Re: Return of the Euler Characteristic of a Category
When my previous paper came out, someone here asked me to write a short summary. Toby beat me to it, so I’ll summarize this one instead.
Given a simplicial set $X$, you’d like to say that the Euler characteristic of $X$ is
$c_0  c_1 + c_2  \cdots,$
where $c_n$ is the number of nondegenerate $n$simplices (which I’m assuming to be finite). Usually this series is divergent. Nevertheless, you can try to sum it in the following way. Consider the power series $f_X(t) = \sum c_n t^n$. If you’re really lucky, this series converges in some neighbourhood of $0$ and has a unique analytic continuation to $1$, and you could then declare the Euler characteristic of $X$ to be the value at $1$ of this analytic continuation.
In particular, you can try this strategy on the nerve $X$ of any finite category, since this has only finitely many simplices of each dimension.
All of this so far was a suggestion of Clemens Berger. As it turns out, it was an excellent suggestion. What happens is this: if $X$ is the nerve of a finite category then $f_X(t)$ is in fact a rational function, whose value at $1$ (if finite) I call the series Euler characteristic of the category. And in many cases  perhaps the most interesting cases  this is equal to the Euler characteristic in my previous sense.
Two problems, though. First, series Euler characteristic isn’t invariant under equivalence! Second, there are examples of categories whose Euler characteristic and series Euler characteristic are different. (This follows from the first point, since the original Euler characteristic is invariant under equivalence.) So personally, I still prefer the original definition.
Overcoming divergence
You guys who post comments to two or three different threads on the
same day  I don’t know how you do it. I’ve been spending the last
few days thinking about John’s suggestion
that there might be a connection between the methods in my paper and a
couple of Tim Silverman’s posts (1,
2).
In fact, I still haven’t made it onto Tim’s second post (possibly the
longest comment that the nCafé has ever seen!). And I
still don’t really know what the connection is. Anyway, here are my thoughts.
In what Tim wrote, the following question is implicit:
Q. What’s the cardinality of a sequence $Y = (Y_0, Y_1,
\ldots)$ of finite sets?
You could say it’s $\sum Y_n$  but however good you might be at
summing divergent series, there are always going to be some awkward
customers that resist being summed. You could say that it’s the
sequence $(Y_0, Y_1, \ldots)$. As I’ll explain (see the
Challenge in my next comment), I think that the right answer is
neither of these.
On the other hand, my paper contains the following question:
Q. What’s the cardinality of a simplicial set $X$ with only
finitely many simplices of each dimension?
You might want to say  and in the paper, I do!  that it’s $\sum
(1)^n X^{nd}_n$, where $X^{nd}_n$ is the number of nondegenerate
$n$simplices. Again, you have the problem that this usually can’t be
summed in any sensible way.
There are some obvious similarities between the two questions. A
simplicial set has, of course, an underlying sequence of sets. But
when it comes to taking the cardinality (a.k.a. Euler
characteristic), you mustn’t ignore the simplicial structure: that
would be like defining the Euler characteristic of a topological space
to be the cardinality of its set of points. That’s why we had an
ordinary sum $\sum Y_n$ in one context and an alternating sum $\sum
(1)^n X^{nd}_n$ in the other.
I want to make some comments about the first question. I’ll begin by
revisiting a result of Tim’s.
The original definition of Euler characteristic ($=$ cardinality) of a
category assumed that the category was finite, and in particular had
finite homsets. It’s not clear what to do with a category with
infinite homsets, because we need to take the cardinality of each
homset, and the things we need to do with those cardinalities don’t
seem to work well with infinite cardinals.
An example of a category with finitely many objects but infinite
homsets is the free category $F(G)$ on a finite directed graph $G$.
Tim observed that although the homsets of $F(G)$ are infinite sets,
these sets are not just bags of indistinguishable elements  they
have some structure. Precisely, each homset in $F(G)$ is graded by
length, and the part in each degree is finite. In other words,
$F(G)$ is really an infinite sequence of finite sets.
Let’s pause for a moment to ‘recall’ that if $V$ is a monoidal
category and we already have a sensible notion of cardinality of
