Categories

For the definition, see Wikipedia.

Here, at this time, we just want to give a small diagram that shows some of the relations between the categories of categories, monoids, (directed) graphs, and sets.
\[\begin{array}{cccccc} \Mon& \subset & \Cat\\ \downarrow && \downarrow\\ \Set & \subset & \Graph\\ \end{array}\] Here the vertical arrows take, respectively, a monoid to its underlying set and a category to its underlying graph (i.e., they forget the composition and the specification of the identities).
The top inclusion takes a monoid into a category with just one object and that monoid as its set of arrows.
The bottom inclusion takes a set into the graph with only one vertex and that set as its set of edges.

Both vertical (downward) arrows have left adjoints going up, taking respectively a set to the free monoid on that set and a graph to the free category on that graph.

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Following is work in progress!
Here are two diagrams, one for ordinals and the other for internal categories:

$\begin{array}{} &&&& \{1\} \\ &&& \swarrow && \searrow \\ && \{0,1\} &&&& \{1,2\} \\ & \nearrow && \searrow && \swarrow && \nwarrow \\ &&&& \{0,1,2\} \\ &&&& \uparrow \\ \{0\} && \rightarrow && \{0,2\} && \leftarrow && \{2\} \\ \end{array}$

$\begin{array}{} &&&& \catC_0 \\ &&& \nearrow && \nwarrow \\ && \catC_1 &&&& \catC_1 \\ & \swarrow && \nwarrow && \nearrow && \searrow \\ &&&& \catC_2 \\ &&&& \downarrow \\ \catC_0 && \leftarrow && \catC_1 && \rightarrow && \catC_0 \\ \end{array}$

Cartesian closed categories

This is not intended to give an introduction to cartesian closed categories (“cccs”), which are discussed in many places, e.g. Wikipedia, nlab.
Rather, the aim of this document is simply to compare two extremely significant cccs: $\bftwo$ which is basic to Boolean logic, and $\Set$ which is, of course, basic to set theory.
More precisely, we wish to compare the notations which are used to express the ccc operations, and how several of the simple but significant theorems valid in these cccs are expressed.
In fact, both $\bftwo$ and $\Set$ are, as well as being cartesian closed, are also cocomplete.
So the names of the operators relevant to cocompletion are also shown, with their related simple theorems.

Description		ccc				Comment
		$\bftwo$		$\Set$
initial object		$\bot$		$\emptyset$
terminal object		$\top$		$1$
binary cartesian product		$\wedge$		$\times$
internal hom		$\Rightarrow$		$[,]$
formulae involving the binary product		$\bot\wedge\objA = \bot$		$\emptyset\times\objA \cong \emptyset$		absorption
		$\top\wedge\objA = \objA$		$1\times\objA \cong \objA$		identity or unit
formulae involving the internal hom		$\bot\Rightarrow\objA = \top$		$[\emptyset,\objA] \cong 1$
		$\top\Rightarrow\objA = \objA$		$[1,\objA] \cong \objA$
the product of $\functionA:\setX\to \text{ccc}$ (where $\setX\in\Set$)		$ \left\{ \begin{array}{}(\forall \eltx\in\setX)\functionA_\eltx\\ \displaystyle\bigwedge_{\eltx\in\setX}\objA_\eltx\\ \end{array} \right\} $		$\displaystyle\prod_{\eltx\in\setX}\functionA_\eltx$
binary coproduct		$\vee$		$+$ or $\coprod$
formulae involving the binary coproduct		$\bot\vee\objA = \objA$		$\emptyset+\objA \cong \objA$		identity or unit
		$\top\vee\objA = \top$		$1+\objA$
the coproduct of $\functionA:\setX\to \text{ccc}$ (where $\setX\in\Set$)		$ \left\{\begin{array}{}(\exists\eltx\in\setX)\functionA_\eltx\\ \displaystyle\bigvee_{\eltx\in\setX}\objA_\eltx\\ \end{array} \right\} $		$\displaystyle\sum_{\eltx\in\setX}\functionA_\eltx$

