Group actions

If ($G$ is a group) and ($\rightcat X$ is a set), (a (right) group action) is
(a function $\boxed{{\rightcat X} \times G \to \rightcat X}$) which is associative and unital,
meaning that $x(gh) = (xg)h$ and $xe = x$,
where $x\in \rightcat X$, $g,h\in G$ and $e$ is the unit (identity) element of $G$.
For (a left group action), just replace ${\rightcat X} \times G$ with $G \times {\rightcat X}$,
and change the equations to $(gh)x = g(hx)$ and $ex = x$.

---

For some important examples of group actions, see the post on
The bimodule of sets, functions and permutations.

<hr/>

For (given $G,\rightcat X$) in (a cartesian closed category), there are two other, equivalent, ways of (giving the structure) and (formulating the axioms), given as follows:

 

\[ \boxed{  \begin{array} {cccccccccc|clc|cccccccc|l} &&&  {} \rlap{ \kern-2em \text{Structure} }  &&&&&&&  {} \rlap{ \kern8em \text{Description} }   &&& {} \rlap{ \kern1.5em \text{Equational Axioms, aka Constraints} }   &&&&&&&&  {} \rlap{  \kern3em \text{Comment}  }     \\   &&& &&& &&& &&& &&   {} \rlap{ \kern-1em \text{Identity Axiom} } &&&&  {} \rlap{ \kern-1.5em \text{Associativity Axiom} }     \\     \hline  {\rightcat X}_{-} & : & G  &  \to  &  [\rightcat X,\rightcat X]  & : & g & \mapsto & {\rightcat X}_g  &&&  \text{the monoid $[\rightcat X,\rightcat X]$ as a $\textit{representation}$ of $G$}  &&&  {\rightcat X}_e & = & \rightcat{1_X} && {\rightcat X}_g{\rightcat X}_h & = & {\rightcat X}_{gh}   &  \rightcat X \text{ is functorial}    \\      \hline   {\rightcat -} \tensor {-} & : & {\rightcat X} \times G  &  \to  &  \rightcat X  & : & \langle x,g \rangle & \mapsto & x \tensor g = xg &&&  \text{$\rightcat X$ as a $\textit{module}$ over $G$, i.e. as a $G{-}\textit{module}$}  &&&  xe & = & x  && (xg)h & = & x(gh)   &   \text{the action is associative}    \\     &&&&&& [x]g & \mapsto & (x)g = xg  &&& \text{Kelly's notation for actions of "clubs"}  && &(x)e & = & x && \big((x)g\big)h & = & (x)(gh)    \\    \hline   \rightcat{  \widehat{ \black{(-)} }  } & \rightcat : & \rightcat X  &  \rightcat\to  & \rightcat{ [\black G, X] }  & : & x & \rightcat\mapsto & \rightcat{ \hat{\black x} } &&&  \text{the $G{-}\textit{orbits}$ of $\rightcat X$} :  &&& e \rightcat{ \hat{\black x} } & = & x && h \rightcat{  \widehat{ \black{( g \rightcat{ \hat{\black x} } )} }  } & = & (gh) \rightcat{ \hat{\black x} }      \\    &&& &&& &&& &&  \boxed{    \rightcat{   {\hat{\black x}} : {\black G} \to  X : {\black g} \mapsto {\black g} \rightcat{ \hat{\black x} } = {\black x}{\black g} = {\black x} X_{\black g}  }    } & &&& &&&  \rightcat{ \hat{\black x} } {\rightcat X}_h & = & G_h \rightcat{ \hat{\black x} }   &   \text{each orbit $\rightcat{\hat{\black x}}$ is $G$-natural}  \\     &&& &&& &&& &&   \rightcat{ \alpha : {\black G} \to X  : {\black g} \mapsto {\black g}\alpha   } \text{    such that  } (\rightcat\alpha \text{ is }  \mathit{natural}) & & &  && &&  {\rightcat\alpha} {\rightcat X}_h & = & G_h {\rightcat\alpha}    \\    \end{array}  }  \]

( The additional line for $[x]g \mapsto (x)g$ under "$\rightcat X$ as a $\textit{module}$") shows (a notation introduced by Kelly in (his papers on clubs) ).

Here of course ($g$ is being viewed as an operator), ($x$ as an operand).

In the event that $G$ has (a distinguished element $e$) 

and (an internal composition operation, here denoted merely by juxtaposition), 

often one is interested in (structures on the pair $\rightcat X,G$) 

which are related to (the internal structure of $G$) by (the equations displayed in the panel at the right).

In the case where ($G$ is a group), this is what is meant by (the phrase "group action").

More generally, (structures satisfying such axioms) are called "algebras".

Precedent for denoting the action operation by $\tensor$ is in Section 3 of Im+Kelly.

<hr />

\[  \boxed{    \begin{array} {}  && &&     &&  x, \gamma  &&&&  \gamma  &  \kern1em    \\     &&  &&   &&  X \times_{\calC_0} \calC_1  &  {} \rlap{\kern-1em \xrightarrow[\kern10em]{} }  &&& \calC_1    \\    \gamma  &&  \ast,\gamma   && x,\gamma   & \rightadj{ \nearrow \rlap{ \text{monic} }  } && \rightadj{ \text{p.b.} }  &&  \rightadj{ \nearrow \rlap{ \text{monic} } }     \\     (c\downarrow \calC)_0  &  \cong  &  1 \times (c\downarrow \calC)_0   &  \xrightarrow{\textstyle x \times 1}  &  X_c \times (c\downarrow \calC)_0    &  {} \rlap{\kern-1em \xrightarrow[\kern10em]{} }  &&&  (c\downarrow \calC)_0    \\     &&    &&     &&  \leftcat{ \llap{\text{projection} \mapsto x} \Bigg\downarrow }  \rightcat{ \Bigg\downarrow  \rlap{ \text{action} \mapsto x\gamma = x X_\gamma} }   &&&&  \leftcat{ \llap{s} \Bigg\downarrow }  \rightcat{ \Bigg\downarrow \rlap{t} }    \\     && && && && &  {} \rlap{  \kern-3em \leftcat{ \text{p.b. for } s }  }    \\  &&   \leftcat{ \llap{\text{projection} \mapsto \ast} \Bigg\downarrow }  \phantom{ \rightcat{ \Bigg\downarrow } }   &&    \leftcat{ \llap{\text{projection} \mapsto x} \Bigg\downarrow }  \phantom{ \rightcat{ \Bigg\downarrow } } &&&&   \leftcat{ \llap{s} \Bigg\downarrow }  \phantom{ \rightcat{ \Bigg\downarrow \rlap{t} } }  \\  &&   &&   &&  X  &  {} \rlap{\kern-1em \xrightarrow[\kern10em]{} }  &&& \calC_0    \\   && &&   &  \rightadj{ \nearrow \rlap{ \text{monic} } } &&  \rightadj{ \text{p.b.} }  &&   \llap{\leftcat c}  \rightadj{ \nearrow \rlap{ \text{monic} } }  \\  &&     1  & {} \rlap{ \kern-0.5em \xrightarrow[\textstyle x]{\kern3em} }  &    X_c  &  {} \rlap{\kern-1em \xrightarrow[\kern10em]{} }  &&& 1     \\     &&  \ast  & \mapsto &  x     \\    \end{array}    }   \]

Given $x \in \rightcat X$, let $c \in \calC_0$ be its image in $\calC_0$.

Then

(the map $\hat x : (c\downarrow \calC)_0 \to \rightcat X : \gamma \mapsto \gamma \hat x = x\gamma$ via the action of $\calC$ on $\rightcat X$)

is (the <i>orbit</i> of $x$ under the action).

Then the following are equivalent statements:

($\hat x$ is an equivariant map of right $\calC$-sets) 

iff $x(\gamma\gamma') =  (x\gamma)\gamma'$ for all compatible $x,\gamma,\gamma'$ 

iff (the associativity axiom holds for the action).

<hr />

Given $\rightcat{   \boxed{  \alpha : { \black{\hom c \calC -} } \Rightarrow X  }   }$, 

a natural transformation from (the covariant regular representation of $\calC$ at $c$) to (an arbitrary covariant $\calC$-set $\rightcat X$), 

we have, for each $d \in \calC$ and $f \in \hom c \calC d$,

\[  \boxed{    \begin{array} {}    1_c  &  &&  \rightcat\mapsto  &&  (1_c)\alpha_c = \boxed{\check\alpha}    \\    c  &&  \hom c \calC c  &  \rightcat{   \xrightarrow[\kern3em]{ \textstyle {\black c} \alpha = \alpha_{\black c} }   }  &  {\rightcat X}_c    \\  \llap{f} \Bigg\downarrow  &&  \llap{\hom c \calC f} \Bigg\downarrow  &  \rightcat{   {\black f} \alpha = \alpha_{\black f}   }  &  \Bigg\downarrow \rlap{{\rightcat X}_f}    \\    d  &&  \hom c \calC d  &  \rightcat{   \xrightarrow[\textstyle {\black d} \alpha =  \alpha_{\black d}]{\kern3em}   }  &  {\rightcat X}_d    \\      (1_c) \hom c \calC f = 1_c f =  f  & &&  \rightcat \mapsto  &&  \begin{array}{}    (1_c) \hom c \calC f {\rightcat\alpha}_d & \xlongequal{\textstyle (1_c){\rightcat\alpha}_f}  &  (1_c) {\rightcat\alpha}_c {\rightcat X}_f    \\ \llap{ \text{defn } \hom c \calC f } \Vert  &&  \Vert \rlap{ \text{defn } \check{\rightcat\alpha} \text{  -- restriction} }  \\    (1_c f) {\rightcat\alpha}_d   &&    {\check{\rightcat\alpha}} {\rightcat X}_f    \\  \llap{ 1_c \text{ is an identity} }  \Vert  &&  \Vert \rlap{ \text{defn } {\hat{()}}_d  \text{  -- extension}  }     \\     f \boxed{{\rightcat\alpha}_d}   && f \boxed{ \big( \hat{ {\check{\rightcat\alpha}} }  {\big)}_d }   & \kern7em   \\   \text{thus:  }  & \boxed{  \boxed{ \rightcat\alpha = \hat{\check{\rightcat\alpha}} }  }    \\  \end{array}     \\     \end{array}   }    \]

On the other hand, given $x \in {\rightcat X}_c$,

\[ \check{ \hat x }  \xlongequal{\textstyle \text{defn }  \check{()} }   1_c \big(\hat x\big)_c  \xlongequal{\textstyle \text{defn } \big(\hat x\big)_c }  x {\rightcat X}_{1_c} \xlongequal{\textstyle {\rightcat X}  \text{ preserves identities} } x \rightcat{ 1_{X_{\black c}} } = x \]

Thus $\boxed{ \check{ \hat x } = x }$.

