$\begin{array}{} \text{$2$-cells} \\ \ssigma,\functionf,\ttau \\ \symsX \mathrel{\source\times} \SetXY \mathrel{\target\times} \symtY \\ \end{array}$ |
$\mkern2em\mathop\rightrightarrows\limits^{\symsX \times \text{proj}}_{\symsX \target{\times \text{comp}}}$ |
$\begin{array}{} \source{\text{v-$1$-cells}} \\ \ssigma,\functionf \\ \ssigma,\functionf\ttau \\ \symsX \mathrel{\source\times} \SetXY \source{{}=V} \\ \end{array}$ |
$\mkern-5em\target{\xrightarrow[\mkern24em]{\displaystyle \symsX \mathrel{\source\times} \functiont}}$ |
$\begin{array}{} \ssigma, \target[\functionf\symtY\target] \\ \symsX \mathrel{\source\times} \target( \SetXY\target/\symtY \target) \\ \end{array}$ |
||
| $\smash{\lower8ex{\llap{\scriptstyle \text{proj}\times\symtY} \downdownarrows \rlap{\scriptstyle \source{\text{comp}}\times\symtY}}}$ | $\llap{\scriptstyle \text{proj}} \searrow\mkern-.6em\searrow \rlap{\scriptstyle \text{comp}}$ | $\llap{\scriptstyle \text{proj}} \downdownarrows \rlap{\scriptstyle \source{\text{comp}}}$ | $\smash{\lower8ex{\llap{\scriptstyle \text{proj}} \downdownarrows \rlap{\scriptstyle \source{\text{action}}}}}$ | |||
$\begin{array}{} \target{\text{h-$1$-cells}} \\ \functionf,\ttau\mkern.5em;\mkern.5em\ssigma\functionf,\ttau \\ \SetXY \mathrel{\target\times} \symtY \target{{}=H}\\ \end{array}$ |
$\mathop\rightrightarrows\limits^{\text{proj}}_{\target{\text{comp}}}$ | $\begin{array}{} && \text{$0$-cells} \\ \scriptstyle \text{proj} \\ \scriptstyle (\symsX \times \text{proj})\text{proj} &&&& \scriptstyle (\symsX \times \target{\text{comp}})\text{proj} \\ \scriptstyle (\text{proj} \times \symtY)\text{proj} &&&& \scriptstyle (\text{proj} \times \symtY)\target{\text{comp}} \\ \functionf & {}\rlap{\mkern-2em \target{\xrightarrow[\mkern16em]{\displaystyle \ttau}}} &&& \functionf\ttau \\ &&&& \llap\ssigma \downarrow \\ \smash{\source{\llap\ssigma \Bigg\downarrow}} && \SetXY && {\displaystyle \ssigma\target(\functionf\ttau\target) \atop \source(\symsX \source\times \target{\text{comp}}\source)\source{\text{comp}}} \\ &&&& \Vert \\ \ssigma\functionf & {}\rlap{\mkern-2em \target{\xrightarrow[\displaystyle \ttau]{\mkern4.5em}}} & \source(\ssigma\functionf\source)\ttau & {}\rlap{\smash{\mkern-2em\xlongequal{\mkern5.5em}}} & \ssigma\functionf\ttau \\ \scriptstyle (\symsX \times \text{proj})\source{\text{comp}} && \scriptstyle (\source{\text{comp}} \times \symtY)\target{\text{comp}} && \scriptstyle \text{comp} \\ \scriptstyle (\source{\text{comp}} \times \symtY)\text{proj} \\ \end{array}$ | $\mkern1em\target{\xrightarrow[\mkern15em]{\displaystyle \functiont}}$ |
$\begin{array}{} \target{\text{$\symtY$-orbits}}\\ \target[\functionf\symtY\target], \ssigma\target[\functionf\symtY\target] = \target[\ssigma\functionf\symtY\target] \\\SetXY\target/\symtY \\ \end{array}$ |
||
| $\smash{\raise8ex{\Bigg\downarrow}}$ | $\smash{\raise6ex{\source{\llap\functions\Bigg\downarrow} \rlap{\mathrel{\target\times} \symtY}}}$ | $\source{\llap\functions\Bigg\downarrow}$ | $\mkern-6em \functionr \searrow $ |
$\begin{array}{} \source{\text{$\symsX$-orbits of }} \target{\text{$\symtY$-orbits}} \\ \source{\big[}\symsX\target[\functionf\symtY\target]\source{\big]} \\ \symsX\source\backslash \target( \SetXY \target{{/}} \symtY \target) \\ \end{array}$ |
||
| $\wr\Vert$ | ||||||
$\begin{array}{} \source[\symsX\functionf\source], \ttau \\ \source( \symsX\source\backslash\SetXY \source) \mathrel{\target\times} \symtY \\ \end{array}$ |