Adjunctions

$\Newextarrow{\xequiv}{10,10}{0x2261}$ For the definition of adjunction (or adjoint functors/1-cells), see Wikipedia, nlab, or your favorite introduction to category theory.
For the definition of 2-category, again see Wikipedia or nlab, or the classic Categories for the Working Mathematician, 2nd Edition.
Here we merely demonstrate some notation and recall a few related definitions.
Due to the importance of the subject, we present some of the diagrams in two different categories:
conventionally in a 2-category,
and in a double category where, for each 2-cell, the vertical arrows are identities (of course these two are isomorphic).

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The most general adjunction is depicted as follows,
using fairly standard notation for the left and right adjoint 1-cells and the unit and counit 2-cells,
but unusual notation for the 0-cells, which are commonly called $\mathcal A, \mathcal B$ or some such.
First we give the definitions in a double category as above,
and consider 0-cells called $\Leftcat = \calL = \catB$, $\Rightcat = \calR = \catA$,
left and right adjoint 1-cells $\leftadj{L=F}$, $\rightadj{R=U}$, and
unit and counit 2-cells $\leftadj\eta$, $\rightadj\epsilon$.

\[\begin{array}{c} \Leftcat & \leftadj{\xrightarrow{\functL=\functF}} & \Rightcat\\ \leftcat{\llap{1_\calL = 1_\Leftcat}\Vert} & \leftadj{\eta\Rightarrow} \qquad \leftcat{\epsilon\Rightarrow} & \rightcat{\Vert\rlap{1_\Rightcat = 1_\calR}}\\ \leftcat{\Leftcat} & \rightadj{\xleftarrow[\functR=\functU]{}} & \rightcat{\Rightcat}\\ \end{array}\] Now for the triangular equations that data are required to satisfy.
For the triangular equation involving $1_\functL$,
start with the $\Leftcat$ at the left of the top row
and compose the top ($\eta$) 2-cell (the unit)
with the lower right counit ($\epsilon$) 2-cell.
For the triangular equation involving $1_\functR$,
start with the $\Rightcat$ at the left of the middle row
and compose the top ($\eta$) 2-cell (the unit again)
with the lower left counit ($\epsilon$) 2-cell. \[\begin{array}{ccccccc} && \Leftcat && \leftcat{\xrightarrow{1_\Leftcat}} && \Leftcat && \\ &&\leftcat\Vert && \leftadj{\Downarrow\rlap\eta} && \leftcat{\Vert} && \\ \Rightcat & \rightadj{\xrightarrow{\functR}} & \Leftcat & \leftadj{\xrightarrow[\functL]{}} & \Rightcat & \rightadj{\xrightarrow[\functR]{}} & \Leftcat & \leftadj{\xrightarrow{\functL}} & \Rightcat \\ \rightcat\Vert && \rightadj{\Downarrow\rlap\epsilon} && \rightcat\Vert && \rightadj{\Downarrow\rlap\epsilon} && \rightcat\Vert \\ \Rightcat && \rightcat{\xrightarrow[1_\Rightcat]{}} && \Rightcat && \rightcat{\xrightarrow[1_\Rightcat]{}} && \Rightcat \\ \end{array}\]

Now for a presentation in a 2-category.
For variety we use a slightly different notation for the 0-cells: $\Leftcat = \calL$, $\Rightcat = \calR$.