Thus finally we have a bijection

\[ \boxed{  \leftcat{ (\check{\rightcat\alpha} = x) \in {\rightcat X}_c} \; {  \leftadj{ \xrightarrow[\text{extension}]{\textstyle \hat{()} }} \atop { \rightadj{ \xleftarrow[\textstyle \check{()}]{\text{restriction}} } }  } \; \rightcat{ { \hom {\hom c \calC -} {[\calC, \Set]} {\rightcat X} } \ni (\alpha = \hat {\leftcat x}) }   } \]

almost surely the simplest and most basic example of the "Yoneda bijection".

<hr />

To prepare for generalizations, and to justify calling $\hat x$ an "extension", 

consider the forgetful functor $\boxed{ \rightadj U \mathrel{\rightadj :} \rightcat{G{-}\Set} \mathrel{\rightadj\to} \leftcat\Set \mathrel{\rightadj :} \rightcat X \mathrel{\rightadj\mapsto} \rightcat X \rightadj U \mathrel{\rightadj =} \leftcat X }$, 

 where we have followed the common practice of using the same letter $X$ to denote both a $G$-set, with its action $X \times G \to X$, and its underlying mere set $\leftcat X$, 

and where $\rightcat{G{-}\Set}$ denotes (the category of $G$-sets) (not to be confused with G-spots! :-).

$G$'s multiplication $\boxed{ G \times G \to G }$ makes $G$ a right (and also a left) $G$-set, the (right) (or left) regular representation of $G$.

The identity element $\boxed{e}$ of $G$ may be specified ("named") via an arrow in $\leftcat\Set$, $\boxed{ \leftadj{   e : \leftcat 1 \to G \rightadj U   } }$.

The content of the most basic Yoneda bijection may then be expressed by the statement

\[ \boxed{   \leftadj {  e : \leftcat 1 \to G \rightadj U \kern1em \text{ is a universal arrow from $\leftcat 1$ to $\rightadj U$.}  }    }  \]

\[  \boxed{    \begin{array} {}      &&  \leftadj G    \\      & \leftadj{ \llap{e} \nearrow}  &  \Vert  & \rightcat{    \searrow \rlap{ \exists! \, \hat{\leftcat x} = \alpha \text{     ;    the } \textit{extension} \leftcat{\text{ of } x}}   }    \\     \leftcat 1 &  \leftcat{      {} \rlap{ \kern-1em \xrightarrow[ {}\rlap{\textstyle \kern-3em \forall \, x = \check{\rightcat\alpha} \text{     ;    the } \textit{restriction} \rightcat{\text{ of } \alpha}} ]{\kern8em} }    }  &&& \rightcat X    &  \kern10em  \\    \end{array}    }    \]

Here the two arrows originating at the set $\leftcat 1$ are both arrows in $\leftcat\Set$, while the arrow $\rightcat{{\leftadj G} \to X}$ is in $\rightcat{G{-}\Set}$.

What "$\leftadj{   e : {\leftcat 1} \to G {\rightadj U}   } \kern1em \text{ is a universal arrow from $\leftcat 1$ to $\rightadj U$}$" means is:

For every arrow $\leftcat{   1 \xrightarrow[\kern1.5em]{\textstyle x} {\rightcat X}   }$ as at bottom (i.e. for every element $\leftcat{x \in X}$), 

there exists a unique morphism of $G$-sets $G \to \rightcat X$ as at right which makes the triangle commute. 

We denote this unique morphism of $G$-sets, whose existence and uniqueness is guaranteed by the universal property, by $\boxed{ \hat x }$.

$\newcommand\GSet{{\rightcat{G{-}\Set}}}$

A familiar example, for $V$ a vector space over $\R$, and $v \in \rightcat V$ a vector in it:

\[ \begin{array} {}      && \R    \\      & \llap{1} \nearrow  &  \Vert  &  \searrow \rlap{ \hat v = \rightcat l \text{     ;    the $\textit{extension}$ of $v$, the parameterized line through $v$}}    \\    \leftcat 1 & {} \rlap{ \kern-1em \xrightarrow[\textstyle v = \check l \rlap{ \text{     ;    the $\textit{restriction}$ of $\rightcat l$}}]{\kern10em} } &&& \rightcat V    \\    \end{array} \]

Here of course "$1$" is being used to denote both a one-element set, say $\leftcat{1=\{\emptyset\}}$, and the real number usually so denoted.

<hr />

Returning to the more general $\rightcat{G{-}\Set}$ context, 

the "bijection between arrows" formulation has a generalization to an adjunction:

Given a group $G$ in $\Set$, there is an adjunction:

\[ \boxed{  \leftcat{Y \in \Set} \; {   { \leftadj{ \xrightarrow[\kern6em]{\textstyle (-\times G)} } } \atop { \rightadj{ \xleftarrow[\textstyle U]{\kern6em} } }   } \; \rightcat{G{-}\Set \ni X}  }  \]

I.e. the arrows in the box above are functors and we have a natural (in $\leftcat Y$ and $\rightcat X$) bijection of sets

\[ \boxed{    \begin{array} {}  \rightcat\alpha  & \rightcat\in  &   \rightcat{  \hom { \leftadj{ ({\leftcat Y} \times G) } } {(G{-}\Set)} X  }    \\  && \red{\wr\Vert}    \\      \leftcat{ x }  &  \leftcat\in  &  \leftcat{ \hom Y \Set {\rightcat X \rightadj U}  } \\    \end{array}  }   \]

We can depict the relation between $\leftcat x$ and $\rightcat\alpha$ with:

\[ \boxed{    \begin{array} {} && \leftadj{ {\leftcat Y} \times G }     \\ &  \leftadj{   \llap{ \leftcat Y \times e }  \nearrow  }  & \Vert & \rightcat{    \searrow \rlap{ \exists! \, \hat {\leftcat x} = \alpha \text{ ; the $\textit{extension}$ of $\leftcat x$}  }}       \\        \leftcat{ Y \cong Y\times 1 }  &  {} \rlap{ \kern-1em \leftcat{   \xrightarrow[\textstyle \forall \, x = \check{\rightcat\alpha} \rlap{ \text{ ; the $\textit{restriction}$ of $\rightcat\alpha$}}]{\kern11em}   }   } &&& \rightcat X  &  \kern10em    \\ \end{array}    }    \]

$\leftadj{ {\leftcat Y} \times G }$ is the free $\rightcat{ G{-}\Set}$ on the set $\leftcat Y$; 

the unit (just a function) is $\leftadj{ \leftcat Y \times e }$.

The counit, a morphism of $\leftadj G$-algebras, at a $\rightcat{ G{-}\Set \; X }$ is just the $\leftadj G$-action $\leftadj{ {\rightcat X \rightadj U} \times G } \mathrel{\rightadj\to} \rightcat X $.

The above relations may be perspicaciously viewed be embedding them in the 2-category $\CAT$ of large 1-categories.

\[    \boxed{   \begin{array} {ccccccccc|l}   &&&&  \calC \calP^\ast &&&&  \calC \calP^\ast   &  \text{Yoneda structure operation }   \calP^\ast  \text{ on } \calC    \\     &&&&  \rightcat\Vert  &&&&  \rightcat\Vert    \\   &&&&  [\calC, \Set]  &&&&  [\calC, \Set]     &    \text{discrete opfibrations (dof) on } \calC   \\    &&&&  \rightcat\Vert  &&&&  \rightcat\Vert    \\    &&&&  [\leftadj G \mathbf B, \Set]  &&&&  [\leftadj G \mathbf B, \Set]  &    \text{discrete opfibrations (dof) on } \leftadj G \mathbf B \\      &&&&  \rightcat\Vert  &&&&  \rightcat\Vert    \\     \I  &  {} \rlap{ \kern-1em \rightcat{  \xrightarrow[\kern11em]{\textstyle X}  }  }   &&&  \GSet  & {} \rlap{ \kern-2em \rightcat{ \xrightarrow[\kern11em]{} } }  &&&  \GSet     \\      &  \leftcat{  \llap Y \searrow  }  &  \rightcat{  \raise1ex{ \smash{ \llap{\exists! \alpha} \Bigg\Uparrow } }  }  &  \leftadj{  \nearrow  \rlap{\scriptstyle \kern-3em ({\leftcat -} \times G)}   }  \leftcat{ \raise.3ex{  \smash{ \Bigg\Uparrow \rlap{ \forall x}  }  }  }  &  \leftadj{  \raise0ex{ \smash{ \Bigg\Uparrow \rlap{\scriptstyle \kern-2em \eta = ({\leftcat -} \times e)}  }  }  }  &  \rightadj{  \searrow \rlap{\kern-1em U}  }  &  \rightadj{  \raise1ex{ \smash{ \Bigg\Uparrow \rlap{\scriptstyle \kern-2em  \epsilon = \text{action}}  }  }  }  &  \leftadj{  \nearrow  \rlap{({\leftcat -} \times G)}   }   &&  \red{\text{the adjointness! :-)} }  \\    &&  \leftcat\Set  &  {} \rlap{ \leftcat{ \kern-1em \xrightarrow{\kern13em} } }  &&&  \leftcat\Set   \\    \end{array}    }    \]

