$\newcommand{\unsim}{{\sim}}$The set of conjugacy classes of a group $G$ is the quotient of $G$ by the equivalence relation $\sim_1$ obtained by declaring $a\sim_1b$ if there exists some $g\in G$ such that $b=g^{-1}ag$. This agrees also with the equivalence relation

• $\unsim_2$ given by declaring $a\sim_2b$ if there exists some $g\in G$ such that $ga=bg$;
• $\unsim_3$ generated by $ab\sim_3 ba$.

Passing from groups to monoids, each of the above relations continue to make sense with a feel modifications; given a monoid $M$, we define $\unsim_1$, $\unsim_2$, and $\unsim_3$ to be the equivalence relation generated by (taking the symmetric and transitive closures of) the relations

• $\unsim'_1$ declaring $a\sim'_1 b$ if there exists some invertible $g\in M$ such that $a=g^{-1}bg$;
• $\unsim'_2$ declaring $a\sim'_2 b$ if there exists some $m\in M$ such that $ma=bm$;
• $\unsim'_3$ declaring $ab\sim'_3 ba$.

Each of the three relations above leads to distinct notions of conjugacy classes for monoids. They also admit the following category-theoretic descriptions: \begin{align*} M/\unsim_1 &\cong \mathsf{Fun}(\mathrm{B}\mathbb{N},\mathrm{B}M)/\{\text{isos}\},\\ M/\unsim_2 &\cong \pi_0(\mathsf{Fun}(\mathrm{B}\mathbb{N},\mathrm{B}M)),\\ M/\unsim_3 &\cong \int^{A\in\mathrm{B}M}\mathrm{Hom}_{\mathrm{B}M}(A,A). \end{align*} (The end $\int_{A\in\mathrm{B}M}\mathrm{Hom}_{\mathrm{B}M}(A,A)$ is also a familiar notion: it is the centre of $M$.)

More generally, we may replace $\mathrm{B}A$ with an arbitrary category $\mathcal{C}$, leading to three sensible notions of "conjugacy classes of categories". Similarly, we also have notions of

• "categories of conjugacy classes of monoidal categories$^\dagger$ and $2$-categories";
• "$\infty$-groupoids of conjugacy classes of $\infty$-categories";
• "$\infty$-categories of conjugacy classes of monoidal $\infty$-categories";
• and so on.

_{$^\dagger$For instance, given a monoidal category $\mathcal{C}$, we may define its category of "$\sim_3$-conjugacy" classes by first delooping it into a bicategory $\mathrm{B}\mathcal{C}$ and then taking the pseudo-bicoend of $\mathsf{Hom}_{\mathrm{B}\mathcal{C}}(-,-)$. (Again, the pseudo-biend is also an interesting object: it is the Drinfeld centre of $\mathcal{C}$.)}

Main Question. Let $\kappa$ be a cardinal. Is there a nice(-ish) description of the set $$\mathrm{Cl}(\mathsf{Sets}_{\leq\kappa})\cong\int^{X\in\mathsf{Sets}_{\leq\kappa}}\mathrm{Hom}_{\mathsf{Sets}_{\leq\kappa}}(X,X)$$ of ($\unsim_3$-)conjugacy classes of the category $\mathsf{Sets}_{\leq\kappa}$ of sets of cardinality $\leq\kappa$?

_{Also, what about the category $\int^{\mathcal{C}\in\mathsf{Cats}}_{\mathsf{ps}}\mathsf{Fun}(\mathcal{C},\mathcal{C})$ of conjugacy classes of ("appropriately small"; e.g. finitely generated) categories, or the $\infty$-groupoid $\int^{X\in\mathcal{S}}\mathrm{Map}(X,X)$ of conjugacy classes of ("appropriately small"; e.g. "$\pi$-finite") $\infty$-groupoids?}

$\newcommand{\unsim}{{\sim}}$The set of conjugacy classes of a group $G$ is the quotient of $G$ by the equivalence relation $\sim_1$ obtained by declaring $a\sim_1b$ if there exists some $g\in G$ such that $b=g^{-1}ag$. This agrees also with the equivalence relation

• $\unsim_2$ given by declaring $a\sim_2b$ if there exists some $g\in G$ such that $ga=bg$;
• $\unsim_3$ generated by $ab\sim_3 ba$.