objects of $V$, then it’s possible to talk about the cardinality
of a finite $V$enriched category. (Simply take the definition
of Euler characteristic of an ordinary category and replace
finite sets by objects of $V$.) In particular, we can do this
when $V = FinSet^{\mathbb{N}}$, with the convolution tensor
product, declaring the cardinality of $Y \in FinSet^\mathbb{N}$
to be the formal power series
$\sum_n Y_n t^n \in \mathbb{Q}[[t]].$
Let $A$ be a finite $FinSet^\mathbb{N}$enriched category. We
have the matrix $\zeta_A$ of cardinalities of homsets, with
entries in $\mathbb{Q}[[t]]$. With luck, $\zeta_A$ has an
inverse $\mu_A$, and in that case, the cardinality of $A$ is the
sum of the entries of $\mu_A$. Thus $A \in \mathbb{Q}[[t]]$.
You can then try your hand at evaluating this formal power series
at $t = 1$ to obtain an actual number.
Now here’s Tim’s result. Take a finite directed graph $G$ with $g_0$
vertices and $g_1$ edges. Write $\gamma$ for the ‘incidence’ matrix
of $G$, that is, the $g_0 \times g_0$ matrix over $\mathbb{N}$ whose
$(i, j)$entry is the number of edges from the $i$th object to the
$j$th object. Let $F(G)$ be the free category on $G$, regarded as a
$FinSet^\mathbb{N}$enriched category as above. Then
$\zeta_{F(G)}(t)
=
1 + \gamma t + \gamma^2 t^2 + \cdots,$
(a matrix over $\mathbb{Q}[[t]]$), which has inverse
$\mu_{F(G)}(t)
=
1  \gamma t.$
The sum of the entries of the identity matrix $1$ is $g_0$, and the sum of
the entries of $\gamma$ is $g_1$, so
$F(G) = g_0  g_1 t \in \mathbb{Q}[[t]].$
But this is a mere polynomial over $\mathbb{Q}$, so it can
certainly be evaluated at $t = 1$, giving an answer of $g_0  g_1$. And this is the Euler characteristic of $G$ itself.
Tadah!
But there’s a fly in the ointment (or sand in the vaseline).
I’ve been flitting between two different opinions on what type of thing the cardinality of a
sequence of finite sets should be  one moment I’m saying it should be a
sequence of numbers (or equivalently, a formal power series), and the
next I’m saying that it should be a single number. I’ll suggest how
to remove the fly/sand in my next comment.
What’s the cardinality of a sequence of finite sets?
For various reasons (some mentioned in my previous comment,
some not), I want to issue a challenge. First let me set the
scene.
We’re interested in defining the ‘cardinality’ of objects of
various kinds. Ultimately it would be good to have a general
definition of the cardinality of objects of any kind, but
let’s put that ambition on hold. Given a new type of
object for which one would like a definition of cardinality, one
can apply the following
General Principle: Don’t try to define the cardinality of
an object. Instead, say when two objects have the same
cardinality.
To see how this works, suppose you’ve come up with a good
equivalence relation $\sim$, ‘has the same cardinality as’, on
the collection of objects. Then the cardinality of an object is
simply its equivalence class, and you might call $\{ objects \} /
\sim$ the collection of cardinals.
Very often the collection of objects has some structure  for
instance, they might be the objects of a rig category. If $\sim$
respects that structure then the cardinals inherit the structure
too  in this case, they’d form a rig.
(It might be that there’s a further map from the rig of cardinals
to some familiar rig such as $\mathbb{Z}$ or $\mathbb{C}$, or
even an isomorphism  but if so, that’s a bonus feature of the
particular type of object you’re considering, and not part of the
general procedure.)
A simple and familiar example is (possibly infinite) sets. It’s
easy to decide when two sets have the same cardinality: $\sim$ is
isomorphism. Because sets form a rig category, cardinals form a
rig.
A more sophisticated example is Schanuel’s Euler characteristic
of a polyhedral set, in his ‘Negative sets…’ paper.
A nonexample is my work on Euler characteristic of categories!
That’s why there’s this unattractive business of Euler characteristic
being undefined for some categories. It would be great if someone
could find an approach that conforms to the General Principle.
Now: I’d like to define the cardinality of a sequence of finite
sets in a way that follows the General Principle. In other
words, I’d like to decide when two sequences of finite sets
should have the same cardinality. This is easier said than done,
and leads to the following challenge.
Terminology: call a sequence $a = (a_n)_{n \in \mathbb{N}}$
of natural numbers summable if $a_n = 0$ for all but
finitely many values of $n$.
Challenge: Find an equivalence relation $\sim$ on the set
$\mathbb{N}^\mathbb{N}$ of sequences of natural numbers such that