The two adjunction triangle equalities are, 

first, that for any set $\leftcat Y$, $\leftcat{ y \in Y}$, and $g \in G$, 

\[ {\leftcat Y} \times {\leftadj G}  \to   {\leftcat Y} \times {\leftadj G}  :  [\leftcat y] \leftadj g  \mathrel{\leftadj\mapsto}  \big[ [\leftcat y] \leftadj e \big]  \leftadj g  \mathrel{\rightadj\mapsto}   [\leftcat y] \leftadj{(e g)} \xlongequal{\text{left identity axiom for group } \leftadj G}  [\leftcat y] \leftadj g \;  , \]

and second, that for any  $G$-set $\rightcat X$ and $x \in \rightcat X \rightadj U$, 

\[   {\rightcat X \rightadj U}  \to {\rightcat X \rightadj U}   :   x \mathrel{\leftadj\mapsto}  [x] \leftadj e  \mathrel{\rightadj\mapsto} (x) \leftadj e  =  x  \leftadj e \xlongequal{\text{identity axiom for action of $\leftadj G$ on $X$}} x  \;  .  \]

Note that $\hat{\leftcat x}$ equals

\[ \begin{array}{}  \leftadj{ {\leftcat Y} \times  G }  &   \xrightarrow[\kern3em]{\leftadj{ \leftcat x \times G }}  &  \leftadj{  {\rightcat X \rightadj U} \times G  }  &  \rightadj{ \xrightarrow[\kern3em]{\rightcat X \rightadj \epsilon}  }  &  \rightcat X    \\   [\leftcat y] \leftadj g  &  \mapsto  &  [\leftcat {yx}] \leftadj g  &  \mapsto  &  (\leftcat {yx}) \leftadj g   \end{array}  \]

<hr />

Here are 2-D, 1-D, and 0-D diagrams for the above situation:

\[   \boxed{   \begin{array} {rcc}  {} \rlap{   \kern-21em \text{The Element/Orbit Bijection $\boxed{\leftadj\eta \leftrightarrow \rightcat\alpha}$ (i.e., the Yoneda Lemma) for a Representation $\rightcat X$ of a Group  $G$}   }    \\     \hline   \\     \text{viewing $\CAT$ as a double category (trivial vertical 1-cells)}  &&     \boxed{     \begin{array} {}   \leftcat\calI   &   \leftadj{  \xrightarrow [\kern2em] {\textstyle e}  }   &    \leftadj G    &    \rightcat{  \xrightarrow [\kern2em] {\textstyle X}  }    &    \Set    \\   \leftcat{\Big\vert}  &  \llap g \smash{\Bigg\Uparrow} &   \leftadj{\Big\vert}  &  \rightcat{  \smash{\Bigg\Uparrow} \rlap{ \kern-1.9em \boxed{\alpha = \hat\eta} }  }  &  \Big\vert    \\   \leftcat\calI   &   \leftadj{  \xrightarrow [\kern2em] {\textstyle e}  }   &   \llap{\smash{ {\ulcorner g \urcorner} \Bigg\Uparrow }} {\leftadj G} \rlap{\smash{  \leftadj{ \Bigg\Uparrow \boxed{\eta = \check\alpha} }  }}   &    \leftadj{  \xrightarrow [\kern2em] {\textstyle \hom e G {\leftcat -}}  }    &    \Set    \\     \leftcat{\Big\vert}  &&  \leftadj{ \llap{1_e} \Big\Uparrow }  &&  \Big\vert    \\    \leftcat\calI  &  \leftcat{  {} \rlap{ \kern-2em \xrightarrow [\textstyle 1] {\kern12em} }   }   &&&  \Set    \\     \end{array}     }      \\       \text{2-D, in the very large 2-category $\CAT$ of large 1-categories}   &   \kern2em   &     \boxed{ \begin{array} {} &&  \leftadj G  &  \leftadj{ \xlongequal{\kern1em} }  &   \leftadj G  &  \leftadj{ \xlongequal{\kern1em} }  &  \leftadj G    \\    &   \leftadj{ \llap{e} \nearrow}  &  \buildrel \textstyle g \over \Leftarrow  & \leftadj{ \llap{e} \nearrow } \rlap{\smash{  \kern-1em \Bigg\Uparrow \rlap{ \ulcorner g \urcorner } }}    &   \smash{     \leftadj{   \llap{1_e} \Big\Uparrow  \rlap{\smash{ \kern1.5em \Bigg\Uparrow \rlap{ \boxed\eta } }}   }     }  & \leftadj{   \searrow \rlap{ \kern-2.3em \hom e G - }  }  &  \rightcat{   \buildrel \textstyle {} \rlap{ \kern-1.6em \boxed{\alpha = \hat\eta} } \over \Rightarrow   }  &  \rightcat{  \searrow \rlap X  }    \\     \leftcat\calI  &  \leftcat{ \xlongequal{\kern1em} }  &  \leftcat\calI  &  {} \rlap{  \kern-1.5em \leftcat{ \xrightarrow[\textstyle 1]{\kern9.5em} }  }  &&&  \leftcat\Set  &  \leftcat{ \xlongequal{\kern1em} }  &  \leftcat\Set   \\      \end{array}     }      \\   \\          \text{1-D, in the large category $\Set$ of small sets}   &&    \boxed{    \begin{array} {}  &&  X_e    \\   &  \rightcat{  \llap{ {\alpha}_{\black e} } \nearrow  }  &&  \rightcat{  \nwarrow \rlap{ X_{\black g} }  }   \\    \hom e G e  &&  g{\rightcat\alpha}  &&  {\rightcat X}_e   &     \\     &  \llap{ \hom e G g } \nwarrow  &&  \rightcat{  \nearrow \rlap{ \alpha_{\black e} }  }  \\    && \hom e G e    \\    & \llap{ \ulcorner g \urcorner } \nwarrow  &  {\big\uparrow} \rlap{1_e}  &  \leftadj{   \nearrow \rlap{  \boxed{ \eta = \rightcat{\check\alpha} }  }   } \\    &&    1    \\  \end{array}    }    \\    \\      \text{0-D, in the small set ${\rightcat X}_{\leftadj e}$ }  &&  \boxed{   g{  \rightcat{ \alpha_{\black e} }  } = {\leftadj\eta} g =  \rightcat{\check\alpha} g = g \big( \rightcat{ \hat{\check\alpha} } {\big)}_e  }  \\   \end{array}   }    \] 

<hr />

\[ \boxed{     \begin{array}  {l|c|c}   \text{action}  &  &  \text{equalities in } \N &  \text{bijections in } \Set   \\   \hline  \text{one object}  &  X \times G \to X &  \text{orbit-stabilizer equation} ; \text{class equation}&  \text{orbit-element bijection} \\  \hline \text{several objects}  & X_c \times {\hom c \calC -} \to X_- &    &  \text{Yoneda lemma}   \\  \end{array}     }      \]

<hr />

$ \leftadj{ \llap{ \text{(quotient map of ($\red{\hat x}$ cokernel))} } \nearrow } $

\[  \boxed{  \begin{array} {}  \rightadj{   \boxed{  \text{Stab}_{\black x} = \text{Aut}_{\black x}  }    }  &  \rightadj\rightarrowtail  & G  &  \leftadj{  \displaystyle \mathop\twoheadrightarrow^{\text{quotient}}_{\text{map}}    }   & \leftadj{   \boxed{  {\black G}{/}\rightadj{ \text{Stab} }_{\black x}  }   }   \\   && \wr\Vert  &&  \red{\wr\Vert} \rlap{  \; (  \leftadj{\text{orbit}} \text{-} \rightadj{\text{stabilizer}} \text{ theorem} )  }    \\    \smash{\rightadj{\Bigg\downarrow}}  &   \smash{     \rightadj{ \raise5ex{\kern-4em \lrcorner} \text{p.b.}  }    }    &   1 \times G  &  \leftadj{  \buildrel \textstyle \black{\hat x} \over \twoheadrightarrow  }  &  \leftadj{   \boxed{  [\black x] = \black{xG}  }   }  \rlap{  \kern0em \xrightarrow[\smash{\kern12em}]{\textstyle !}  }  &&  1  & \kern0em   \\     &&  \llap{  {\ulcorner x \urcorner} \times G  } \Bigg\downarrow   &  \llap{ \scriptstyle  \text{defn. } \red{\hat x} \kern-.6em } \smash{   \red{   \buildrel \textstyle \kern1em \boxed{\hat x} \over \searrow   }   } \rlap{ \scriptstyle \kern-.5em \text{image fact.} }    &  \Bigg\downarrow  &  \smash{     \rightadj{ \raise3ex{\kern-2em \lrcorner} \text{p.b.}  }    }  &  \Bigg\downarrow \rlap{ \ulcorner \leftadj{ [\black x] } \urcorner }  \\   \rightadj{  \boxed{   \black{ (\displaystyle \mathop\rightrightarrows^\tensor_{\pi_X}) } \, \text{Equalizer}   }    }    &  \rightadj\rightarrowtail  &  X \times G  & \displaystyle \mathop\rightrightarrows^\tensor_{\pi_X}  & X {} \rlap{ \kern0em \leftadj{\xrightarrow{\kern5em}}   }  && \leftadj{  \boxed{   \black{ (\displaystyle \mathop\rightrightarrows^\tensor_{\pi_X}) } \, \text{Coeq} = \black X {/} \black G = (\black{X//G})\pi_0   }    }  \\  &&&&  x & \leftadj\mapsto & {} \rlap{   \kern-4em  \leftadj{ [\black x] } = xG = G\hat x = \hat x \, {  \leftadj{ \text{Image} }  }   }     \\    &&&&  X {} \rlap{  \rightadj{ \xleftarrow[\textstyle a]{ \kern1em \text{section} \kern1em } }  }    &&    \leftadj{  \boxed{   \black{ (\displaystyle \mathop\rightrightarrows^\tensor_{\pi_X}) } \, \text{Coeq} = \black X {/} \black G = (\black{X//G})\pi_0   }    }   \\ \end{array}  }  \]

\[ \boxed{ \displaystyle    X  \buildrel \Gsets \over \cong  \leftadj{ \sum_{[\black x] \in {\black X}/{\black G}} }  \leftadj{ [\black x] }  \buildrel \Gsets \over \cong   \leftadj{  \sum_{ [\black x] \in {\black X}/{\black G}}  } {\black G}{/}\rightadj{ \text{Stab} _{a_{\leftadj{[\black x]}}} }    }     \]

