The content starts on a new page, to keep the rather large diagram below on one page when printing in landscape mode.
| $\equ$ | $\begin{array}{} \boxed{ \begin{array}{} \boxed{ \begin{array}{} \setX\times_{(\setX\times\setY)}\setX \\ \eltx,\eltxp \mid \langle \eltx,\eltx\functionf \rangle \xlongequal{\setX\times\setY} \langle \eltxp,\eltxp\functiong \rangle \\ \eltx,\eltxp \mid \eltx \xlongequal{\setX} \eltxp \land \eltx\functionf \xlongequal{\setY} \eltxp\functiong \\ \eltx \mid \eltx\functionf \xlongequal{\setY} \eltx\functiong \\ \boxed{ \langle \functionf,\functiong \rangle \Equ} \\ \end{array} } & \xrightarrow{} & \setX \\ \llap{\langle \functionf,\functiong \rangle \equ} \Big\downarrow & \mkern-2em\smash{\raise2ex\hbox{$\lrcorner$}} & \Big\downarrow \rlap{\langle \setX,\functiong \rangle} \\ \setX & \xrightarrow{\smash{ \textstyle \langle \setX,\functionf \rangle }} & \setX\times\setY \\ \end{array} \mkern1.5em } \\ \boxed{ \begin{array}{} \boxed{ \begin{array}{} \eltx,\elty \mid \eltx\functionf = \elty = \eltx\functiong \\ \eltx \mid \eltx\functionf = \eltx\functiong \\ \boxed{ \langle \functionf,\functiong \rangle \Equ} \\ \end{array} } & \xrightarrow{} & \boxed { \begin{array}{} \eltx,\elty,\eltxp \mid \eltx\functionf = \elty = \eltxp\functiong \\ \eltx,\eltxp \mid \eltx\functionf = \eltxp\functiong \\ \end{array} } & \xrightarrow{} & \setY \\ \llap{\langle \functionf,\functiong \rangle \equ} \Big\downarrow & \mkern-2em\smash{\raise2ex\hbox{$\lrcorner$}} & \Big\downarrow & \mkern-2em\smash{\raise2ex\hbox{$\lrcorner$}} & \Big\downarrow \rlap{\Delta_\setY} \\ \setX & \xrightarrow{\smash {\textstyle \Delta_\setX}} & \setX\times\setX & \xrightarrow{\smash {\textstyle \functionf\times\functiong}} & \setY\times\setY \\ \Vert &&&& \Vert \\ \setX & {}\rlap{\mkern-2em\xrightarrow[\textstyle \mkern17em]{\smash {\textstyle \langle \functionf,\functiong \rangle }}} &&& \setY\times\setY \\ \end{array} } \\ \end{array} $ | $ \boxed { \begin{array}{} \big(\setX \xrightrightarrows[\functiong]{\functionf}\setY\big)\Limit \\ \boxed{ \langle \functionf,\functiong \rangle \Equ } \\ \eltx \mid \eltx\functionf = \eltx\functiong \\ \end{array} } \xrightarrow{\textstyle \boxed{\langle \functionf,\functiong \rangle \equ}} \setX \xrightrightarrows[\textstyle\functiong]{\textstyle\functionf} \setY $ | |
|---|---|---|---|
| $\kp$ | $\begin{array}{} \boxed{ \begin{array}{} \boxed{ \functionf\Kp} && \xrightrightarrows[\smash {\textstyle \functionf\kp}]{} && \setX \\ \Vert &&&& \Vert \\ \boxed{ \begin{array}{} \eltx,\elty,\eltxp \mid \eltx\functionf = \elty = \eltxp\functionf \\ \eltx,\eltxp \mid \eltx\functionf = \eltxp\functionf \\ \boxed{ \functionf\Kp} \\ \end{array} } & \xrightarrow{} & \setX\times\setX & \xrightrightarrows[\textstyle \pi_1]{\textstyle \pi_0} & \setX \\ \big\downarrow & \mkern-2em\smash{\raise2ex\hbox{$\lrcorner$}} & \smash{\Bigg\downarrow\rlap{\mkern-1.5em \functionf\times\functionf}} && \smash{\Bigg\downarrow \rlap\functionf} \\ \setY & \xrightarrow{\smash {\textstyle\Delta_\setY}} & \setY\times\setY & \xrightrightarrows[\textstyle \pi_1]{\smash{\textstyle \pi_0}} & \setY \\ \end{array} \mkern.5em} \\ \boxed{ \begin{array}{} \boxed{ \begin{array}{} \eltx,\eltxp \mid \eltx\functionf = \eltxp\functionf \\ \boxed{ \functionf\Kp} \\ \end{array} } & \xrightarrow{} & \setX \\ \Big\downarrow & \mkern-2em\smash{\raise2ex\hbox{$\lrcorner$}} & \Big\downarrow \rlap{\functionf} \\ \setX & \xrightarrow{\smash {\textstyle \mkern.5em\functionf\mkern.5em}} & \setY \\ \end{array} \mkern1.5em } \\ \end{array} $ | $\begin{array}{} \boxed{ \begin{array}{} \big(\setX \xrightarrow{\functionf} \setZ \xleftarrow{\functionf} \setX\big)\Limit \\ \boxed{\functionf\Kp} \\ \eltx,\eltxp \mid \eltx\functionf = \eltxp\functionf \\ \end{array} } & \xrightrightarrows{\textstyle \boxed{\functionf\kp}} & \setX & \xrightarrow{\textstyle\functionf} & \setY \\ \end{array}$ | $\begin{array}{} \boxed{ \begin{array}{} \langle \pi_0\functionf,\pi_1\functionf \rangle \Equ \\ \eltx,\eltxp \mid \langle \eltx,\eltxp \rangle \pi_0\functionf = \langle \eltx,\eltxp \rangle \pi_1\functionf \\ \eltx,\eltxp \mid \eltx\functionf = \eltxp\functionf \\ \boxed{\functionf\Kp} \\ \end{array} } \\ & \llap{\langle \pi_0\functionf,\pi_1\functionf \rangle \equ} \searrow \\ && \setX\times\setX & \xrightarrow{\textstyle \pi_1} & \setX \\ && \llap{\pi_0} \big\downarrow & \not= & \big\downarrow \rlap\functionf \\ && \setX & \xrightarrow[\textstyle \functionf]{} & \setY \\ \end{array}$ |
| $\pb$ | $\begin{array}{} \boxed{\begin{array}{} \big(\setX \xrightarrow{\functionf} \setZ \xleftarrow{\functiong} \setY\big)\Limit \\ \boxed{\setX\times_\setZ\setY} \\ \eltx,\eltz,\elty \mid \eltx\functionf = \eltz = \elty\functiong \\ \eltx,\elty \mid \eltx\functionf = \elty\functiong \\ \end{array} } & \xrightarrow{} & \setY \\ \bigg\downarrow & \mkern-2em\smash{\raise2ex\hbox{$\lrcorner$}} & \bigg\downarrow \rlap{\functiong} \\ \setX & \xrightarrow[\textstyle \mkern.5em\functionf\mkern.5em]{} & \setZ \\ \end{array} $ | $\begin{array}{} \boxed{ \begin{array}{} \langle \pi_0\functionf,\pi_1\functiong \rangle \Equ \\ \eltx,\elty \mid \langle \eltx,\elty \rangle \pi_0\functionf = \langle \eltx,\elty \rangle \pi_1\functiong \\ \eltx,\elty \mid \eltx\functionf = \elty\functiong \\ \boxed{\setX\times_\setZ\setY} \\ \end{array} } \\ & \llap{\langle \pi_0\functionf,\pi_1\functiong \rangle \equ} \searrow \\ && \setX\times\setY & \xrightarrow{\textstyle \pi_1} & \setY \\ && \llap{\pi_0} \big\downarrow & \not= & \big\downarrow \rlap\functiong \\ && \setX & \xrightarrow{\smash {\textstyle \functionf}} & \setZ \\ \end{array}$ | |
| $\pb$ | $\kp$ | $\equ$ |
What is below is a draft.
$\kp$ as a weighted limit
Let $\calK = \bftwo$ be a category with two objects, say $\objectK$ and $\objectL$, and only one non-identity arrow: $\objectK \xrightarrow{\textstyle \arrowk} \objectL$.