Passing from groups to monoids, each of the above relations continue to make sense with a feel modifications; given a monoid $M$, we define $\unsim_1$, $\unsim_2$, and $\unsim_3$ to be the equivalence relation generated by (taking the symmetric and transitive closures of) the relations

• $\unsim'_1$ declaring $a\sim'_1 b$ if there exists some invertible $g\in M$ such that $a=g^{-1}bg$;
• $\unsim'_2$ declaring $a\sim'_2 b$ if there exists some $m\in M$ such that $ma=bm$;
• $\unsim'_3$ declaring $ab\sim'_3 ba$.

Each of the three relations above leads to distinct notions of conjugacy classes for monoids. They also admit the following category-theoretic descriptions: \begin{align*} M/\unsim_1 &\cong \mathsf{Fun}(\mathrm{B}\mathbb{N},\mathrm{B}M)/\{\text{isos}\},\\ M/\unsim_2 &\cong \pi_0(\mathsf{Fun}(\mathrm{B}\mathbb{N},\mathrm{B}M)),\\ M/\unsim_3 &\cong \int^{A\in\mathrm{B}M}\mathrm{Hom}_{\mathrm{B}M}(A,A). \end{align*} (The end $\int_{A\in\mathrm{B}M}\mathrm{Hom}_{\mathrm{B}M}(A,A)$ is also a familiar notion: it is the centre of $M$.)

More generally, we may replace $\mathrm{B}A$ with an arbitrary category $\mathcal{C}$, leading to three sensible notions of "conjugacy classes of categories". Similarly, we also have notions of

• "categories of conjugacy classes of monoidal categories$^\dagger$ and $2$-categories";
• "$\infty$-groupoids of conjugacy classes of $\infty$-categories";
• "$\infty$-categories of conjugacy classes of monoidal $\infty$-categories";
• and so on.

_{$^\dagger$For instance, given a monoidal category $\mathcal{C}$, we may define its category of "$\sim_3$-conjugacy" classes by first delooping it into a bicategory $\mathrm{B}\mathcal{C}$ and then taking the pseudo-bicoend of $\mathsf{Hom}_{\mathrm{B}\mathcal{C}}(-,-)$. (Again, the pseudo-biend is also an interesting object: it is the Drinfeld centre of $\mathcal{C}$.)}

Main Question. Let $\kappa$ be a cardinal. Is there a nice(-ish) description of the set $$\mathrm{Cl}(\mathsf{Sets}_{\leq\kappa})\cong\int^{X\in\mathsf{Sets}_{\leq\kappa}}\mathrm{Hom}_{\mathsf{Sets}_{\leq\kappa}}(X,X)$$ of ($\unsim_3$-)conjugacy classes of the category $\mathsf{Sets}_{\leq\kappa}$ of sets of cardinality $\leq\kappa$? In particular, what are the answers for the cases $\kappa=\aleph_0$ and $\kappa=2^{\aleph_0}$?

_{Also, what about the category $\int^{\mathcal{C}\in\mathsf{Cats}}_{\mathsf{ps}}\mathsf{Fun}(\mathcal{C},\mathcal{C})$ of conjugacy classes of ("appropriately small"; e.g. finitely generated) categories, or the $\infty$-groupoid $\int^{X\in\mathcal{S}}\mathrm{Map}(X,X)$ of conjugacy classes of ("appropriately small"; e.g. "$\pi$-finite") $\infty$-groupoids?}

$\newcommand{\unsim}{{\sim}}$The set of conjugacy classes of a group $G$ is the quotient of $G$ by the equivalence relation $\sim_1$ obtained by declaring $a\sim_1b$ if there exists some $g\in G$ such that $b=g^{-1}ag$. This agrees also with the equivalence relation

• $\unsim_2$ given by declaring $a\sim_2b$ if there exists some $g\in G$ such that $ga=bg$;
• $\unsim_3$ generated by $ab\sim_3 ba$.

Passing from groups to monoids, each of the above relations continue to make sense with a feel modifications; given a monoid $M$, we define $\unsim_1$, $\unsim_2$, and $\unsim_3$ to be the equivalence relation generated by (taking the symmetric and transitive closures of) the relations

• $\unsim'_1$ declaring $a\sim'_1 b$ if there exists some invertible $g\in M$ such that $a=g^{-1}bg$;
• $\unsim'_2$ declaring $a\sim'_2 b$ if there exists some $m\in M$ such that $ma=bm$;
• $\unsim'_3$ declaring $ab\sim'_3 ba$.