(E.g., let $G = \langle \{\pm1\}, \times, +1\rangle$. Then, with (the evident action given algebraically by multiplication or geometrically by reflection), 

\[   \boxed{  X = \{-2,-1,0,1,2\}  \cong  \{\pm 2\} \leftadj+ \{\pm1\} \leftadj+ \{0\}  =   [2]  \leftadj+ [1] \leftadj+ [0]  \cong   \{\pm1\}/\rightadj{\{+1\}} \leftadj+ \{\pm1\}/\rightadj{\{+1\}} \leftadj+ \{\pm1\}/\rightadj{\{\pm1\}}    }   \]  To see a picture of this $\{\pm1\}$-set,  visit https://golem.ph.utexas.edu/category/2021/07/diversity_and_the_mysteries_of.html   )

The last general result yields, via $|\cdot| : \FinSet_0 \to \N$, (the "<a href="https://ncatlab.org/nlab/show/class+equation">class equation</a>",  an equality of natural numbers in $\N$):  \[ \boxed{  |X| = \Big|\sum_{[\black x] \in {\black X}/{\black G}} G/{ \rightadj{ \text{Stab} }_{\black x} } \Big|  = \sum_{[\black x] \in {\black X}/{\black G}} |G/{ \rightadj{ \text{Stab} }_{\black x} }|  = \sum_{[\black x] \in {\black X}/{\black G}} |G|/{ \rightadj{ |\text{Stab} }_{\black x} | } }   \] 

which is numerically equivalent to the equality of rational numbers in $\Q$:   \[ \boxed{   |X|/|G|   \buildrel \text{above} \over =   \sum_{ [\black x] \in {\black X}/{\black G} } 1/{ \rightadj{ | \text{Stab} }_{\black x} | }  \buildrel \text{defns} \over =   \sum_{ [\black x] \in \pi_0({\black X}//{\black G}) } 1/| \Aut{x} |    \buildrel \text{defn.} \over \equiv  \boxed{ |X//G| }     }  \] , this last (the "<a href="https://ncatlab.org/nlab/show/groupoid+cardinality"><i>groupoid cardinality</i></a>" of the <a href="https://ncatlab.org/nlab/show/action+groupoid"><i>action groupoid</i></a> $\boxed{X//G}$ associated with (the action $X_-$)). Note the usage of (the full-up equality $\boxed{X/G = \pi_0(X//G)}$), and that (the groupoid cardinality) can be defined for (any groupoid), not just (action groupoids).

<hr />

<h3>References</h3>

Im+Kelly, A Universal Property of the Convolution Monoidal Structure

https://doi.org/10.1016/0022-4049(86)90005-8

For possible future use: $\rightadj{\text{counit of }} ( \leftadj{\text{cokernel}} \text{-} \rightadj{\text{kernel}} \text{ adjunction} ) \rightadj{\text{ is here monic}} ) }$

The bimodule of sets, functions and permutations, and its quotients

1. Introduction to the bimodule of sets, functions, and permutations
2. Quotients: the “twelvefold way”; the twelvefold way diagram
3. An example: $\homst {[4]} \Set {\{a,b\}}$
4. Division (quotient) as (tensor product) with (the terminal bimodule)

This situation is worth careful consideration because, not only is it fundamental to combinatorics,
but also because it is a model for many situations more complex than merely the category of sets $\Set$.

---

1. Introduction to the bimodule of sets, functions, and permutations

Let $\source \setX$ and $\target \setY$ be arbitrary sets in the category of sets $\Set$.
Consider the groups of permutations $\source{\boxed{\symsX}}$ and $\target{\boxed{\symtY}}$ of $\source \setX$ and $\target \setY$ respectively,
and also the set $\boxed{\SetXY}$ of all $|\settY|^{|\setsX|}$ functions from $\source \setX$ to $\target \setY$.
There are two actions:
The group $\symsX$ of permutations of $\setX$ acts on $\SetXY$ by precomposition, while
the group $S_\setY$ of permutations of $\setY$ acts on it by postcomposition: \[\begin{array}{c} {\source{S_\setX}} \times \SetXY & \source\to & \SetXY\\ \langle \ssigma ,f\rangle & \source\mapsto & \ssigma f\end{array} \mkern6em \hbox{and} \mkern6em \begin{array}{c} \SetXY \times \target{S_\setY} & \target\to & \SetXY\\ \langle f,\ttau \rangle & \target\mapsto & f\ttau . \end{array}\] These actions are each associative and unital.
Also they commute with each other:
mixed pre-post actions are of the form

$\source{\setX \xrightarrow{\sigma} \setX} \xrightarrow{f} \target{\setY \xrightarrow{\tau} \setY}$

which yields

$\smash{\begin{array}{} && \setsX & {} \rlap{\mkern-1em \xrightarrow[\smash{\mkern8em}]{\displaystyle \functionf\ttau}} &&& \settY \\ & \source{ \llap{\symsX\ni\sigma} \nearrow } & \Vert & \searrow \rlap{\mkern-1em\functionf} & \Vert & \target{ \nearrow \rlap{\tau\in\symtY} } \\ \setsX & {}\rlap{\mkern-1em \xrightarrow[\displaystyle \ssigma\functionf]{\mkern8em}} &&& \settY \\ \end{array}}$

yielding

$\setsX \xrightarrow{\displaystyle \ssigma\functionf\ttau} \settY$

where $\source{\sigma\in S_\setX}$, $f\in \SetXY$, and $\target{\tau\in {S_\setY}}$.
Since (function composition is associative), (the two actions commute): $(\ssigma f)\ttau = \ssigma (f\ttau)$, yielding $\ssigma\functionf\ttau$.

So in addition to (the two one-sided, single actions)
there is (a double, simultaneous action from both sides),
which may be represented either
as a triple product, with (the action actors coming from each side simultaneously): \[\begin{array}{c} \source{S_\setX \times {}} \SetXY \target{{} \times S_\setY} & \to & \SetXY\\ \langle \ssigma, f, \ttau \rangle &\mapsto &\ssigma f \ttau\\ \end{array}\] or as a binary product, with (one factor itself being a product), and with (the joint, combined, again simultaneous, action coming from the right): \[\begin{array}{c} \SetXY \times (\source{S_\setX\op} \times \target{S_\setY}) &\to &\SetXY\\ \bigl\langle f, \langle \ssigma,\ttau \rangle \bigr\rangle &\mapsto & \ssigma f \ttau .\\ \end{array}\] (Associativity of the two-sided action) is (the vertical left hand equality) in \[\begin{array}{cl} \bigl\langle \source{\sigma'}, \langle \ssigma, f, \ttau \rangle, \target{\sigma'} \bigr\rangle & \xlongequal{\text{action}} \bigl\langle \source{\sigma'}, \ssigma f \ttau , \target{\sigma'} \bigr\rangle \xlongequal{\text{action}} \source{\sigma'}(\ssigma f \ttau) \target{\tau'}\\ \llap{\scriptstyle \text{action assoc }}\Vert\rlap{\text{ ???}}\\ \bigl\langle \source{\sigma'\sigma}, f, \target{\tau\tau'} \bigr\rangle & \xlongequal{\text{action}} \source{(\sigma'\sigma)} f \target{(\tau\tau')} .\end{array}\] (Equality of the right-hand sides) follows from (the associativity of function composition), and associativity of the action follows.

In the action where (the action actors both come from the right),
the ${}^{\rm op}$ is necessary (to make the action associative),
as shown by the following, where (the desired associativity of the action) is again (the vertical equality on the left).

$\smash{\begin{array}{} \setsX & \xrightarrow[\smash{\mkern3em}]{\displaystyle \functionf} & \settY \\ \source{\llap\ssigma \uparrow} && \target{\downarrow\rlap\ttau} \\ \setsX && \settY \\ \source{\llap{\ssigma'} \uparrow} && \target{\downarrow \rlap{\ttau'}} \\ \setsX && \settY \\ \end{array}}$

$$\begin{array}{cl} \bigl(f\langle\ssigma,\ttau \rangle\bigr)\langle\source{\sigma'},\target{\tau'}\rangle &\xlongequal{\text{action}} (\ssigma f \ttau)\langle\source{\sigma'},\target{\tau'}\rangle \xlongequal[\class{red}{*}]{\text{action}} \source{\sigma'}(\ssigma f \ttau)\target{\tau'}\\ \llap{\scriptstyle \text{action assoc }}\Vert\rlap{\text{ ???}}\\ f \bigl(\langle \ssigma,\ttau \rangle \langle\source{\sigma'},\target{\tau'}\rangle \bigr) & \xlongequal[\class{red}{*}]{\source{\text{composition in $\symsX\op$}}} f \langle \source{\sigma' \sigma}, \target{\tau \tau'} \rangle \xlongequal{\text{action}} \source{(\sigma'\sigma)} f \target{(\tau\tau')} \\ \end{array}$$ (The right-hand sides are again equal) by (the associativity of function composition in the diagram at right);
the desired associativity of (the action with right actors) follows.
Note that (the equalities where $\source{\sigma'}$ skips over $\ssigma$) are marked by (a $\class{red}{*}$).

