Re: the mysteries of counting

In article <d9s1eu$5t8$1@xxxxxxxxxxxxxxxx>,
John Baez <baez@xxxxxxxxxxxxxx> wrote:

>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.

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...

>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 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.

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.

>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).

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 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.

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.

>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.

Yes!

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

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

>OK, but this space doesn't have the property that its higher homotopy
>groups vanish, and so it doesn't come from a category.

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.

>It comes from an infinity-category! I don't know what that means!
>No one does.

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.

>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.

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

>So, my question is,

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

>... 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,

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

>that's a tough question. So I'll ask an easier question:

Okay.

>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.

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

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

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???


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