Each of the three relations above leads to distinct notions of conjugacy classes for monoids. They also admit the following category-theoretic descriptions: \begin{align*} M/\unsim_1 &\cong \mathsf{Fun}(\mathrm{B}\mathbb{N},\mathrm{B}M)/\{\text{isos}\},\\ M/\unsim_2 &\cong \pi_0(\mathsf{Fun}(\mathrm{B}\mathbb{N},\mathrm{B}M)),\\ M/\unsim_3 &\cong \int^{A\in\mathrm{B}M}\mathrm{Hom}_{\mathrm{B}M}(A,A). \end{align*} (The end $\int_{A\in\mathrm{B}M}\mathrm{Hom}_{\mathrm{B}M}(A,A)$ is also a familiar notion: it is the centre of $M$.)

More generally, we may replace $\mathrm{B}A$ with an arbitrary category $\mathcal{C}$, leading to three sensible notions of "conjugacy classes of categories". Similarly, we also have notions of

• "categories of conjugacy classes of monoidal categories$^\dagger$ and $2$-categories";
• "$\infty$-groupoids of conjugacy classes of $\infty$-categories";
• "$\infty$-categories of conjugacy classes of monoidal $\infty$-categories";
• and so on.

_{$^\dagger$For instance, given a monoidal category $\mathcal{C}$, we may define its category of "$\sim_3$-conjugacy" classes by first delooping it into a bicategory $\mathrm{B}\mathcal{C}$ and then taking the pseudo-bicoend of $\mathsf{Hom}_{\mathrm{B}\mathcal{C}}(-,-)$. (Again, the pseudo-biend is also an interesting object: it is the Drinfeld centre of $\mathcal{C}$.)}

Is there a "nice(-ish)" description of the set $$\mathrm{Cl}(\mathsf{Sets})\cong\int^{X\in\mathsf{Sets}}\mathrm{Hom}_{\mathsf{Sets}}(X,X)$$ of ($\unsim_3$-)conjugacy classes of sets?

What about the category $\int^{\mathcal{C}\in\mathsf{Cats}}_{\mathsf{ps}}\mathsf{Fun}(\mathcal{C},\mathcal{C})$ of conjugacy classes of categories, or the $\infty$-groupoid $\int^{X\in\mathcal{S}}\mathrm{Map}(X,X)$ of conjugacy classes of $\infty$-groupoids?

$\newcommand{\unsim}{{\sim}}$The set of conjugacy classes of a group $G$ is the quotient of $G$ by the equivalence relation $\sim_1$ obtained by declaring $a\sim_1b$ if there exists some $g\in G$ such that $b=g^{-1}ag$. This agrees also with the equivalence relation

• $\unsim_2$ given by declaring $a\sim_2b$ if there exists some $g\in G$ such that $ga=bg$;
• $\unsim_3$ generated by $ab\sim_3 ba$.

Passing from groups to monoids, each of the above relations continue to make sense with a feel modifications; given a monoid $M$, we define $\unsim_1$, $\unsim_2$, and $\unsim_3$ to be the equivalence relation generated by (taking the symmetric and transitive closures of) the relations

• $\unsim'_1$ declaring $a\sim'_1 b$ if there exists some invertible $g\in M$ such that $a=g^{-1}bg$;
• $\unsim'_2$ declaring $a\sim'_2 b$ if there exists some $m\in M$ such that $ma=bm$;
• $\unsim'_3$ declaring $ab\sim'_3 ba$.

Each of the three relations above leads to distinct notions of conjugacy classes for monoids. They also admit the following category-theoretic descriptions: \begin{align*} M/\unsim_1 &\cong \mathsf{Fun}(\mathrm{B}\mathbb{N},\mathrm{B}M)/\{\text{isos}\},\\ M/\unsim_2 &\cong \pi_0(\mathsf{Fun}(\mathrm{B}\mathbb{N},\mathrm{B}M)),\\ M/\unsim_3 &\cong \int^{A\in\mathrm{B}M}\mathrm{Hom}_{\mathrm{B}M}(A,A). \end{align*} (The end $\int_{A\in\mathrm{B}M}\mathrm{Hom}_{\mathrm{B}M}(A,A)$ is also a familiar notion: it is the centre of $M$.)

