baez

Some of you might enjoy this:

The Mysteries of Counting:
Euler Characteristic versus Homotopy Cardinality

We all know what it means for a set to have 6 elements, but what sort
of thing has -1 elements, or 5/2? Believe it or not, these questions
have nice answers. The Euler characteristic of a space is a generalization
of cardinality that admits negative integer values, while the homotopy
cardinality is a generalization that admits positive real values. These
concepts shed new light on basic mathematics. For example, the space of
finite sets turns out to have homotopy cardinality e, and this explains
the key properties of the exponential function. Euler characteristic and
homotopy cardinality share many properties, but it's hard to tell if they
are the same, because there are very few spaces for which both are
well-defined. However, in many cases where one is well-defined, the
other may be computed by dubious manipulations involving divergent series -
and the two then agree! The challenge of unifying them remains open.

For details see:

http://math.ucr.edu/home/baez/counting/

apollonius de tyane

I have something like a question for you, maybe related to this. Do
you mind if, while asking it, I do an impression of you, and load my
question down with a lot of expository chatter? Impressions are a form
of flattery. The highest!

The natural numbers categorify to finite sets (is how you would put
it.)
This means that there is a nice way to associate a natural number to a
finite set, and under this association certain operations on finite
sets are...associated...with certain operations on natural numbers.

I learned this in grade school. The reason you (OK, I admit it (but
*I'm* anonymous): we) like to use the word "categorify" when talking
about this is because it gives us a glamorous and mysterious question
to ask about almost any mathematical thing: what does the thing
categorify to? Glamorous because there is a richer theory of the
categorified thing than of the original thing ("finite sets" is short
for "finite sets and functions between them"; these are more
interesting than natural numbers, which are interesting), and
mysterious because the question is hard to answer.

Like, the integers? Is a mathematical thing. What does it categorify
to?

There are probably all kinds of answers to this question, in varying
degrees of half-bakedery. I will sketch the one I know best, which is
my favorite.

The integers are fake! Or, at least, fabricated. You fabricate them
by pretending that you can subtract a big number from a small one. The
idea is to fabricate the categorification of the integers from the
categorification of the natural numbers.

I know how to do this because I read a paper by Graeme Segal called
"categories and cohomology theories". But I have to bamboozle you in
two different ways; the first way is serious, and is what my question
is about. The second isn't, at least if you're a true believer in
infinity-categories.

You're still reading!

The first bamboozle is that I don't actually know how to do this for
"finite sets and functions between them," but only for "finite sets and
bijections between them." The reason is that bijections can be
inverted, and other functions can't be. The reason I care about that
is the second bamboozle.

The second bamboozle is that part of the glamorous, mysterious
hocus-pocus I'm going on about allows me to replace categories where
everything can be inverted with topological spaces. For real!
Basically, I will build the space by giving it, to start, as many
points as there are finite sets. Then I'll draw as many paths between
two points as there are bijections between the sets. Then I will do
whatever's necessary to make sure that the "higher homotopy groups
vanish." A similar thing doesn't work for functions that aren't
bijections, because I don't know how to draw a path you can't invert
(draw backwards).

So now I have a space that I'll call X. It is a satisfactory answer to
the question "what do natural numbers categorify to" if you only care
about bijections and you're a true believer. Now, I'm going to
fabricate what the integers categorify to, by pretending that I can
always subtract one element of X from another.

Oops: first I should tell you how I *add* two points of X. The answer
is disjoint union. Some points of X are finite sets. Some others are
points that lie on paths which are bijections between finite sets. I
can add these kinds of points by saying "disjoint union." It's a
tedious-to-verify, but I hope believable, fact that I can make sense of
this for any two points on X.

Now, if you know how to build the integers from the natural numbers,
you know how to build a "categorification" of the integers from X. The
answer will be another topological space Y; it's a very famous space
whose homotopy groups are the stable homotopy groups of spheres.

That deserves a dramatic pause, and an attribution. It's called the
"Barrat-Priddy/Quillen" theorem.

