TITLE: How does it End?
QUESTION [21 upvotes]: A recent project has forced my colleague and me to take a rather abstract approach to dynamical systems, and the following definition arose naturally in that context.
Let $\mathcal{C}$ be a category. Its endomorphism category $\text{End}(\mathcal{C})$ is defined as follows. The objects are the endomorphisms $f:x \to x$ of $\mathcal{C}$ and a morphism from $f:x \to x$ to $g: y \to y$ is a morphism $h:x \to y$ in $\mathcal{C}$ so that the commuting relation $h \circ f \equiv g \circ h$ holds in $\mathcal{C}$.
That's all we really need for the project, but now a perverse question springs to mind. Note that $\text{End}$ itself defines a dynamical system. After all, we have $\text{End}(\text{End}(\mathcal{C}))$ and so on by iterating the process. I'm interested in the asymptotic behavior. So, here are some questions:

Which categories are fixed points of $\text{End}$, i.e., for which $\mathcal{C}$ is $\text{End}(\mathcal{C})$ equivalent to $\mathcal{C}$?

Slightly more interesting is the question of when $\text{End}(\mathcal{C})$ is homotopy equivalent to $\mathcal{C}$. And while I suspect that the answer to the following question is no, I was unable to prove it:

Is it possible to have non-trivial periodic orbits, for instance, non-equivalent categories $\mathcal{C}$ and $\mathcal{D}$ so that $\text{End}(\mathcal{C})$ is equivalent to $\mathcal{D}$ and vice-versa?

REPLY [8 votes]: Here are some easy examples within groupoids.$\DeclareMathOperator\End{End}$
For a group $G$, $\End(BG)$ is equivalent to $\coprod_{x\in I} BC_x$ (which is skeletal) where $I$ is a set containing one representative of each conjugacy class of $G$ and $C_x$ denotes the centralizer of $x$ in $G$ (in fact, $\End(BG)$ is isomorphic to $\coprod_{x \in I} BC_x \times E_{[G:C_x]}$ where $E_n$ is the groupoid on $n$ objects and a unique isomorphisms between every pair of them). From this we see, for example, that:

The only finite groupoids that are fixed points for $\End$, or even belong to an $\End$-cycle, are the finite discrete categories: when you apply $\End$ to a groupoid $C$ you always get a copy of $C$ in the result and, unless, $C$ is discrete, you get a groupoid with a strictly larger skeleton.
Among groupoids there are no cycles that are not simply fixed points (again, because $BC$ always contains a copy of $C$, so the multiplicity of any given group in $\End^n(C)$ is a monotone function of $n$).
Given any $\mathcal{G}$ is any class of groups such that whenever $x\in G \in \mathcal{G}$ then also $C_x \in \mathcal{G}$, we can form the groupoid $C := \coprod_{G \in \mathcal{G}} \coprod_{\mathbb{N}} BG$ and this will be an $\End$-fixed point. In particular, for any Abelian group $G$, $\coprod_{\mathbb{N}}BG$ is a fixed point. (And notice too, that the equivalence $C \simeq \End(C)$ is not canonical in these examples.)
As a simple, explicit example, $\End(BS_3) = BS_3 \sqcup B\mathbb{Z}/2 \sqcup B\mathbb{Z}/3$, and $\End(BG) = |G|\cdot BG$ for any Abelian $G$ (as was already mentioned by Benjamin Steinberg --and where $n \cdot C$ is the coproduct of $n$ copies of the category $C$), so we get that $\End^n(BS_3) = BS_3 \sqcup (2^n-1)\cdot B\mathbb{Z}/2 \sqcup \frac{1}{2}(3^n-1) \cdot B\mathbb{Z}/3$. (And similarly, for any fixed group, working out the formula for the $n$-th iterate $\End^n(BG)$ is just a a matter of patience.)

ADDED: It might be worth mentioning that when you stick to just groupoids, the question is very homotopy theoretical: equivalence is the same as homotopy equivalence and $\End$ is just the free loop space functor. In other words, the question for groupoids becomes "study the dynamical system given by the free loop space functor on the homotopy category of 1-types". A reasonable analogue of this question for $\infty$-groupoids would be to study the free loop space functor on the homotopy category (of all spaces) as a dynamical system. An analogue of the above description of $\End(BG)$ works for topological groups too: the free loop space $\mathop{Map}(S^1,BG)$ is the homotopy quotient $G/G$ where $G$ acts on itself by conjugation — this is analogous because the description given above of (the isomorphism type of) $\End(BG)$ for discrete $G$ is just the action groupoid of $G$ acting on itself by conjugation.