$\calK$ is purely an indexing category, with its objects and arrows having no significance beyond their use as indices.
Think of $\objectK,\objectL$ as analogous to $i,j$ as indices into sequences, and $\objectK \xrightarrow{\textstyle \arrowk} \objectL$ as analogous to $i \lt j$.
Let $2$ be a set with two elements, $0$ and $1$.
Let ($\setX\xrightarrow{\functionf}\setY$ be an arbitrary function in $\Set$), and ($2\xrightarrow{!_2}1$ be the only possible function).
Each of these (functions in $\Set$) defines (a functor $\bftwo\to\Set$), the "name" of (the function),
just as each (set in $\Set$) defines (a functor $\bfone\to\Set$), called the "name" of (the set).
with (the first being the "weight") and (the second being that whose limit is being taken),
is (per (3.7) of Max Kelly's Basic Concepts of Enriched Category Theory) {the set of natural transformations from $2\xrightarrow{!_2}1$ to $\setX\xrightarrow{\functionf}\setY$}
i.e. the set $\hom {(2\xrightarrow[!_2]{}1)} {[\bftwo,\Set]} {( \setX \xrightarrow[\functionf]{} \setY )}$ of all possible ways of selecting vertical (downward) arrows (i.e., functions) to make the square at right commute.
But (arrows $2\to\setX$ on the left of the square) correspond to (pairs of elements $\eltx_0,\eltx_1$ of $\setX$),
while (arrows $1\to\setY$ on the right) correspond to (single elements $\elty$ of $\setY$).
The requirement that (the diagram commute) is that ($\eltx_0\functionf = \elty$ and $\eltx_1\functionf = \elty$).
But that is just the condition defining (the kernel pair of $\functionf$, $\functionf\Kp$).
Thus (the kernel pair) is a simple example of (a weighted limit).
Comparison of conical limits and weighted limits in $\Set$
| conical limit | weighed limit | ||
|---|---|---|---|
| for specific weight and target functors | for general weight and target functors | ||
| commuting triangles (two) | commuting squares (one) | ||
| indexing category $\calK$ | $\objectK_0 \xrightarrow{\arrowk_0} \objectL \xleftarrow{\arrowk_1} \objectK_1$ | $\objectK \xrightarrow{\arrowk} \objectL$ | |
| weighting functor $\functorF:\calK\to\Set$ |
$\big( 1 \xrightarrow{1_1} 1 \xleftarrow{1_1} 1 \big) = \big(!_\calK1\big)$ | $2 \xrightarrow{!_2} 1$ | $\objectK\functorF \xrightarrow{\arrowk\functorF} \objectL\functorF$ |
| target functor $\functorG:\calK\to\Set$ |
$\setX \xrightarrow{\functionf} \setY \xleftarrow{\functionf} \setX$ | $\setX \xrightarrow{\functionf} \setY$ | $\objectK\functorG \xrightarrow{\arrowk\functorG} \objectL\functorG$ |
| Diagrams | $ \begin{array}{} \cdot & \xrightarrow{} & \cdot & \xleftarrow{} & \cdot \\ \\ \\ && 1 \\ & \llap{\eltx_0} \swarrow & \llap\elty \downarrow & \searrow \rlap{\eltx_1} \\ \setX & \xrightarrow[\textstyle \functionf]{} & \setY & \xleftarrow[\textstyle \functionf]{} & \setX \\ \end{array} $ | $ \begin{array}{} && \cdot & \xrightarrow{} & \cdot \\ \\ 1 & \xrightrightarrows[\textstyle 1]{\textstyle 0} & 2 & \xrightarrow{\textstyle !_2} & 1 \\ \Vert && \llap\eltx \downarrow && \downarrow \rlap\elty \\ 1 & \xrightrightarrows[\eltx_1 \xlongequal{\setX} 1\eltx ]{\eltx_0 \xlongequal{\setX} 0\eltx} & \setX & \xrightarrow[\textstyle \functionf]{} & \setY \\ \end{array} $ | $ \begin{array}{} & \mkern1em & \objectK & \xrightarrow{\textstyle \arrowk} & \objectL \\ \\ \functorF && \objectK\functorF & \xrightarrow{\textstyle \arrowk\functorF} & \objectL\functorF \\ \llap\nattransalpha \Bigg\Downarrow && \llap{\objectK\nattransalpha} \Bigg\downarrow && \Bigg\downarrow \rlap{\objectL\nattransalpha} \\ \functorG && \objectK\functorG & \xrightarrow[\textstyle \arrowk\functorG]{} & \objectL\functorG \\ \end{array} $ |
