% Copyright (c) 2004 Miguel Carrion Alvarez
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% under the terms of the GNU Free Documentation License, Version 1.2
% or any later version published by the Free Software Foundation;
% with no Invariant Sections, no Front-Cover Texts, and no Back-Cover
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% the license.
\nopagenumbers
\input xy
\xyoption{all}
\def\aut{\rm Aut}
\def\from{\colon}
\def\To{\Rightarrow}
\def\Z{{\bf Z}}
\centerline{\bf Math 260: Categorifying the Riemann Zeta Function\strut}
\centerline{Miguel Carri\'on \'Alvarez\strut}
\proclaim 1. Subcategories have faithful inclusions.
Assume that~$C$ is a subcategory of~$D$, that is, there is a
functor~$i\from C\to D$ which is an inclusion of objects and
morphisms, preserving units and composition. To show that~$i$ is
faithful, we need to show that~$i$ is one-to-one on morphisms, but
this is automatically true since~$i$ is an inclusion of morphisms.
\proclaim 2. Full subcategories have full, faithful inclusions.
If $C$ is a full subcategory of~$D$, then the functor~$i\from C\to D$
induces a bijection between~$\hom_C(c,c')$
and~$\hom_D\bigl(i(c),i(c')\bigr)$, so~$i$ is clearly onto for
morphisms, and hence full in addition to faithful.
\proclaim 3. Skeletons have essentially surjective, full and faithful
inclusions.
Assume now that, in addition,~$C$ is a skeleton of~$D$, and let~$d\in
D$. There is then exactly one object~$c\in C$ in the isomorphism class
of~$d$, that is,~$i(c)\cong d$ and~$i$ is essentially surjective, in
addition to full and faithful.
\proclaim 4. Groupoid cardinality and equivalence.
If~$C$ is a groupoid, then its cardinality is defined by
$$
|C|\colon=\sum_{[c]}{1\over|\aut(c)|}.
$$
Suppose that~$F\from C\to D$ is an equivalence of categories. Then,
because~$F$ is full and faithful, if~$f\from c\to c'$ is an
isomorphism then~$F(f)\from F(c)\to F(c')$ is, too. Conversely,
Let~$g\from d\to d'$ be an isomorphism in~$D$. Because~$F$ is
essentially surjective, there are isomorphisms~$\alpha_d\from F(c)\to
d$ and~$\alpha_{d'}\from F(c')\to d'$, and so there is an
isomorphism~$\alpha_dg\alpha_{d'}^{-1}\from F(c)\to F(c')$. Since~$F$
is full,~$\alpha_d g\alpha_{d'}=F(f)$ for some~$f\from c\to c'$, which
is an isomorphisms because~$F$ is faithful. It follows that
$$
|D|=\sum_{[d]}{1\over|\aut(d)|}=\sum_{[c]}{1\over\bigl|\aut F(c)\bigr|}=\sum_{[c]}{1\over\bigl|\aut(c)\bigr|}=|C|,
$$
and~$|D|$ is finite if, and only if,~$|C|$ is finite.
\proclaim 5, 6. Cyclic sets.
A skeleton of~${\rm Cyc}_n$ consists of a single cyclically
ordered~$n$-element set (since all such are mutually isomorphic) and
all bijections of it preserving the cyclic order, which is isomorphic
to~$\Z_n$. It follows that
$$
{\rm Cyc}_n\simeq 1//\Z_n.
$$
Clearly, then
$$
|{\rm Cyc}_n|=|1//\Z_n|={1\over n},
$$
since~${\rm Cyc}_n$ consists of exactly one isomorphism class and the
automorphism group~$\Z_n$ has order~$n$.
\proclaim 7,8. The Riemann~$\zeta$ groupoid.
The groupoid of~$k$-tuples of cyclically ordered~$n$-element sets
is~$({\rm Cyc}_n)^k\simeq (1//\Z_n)^k\simeq 1//(\Z_n)^k$. This means
that
$$
Z(k)\simeq\sum_{n>0}{1//(\Z_n)^k}.
$$
Then,
$$
|Z(k)|=\sum_{n\ge 0}{1\over n^k}=\zeta(k),
$$
where~$\zeta$ is the Riemann zeta function. It follows that~$Z(k)$ is
tame for all~$k>1$. That~$\zeta(2)={\pi^2\over 6}$ is well-known.
\proclaim 9. The~$\zeta$ structure type.
The generating function of $F\from Z(k)\to E$ is
$$
|F|(z)=\sum_{n\ge 0}{z^n\over n^k},
$$
which is sometimes denoted~$\zeta_k(z)$ in applications to
mathematical physics.
\bye