OK, but this space doesn't have the property that it's higher homotopy
groups vanish, and so it doesn't come from a category. It comes from
an infinity-category! I don't know what that means! No one does. But
they do know that an infinity category has things called 1-morphisms,
2-morphisms, 3-morphisms,... and they know that it should be possible
to tell when one of these things is invertible, and that if every
single one of them is invertible then you should be able to replace it
with a topological space, and that the topological space is a
satisfactory replacement for the infinity-category.

So, my question is, what if I want to categorify functions that aren't
bijections? The right answer should be an infinity-category Z whose
invertible morphisms form the
infinity-category-that-is-a-topological-space Y. Since I don't know
what an infinity-category is, and neither do you, that's a tough
question. So I'll ask an easier question:

Start with Z. If you throw out all 1-morphisms, 2-morphisms,
3-morphisms, ... that aren't invertible, then you get something you can
replace with a topological space. But if you keep all the 1-morphisms,
and only throw away non-invertible 2-morphisms, 3-morphisms, etc., you
get something you can replace with a "topological category." I *do*
know what that means! It means a category whose set of arrows is a
topological space.

My question is: *which* topological space is it? Graeme Segal doesn't
give a good answer...

Thanks!
A. de T.

baez

Hmm. Saying I'm "chattering" doesn't sound flattering. But,
since you actually do some nice exposition yourself, you must
know that math is infinitely more interesting when it goes hand
in hand with a bit of what Douxadis calls "paramathematics":

Apostolous Douxadis, Embedding mathematics in the soul:
narrative as a force in mathematics education,
http://www.apostolosdoxiadis.com/files/essays/embeddingmath.pdf

As he says:

"Of course: abstraction, irrelevance, purity, formalism make for
good mathematics.... But sadly, they make for bad mathematics
education. Each one of these concepts - abstract, irrelevance,
purity, formalism - pushes mathematics further away from a growing
human being, a being whose psyche is in the phase of it development
that no soft-brained psychologist but a great mathematician, Alfred
North Whitehead, calls the Romantic Phase."

Right. The set of natural numbers is the decategorification of the
category of finite sets.


"Associated", especially guarded by ellipses like that, sounds a bit
spooky and vague. But in fact decategorification is a completely
systematic process! In fact, it's a 2-functor:
Decat: Cat -> Set

assigning to each category C its set Decat(C) of isomorphism classes, to each
functor F: C -> D the corresponding function Decat(F): Decat(C) -> Decat(D),
and to each natural transformation the identity. (The category Set becomes
a 2-category in a trivial way, with only identity 2-morphisms.)

When we apply this to FinSet we get N, the set of natural numbers.
The product and coproduct on FinSet give x and + for natural numbers, and so on.


Wow! Good education. But yes, we learn in grade school how natural
numbers are stand-ins for isomorphism classes of finite sets. We don't
learn all the formalism lurking behind this process, but we do
learn how it works - or should: if we don't, numbers are meaningless.


Right! While decategorification is a systematic, turn-the-crank
affair, in practice we often start with some math that's been developed
using sets, and are trying to guess what it could be the decategorification
*of*. This is an *unsystematic* and therefore mysterious process.
And yes, it's very glamorous, because when one succeeds one taps into
a deeper world of meaning, full of its own interesting questions and theorems.

I have various tentative answers to this fascinating puzzle - only one
of which appears here:

http://math.ucr.edu/home/baez/counting/

but I'll give them in a separate post, because I have to go pick
up my new glasses from the optician. And then I'll also take a stab
at the more *precise* question at the end of your post - I need to
think about it a bit.

Indeed!

Best,
jb

baez

Sorry: it assigns to each natural *isomorphism* the identity.
Decategorification takes a category, promotes the isomorphisms
to equations, and throws out the rest of the morphisms - so it
doesn't know what to do with natural transformations that aren't
natural isomorphisms.

So above "Cat" must stand for the 2-category of categories, functors
and natural *isomorphisms*. And, as I said before, "Set" stands for
the 2-category of sets, functions, and identity 2-morphisms.