| Equations | $\eltx_0\functionf \xlongequal{\setY} \elty \xlongequal{\setY} \eltx_1\functionf$ | $ \begin{array}{} && \eltx\functionf & \xlongequal{\textstyle [2,\setY]} & !_2\elty \\ \eltx_0\functionf & \xlongequal{\setY} & 0\eltx\functionf & \xlongequal{\setY} & 0!_2\elty & \xlongequal{\setY} & \elty \\ \eltx_1\functionf & \xlongequal{\setY} & 1\eltx\functionf & \xlongequal{\setY} & 1!_2\elty & \xlongequal{\setY} & \elty \\ \end{array} $ | $ \begin{array}{} \objectK\nattransalpha \cdot \arrowk\functorG & \xlongequal{\textstyle [\objectK\functorF,\objectL\functorG]} & \arrowk\functorF \cdot \objectL\nattransalpha \\ \end{array} $ |
| Notation | $\begin{array}{} \big( \setX \xrightarrow{\textstyle \functionf} \setY \xleftarrow{\textstyle \functionf} \setX \big)\Limit \\ \setX\times_\setY\setX \\ \end{array}$ | $\big\{(2 \xrightarrow{\textstyle !_2} 1),\, (\setX \xrightarrow{\textstyle \functionf} \setY) \big\}$ | $\big\{\functorF,\, \functorG \big\}$ |
| as a set of natural transformations |
$ \hom {(!_\calK1)} {[\bftwo\vee\bftwo,\Set]} {( \setX \xrightarrow[\functionf]{} \setY \xleftarrow[\functionf]{} \setX )} $ | $ \hom {(2\xrightarrow[!_2]{}1)} {[\bftwo,\Set]} {( \setX \xrightarrow[\functionf]{} \setY )} = \hom {!_2} {[\bftwo,\Set]} {\functionf} $ | $ \hom \functorF {[\calK,\Set]} \functorG $ |
In ordinary mathematics first we give explicit definitions of various categorical limits, most basically the product of two sets.
Then we show that (these definitions) satisfy (universal properties).
Analogous to that approach, for $\functorF,\functorG:\calK\to\Set$, we have given above an explicit definition of $\big\{\functorF,\, \functorG \big\}$ as $\hom \functorF {[\calK,\Set]} \functorG$.
Now we show that (that definition) satisfies (the universal property that Kelly uses, in (3.1), to characterize $\big\{\functorF,\, \functorG \big\}$).
\[\begin{array}{}
\big[\objectT, \hom \functorF {[\calK,\Set]} \functorG \big] &&&& \hom \functorF {[\calK,\Set]} {[\objectT,\functorG]} & \buildrel \text{(3.1)} \over \cong & \big[\objectT,\{\functorF,\functorG\}\big]\\
\llap{(2.10)} \Vert &&&& \Vert \rlap{(2.10)} \\
\big[\objectT, \int_\objectK [\objectK\functorF,\objectK\functorG] \big] & \mathop\cong\limits^{\text{(2.3)}}_{\text{right adjoints preserve limits}} & \displaystyle\int_\objectK \big[\objectT, [\objectK\functorF,\objectK\functorG] \big] & \buildrel{\text{sym.}} \over \cong & \displaystyle\int_\objectK \big[\objectK\functorF, [\objectT,\objectK\functorG] \big] \\
\\
\text{$\objectT$-tuples} &&&& \text{matching pairs} \\
\text{of matching pairs} &&&& \text{of $\objectT$-tuples} \\
\end{array}\]
Thus $\hom \functorF {[\calK,\Set]} \functorG$ satisfies (Kelly’s definition (3.1) of $\big\{\functorF,\, \functorG \big\}$), justifying $\big\{\functorF,\, \functorG \big\} \buildrel \text{(3.7)} \over \cong \hom \functorF {[\calK,\Set]} \functorG$.