More generally, we may replace $\mathrm{B}A$ with an arbitrary category $\mathcal{C}$, leading to three sensible notions of "conjugacy classes of categories". Similarly, we also have notions of

• "categories of conjugacy classes of monoidal categories$^\dagger$ and $2$-categories";
• "$\infty$-groupoids of conjugacy classes of $\infty$-categories";
• "$\infty$-categories of conjugacy classes of monoidal $\infty$-categories";
• and so on.

_{$^\dagger$For instance, given a monoidal category $\mathcal{C}$, we may define its category of "$\sim_3$-conjugacy" classes by first delooping it into a bicategory $\mathrm{B}\mathcal{C}$ and then taking the pseudo-bicoend of $\mathsf{Hom}_{\mathrm{B}\mathcal{C}}(-,-)$. (Again, the pseudo-biend is also an interesting object: it is the Drinfeld centre of $\mathcal{C}$.)}

Main Question. Let $\kappa$ be a cardinal. Is there a nice(-ish) description of the set $$\mathrm{Cl}(\mathsf{Sets}_{\leq\kappa})\cong\int^{X\in\mathsf{Sets}_{\leq\kappa}}\mathrm{Hom}_{\mathsf{Sets}_{\leq\kappa}}(X,X)$$ of ($\unsim_3$-)conjugacy classes of the category $\mathsf{Sets}_{\leq\kappa}$ of sets of cardinality $\leq\kappa$?

_{Also, what about the category $\int^{\mathcal{C}\in\mathsf{Cats}}_{\mathsf{ps}}\mathsf{Fun}(\mathcal{C},\mathcal{C})$ of conjugacy classes of ("appropriately small"; e.g. finitely generated) categories, or the $\infty$-groupoid $\int^{X\in\mathcal{S}}\mathrm{Map}(X,X)$ of conjugacy classes of ("appropriately small"; e.g. "$\pi$-finite") $\infty$-groupoids?}

$\newcommand{\unsim}{{\sim}}$The set of conjugacy classes of a group $G$ is the quotient of $G$ by the equivalence relation $\sim_1$ obtained by declaring $a\sim b$ iff there exists $g\in G$ such that $b=g^{-1}ag$. This agrees also with the equivalence relations $\sim_2$ and $\sim_3$ on $G$ generated by [$ma\sim_2 bm$, $m\in G$] and $ab\sim_3 ba$.

These three equivalence relations make sense also for monoids, but now they are inequivalent notions: for a general monoid $M$, we can have $M/\unsim_1\ncong M/\unsim_2\ncong M/\unsim_3$. Each of these also admit nice descriptions in terms of category-theoretic notions: \begin{align*} M/\unsim_1 &\cong \mathsf{Fun}(\mathrm{B}\mathbb{N},\mathrm{B}M)/\{\text{isos}\},\\ M/\unsim_2 &\cong \pi_0(\mathsf{Fun}(\mathrm{B}\mathbb{N},\mathrm{B}M)),\\ M/\unsim_3 &\cong \int^{A\in\mathrm{B}M}\mathrm{Hom}_{\mathrm{B}M}(A,A). \end{align*} (The end $\int_{A\in\mathrm{B}M}\mathrm{Hom}_{\mathrm{B}M}(A,A)$ is also a familiar notion: it is the centre of $M$.)

Now, each of the above expressions makes perfect sense if instead of $\mathrm{B}A$ we started with a general category $\mathcal{C}$. This leads to three sensible notions of "conjugacy classes of a category", and there are also similar notions of "categories of conjugacy classes of monoidal categories$^\dagger$ and $2$-categories", "$\infty$-groupoids of conjugacy classes of $\infty$-categories", "$\infty$-categories of conjugacy classes of monoidal $\infty$-categories", and so on.