It's sort of awkward, but decategorification is inevitably an
ugly business. Getting rid of the top-level morphisms is like
cutting off someone's head. Or those nasty jobs some people do
of pruning trees, where they lop off all the branches over a
certain height, leaving sad stumps.

Anyway, on to the question of categorifying the integers...

This isn't a bamboozle. The real bamboozle is switching from the
majestic world of infinity-categories to the less majestic but far
better understood world of infinity-groupoids... which can be treated as topological spaces.


This famous trick is called "the geometric realization of the nerve
of a groupoid". But, you can also take the geometric realization of
the nerve of a category! The big difference is that the higher
homotopy groups may not vanish. But, you can do both groupoids
and more general categories using the same method.

Draw a dot for each object:

x

Draw an edge for each morphism:
x---f-->y

Draw a triangle for each composable pair of morphisms:

y
/ \
f g / \
x---fg-->z


and so: an n-simplex for each length-n composable string of morphisms!


So, in technojargon: you've taken the groupoid of finite sets
and formed the geometric realization of its nerve. The result
is a space with one connected component for each natural number
n. This connected component is called K(S_n,1), but what it *is*
is the space of all n-element subsets in R^infinity, with the obvious
topology on it.

In my lecture:

http://math.ucr.edu/home/baez/counting/

I call your space X "the space of finite sets". It's the space of all
finite subsets of R^infinity. I like to claim that R^infinity is the
space mathematicians are working in when they're drawing pictures of
sets as bunches of dots. They don't really mean these dots are on the
*plane* - that's just a limitation of current-day blackboards!

And, these days I call your space not X but "E", because (as I show in the
talk) its cardinality is e, the base of the natural logarithms. Homotopy
cardinality, that is!!! This is a funny sort of cardinality that applies
to a large class of topological spaces... and E has homotopy cardinality e.

Anyway, this is all starting from the *groupoid* of finite sets.
You could also take the geometric realization of the nerve of
the *category* of finite sets if you wanted! I leave it as a puzzle
to work out what it is - or more precisely, what it's homotopy equivalent to.

Wow! Drama! Andre Joyal has a suitably dramatic name for this space:
he calls it the "true integers". Most homotopy theorists call it S,
or the "sphere spectrum".

I explain it here: http://math.ucr.edu/home/baez/week102.html

Wait a minute... *every* space is homotopy equivalent to the geometric
realization of the nerve of some *category*. It's just *groupoids* that
give spaces with vanishing higher homotopy groups.

Actually lots of people know what "infinity-category" means - the
problem is, they don't all agree! Various definitions of infinity-category
have been proposed:

http://arxiv.org/abs/math.CT/0107188

and a bunch of them are probably "right" - for example, Batanin's globular
definition and Street's simplicial definition. The problem is that we
don't know how to relate these definitions, much less do really interesting
things with them.

BUT, you would be happy with infinity-groupoids here, and those are
infinitely better understood: Kan complexes are a perfectly good
simplicial approach to those.


Right. A topological space, or for that matter a Kan complex.


Whew - I was beginning to wonder when that was coming!

I actually do, but it doesn't help me much.

And of course the space of objects, too...


Let's see if I can understand you. You're trying to create
a topological category where the space of objects is just what
I call E, the space of finite subsets of R^infinity... but what
should its space of morphisms be?

Is that the question???

urs schreiber

"Apollonius de Tyane" <apolloniusdetyane@yahoo.com> schrieb im Newsbeitrag
news:d9q972$lbs$1@news.ks.uiuc.edu...


I don't know if it matters for what you have in mind, but I would like to
say that I have thought about this question a little recently in a
physics-motivated context and believe that there is something rather
interesting going on.

I am interested in knowing not the categorification of the integers, but
more generally the categorification of the set of K-valued functions on some
"space" M, where K is the natural numbers, or the integers, or the rationals
or, the reals or, ultimately, the complex numbers.

The reason for being interested in this is that a good categorification of
the set of complex-valued functions on some space should go a long way
towards a categorification of spectral geometry and of (supersymmetric)
quantum mechanics and might have to say something about string theory.