(The counits $\boxed{ \calK \twocellctb \mu \functorF {\big[\{\functorF,\functorG\},\functorG {\big]}} \Set }$) of (the representation (3.1)),
for (the specific example above of $\big\{!_2,\,\functionf \big\} \cong \hom {!_2} {[\bftwo,\Set]} {\functionf}$) at the left,
and (the general case $\big\{\functorF,\, \functorG \big\} \cong \hom \functorF {[\calK,\Set]} \functorG$) at the right, are then:
\[\begin{array}{}
2 & \xrightarrow{\textstyle !_2} & 1 & \mkern8em & \objectK\functorF & \xrightarrow{\textstyle \arrowk\functorF} & \objectL\functorF \\
\llap{\objectK\mu} \bigg \downarrow \rlap{{} = \overline{\objectK\pi} = \overline{\objectK\lambda}} && \llap{\objectL\mu} \bigg \downarrow \rlap{{} = \overline{\objectL\pi} = \overline{\objectL\lambda}} && \llap{\objectK\mu} \bigg \downarrow \rlap{{} = \overline{\objectK\pi} = \overline{\objectK\lambda}} && \llap{\objectL\mu} \bigg \downarrow \rlap{{} = \overline{\objectL\pi} = \overline{\objectL\lambda}} \\
\big[ \hom {!_2} {[\bftwo,\Set]} {\functionf}, \setX \big] & \xrightarrow[\textstyle \big[ \hom {!_2} {[\bftwo,\Set]} {\functionf}, \functionf {\big]}]{} & \big[ \hom {!_2} {[\bftwo,\Set]} {\functionf}, \setY \big] && \big[ \hom {\functorF} {[\calK,\calV]} {\functorG}, \objectK\functorG \big] & \xrightarrow[\textstyle \big[ \hom {\functorF} {[\calK,\calV]} {\functorG}, \arrowk\functorG{\big]}]{} & \big[ \hom {\functorF} {[\calK,\calV]} {\functorG}, \objectL\functorG \big] \\
\end{array}\]
$\objectK\mu$ sends $0\in2$ to the projection $\langle \eltx_0,\eltx_1,\elty \rangle \mapsto \eltx_0$, and $1\in2$ to the projection $\langle \eltx_0,\eltx_1,\elty \rangle \mapsto \eltx_1$.
$\objectL\mu$ sends the unique element of $1$ to the projection $\langle \eltx_0,\eltx_1,\elty \rangle \mapsto \elty$.
The commuting of (the naturality square for $\mu$) follows from (the naturality of each transformation $!_2 \xRightarrow{\textstyle \langle \eltx_0,\eltx_1,\elty \rangle } \functionf$); see (the diagram for it above).
The above has concerned the weighted limit of ordinary functors $\functorF,\functorG:\calK\to\Set$.
Perhaps unsurprisingly, weighted limits may be defined on more general classes of functors.
To wit, we have the following four possibilities for the targets of our weight ($\functorF$) and target ($\functorG$) functors,
from (the least general at the upper left) (the case $\calB=\calV=\Set$ already considered) to (the most general at the lower right):
| $\functorF:\calK\to\calV$ | |||
|---|---|---|---|
| $\calV=\Set$ (functors are $\Set$-functors) | $\calV$ (functors are $\calV$-functors) | ||
| $\functorG: \calK\to\calB$ | $\calB=\calV$ | $\begin{array}{}\functorF & : & \calK & \to & \Set \\ \functorG & : & \calK & \to & \Set \end{array}$ $\big\{\functorF,\functorG\} = \hom \functorF {[\calK,\Set]} \functorG$ |
$\begin{array}{}\functorF & : & \calK & \to & \calV \\ \functorG & : & \calK & \to & \calV \end{array}$ $\big\{\functorF,\functorG\} = \hom \functorF {[\calK,\calV]} \functorG$ |
| $\calB$ | $\begin{array}{}\functorF & : & \calK & \to & \Set \\ \functorG & : & \calK & \to & \calB \end{array}$ | $\begin{array}{}\functorF & : & \calK & \to & \calV \\ \functorG & : & \calK & \to & \calB \end{array}$ | |
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