$\begin{array}{} \functionf & \target{\xrightarrow[\mkern3em]{\displaystyle \ttau}} & \functionf\ttau \\ \llap\ssigma \big\downarrow && \llap\ssigma \big\downarrow \\ \big( \functiong = \ssigma\functionf \big) & \target{\xrightarrow[\displaystyle \ttau]{\mkern3em}} & \big( \functiong\ttau = (\ssigma\functionf)\ttau = \ssigma(\functionf\ttau) \big) \\ \end{array}$

Since the actions commute,
each action respects ((the equivalence relation) determined by (the other action)).
For example, if $f \source{{} \equiv_{S_\setX}} g$, then $\source{(\exists \sigma\in S_\setX)}(g= \ssigma f)$,
so, for all $\tau \in S_\setY$, and using the same $\sigma\in S_\setX$,
we have $g\ttau = (\ssigma f)\ttau = \ssigma(f\ttau)$, so $f\ttau \source{{} \equiv_{S_\setX}} g\ttau$.
The diagram, in the associated double groupoid, at right illustrates the situation.
And similarly if the roles of $\source \setX$ and $\target \setY$ are reversed.

Thus we have action maps $\source{\big(\symsX \backslash} \SetXY\source{\big)} \times \symtY \mathrel{\target{\xrightarrow{\text{action}}}} \source{\big(\symsX \backslash} \SetXY\source{\big)}$ and $\symsX \times \target{\big(} \SetXY \target{{} / \symtY \big)} \mathrel{\source{\xrightarrow{\text{action}}}} \target{\big(} \SetXY \target{{} / \symtY \big)}$,
and may form the two iterated quotients (quotients of quotients): $\boxed{ \source{( S_\setX \backslash } \SetXY \source) \target{/ S_\setY} }$ and $\boxed{ \source{S_\setX \backslash } \target( \SetXY \target{/ S_\setY )} }$ .

The five quotients of $\SetXY$ : two single, two iterated, one double

$\begin{array}{} \SetXY & {} \rlap{ \mkern-.5em\target{\xrightarrow[\mkern18em]{\displaystyle \functiont}} } &&& \mkern1em \SetXY \target{ / S_\setY} \\ &&&& \source{\big\downarrow} \\ \smash{\source{\llap\functions \Bigg\downarrow}} && \searrow \rlap\functionr && \source{S_\setX \backslash } \target( \SetXY \target{/ S_\setY )} \\ &&&& \wr\Vert \\ \source{S_\setX \backslash} \SetXY & {} \rlap{\mkern-.5em\target\longrightarrow} & \mkern1em \source{( S_\setX \backslash } \SetXY \source) \target{/ S_\setY} & \cong & \source{S_\setX \backslash} \SetXY \target{/ S_\setY} \\ \end{array}$

From either the two-sided action, $\langle \ssigma, f, \ttau \rangle \mapsto \ssigma f \ttau ,$
or the action of the product group ${\source{S_\setX}^{\text{op}}} \times {\target{S_\setY}}$, $\bigl\langle f, \langle \ssigma,\ttau \rangle \bigr\rangle \mapsto \ssigma f \ttau, $
on $\SetXY$
we have a third quotient, which we denote by $\boxed{\source{S_\setX \backslash} \SetXY \target{/ S_\setY}}$,
representing the result of dividing out simultaneously by the two actions,
with its quotient map $\SetXY \longrightarrow \source{S_\setX \backslash} \SetXY \target{/ S_\setY}$.
There are canonical bijections between all three quotient sets
which commute with their canonical quotient maps:

The surjections $\homst \setX \Surj \setY$ and injections $\homst \setX \Inj \setY$ form submodules of $\SetXY$,
with analogous diagrams concerning their quotients.

---

2. The “twelvefold way”

The double quotient square for $\SetXY$

$ \begin{array}{} \text{\{ ordered weak $\target \setY$-partitions of $\source \setX$ \}} && \text{\{ unordered weak $\target{|\setY|}$-partitions of $\source \setX$ \}} \\ {\wr\Vert} && \wr\Vert \\ \SetXY & {} \rlap{\mkern-7em\xrightarrow[\mkern16em]{\displaystyle \functiont}} & \mkern1.6em \SetXY \target/ \symtY \\ \smash{\source{\llap\functions \Bigg\downarrow}} & \searrow \rlap\functionr & \Bigg\downarrow \\ \symsX \source\backslash \SetXY \mkern2em & {} \rlap{\mkern-7em\xrightarrow{\mkern16em}} & \symsX \source\backslash \SetXY \target/ \symtY \\ \wr\Vert && \wr\Vert \\ \text{\{ ordered weak $\target{\setY}$-partitions of $\source{|\setX|}$ \}} && \text{\{ unordered weak $\target{|\setY|}$-partitions of $\source{|\setX|}$ \}} \\ \wr\Vert \\ \text{\{ weak $\source{|\setX|}$-multisets (values $\ge0$) on $\target \setY$ \}} \\ \end{array} $

The double quotient square for $\homst \setX \Surj \setY$

$ \begin{array}{} \text{\{ ordered strong $\target{\setY}$-partitions of $\source \setX$ \}} && \text{\{ unordered strong $\target{|\setY|}$-partitions of $\source \setX$ \}} \\ \wr\Vert && \wr\Vert \\ \homst \setX \Surj \setY & {} \rlap{\mkern-7em\xrightarrow[\mkern16em]{\displaystyle \functiont}} & \mkern1.6em \homst \setX \Surj \setY \target/ \symtY \\ \smash{\source{\llap\functions \Bigg\downarrow}} & \searrow \rlap\functionr & \Bigg\downarrow \\ \symsX \source\backslash \homst \setX \Surj \setY \mkern1.6em & {} \rlap{\mkern-7em\xrightarrow{\mkern16em}} & \symsX \source\backslash \homst \setX \Surj \setY \target/ \symtY \\ \wr\Vert && \wr\Vert \\ \text{\{ ordered strong $\target{\setY}$-partitions of $\source{|\setX|}$ \}} && \text{\{ unordered strong $\target{|\setY|}$-partitions of $\source{|\setX|}$ \}} \\ \wr\Vert \\ \text{\{ strong $\source{|\setX|}$-multisets (values $\ge1$) on $\target \setY$ \}} \\ \end{array} $

The double quotient square for $\homst \setX \Inj \setY$

$ \begin{array}{} \text{ \{ injections from $\source \setX$ to $\target \setY$ \}} && \text{$1$ or $\emptyset$ as $\source{|\setX|} \le {}$ or $\gt \target{|\setY|}$} \\ \Vert && \wr\Vert \\ \homst \setX \Inj \setY & {} \rlap{\mkern-10em\xrightarrow[\mkern16em]{\displaystyle \functiont}} & \mkern1.4em \homst \setX \Inj \setY / {\target{S_\setY}} \\ \smash{\source{\llap\functions \Bigg\downarrow}} & {} \rlap{\mkern-4em \searrow\rlap\functionr} & \Bigg\downarrow \\ {\source{S_\setX}} \backslash \homst \setX \Inj \setY \mkern1.6em & {} \rlap{\mkern-10em\xrightarrow{\mkern16em}} & {\source{S_\setX}} \backslash \homst \setX \Inj \setY / {\target{S_\setY}} \\ \wr\Vert && \wr\Vert \\ \text{\{ $\source{|\setX|}$-subsets ($\source{|\setX|}$-multisets with values $0$ or $1$) of $\target \setY$ \}} && \text{$1$ or $\emptyset$ as $\source{|\setX|} \le {}$ or $\gt \target{|\setY|}$ } \\ \end{array} $

Putting this all together, we have the three diagrams at the right for

• $\SetXY$ = all functions from $\source \setX$ to $\target \setY$,
• $\homst \setX \Surj \setY$ = surjections from $\source \setX$ to $\target \setY$, and
• $\homst \setX \Inj \setY$ = injections from $\source \setX$ to $\target \setY$,

showing the source, target, and double quotients
in each of the bimodules of functions, surjections, and injections.
These diagrams underlie
the “twelvefold way” of combinatorics.

Remark:
It is common (e.g., Stanley, EC, §1.9) in combinatorics to call
(the elements of the source set $\setX$) balls, and
(the elements of the target set $\setY$) boxes or urns.
Then (a function $\source\setX \to \target\setY$) is
(an assignment of balls to boxes, ball $\mapsto$ box),
and can be visualized physically,
as actually placing (physical balls) in (physical boxes or urns).

(The kernel-partition (or fibre) form)
used below to describe functions
amounts to showing, for each “box” its contents,
i.e., which “balls” are placed in it;
thus the $\{\elta,\eltb\}$-ordered partition $0,2,3 \mid 1$ means
balls $0,2,3$ go into box $\elta$,
while ball $1$ goes into box $\eltb$;
the word form for this function is $\elta\eltb\elta\elta$.
For pictorial examples, again see EC, §1.9.