John Baez had argued in

J. Baez
Higher Dimensional Algebra II: 2-Hilbert Spaces
Adv. Math. 127 (1997) 125-189
http://math.ucr.edu/home/baez/2hilb.ps

that a good categorification of the set of complex valued functions is the
functor category of functors from a base 2-space (base category) to Hilb,
the category of Hilbert spaces. A similar idea has appeared for instance in
the work by Baas, Dundas and Rognes.

Now, in Hilb we cannot "subtract" and "divide". But there is a more or less
straightforward way to enhance the above functor category in such a way that
a notion of subtraction and division does appear. Interestingly to me, the
resulting structure does indeed have a relation to string theory.


http://golem.ph.utexas.edu/string/archives/000578.html .

apollonius de tyane

Hmm. The "true integers" is a dramatic name, and gets to the point,
but I think it's jumping the gun a little. I think the TRUE true
integers are yet-to-be constructed; they should be an infinity-category
that I called Z last time (because it comes after X and Y, not because
it's the first letter of "zahl"), that bears the same relationship to
the sphere spectrum (what I called Y, and homotopy theorists call S (or
QS^0, or Omega^infinity S^infinity, or lots of other things) and Joyal
calls the "true integers") that the "category of finite sets" bears to
the "groupoid of finite sets."

Wow. There are more complicated names in math than there are
complicated ideas. Let me restate that as an SAT ****ogy (recently
extinct!):

Z::Y as functions::bijections


I think this is a red herring. The nerve construction does turn
categories into spaces, and every space comes from a category in this
way, but the space is not a satisfactory replacement for the category
unless the category was a groupoid. E.g., because of the answer to
your "nerve of the category of finite sets" puzzle. Besides, there are
some technical things that come up that probably aren't worth talking
about unless you actually have to write a proper math paper, like the
fact that Segal actually needs a subtle variation of the nerve
construction...

....so let's just say that groupoids are the same thing as topological
spaces with vanishing higher homotopy groups, and that
infinity-groupoids are the same thing as topological spaces. And
please don't tease me for playing fast-and-loose with the word "same".


Oops, right. But I mean something with a discrete set of objects.

No!

I'm trying to create a topological category, with a discrete set of
objects, which decategorifies to the integers, but is a better
approximation to the true true integers than Y is. One way to measure
the fact that it's a better approximation is that there should be a
special functor from the (totally discrete) category of finite sets to
it, in the same way that there's a special continuous map from X to Y.

I'll expound. Maybe it's hyperbole to say no one knows what an
infinity category is, but it's not to say that *I* don't know what one
is. But I do know that a topological space is a special kind of one.
I'm arguing that a topological category (with a discrete set of
objects) is a *less* special kind of one.

Why? Well, an infinity category has a (discrete) set of objects.
Between any two objects there's an infinity-category of morphisms.
Those infinity-categories contain infinity-groupoids (everthing that's
invertible). We can replace those infinity-groupoids by topological
spaces, yielding a topological category.

(When speculating about infinity-categories, it's nice to know that
topological spaces are examples. But any phenomena you notice this way
have probably been noticed by some homotopy theorist, somewhere.
Topological categories (with a discrete set of objects) are also
examples, and sometimes you can spot NEW phenomena in this way. Like
Jacob Lurie's paper on the arxiv about infinity-topoi...)

Dig?

A. de T.

apollonius de tyane

I can't resist pointing this out before I look at your links; sorry in
advance if it's redundant or irrelevant:

I'm not sure that the rationals, reals, and complex numbers will have
very interesting categorifications. The reason is that the stable
homotopy groups of spheres are not very interesting when you tensor
them with these fields.

What I mean is: if you look at the process of building the rational
numbers from the integers, and mimic that process starting with Joyal's
"true integers" that John was talking about, you just get the ordinary
rational numbers back.

On the other hand, maybe something interesting would happen if we had
the TRUE true integers that I talked about in my other post (maybe yet
to appear). Or maybe there are other categorifications of the integers
to start with, which is maybe what you're getting at below. John
points out in the counting paper that started the thread that
*positive* rational numbers *do* have an interesting categorification.
Finite groupoids! Cool! But maybe subtraction is really important to
you in string theory...

baez

Okay, that sounds good.