$\mkern2em\mathop\rightrightarrows\limits^{\text{proj}}_{\target{\text{action}}}$ |
$\begin{array}{}\source{\text{$\symsX$-orbits}}\\ \source[\symsX\functionf\source] \\ \source[\symsX\functionf\source]\ttau = \source[\symsX\functionf\ttau\source] \\ \symsX\source\backslash\SetXY \\ \end{array}$ |
$\mkern-6.5em\xrightarrow[\mkern7em]{}$ |
$\begin{array}{} \target{\text{$\symtY$-orbits of }} \source{\text{$\symsX$-orbits}}\\ \target{\big[}\source[\symsX\functionf\source]\symtY\target{\big]} \\ \source( \symsX\source\backslash\SetXY \source) \target{{/}} \symtY \\ \end{array}$ |
$\mkern1em\cong$ |
$\begin{array}{} \text{$\symsX, \symtY$-orbits}\\ \source[\symsX\functionf\symtY\source] \\ \symsX \source\backslash \SetXY \target{{/}} \symtY \\ \end{array}$ |
The above diagram is constructed in a step-by-step fashion:
- (The single actions of $\symsX$ and $\symtY$ on $\SetXY$), and (the joint, simultaneous, action of $\symsX$ and $\symtY$ on it) (all given by composition),
together with (projection maps out of the products), give (the three graphs into $\SetXY$). - Applying (the quotient functor $\ladjQ$) (see “The quotient-kernel adjunction”) to (the graphs for (the actions of $\symsX$, $\symtY$, and $(\symsX,\symtY$)))
gives (the quotient maps $\source\functions, \target\functiont, \functionr$) for (the middle column, middle row, and diagonal). - Consider (the middle column, for the $\symsX$-action).
Applying (the functor $-\times\symtY$) to it, and using (the associativity of cartesian product in $\Set$), gives (the column to its left).
Likewise, applying (the functor $\symsX\times-$) to (the middle row, for the $\symtY$ action) gives (the row above it). - This yields (the four non-diagonal graphs) in (the upper left quadrant).
That (the square diagram of parallel pairs formed by (those four graphs)) serially commutes follows from (the properties of the projection maps)
and from the fact that (the $\symsX$ and $\symtY$ actions on $\SetXY$ commute), i.e., from (the associativity of left and right actions in the bimodule). - Recall that $\langle \Set, \times \rangle$ is a cartesian closed category, thus for each set $\setZ$,
$\setZ\times- \dashv [\setZ,-] = \homst \setZ \Set -$ and $-\times\setZ \dashv [\setZ,-] = \homst \setZ \Set -$.
Left adjoints preserve colimits, hence preserve quotients.
Hence (the arrows $\functions\times\symtY$ and $\symsX\times\functiont$) are (quotient maps for (the left column and top row)), thus are in (the image of $\ladjQ$). - That (the upper left quadrant) serially commutes then gives (the action maps at (lower left and upper right)).
(The projection maps parallel to (those action maps)) are part of (the definitions of the products).
Thus we have (the graphs at (the lower left and upper right)). - Taking quotients of (those graphs) gives ((the two remaining quotient maps) in (the lower right quadrant)).
- The canonical bijections in (the lower right quadrant) are as shown; all are well-defined.
- (The final arrow in each row) is (a quotient for (the graph to its left)),
and (the final arrow in each column) is (a quotient for (the graph above it)).
This, by definition, makes each of (the three rows and the three columns) “right exact” (per the definition in [Bourn 2003]). - However, (the six graphs) to the left of, or above, (these quotient maps) are not in general (the kernel relations) for (those quotients).
Indeed, they are not in general even relations (i.e., jointly monic).