---

The twelvefold way

$\bbox[navajowhite,10px,border:4px groove red]{ \begin{array}{} && & {} \mkern5em \rlap{\text{Injections}} &&&& {} \mkern7em \rlap{\text{All functions}} &&&& {} \mkern8em \rlap{\text{Surjections}} \\ && & \settY & & \target{|\setY|} & \mkern4em & \settY & & \target{|\setY|} & \mkern4em & \settY & & \target{|\setY|} \\ \setsX,\settY && & \boxed{\begin{array}{} \cattwo, \setsX, \settY\\ \homst \setX \Inj \setY\\ {\target{|\setY|}}^{\source{\underline{|\setX|}}} \\ \end{array} } && && \boxed{\begin{array}{} \text{weak}, \setsX, \settY\\ \SetXY\\ {\target{|\setY|}}^{\source{|\setX|}} = \displaystyle\target{\sum_{i=0}^{|\setY|}} \source{{|\setX| \brace \target i}}\target{i!}\target{{|\setY| \choose i}}\\ \end{array} } && && \boxed{\begin{array}{} \text{strong}, \setsX, \settY\\ \homst \setX \Surj \setY\\ \displaystyle{\source{|\setX|} \brace \target{|\setY|}}\target{|\setY|!}\\ \end{array} } \\ & \setsX & & & \target\searrow &&& & \target\searrow && {} \rlap{\mkern3em \source{\boxed{\boxed{\text{U = $\setsX$}}}}} & & \target\searrow \\ && \setsX,\target{|\setY|} & \raise2ex\hbox{$\smash{\source{\Bigg\downarrow}}$} && \llap{\lower6ex\hbox{$\boxed{\boxed{\text{L} = \cattwo}}\mkern.1em$}} \boxed{\begin{array}{} \cattwo, \setsX, \target{|\setY|} \\ \homst \setX \Inj \setY \target/ \symtY\\ [\source{|\setX|}\leq\target{|\setY|}] \\ \end{array} } && \raise2ex\hbox{$\smash{\source{\Bigg\downarrow}}$} && \llap{\lower6ex\hbox{$\boxed{\boxed{\text{C = weak}}} \mkern.2em$}} \boxed{\begin{array}{} \text{weak}, \setsX, \target{|\setY|} \\ \SetXY \target/ \symtY\\ \displaystyle\target{\sum_{i=0}^{|\setY|}} {\source{|\setX|} \brace \target i}\\ \end{array} } \rlap{\mkern.5em \target{\boxed{\boxed{\text{B = $\settY$}}}} } && \raise2ex\hbox{$\smash{\source{\Bigg\downarrow}}$} && \llap{\lower6ex\hbox{$\boxed{\boxed{\text{R = strong}}} \mkern.2em$}} \boxed{\begin{array}{} \text{strong}, \setsX, \target{|\setY|} \\ \homst \setX \Surj \setY \target/ \symtY\\ \displaystyle {\source{|\setX|} \brace \target{|\setY|}} \end{array} } \\ \source{|\setX|},\settY && & \boxed{\begin{array}{} \cattwo, \source{|\setX|}, \settY\\ \symsX \source\backslash \homst \setX \Inj \setY\\ \displaystyle {\target{|\setY|} \choose \source{|\setX|}}\\ \end{array} } && \lower3ex\hbox{$\smash{\source{\Bigg\downarrow}}$} && \boxed{\begin{array}{} \text{weak}, \source{|\setX|}, \settY\\ \symsX \source\backslash \SetXY\\ \displaystyle \bigg(\mkern-.35em {\target{{|\setY|}} \choose \source{|\setX|}} \mkern-.35em\bigg) = {\source{|\setX|}+\target{|\setY|}-1 \choose \target{|\setY|-1}} \end{array} } && \lower3ex\hbox{$\smash{\source{\Bigg\downarrow}}$} && \llap{\target{\boxed{\boxed{\text{F = } \target{|\setY|}}}} \; } \boxed{\begin{array}{} \text{strong}, \source{|\setX|}, \settY\\ \symsX \source\backslash \homst \setX \Surj \setY\\ \displaystyle \bigg(\mkern-.35em {\target{{|\setY|}} \choose \source{|\setX|} - \target{|\setY|}} \mkern-.35em\bigg) = \displaystyle {\source{|\setX|-1} \choose \target{|\setY|-1}} \end{array} } && \lower3ex\hbox{$\smash{\source{\Bigg\downarrow}}$} \\ & |\setsX| & & & \target\searrow &&& & \target\searrow && {} \rlap{\mkern3em \source{\boxed{\boxed{\text{D = } \source{|\setX|}}}}} & & \target\searrow \\ && \source{|\setX|},\target{|\setY|} & && \boxed{\begin{array}{} \cattwo, \source{|\setX|}, \target{|\setY|} \\ \symsX \source\backslash \homst \setX \Inj \setY \target/ \symtY\\ [\source{|\setX|}\leq\target{|\setY|}]\\ \end{array} } &&& & \boxed{\begin{array}{} \text{weak}, \source{|\setX|}, \target{|\setY|} \\ \symsX \source\backslash \SetXY \target/ \symtY\\ \displaystyle\target{\sum_{i=0}^{|\setY|}} \functionp_{\target i} (\source{|\setX|})\\ \end{array} } &&& & \boxed{\begin{array}{} \text{strong}, \source{|\setX|}, \target{|\setY|} \\ \symsX \source\backslash \homst \setX \Surj \setY \target/ \symtY\\ \functionp_{\target{|\setY|}} (\source{|\setX|})\\ \end{array} } & \\ \end{array} }$

The numerical formulae above are valid when $\setX$ and $\setY$ are finite nonempty sets, so that $\target{|\settY|-1\geq 0}$.
For slight variants which are valid even when $\target{|\settY|} = 0$ (i.e., when $\settY = \emptyset$), see the Wikipedia article.

The $\langle \source{|\setX|}, \settY \rangle$-cases: $\settY$-ordered partitions of the finite positive integer $\source{\numbern=|\setX|}$

Assume $\settY$ is linearly ordered with $\target{|\setY|=\numberk}$, a finite positive integer.

Strong $\numberk$-ordered partitions of $\numbern$, $1\leq\numberk\leq\numbern$.
Write ($\source\numbern$ dots in a row).
There are ($\source{\numbern-1}$ spaces) between (the $\source\numbern$ dots).
Choose $\target{\numberk-1}$ of (those $\source{\numbern-1}$ spaces), and place (a divider “$|$” (a “bar”)) in each of (the chosen $\target{\numberk-1}$ spaces).
The result is that (the $\source\numbern$ dots) are divided into ($\target\numberk$ non-empty groups).
There are ${\source{\numbern-1} \choose \target{\numberk-1}} = {\source{|\setX|-1} \choose \target{|\setY|-1}}$ ways of making (the choices described above).
For each ($\numberk$-ordered strong partition of $\numbern$) there is (a unique such choice) which (delivers the partition).
And note that any choice from the $\source{\numbern-1}$ inter-dot spaces yields a strong partition, thus there are $2^{\source{\numbern-1}}$ strong partitions in all.

Weak $\numberk$-ordered partitions of $\numbern$, arbitrary finite positive $\numberk$.
In this case, write ($\source\numbern+\target\numberk-1$ dots in a row).
Choose $\target{\numberk-1}$ of those dots, and replace (those chosen $\target{\numberk-1}$ dots) with (the divider symbol “$|$” (the “bar”)).
The result is again a division of (the $\source\numbern$ dots) into ($\target\numberk$ groups), but now (some of those groups may be empty).
There are ${\source\numbern+\target{\numberk-1} \choose \target{\numberk-1}} = {\source{|\setX|}+\target{|\setY|}-1 \choose \target{|\setY|-1}}$ ways of making (the choices described), yielding all possible (weak $\target\numberk$-ordered partitions of $\source\numbern$).

---

3. An example: $\homst {[4]} \Set {\{a,b\}}$

The double quotient square for $\homst {(\setX=[4])} \Set {(\setY=\{a,b\})}$

$ \begin{array}{} \text{\{ ordered weak $\target{\{\elta,\eltb\}}$-partitions of $\source {[4]}$ \}} && \text{\{ unordered weak $\target 2$-partitions of $\source {[4]}$ \}} \\ \llap{(\#=16)\;} {\wr\Vert} && \wr\Vert \rlap{\;(\#=8)} \\ \mkern.5em \homst {[4]} \Set {\{a,b\}} & {} \rlap{\mkern-6em\xrightarrow{\mkern14em}} & \mkern3em \homst {[4]} \Set {\{a,b\}} / \target{S_{\{a,b\}}} \\ \Bigg\downarrow && \Bigg\downarrow \\ \mkern.5em {\source{S_{[4]}}} \backslash \homst {[4]} \Set {\{a,b\}} \mkern2em & {} \rlap{\mkern-6em\xrightarrow{\mkern14em}} & \mkern1.5em {\source {S_{[4]}}} \source\backslash \homst {[4]} \Set {\{a,b\}} \target/ {\target{S_{\{a,b\}}}} \\ \llap{(\#=5)\;} {\wr\Vert} && \wr\Vert \rlap{\;(\#=3)} \\ \text{\{ ordered weak $\target{\{\elta,\eltb\}}$-partitions of $\source 4$ \}} && \text{\{ unordered weak $\target 2$-partitions of $\source 4$ \}} \\ \Vert && \Vert \\ \source{\{4+0, 3+1, 2+2, 1+3, 0+4 \}} && \source{\{4+0, 3+1, 2+2\}} \\ \wr\Vert \\ \text{\{ weak $\source 4$-multisets on $\target {\{a,b\}}$ \}} \\ \Vert \\ \{\target a^4 \target b^0, \target a^3 \target b^1, \target a^2 \target b^2, \target a^1 \target b^3, \target a^0 \target b^4\} \\ \end{array} $

There is a certain abstractness about
talking about arbitrary sets $\source \setX$ and $\target \setY$.
Let's see what all this means in a case where we can easily
both represent and count
{ the elements of the function sets $\SetXY$ and $\homst \setX \Surj \setY$ }.
In particular,
we take ($\source{\setX = [4] \equiv \{0,1,2,3\}}$) and ($\target{\setY = \{a,b\}}$),
so functions $\functionf : \source{[4]} \to \target{\{\elta,\eltb\}}$
can be represented as
four-letter words in the alphabet $\target{\{\elta,\eltb\}}$, e.g. $\target{\elta\eltb\elta\elta}$;
then $|\SetXY| = \target{|\setY|}^{\source{|\setX|}} = {\target 2}^{\source 4} = 16$
and $|\homst \setX \Surj \setY| = 14$ (all but the two constant functions).
Then (the double quotient square for $\homst {(\setX=[4])} \Set {(\setY=\{a,b\})}$)
becomes the diagram at right.
The contents of the various sets in the diagram
are described in detail below.

