C?
The answer to the question is a category.
>From: james dolan <jdo...@math.ucr.edu>
>Subject: Re: Just Categories now
>Date: 05 Jan 1999 00:00:00 GMT
>Toby Bartels wrote:
>>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.
For example, is a Riemannian manifold,
which is naturally equipped with a connection on the tangent bundle,
a space-with-connection with extra structure
(to wit the structure of a Riemannian metric that reproduces the connection)?
We have discussed this on this board before.
>From the groupoid POV, yes.
This is, in part, because isometries preserve the connection.
>From the category POV, no.
This is because uninvertible unitary maps need not preserve the connection,
so there is no functor.
The problem here has light shed on it by the following example:
A _semigroup_ is a set with a binary operation;
a _monoid_ is a semigroup with an identity element.
(A group is a monoid with inverses.)
Now, is a monoid a semigroup with extra property?
The definition as phrased above suggests so,
and from the groupoid POV that is entirely correct.
But the functor from _Mon_ to _SGrp_ is not full on *all* morphisms,
because monoid homomorphisms are required to preserve the identity.
Thus, from the *morphisms'* POV, the identity is a structure to be preserved.
Had we not made this requirement on monoid homomorphisms,
then a monoid would indeed be a semigroup with extra property, from any POV.
I like to describe the situation in which
the groupoid POV says that the functor is extra property
but the category POV says only that the functor is extra structure
by saying that there *is* extra structure
but the structure is *definable* in terms of the old structure
(in this case, the structure of the semigroup alone, the binary operation).
Jim doesn't like this way of looking at it,
but we do agree that the phenomenon is there
(and has something to do with definability in a logical sense).
Now we see what went wrong with
the functor from _RiemMan_ to _ManW/Conn_,
the functor that was extra structure from the groupoid POV
but didn't even exist from the category POV.
The problem is that the morphisms in _RiemMan_
did not preserve the Riemannian connection.
We can create a new category, _RiemManW/Conn_,
whose morphisms *are* required to preserve this
(in addition to the metric itself).
Then the picture looks like this:
RiemManW/Conn
/ \
/ \
RiemMan ManW/Conn
The functor on the right is the functor of extra structure, from any POV.
The functor on the left is *also* a functor of extra structure,
in fact a functor of *only* extra structure, from the category POV.
However, this extra structure (the Levi Civita connection)
is definable in terms of the structure in _RiemMan_ (the metric).
So from the groupoid POV, it's actually a functor of extra property.
However, there is no extra property in this structure --
*only* extra structure, because every metric has a Levi Civita connection --
so in fact the extra property from the groupoid POV is the vacuous property
and the functor on the left is an equivalence of groupoids.
This makes it invertible, so -- from the groupoid POV --
the bottom row can be filled in with a functor from _RiemMan_ to _ManW/Conn_
that, like the functor on the right, is of extra structure.
But from the category POV, no such functor exists.
All of this -n-category stuff suggests a creation story that I'm developing.
In the beginning, there was nothing.
But with nothing came nothingness, the vacuity, which was something.
So now there was nothingness and somethingness,
vacuity and triviality, falsehood and truth,
which were 2 things.
So now there was a set of nothingness and somethingness,
and the whole realm of sets of elements
sprang up out of the set of nothingness and somethingness,
and sprouted functions between them
to relate them back to the set of nothingness and somethingness.
So now there was a category of sets and functions,
and the whole realm of categories of objects and morphisms
sprang up out of the category of sets and functions,
and sprouted functors between them,
to relate them back to the category of sets and functions,
which sprouted natural transformations between *them*,
to relate the relationships.
So now there was ...
-- Toby
to...@math.ucr.edu
Christopher Simmons
and something occurred to me, which may or may not be relevant. I hope that
you'll bare with me.
Firstly, suppose we forget about associativity and identities, back to
basics time. Now, suppose we say a 2-oid is just a category, without
associativity and identities. That is, we have a set (I'm sticking to all
set based for now) of objects O and arrows A and maps m_1, m_2, and a
partial operation . : A \times A \to A, where p.q is defined only when
p m_2 = q m_1. Now, suppose we have, as above, but now we have three maps
m_1, ..., m_3 and
pqr exists only when
p m_1 p m_2 p m_3
q m_1 q....
r m_1 r...
is symmetric, with pqr : p m_1 \to q m_2 \to r m_3. In general an n-oid has
maps m_i (i=1, .., n) and the n-product of
p_i (i = 1, .., n) : p_1 m_1 \to ...\to p_n m_n.
All this makes me think one could just paste together general
structures..... Anyway. Remember how, in a groupoid, we may take p : e \to
f to an arrow p^{-1} : f \to e. notice that S_{e,f} \cong {1,-1} as a
multiplicative group; thus, S_{e,f} acts on arrows p : e \to f. If we were
dealing with a three-oid, we can instead make S_{e,f,g} act on an arrow p :
e\to f \to g, by a homomorphism; ie so that if, a,b,c \in S_{e,f,g} then p^a
: ea \to fa \to ga and p^{ab} = (p^{a})^{b} and p^id = p. And so forth.
I'm not sure that this is going anywhere, but I'm finding it all quite
compelling and just needed to tell someone. :) thanks.
Chris Simmons.
Christopher Simmons
"Christopher Simmons" <Cp_Si...@btopenworld.com> wrote in message
news:a9vim6$9hn$1...@knossos.btinternet.com...
down to the following. Suppose we had an n-oid, defined as above, and a
group G which acts on the right of {1, .., n} by bijection - and we have an
action, exponentiation, of arrows by elements of G in such a way that, if
a,b \in G and p : (e_j)_{j=1}^{j=n}, then
p^a : (e_{ja}), p^1 = p and (p^a)^b = p^(ab) and, if p_i : (e_{i,j}), then
(\pi p_i)^a = \pi (p_{ia}^a) : (e_{ka,ka}) (taking all the i's, j's and k's
from 1 to n). (**)
If you're having trouble getting your head round (**), look below to the
2-oid example, then put n-3, pick some element of S_3. Draw a picture with
three terms with pqr defined, apply the permutation...
This is well-defined, since if we put q_i = p_{ia}^a and f_{i,j} =
e_{ia,ja}, then since the matrix E = (e_{i,j}) is symmetric,
so is F = (f_{i,j}) so that the product of the q_i's exists. To phrase
what's going on intuitively, if we have some permutation a of {1, .., n}
then applying this permutation to a product of n n-topes is equivalent to
applying it to each n-tope in the product, and computing their product in a
new order, as per the said permutation.
This firstly clarifies what happens in a `non associative groupoid' (the
thing defined at the start); consider the multiplicative group {1, -1}. We
define an action on arrows (in a groupoid), if p : e \to f then
p^1 = p and
p^{-1} : f \to e, as per usual. Also, 1 acts on {1,2} as the identity,
and -1 as the permutation (12).
The rule (**) then translates as saying if p : e \to f and q: f \to g then
(pq)^{-1} = q^{-1}p^{-1}.
The way of reaching this rule through such an large generalisation just
seems too beautiful to be ignored to me.
Questions which immediately pop to mind are:
- What happens if M is a monoid?
- What if G is a groupoid?
- What, for that matter, if we use a category C instead? And...
- Could this give a means of pasting general topes (ie number of objects may
vary between arrows)?