\[\begin{array}{} \source\calL && \source\longrightarrow && \source\calL && \source\longrightarrow && \source\calL \\ & \leftadj{ \llap \functL \searrow } & \leftadj{ \big\Downarrow \rlap\eta } & \rightadj{ \nearrow \mkern{-24mu} \functR } & \rightadj{ \big\Downarrow \rlap\epsilon } & \leftadj{ \searrow \mkern{-20mu} \functL } & \leftadj{ \big\Downarrow \rlap\eta } & \rightadj{ \nearrow \mkern{-24mu} \functR } & \rightadj{ \big\Downarrow \rlap\epsilon } & \leftadj{ \searrow \rlap \functL } \\ && \rightadj{ \target\calR } && \rightadj{ \target\longrightarrow } && \target\calR && \target\longrightarrow && \target\calR \\ \\ \mkern{-10mu} \rlap{\text{while the triangular equations are:}} \\ \\ & \leftadj \functL & \leftadj = &\leftcat{1_\calL} \leftadj \functL && \rightadj \functR \leftcat{1_\calL} & \rightadj = & \rightadj \functR \\ &&& \leftadj{ \llap\eta \Big\Downarrow \rlap\functL } && \llap{\rightadj\functR} \leftadj { \Big\Downarrow \rlap\eta } \\ \llap{\text{in 1-D} \mkern50mu} {} & \leftadj{ \llap{1_\functL} \Bigg\Downarrow } & \leftadj{ \xequiv[(\text{left }\bigtriangleup)]{\hom \functL {[\calL,\calR]} \functL} } & \leftadj \functL \rightadj \functR \leftadj \functL && \rightadj \functR \leftadj \functL \rightadj \functR & \rightadj{ \xequiv[(\text{right }\bigtriangleup)]{\hom \functR {[\calR,\calL]} \functR} } & \rightadj{ \Bigg\Downarrow \rlap{1_\functR} }\\ &&& \llap{\leftadj\functL} \rightadj{\Big\Downarrow \rlap\epsilon} && \rightadj{ \llap\epsilon \Big\Downarrow \rlap\functR } \\ & \leftadj \functL & \leftadj = &\leftadj \functL \rightcat{1_{\calR}} && \rightcat{1_{\calR}} \rightadj \functR & \rightadj = & \rightadj \functR \\ \\ \\ \llap{\text{in 0-D} \mkern50mu} {} & \leftadj{1_\functL} \rlap{ {} \mathrel{\leftadj\equiv} (\leftadj\eta \ncomp0 \leftadj \functL) \ncomp1 (\leftadj \functL \ncomp0 \rightadj\epsilon) } &&&&&& \llap{ ( \rightadj \functR \ncomp0 \leftadj\eta ) \ncomp1 ( \rightadj\epsilon \ncomp0 \rightadj \functR ) \mathrel{\rightadj\equiv} {} } \rightadj{1_\functR} \\ \end{array}\]

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Work in progress:

\[\begin{array}{} \catI && \leftcat{ \xrightarrow[]{\textstyle \mkern{12mu} \objb \mkern{12mu}} } && \leftcat\catB && \\ & \rightcat { \llap{\objap \searrow \buildrel \alpha \over \Leftarrow \leftcat\objb\leftadj\functF \mkern{-20mu} } \searrow } & \leftadj{\Downarrow \rlap { \hom {\leftcat\objb} {\leftadj\eta} {} } } & \rightadj{ \nearrow \rlap{\mkern-20mu\functU} } & \Uparrow \rlap{\hat{ \hom {\leftcat\objb} {\leftadj\eta} {} }} & \leftcat{ \searrow \rlap{ \hom \objb \catB {?'}} } \\ && \rightcat\catA && \rightcat{ \xrightarrow[\textstyle \hom {\leftcat\objb\leftadj\functF} {\rightcat\catA} {-'}]{} } && \Set\\ \end{array}\]