The source quotient: ${\source{S_{[4]}}} \backslash \homst {[4]} \Set {\{a,b\}} \cong \catI \tensor_{\source{S_{[4]}}} \homst {[4]} \Set {\{a,b\}}$

$\bbox[navajowhite,10px,border:4px groove red]{\begin{array}{lcccccccc} &&& \displaystyle { \source{0,1 \mid 2,3} \brack \target{aabb} } \\ && \displaystyle { \source{0,1,2 \mid 3} \brack \target{aaab} } & \displaystyle { \source{0,2 \mid 1,3} \brack \target{abab} } & \displaystyle { \source{0 \mid 1,2,3} \brack \target{abbb} }\\ && \displaystyle { \source{0,1,3 \mid 2} \brack \target{aaba} } & \displaystyle { \source{0,3 \mid 1,2} \brack \target{abba} } & \displaystyle { \source{1 \mid 0,2,3} \brack \target{babb} }\\ & \displaystyle { \source{0,1,2,3 \mid {}} \brack \target{aaaa} } && \class{red}\bullet && \displaystyle { \source{{} \mid 0,1,2,3} \brack \target{bbbb} }\\ && \displaystyle { \source{0,2,3 \mid 1} \brack \target{abaa} } & \displaystyle { \source{1,2 \mid 0,3} \brack \target{baab} } & \displaystyle { \source{2 \mid 0,1,3} \brack \target{bbab} }\\ && \displaystyle { \source{1,2,3 \mid 0} \brack \target{baaa} } & \displaystyle { \source{1,3 \mid 0,2} \brack \target{baba} } & \displaystyle { \source{3 \mid 0,1,2} \brack \target{bbba} }\\ &&& \displaystyle { \source{2,3 \mid 0,1} \brack \target{bbaa} }\\[2ex] \hline \source{\text{size of $S_{[4]}$-orbit}} & \source 1 & \source 4 & \source 6 & \source 4 & \source 1\\ \source{\text{size of $S_{[4]}$-stabilizer}} & \source{4!0!} & \source{3!1!} & \source{2!2!} & \source{1!3!} & \source{0!4!}\\ \source{\text{template (set form)}} & \source{\{\cdotp\cdotp\cdotp\cdotp\} \{\}} & \source{\{\cdotp\cdotp\cdotp\} \{\cdotp\}} & \source{\{\cdotp\cdotp\} \{\cdotp\cdotp\}} & \source{\{\cdotp\} \{\cdotp\cdotp\cdotp\}} & \source{\{\} \{\cdotp\cdotp\cdotp\cdotp\}} \\ \source{\text{template (divider form)}} & \source{\cdotp\cdotp\cdotp\cdotp \mid {}} & \source{\cdotp\cdotp\cdotp \mid \cdotp} & \source{\cdotp\cdotp \mid \cdotp\cdotp} & \source{\cdotp \mid \cdotp\cdotp\cdotp} & \source{{} \mid \cdotp\cdotp\cdotp\cdotp} \\ \text{partition (binary form)} & \source{0000}\target1 & \source{000}\target1\source0 & \source{00}\target1\source{00} & \source0\target1\source{000} & \target1\source{0000}\\ \text{ordered weak $\target2$-partitions of }\source4 & \source4 \target+ \source0 & \source3 \target+ \source1 & \source2 \target+ \source2 & \source1 \target+ \source3 & \source0 \target+ \source4\\ \text{weak $\source 4$-multisets on }\target{\{a,b\}} & \target a^{\source 4}\source b^{\target 0} & \target a^{\source 3}\target b^{\source 1} & \target a^{\source 2}\target b^{\source 2} & \target a^{\source 1}\target b^{\source 3} & \target a^{\source 0}\target b^{\source 4}\\[2ex] \target{\text{size of BLOCK $S_{\{a,b\}}$-stabilizer}} & \target 1 & \target 1 & \target 2 & \target 1 & \target 1 \\ \target{\text{size of BLOCK $S_{\{a,b\}}$-orbit}} & \target 2 & \target 2 & \target 1 & \target 2 & \target 2 \\ \end{array}}$

Dividing out by
(permutations of the source $\setX = \source{[4]}$),
$\SetXY \longrightarrow {\source{S_\setX\backslash{}}} \SetXY$,
i.e., indistinguishable source elements or balls,
can be visualized via the figure at right,
which shows all ${\target 2}^{\source 4}=16$ functions
from $\source{[4]=\{0,1,2,3\}}$ to $\target{\{a,b\}}$,
i.e., the elements of $\homst {[4]} \Set {\{a,b\}} = {\target{\{a,b\}}}^{\source{[4]}}$.
It shows each function
first in ordered kernel-partition (or fibre) form
($\target a$ component of the partition (or fibre)
to the left of the divider “$\mid$” (the “bar”),
$\target b$ component on the right),
then on the next line
the representation of the function
as a four letter word in the alphabet $\{a,b\}$.
(The columns) are precisely
{the orbits of $\homst {[4]} \Set {\{a,b\}}$ under
the left (pre-) $\source{S_{[4]}}$-action},
thus are precisely
{the elements of
the left quotient set $\source{S_{[4]} \backslash} \homst {[4]}\Set {\{a,b\}}$}.
Thus the $\symsX$-quotient map
($\SetXY \longrightarrow {\source{S_\setX\backslash{}}} \SetXY$)
takes (each function) into
(its column in the table).
(The bottom rows of the table) show
some information about each column/$\symsX$-orbit,
namely, its size and
the size of the stabilizers of its elements,
and some alternative ways of describing
the column/$\symsX$-orbit, i.e.,
some total invariants of the column/$\symsX$-orbit:
(the ordered weak $\target{\{\elta,\eltb\}}$-partitions of $\source 4$) and (the weak $\source 4$-multisets on $\target{\{a,b\}}$).

For surjections, simply omit the two outer columns.

Now consider (the action of $\symtY = \target{S_{\{a,b\}}}$) on both (the total space $\homst {[4]} \Set {\{a,b\}}$) and (its source quotient ${\source{S_{[4]}\backslash{}}} \homst {[4]} \Set {\{a,b\}}$).
(The columns of the table) have been so ordered that (the $\source{S_{\{a,b\}}}$-action) on (the total space $\homst {[4]} \Set {\{a,b\}}$)
amounts to reflection in (the dot $\class{red}\bullet$ which is at the center of the table).
The key point is that (reflection in the center dot $\class{red}\bullet$) (respects the columns), i.e., it takes (functions\words in the same column) into (functions\words in the same column).
Thus (the central reflection on the 16 points) passes to (a reflection on the five columns), namely, (reflection in (the vertical axis through the center of the table)).
That reflection (swaps the four outer columns), but (fixes the center column).
Thus (the resulting iterated quotient $\source{( S_{[4]} \backslash } \homst {[4]} \Set {\{a,b\}} \source) / \target{S_{\{a,b\}}}$) has three elements ($\symtY$-orbits of $\symsX$-orbits):
{ the pair of outer columns, the pair of next-to-outer columns, and the center column }.

---

The target quotient:
$\target( \homst {[4]} \Set {\{a,b\}} \target{/ S_{\{a,b\}} )} \cong \homst {[4]} \Set {\{a,b\}} \tensor_{\target{S_{\{a,b\}}}} \catI$

$$\begin{array}{cc} \displaystyle { \source{0,1,2,3 \mid {} } \brack \target{aaaa} } \displaystyle { \source{ {} \mid 0,1,2,3} \brack \target{bbbb} } \end{array}$$		$\{\,4+0,0+4\,\}$	$\{\,4,0\,\}$
$$\begin{array}{cc} \displaystyle { \source{0,1,2 \mid 3} \brack \target{aaab} } & \displaystyle { \source{3 \mid 0,1,2} \brack \target{bbba} }\\ \displaystyle { \source{0,1,3 \mid 2} \brack \target{aaba} } & \displaystyle { \source{2 \mid 0,1,3} \brack \target{bbab} }\\ \displaystyle { \source{0,2,3 \mid 1} \brack \target{abaa} } & \displaystyle { \source{1 \mid 0,2,3} \brack \target{babb} }\\ \displaystyle { \source{1,2,3 \mid 0} \brack \target{baaa} } & \displaystyle { \source{0 \mid 1,2,3} \brack \target{abbb} }\\ \end{array}$$		$\{\,3+1,1+3\,\}$	$\{\,3,1\,\}$
$$\begin{array}{cc} \displaystyle { \source{0,1 \mid 2,3} \brack \target{aabb} } & \displaystyle { \source{2,3 \mid 0,1} \brack \target{bbaa} }\\ \displaystyle { \source{0,2 \mid 1,3} \brack \target{abab} } & \displaystyle { \source{1,3 \mid 0,2} \brack \target{baba} }\\ \displaystyle { \source{0,3 \mid 1,2} \brack \target{abba} } & \displaystyle { \source{1,2 \mid 0,3} \brack \target{baab} }\\ \end{array}$$		$\{\,2+2\,\}$	$\{\,2,2\,\}$

For dividing out by (permutations of the target $\target{\setY = \{a,b\}}$),
$\SetXY \mathrel{\target\longrightarrow} \SetXY {\target{{}/S_\setY}}$, i.e., indistinguishable target elements or boxes,
we have a similar figure, at right.
Here (the $\target{S_\setY}$-orbits) are arranged in rows;
thus ((the $\target{S_{\setY}}$-quotient map) onto (the $\target{S_{\setY}}$-orbits)) is now simply (taking the row).