But, how can we get our hands on these TRUE true integers?
I came up with an idea about this. But, not for the integers:
just for the natural numbers. I'll describe an omega-categorified
version of the category of finite sets.

Let E be the space of finite subsets of R^infinity.

For starters, there should be an omega-groupoid where the objects
are points in E, where the morphisms are paths in E, where the
2-morphisms are paths-of-paths in E, and so on. This is an
omega-categorified version of the *groupoid* of finite sets,
and it's nothing new - we've talked about it before.

But, there should also be an omega-category where the objects
are points in E, and the morphisms are paths in E that allow
two or more points in a finite subset of R^infinity to COLLIDE AND FUSE!
We also allow points to POP INTO EXISTENCE.

These paths aren't continuous in the usual topology on E, but
we can demand that they're continous except at these catastrophes,
and I can guess we should demand that only finitely many of these
catastrophes occur.

Note: there's a built-in "arrow of time" in this setup.
We don't allow points in our finite subset of R^infinity to SPLIT,
and we don't allow them to DISAPPEAR!

This reflects the fact that a function can map many points to one,
but not one to many. It can also map no points to one, but not
one point to none.

We can also define 2-morphisms and higher morphisms by considering
paths of paths, paths of paths of paths, etc - but I admit I haven't
worked out how this should go.

If we massage this thing a bit we can probably make it into a
category enriched over Top. In other words: right now composition
is not associative, but there's probably a topology on each hom-set
for which composition is associative up to coherent homotopies, so
I think we get an A_infinity category. But, these things can always
be strictified to make composition associative on the nose, so we
get a "category enriched over Top" - which is what you were calling
a "topological category".

("Topological category" is also used to mean a "category internal
to Top", where there is a topology on the set of all objects and a
topology on the set of all morphisms. These include "categories
enriched over Top" as a special case, namely those for which the
space of objects is discrete.)

apollonius de tyane

OK, I like this. Let me see if I understand.

An object is a bunch of points:

x x x

which, like you've said, live in R^infinity.

An arrow is a tree.

I mean: one way to go from

x x x (call this set S, a subset of R^infinity)

to

x x x (call this set T, another subset of R^infinity)

is

x x x (source)
x x x
x x (collide)
x x
x x x (wink into existence)
x x x
x x x
x x x (target)


Where really the picture (which is a "tree" in the sense that it's an
acyclic one-dimensional figure) is somehow mapped into R^infinity, so
that the source and target agree with S and T.

Now, it's pretty clear what it means for one map of a tree into
R^infinity to be close to another one -- but somehow be careful about
what this means for two different trees -- and so the hom sets are
topological spaces.

Composition, etc, is almost obvious...

....but if you work it out it will only be associative up to homotopy,
whatever that means. But one thing it means is that we can strictify
it without changing the homotopy types of the hom sets, so, up to
homotopy, we
already have the right answer.

Wait, but what's that answer?

I think it's this: every connected component of every hom set is
contractible, and there are as many connected components as there are
functions from the source set to the target set. I won't explain why
unless someone starts arguing with me.

So, in some sense, this infinity-category is equivalent to the ordinary
category of finite sets, and it's nothing new.

I *still* like the answer, because the category you're describing is
really, really flabby. For every function between discrete, finite
sets, you've given be great big infinite-dimensional contractible blob.
This kind of thing is called "fibrant replacement," in other contexts.
If there is a construction you have for ordinary things (like taking
the fixed points of a group action, or the global sections of a sheaf,
or, more to the point, going from the natural numbers to the integers),
and you want some kind of categorified version of it (maybe more
usually called a "homotopy version" or "derived version"), sometimes
the following recipe is successful: replace the ordinary thing with a
flabby-enough thing equivalent to it (in some sense), and then apply
your original construction.

So, now I want to know, if we apply the original construction (natural
numbers to the integers) to this flabby category of finite sets, what
do we get? But I think one has to be a little creative to interpret
the "original construction" in this context...I mean a little more
creative than me.

Thoughts?

A. de T.