Definition. The centre of a category $\mathcal{C}$ is the set $\mathrm{Z}(\mathcal{C})$ defined by
\begin{align*}
\mathrm{Z}(\mathcal{C}) &\mathbin{\overset{\mathrm{def}}{=}} \int_{A\in\mathcal{C}}\mathrm{Hom}_{\mathcal{C}}(A,A)\\
&\cong \mathrm{Nat}(\mathrm{id}_{\mathcal{C}},\mathrm{id}_{\mathcal{C}}),
\end{align*}
while the trace of $\mathcal{C}$ is the set $\mathrm{Tr}(\mathcal{C})$ defined by
\begin{align*}
\mathrm{Tr}(\mathcal{C})
&\mathbin{\overset{\mathrm{def}}{=}}
\int^{A\in\mathcal{C}}\mathrm{Hom}_{\mathcal{C}}(A,A)\\
&\cong \mathrm{End}(\mathcal{C})/\mathord{\sim},
\end{align*}
where $\mathrm{End}(\mathcal{C})$ is the set of all endomorphisms of $\mathcal{C}$ and $\mathord{\sim}$ is the equivalence relation on $\mathrm{End}(\mathcal{C})$ generated by $g\circ f\sim f\circ g$. Here, the quotient map
$$\mathrm{tr}\colon\mathrm{End}(\mathcal{C})\to\mathrm{Tr}(\mathcal{C})$$
is called the trace map.
Monoid Structure. The center of $\mathcal{C}$ comes with a natural monoid structure given by composition of natural transformations, and every monoidal category structure on $\mathcal{C}$ descends to a monoid structure on $\mathrm{Z}(\mathcal{C})$ (making it a commutative monoid by Eckmann–Hilton) and $\mathrm{Tr}(\mathcal{C})$.
Example. Take a group $G$ and view it as a one-object category $\mathrm{B}G$. Then, we can show that the centre of $\mathrm{B}G$ is the usual group-theoretic centre $\mathrm{Z}(G)$ of $G$, while the trace of $\mathrm{B}G$ is the set of conjugacy classes of $G$.
Question. What is the centre and trace of the simplex category $\Delta$?
What about the centre and trace of the augmented simplex category $\Delta_+$, which comes with a monoidal structure $\oplus$, thus making its centre a commutative monoid and its trace a monoid?
Best Answer
The center is trivial : as Benjamin said in the comments, an endomorphism of the identity is the identity on $\Delta^0$, and using the maps $\Delta^0\to \Delta^n$ you find that any endomorphism of the identity is the identity.
For the trace, you can use the following fact coming from unwinding the definition : if $f,g$ are composable in both orders in $C$, then $tr(fg) = tr(gf) \in Tr(C)$ where I let $tr$ of an endomorphism be the class it defines in $Tr(C)$.
Using this, for any endomorphism $f\in \Delta$, writing it as a surjection followed by an injection and inducting on size shows that every $tr(f)$ is $tr(id_S)$ for some $S\in \Delta$ (note that any automorphism is the identity). Furthermore, $f\mapsto $ the cardinality of the fixed point set of $f$ defines a morphism $Tr(\Delta)\to \mathbb N$ (this is not obvious, but a fun exercise in combinatorics) which distinguishes all the $tr(id_S), S\in \Delta$ so that $Tr(\Delta)\cong \mathbb N_{>0}$