(The source permutation group $\source{S_\setX = S_{[4]}}$) acts, not only on (the elements of $\SetXY$),
but also on (their $\symtY$-orbits): $\symsX \times \target{\big(} \SetXY \target{{} / \symtY \big)} \mathrel{\source\longrightarrow} \target{\big(} \SetXY \target{{} / \symtY \big)}$.
{The orbits of $\SetXY \target/ \symtY $ under that ($\source{\symsX = S_{[4]}}$)-action}
are {the three blocks of rows separated by horizontal rules} —
these are the three elements of the set of $\symsX$-orbits of $\symtY$-orbits, $\source{S_\setX \backslash} \target{\big(} \SetXY \target{{} / S_\setY \big)}$.
They correspond to the three unordered weak $\target 2$-partitions of $\source 4$: $\source{4+0, 3+1, 2+2}$.

The key point again is that
there are bijections between (the iterated quotients) and (the double quotient)
i.e., between {blocks of columns}, {blocks of rows}, and (the orbits of the two-sided action),
rendering commutative the five-quotient diagram $$\begin{array}{} \smash{\SetXY} & {} \rlap{ \mkern-.5em \smash{\target{\xrightarrow[\mkern18em]{\displaystyle \functiont}} } } &&& \mkern1em \SetXY \target{ / S_\setY} \\ &&&& \source{\big\downarrow} \\ \smash{\source{\llap{\functions} \Bigg\downarrow}} && \searrow \rlap{\raise1ex\functionr} && \smash{ \source{S_\setX \backslash } \target( \SetXY \target{/ S_\setY )} } \\ &&&& \wr\Vert \\ \source{S_\setX \backslash} \SetXY & {} \rlap{\mkern-.5em\target\longrightarrow} & \mkern1em \source{( S_\setX \backslash } \SetXY \source) \target{/ S_\setY} & \cong & \source{S_\setX \backslash} \SetXY \target{/ S_\setY} & , \\ \end{array}$$ namely the evident bijections between (the three-element sets) listed on (three rows below).
Note how (the double quotient), in the middle row, flattens (the iterated quotients),
replacing (orbits of orbits) with just (orbits).

$\mkern-3em \begin{array}{cccccccccl} {} \rlap{\mkern19em \text{The quotients as sets of sets (of sets), and some invariants of those sets}} \\ && \Big\{ & \source{\big\{0123\mid{}\big\}}& , & \source{\big\{\mkern.5em 012\mid3 \mkern1em,\mkern1em 013\mid2 \mkern1em,\mkern1em 023\mid1 \mkern1em,\mkern1em 123\mid0\mkern.5em\big\}} & , & \source{\big\{\mkern1em 01\mid23 \mkern1em,\mkern1em 02\mid13 \mkern1em,\mkern1em 03\mid12 \mkern1em\big\}} & \Big\} & \text{unordered parts. of $[4]$} \\ \source{S_\setX \backslash } \target( \SetXY \target{/ S_\setY )}& = & \Big\{ & \big\{ \{\elta^4,\eltb^4\} \big\} & , & \big\{ \{\elta^3\eltb,\eltb^3\elta\}, \{\elta^2\eltb\elta,\eltb^2\elta\eltb\}, \{\elta\eltb\elta^2,\eltb\elta\eltb^2\}, \{\eltb\elta^3,\elta\eltb^3\} \big\} & , & \big\{ \{\elta^2\eltb^2,\eltb^2\elta^2\}, \{\elta\eltb\elta\eltb,\eltb\elta\eltb\elta\}, \{\elta\eltb^2\elta, \eltb\elta^2\eltb\} \big\} & \Big\} & \text{$\symsX$-orbits of $\symtY$-orbits} \\ \source{S_\setX \backslash} \SetXY \target{/ S_\setY} & = & \Big\{ & \big\{ \elta^4,\eltb^4 \big\} & , & \big\{ \elta^3\eltb, \elta^2\eltb\elta, \elta\eltb\elta^2, \eltb\elta^3, \elta\eltb^3, \eltb\elta\eltb^2, \eltb^2\elta\eltb, \eltb^3\elta \big\} & , & \big\{ \elta^2\eltb^2, \elta\eltb\elta\eltb, \elta\eltb^2\elta, \eltb\elta^2\eltb, \eltb\elta\eltb\elta, \eltb^2\elta^2 \big\} & \Big\} & \text{$\source{\symsX\op} \rightadj\times \symtY$-orbits} \\ \source{( S_\setX \backslash } \SetXY \source) \target{/ S_\setY} & = & \Big\{ & \big\{ \{\elta^4\},\{\eltb^4\} \big\} & , & \big\{ \{\elta^3\eltb, \elta^2\eltb\elta, \elta\eltb\elta^2, \eltb\elta^3\}, \{\elta\eltb^3, \eltb\elta\eltb^2, \eltb^2\elta\eltb, \eltb^3\elta\} \big\} & , & \big\{ \{\elta^2\eltb^2, \elta\eltb\elta\eltb, \elta\eltb^2\elta, \eltb\elta^2\eltb, \eltb\elta\eltb\elta, \eltb^2\elta^2\} \big\} & \Big\} & \text{$\symtY$-orbits of $\symsX$-orbits} \\ && \Big\{ & \source{\big\{4+0,0+4\big\}} & , & \source{\big\{\mkern3em 3+1 \mkern3em , \mkern3em 1+3 \mkern3em\big\}} & , & \source{\big\{\mkern4em 2+2 \mkern4em\big\}} & \Big\} & \text{ordered parts. of $4$} \\ && \Big\{ & \source{4+0} & , & \source{3+1} & , & \source{2+2} & \Big\} & \text{unordered parts. of $4$} \\ \end{array}$

For further information, see the post “The fiber product at work (in elementary combinatorics)”.

---

References

EC, Stanley, Richard P., Enumerative Combinatorics I, 1986/2011
W, Wikipedia, “Twelvefold way”
FPAW, “The fiber product at work (in elementary combinatorics)”, 2019
DG12W, “Double groupoids and the twelvefold way”, 2016
PF, “The parts of a function”, 2016
CFP, “Classifying functions by their parts”, 2016
QKA, “The quotient-kernel adjunction”, 2016
KQFC, “Kernels, quotients, and function composition”, 2019

Modules in combinatorics

Consider (the totally disconnected groupoid $\boxed\Perm$ of permutations of sets),
which (considered as a category) is a subcategory of (the category $\Set$): $\quad \Perm \subset \Set$.
Recall that any category, say $\calC$, is a left-right bimodule over itself, via its hom bifunctor: $\smash{ \quad \calC\op \times \calC \xrightarrow{\textstyle \hom - \calC -} \Set }\;$.
In particular (the category $\Set$) is a left-right bimodule over itself: $(\functionf,\functiong,\functionh) \mapsto \functionf\functiong\functionh$.
By restriction of scalars, $\Set$ is a bimodule over (its groupoid subcategory $\Perm$): $\smash{ \quad \Perm\op\times\Perm \xrightarrow{\textstyle \hom - \Set -} \Set \; : \; (\sigma, \tau) \mapsto ( \hom \sigma \Set \tau \; : \; \functionf \mapsto \sigma \functionf \tau ) } \;$.
(Analogy: In algebra, any ring is a left-right bimodule (Wikipedia, nLab) over each of its subrings.)

Take the left, right, and left-right quotients of $\Set$ by $\Perm$.
This yields [(the first four parts) of (the "twelvefold way")] .

Now consider the subcategories of surjections, $\boxed\Surj$, and, of injections, $\boxed\Inj$, of the category $\Set$.
Each is closed under both left and right actions of $\Perm$, thus is a sub-$\Perm$-bimodule.
Applying the same (quotient by $\Perm$) constructions to (those two submodules) gives [(the other eight parts) of (the twelvefold way)].

We can present the twelvefold way as a table:

The Twelvefold Way

Module, with scalars	Module, without scalars	Left Quotient indistinguishable balls	Right Quotient indistinguishable boxes	Left-Right Quotient indistinguishable balls and boxes
$(\Perm, \Set)$	$\Set$	$\Perm\backslash\Set$	$\Set/\Perm$	$\Perm\backslash\Set/\Perm$
$(\Perm, \Surj)$	$\Surj$	$\Perm\backslash\Surj$	$\Surj/\Perm$	$\Perm\backslash\Surj/\Perm$
$(\Perm, \Inj)$	$\Inj$	$\Perm\backslash\Inj$	$\Inj/\Perm$	$\Perm\backslash\Inj/\Perm$

---

Consider the intersection of the submodules $\Surj$ and $\Inj$, the groupoid $\smash{\boxed\Bij}$ of bijections of sets
(which of course contains $\Perm$ as a subgroupoid: $\Perm \subset \Bij \subset \Surj,\Inj \subset \Set$).
Given any three equipotent sets, say $\setX$, $\setY$, and $\setZ$,
consider commutative triangles in $\Bij$ with those fixed sets as vertices.
Fixing one side of such a triangle to a specific (internal) bijection
sets up (an external bijection between the possible fill-ins for the other two sides).
Thus, for example, choosing (an internal bijection from $\setX$ to $\setY$)
sets up (an external bijection from $\hom \setY \Set \setZ$ to $\hom \setX \Set \setZ$),
which restricts to (an external bijection from $\hom \setY \Bij \setZ$ to $\hom \setX \Bij \setZ$).

If we have an arbitrary finite set $\setY$, of size $n$,
let $\setX = \mathbf n$, the canonical linear order of size $n$.
Then

choosing a particular linear ordering of $\setY$ (a specific element of $\hom {\mathbf n} \Bij \setY$) sets up
an external bijection from {permutations of $\setY$ ($\hom \setY \Perm \setY = \hom \setY \Bij \setY$)} to {linear orders on $\setY$ ($\hom {\mathbf n} \Bij \setY$)},

a result which is frequently used, e.g., rather extensively in Stanley, Enumerative Combinatorics.

---

References

[CKVW] Carboni, A.; Kelly, G. M.; Verity, D.; Wood, R. J. (1998). "A 2-Categorical Approach To Change Of Base And Geometric Morphisms II". Theory and Applications of Categories. 4 (5): 82–136.