E.g., for (a constant function), (every element of $\symsX$) is a loop (stabilizer),
as is (any element of $\symtY$ which does not move, i.e., stabilizes, the constant value that is the image).
Hence (the rows and columns) are not “left exact” (again per the definition in [Bourn 2003]).
For a relevant version of the $3\times3$ lemma, see section 2 of “The denormalized 3×3 lemma” by Dominique Bourn, 2003
References
“The bimodule of sets, functions and permutations, and its quotients”, 2019
FPAW, “The fiber product at work (in elementary combinatorics)”, 2019
DG12W, “Double groupoids and the twelvefold way”, 2016
[Bourn 2003] Dominique Bourn, “The denormalized 3×3 lemma”, 2003
Note: For what Bourn calls a “denormalized exact sequence”,
it seems to me to be preferable to use the term “exact fork”, which is both simpler and more descriptive.
For a precedent for calling such diagrams “forks”, see CWM.
As mentioned, the above diagram is right exact but not left exact.
There are two methods by which we can make it both right and left exact:
either by going forward or by going backward:
- Going forward:
First, note that (the six quotient maps in the diagram) are all images of (the quotient functor $\ladjQ$).
Now apply (the kernel relation functor $\radjK$) to (the six quotient maps).
This gives (six kernel relations), obtained by applying (the composite functor $\ladjQ\radjK$) to (the six original graphs).
Now recall that (the triple composite $\ladjQ\radjK\ladjQ) \cong \ladjQ$.
Hence (each row and column) is now (both left and right exact). - Going backward:
Examine more closely the construction of the diagram given above.
Note that (the quotient functor $\ladjQ$), applied to (a graph), is in fact defined in two stages:
First take (the equivalence relation generated by (the graph)), then take (the quotient of (that equivalence relation)).
When (the graph )is that determined by (the action of a group on a set), as in our three cases here,
the operation of computing (the equivalence relation the graph generates) is particularly simple:
it is just (the image of the graph), as (that image) is already (an equivalence relation).
In this situation, the equivalence relation and its quotient form a (kernel relation/quotient) pair, i.e., (an exact fork), one (both left and right exact).
For the application of this process to the graphs determined by $\symsX$, $\symtY$, and $(\symsX,\symtY$),
see the equivalence relations $\relationsS$, $\relationtT$, and $\relationR$ defined below, and their quotients.
Bits and pieces of some further analysis:
Equivalence relations
For a more detailed analysis, it is useful to state the explicit definitions of
(the equivalence relations $\relationsS, \relationtT, \relationR$) determined by (the three actions), and examine (the relations) between (those equivalence relations).
(The following diagrams) show how (the three actions) generate (the three equivalence relations).
(The diagonal arrows) are (the induced maps) into (the cartesian product);
(the equivalence relations $\relationsS, \relationtT, \relationR$) are (the images) of (those maps).
Thus, e.g., $\langle \functionf,\functiong \rangle \in \relationsS \Longleftrightarrow \source{(\exists\ssigma\in\symsX)}(\functiong=\ssigma\functionf)$.
$\relationsS\relationtT = \relationR = \relationtT\relationsS$