_{$^\dagger$E.g. given a monoidal category $\mathcal{C}$, we may define its category of "$\sim_3$-conjugacy" classes by first delooping it into a bicategory $\mathrm{B}\mathcal{C}$ and then taking the pseudo-bicoend of $\mathsf{Hom}_{\mathrm{B}\mathcal{C}}(-,-)$. (Again, the pseudo-biend is also an interesting object: it is the Drinfeld centre of $\mathcal{C}$.)}

Is there a "nice(-ish)" description of the set $$\mathrm{Cl}(\mathsf{Sets})\cong\int^{X\in\mathsf{Sets}}\mathrm{Hom}_{\mathsf{Sets}}(X,X)$$ of ($\unsim_3$-)conjugacy classes of sets?

What about the category $\int^{\mathcal{C}\in\mathsf{Cats}}_{\mathsf{ps}}\mathsf{Fun}(\mathcal{C},\mathcal{C})$ of conjugacy classes of categories, or the $\infty$-groupoid $\int^{X\in\mathcal{S}}\mathrm{Map}(X,X)$ of conjugacy classes of $\infty$-groupoids?

$\newcommand{\unsim}{{\sim}}$The set of conjugacy classes of a group $G$ is the quotient of $G$ by the equivalence relation $\sim_1$ obtained by declaring $a\sim_1b$ if there exists some $g\in G$ such that $b=g^{-1}ag$. This agrees also with the equivalence relation

• $\unsim_2$ given by declaring $a\sim_2b$ if there exists some $g\in G$ such that $ga=bg$;
• $\unsim_3$ generated by $ab\sim_3 ba$.

Passing from groups to monoids, each of the above relations continue to make sense with a feel modifications; given a monoid $M$, we define $\unsim_1$, $\unsim_2$, and $\unsim_3$ to be the equivalence relation generated by (taking the symmetric and transitive closures of) the relations

• $\unsim'_1$ declaring $a\sim'_1 b$ if there exists some invertible $g\in M$ such that $a=g^{-1}bg$;
• $\unsim'_2$ declaring $a\sim'_2 b$ if there exists some $m\in M$ such that $ma=bm$;
• $\unsim'_3$ declaring $ab\sim'_3 ba$.

Each of the three relations above leads to distinct notions of conjugacy classes for monoids. They also admit the following category-theoretic descriptions: \begin{align*} M/\unsim_1 &\cong \mathsf{Fun}(\mathrm{B}\mathbb{N},\mathrm{B}M)/\{\text{isos}\},\\ M/\unsim_2 &\cong \pi_0(\mathsf{Fun}(\mathrm{B}\mathbb{N},\mathrm{B}M)),\\ M/\unsim_3 &\cong \int^{A\in\mathrm{B}M}\mathrm{Hom}_{\mathrm{B}M}(A,A). \end{align*} (The end $\int_{A\in\mathrm{B}M}\mathrm{Hom}_{\mathrm{B}M}(A,A)$ is also a familiar notion: it is the centre of $M$.)

More generally, we may replace $\mathrm{B}A$ with an arbitrary category $\mathcal{C}$, leading to three sensible notions of "conjugacy classes of categories". Similarly, we also have notions of

• "categories of conjugacy classes of monoidal categories$^\dagger$ and $2$-categories";
• "$\infty$-groupoids of conjugacy classes of $\infty$-categories";
• "$\infty$-categories of conjugacy classes of monoidal $\infty$-categories";
• and so on.

_{$^\dagger$For instance, given a monoidal category $\mathcal{C}$, we may define its category of "$\sim_3$-conjugacy" classes by first delooping it into a bicategory $\mathrm{B}\mathcal{C}$ and then taking the pseudo-bicoend of $\mathsf{Hom}_{\mathrm{B}\mathcal{C}}(-,-)$. (Again, the pseudo-biend is also an interesting object: it is the Drinfeld centre of $\mathcal{C}$.)}

Is there a "nice(-ish)" description of the set $$\mathrm{Cl}(\mathsf{Sets})\cong\int^{X\in\mathsf{Sets}}\mathrm{Hom}_{\mathsf{Sets}}(X,X)$$ of ($\unsim_3$-)conjugacy classes of sets?

What about the category $\int^{\mathcal{C}\in\mathsf{Cats}}_{\mathsf{ps}}\mathsf{Fun}(\mathcal{C},\mathcal{C})$ of conjugacy classes of categories, or the $\infty$-groupoid $\int^{X\in\mathcal{S}}\mathrm{Map}(X,X)$ of conjugacy classes of $\infty$-groupoids?