if $a$ and $b$ are summable then $\sum a_n = \sum b_n$ iff $a
\sim b$

if $a \sim a'$ and $b \sim b'$ then $a + b \sim a' + b'$.
The idea is that the cardinality of a sequence $(X_n)$ of finite
sets would then be the equivalence class of $(X_n)_{n \in
\mathbb{N}}$ in $\mathbb{N}^\mathbb{N}/\sim$.
I probably haven’t succeeded in asking exactly the right
question. (If I knew the right question, I’d be halfway to the
right answer.) For example, there might be some unhelpful
solution to the Challenge involving the Axiom of Choice. A
satisfactory solution is meant to give us the right
notion of cardinality of a sequence of finite sets.
There are many variants of the Challenge. Some might
be easier. For example, you could replace sequences of
natural numbers by sequences of complex numbers.
And you could then replace ‘summability’ by any of the following:

Absolute convergence (of the series $\sum a_n$)

Convergence

‘Continuation summability’, i.e. summability by the method of
analytic continuation discussed at the beginning of this thread

‘Eulerian summability’, i.e. summability by the sort of magic
that Euler used. For example, by trusting his pen, Euler found that
$1 + 1 + 1 + \cdots = 1/2$
and
$1 + 2 + 3 + \cdots = 1/12,$
both of which were later confirmed by Riemann: his
zeta function satisfies $\zeta(0) = 1/2$ and $\zeta(1) = 1/12$.
You can also add to the list of conditions that $\sim$ should
satisfy. For instance, you might want to include:

Invariance under some group of permutations of $\mathbb{N}$
(permuting the sequence should perhaps leave the sum unchanged)

Invariance under rightshift: $(a_0, a_1, \ldots) \sim (0,
a_0, a_1, \ldots)$

Invariance under grouping, e.g. $a \sim (a_0 + a_1, 0, a_2 + a_3
+ a_4, \ldots)$

Multiplicativity: if $a \sim a'$ and $b \sim b'$ then $a \cdot a'
\sim b \cdot b'$, where $\cdot$ is convolution product.
I should maybe point out that some of these conditions are incompatible with some of the types of summability. For
example, if you want invariance under all permutations of
$\mathbb{N}$ then you have to stick to absolute convergence. If
you want invariance under grouping then you’d better not use
continuation summability: for that method gives
$1  1 + 1  1 + \cdots = 1/2$
(because $1  x + x^2  \cdots = 1/(1 + x)$), whereas grouping as
$(1  1) + (1  1) + \cdots$
gives an answer of $0$. If you want multiplicativity then you can’t use
Eulerian summability, since we have a convolution
$(1, 1, 1, \ldots ) \cdot (1, 1, 1, \ldots)
=
(1, 2, 3, \ldots)$
but
$(1 + 1 + 1 + \cdots ) \cdot (1 + 1 + 1 + \cdots )
=
(1/2)(1/2)
\neq
1/12
=
1 + 2 + 3 + \cdots.$
I’d be very happy to see a solution to any version of the
Challenge!
Re: Return of the Euler Characteristic of a Category
When my previous paper came out, someone here asked me to write a short summary. Toby beat me to it, so I’ll summarize this one instead.
Given a simplicial set $X$, you’d like to say that the Euler characteristic of $X$ is $c_0  c_1 + c_2  \cdots,$ where $c_n$ is the number of nondegenerate $n$simplices (which I’m assuming to be finite). Usually this series is divergent. Nevertheless, you can try to sum it in the following way. Consider the power series $f_X(t) = \sum c_n t^n$. If you’re really lucky, this series converges in some neighbourhood of $0$ and has a unique analytic continuation to $1$, and you could then declare the Euler characteristic of $X$ to be the value at $1$ of this analytic continuation.
In particular, you can try this strategy on the nerve $X$ of any finite category, since this has only finitely many simplices of each dimension.
All of this so far was a suggestion of Clemens Berger. As it turns out, it was an excellent suggestion. What happens is this: if $X$ is the nerve of a finite category then $f_X(t)$ is in fact a rational function, whose value at $1$ (if finite) I call the series Euler characteristic of the category. And in many cases  perhaps the most interesting cases  this is equal to the Euler characteristic in my previous sense.
Two problems, though. First, series Euler characteristic isn’t invariant under equivalence! Second, there are examples of categories whose Euler characteristic and series Euler characteristic are different. (This follows from the first point, since the original Euler characteristic is invariant under equivalence.) So personally, I still prefer the original definition.