\[ \begin{array}{} \begin{array}{} & \xrightarrow{\displaystyle\functionh} & \\ \llap{\ssigma} \downarrow & \searrow \rlap{\mkern-1em\ssigma\functionf} & \uparrow \rlap{\ttau} \\ & \xrightarrow[\displaystyle\functionf]{} & \\ \end{array} & \mkern2em\Longleftarrow\mkern2em & \begin{array}{} & \xrightarrow{\displaystyle\functionh} & \\ \llap{\ssigma} \downarrow && \uparrow \rlap{\ttau} \\ & \xrightarrow[\displaystyle\functionf]{} & \\ \end{array} & \mkern2em\Longrightarrow\mkern2em & \begin{array}{} & \xrightarrow{\displaystyle\functionh} & \\ \llap{\ssigma} \downarrow & \nearrow \rlap{\mkern-1.2em\functionf\ttau} & \uparrow \rlap{\ttau} \\ & \xrightarrow[\displaystyle\functionf]{} & \\ \end{array} \\ \Downarrow && \Updownarrow && \Downarrow \\ \langle \functionf,\ssigma\functionf \rangle \in \relationsS \land \langle \ssigma\functionf,\functionh \rangle \in \relationtT && \langle \functionf,\functionh \rangle \in \relationR && \langle \functionf,\functionf\ttau \rangle \in \relationtT \land \langle \functionf\ttau,\functionh \rangle \in \relationsS \\ \Downarrow &&&& \Downarrow \\ \langle \functionf,\functionh \rangle \in \relationsS\relationtT &&&& \langle \functionf,\functionh \rangle \in \relationtT\relationsS \\ \end{array} \]
\[ \begin{array}{} \begin{array}{} & \xrightarrow{\displaystyle\functionh} & \\ \llap{\ssigma} \downarrow & \searrow \rlap{\mkern-1em\functiong} & \uparrow \rlap{\ttau} \\ & \xrightarrow[\displaystyle\functionf]{} & \\ \end{array} & \mkern2em\Longrightarrow\mkern2em & \begin{array}{} & \xrightarrow{\displaystyle\functionh} & \\ \llap{\ssigma} \downarrow && \uparrow \rlap{\ttau} \\ & \xrightarrow[\displaystyle\functionf]{} & \\ \end{array} & \mkern2em\Longleftarrow\mkern2em & \begin{array}{} & \xrightarrow{\displaystyle\functionh} & \\ \llap{\ssigma} \downarrow & \nearrow \rlap{\mkern-1.2em\functiong} & \uparrow \rlap{\ttau} \\ & \xrightarrow[\displaystyle\functionf]{} & \\ \end{array} \\ \Uparrow && \Updownarrow && \Uparrow \\ (\exists\functiong)\big(\langle \functionf,\functiong \rangle \in \relationsS \land \langle \functiong,\functionh \rangle \in \relationtT\big) && \langle \functionf,\functionh \rangle \in \relationR && (\exists\functiong)\big(\langle \functionf,\functiong \rangle \in \relationtT \land \langle \functiong,\functionh \rangle \in \relationsS\big) \\ \Uparrow &&&& \Uparrow \\ \langle \functionf,\functionh \rangle \in \relationsS\relationtT &&&& \langle \functionf,\functionh \rangle \in \relationtT\relationsS \\ \end{array} \]
$(\relationsS\relationtT = \relationR = \relationtT\relationsS) \Longrightarrow (\relationR = \relationsS \lor \relationtT$ in the poset of equivalence relations on $\SetXY)$.
First, clearly $\relationsS\subseteq\relationR$ and $\relationtT\subseteq\relationR$.
Now suppose $\relationsS\subseteq\relationU, \relationtT\subseteq\relationU$, $\relationU$ an equivalence relation on $\SetXY$.
Then \[\begin{array}{} && \relationsS & \subseteq & \relationU \\ &&& \Downarrow \\ \relationR & = & \relationsS\relationtT & \subseteq & \relationU\relationtT & \subseteq & \relationU^2 & \subseteq & \relationU \\ &&&&& \Uparrow \\ &&&& \relationtT & \subseteq & \relationU \end{array}\]
\[\begin{array}{} \relationR & \supseteq & \relationS \\ \cup\mid & \searrow\mkern-.6em\searrow & \downdownarrows \\ \relationT & \rightrightarrows & \objectA & \xrightarrow{\displaystyle\functiont} & \objectA{/}\relationT \\ && \llap\functions \downarrow & \searrow \rlap{\raise1ex\functionr} & \downarrow \\ && \relationS\backslash\objectA & \xrightarrow{} & \relationS\backslash\objectA{/}\relationT \\ \end{array}\]
\[\begin{array}{} (\functionr=\relationR\ladjQ) & \to & \functionu && \big((\functions=\relationsS\ladjQ) \to \functionu\big) & \land & \big((\functiont=\relationtT\ladjQ) \to \functionu\big) \\ & \Updownarrow &&&& \Updownarrow \\ & \relationR \forktwoone \functionu &&& \big(\relationS\forktwoone\functionu\big) & \land & \big(\relationT\forktwoone\functionu\big) \\ & \Updownarrow &&&& \Updownarrow \\ \relationR & \subseteq & (\functionu\radjK=\relationU) & \mkern2em \Longleftrightarrow \mkern2em & \big(\relationS\subseteq (\functionu\radjK=\relationU)\big) & \land & \big(\relationT\subseteq (\functionu\radjK=\relationU)\big) \end{array}\]
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