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$\bbox[3ex,border:4px groove black]{\begin{array}{} \catI & \xrightarrow{\textstyle 1} & \Set \\ \llap{\leftcat\objb\leftadj\functF} \rightcat{ \Bigg\downarrow } & \llap{\leftadj{ \lower14pt\hbox{$\llap{{ \hom {\leftcat\objb} {\leftadj\eta} {} }\mkern-5mu} \Downarrow$} } \mkern-12mu} \leftcat{\searrow \rlap{\mkern-20mu \objb \raise10pt\hbox{$\mkern-10mu \Downarrow \mkern-5mu 1_\objb$}} } & \leftcat{ \Bigg\uparrow \rlap{\mkern-20mu \hom \objb \catB {?'}} } \\ \rightcat\catA & \rightadj{ \xrightarrow[\textstyle \mkern10mu \functU \mkern10mu]{} } & \leftcat\catB \\ \end{array}}$	$\xlongequal[\begin{array}{} b \\ \text{left} \\ \text{lift} \end{array}]{\text {YS2}}$	$\bbox[3ex,border:4px groove black]{\begin{array}{} \catI & \xrightarrow{\textstyle 1} & \Set \\ \llap{\leftcat\objb\leftadj\functF} \rightcat{ \Bigg\downarrow } & \leftadj{ \Bigg\Downarrow \rlap{\mkern-26mu \name{ \hom {\leftcat\objb} {\leftadj\eta} {} }} } & \leftcat{ \Bigg\uparrow \rlap{\mkern-20mu \hom \objb \catB {?'}} } \\ \rightcat\catA & \rightadj{ \xrightarrow[\textstyle \mkern10mu \functU \mkern10mu]{} } & \leftcat\catB \\ \end{array}}$	$\xlongequal[\begin{array}{} \hom {\leftcat\objb\leftadj\functF} {\rightcat\catA} {-'} \\ \text{left} \\ \text{extension}\end{array}]{\text {YS1}}$	$\bbox[3ex,border:4px groove black]{\begin{array}{} \catI & \xrightarrow{\textstyle 1} & \Set \\ \llap{\leftcat\objb\leftadj\functF} \rightcat{ \Bigg\downarrow } & \llap{\leftadj{ \raise10pt\hbox{$\llap{{\rightcat 1}_{\leftcat\objb\leftadj\functF} \mkern-8mu} \Downarrow$} } \mkern-16mu} \leftcat{\nearrow \rlap{\mkern-36mu \lower3pt\hbox{$\hom {\leftcat\objb\leftadj\functF} {\rightcat\catA} {-'}$} \lower14pt\hbox{$\mkern-26mu \Downarrow \mkern-5mu \hom \objb {\leftadj{\hat\eta}} {}$}} } & \leftcat{ \Bigg\uparrow \rlap{\mkern-20mu \hom \objb \catB {?'}} } \\ \rightcat\catA & \rightadj{ \xrightarrow[\textstyle \mkern10mu \functU \mkern10mu]{} } & \leftcat\catB \\ \end{array}}$
$1 \leftcat {{} \xrightarrow[]{1_\objb} \hom \objb \catB \objb} \leftadj{ {} \xrightarrow[]{\leftcat{\hom \objb \catB {\hom \objb {\leftadj\eta} {}}}} {} } \leftcat { \hom \objb \catB {\objb\leftadj\functF\rightadj\functU} }$		$ 1 \leftadj { {} \xrightarrow[]{\name{\hom {\leftadj\objb} \eta {}}} {} } \leftcat { \hom \objb \catB {\objb\leftadj\functF\rightadj\functU} }$		$1 \rightcat {{} \xrightarrow[]{1_{\leftcat\objb\leftadj\functF}} \hom {\leftcat\objb\leftadj\functF} \catA {\leftcat\objb\leftadj\functF}} \leftadj{ {} \xrightarrow[]{ \hom {\leftcat\objb} {\leftadj{\hat\eta}} {\leftcat\objb\leftadj\functF} } \leftcat { \hom \objb \catB {\objb\leftadj\functF\rightadj\functU} }}$

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Here are some standard definitions using the concept of adjunction:

isomorphism (in a 2-category)

The unit and counit are both identities: $\leftadj\eta \mathrel{\leftadj=} \leftcat{1_{1_\Leftcat}}$ and $\rightadj\epsilon \mathrel{\rightadj=} \rightcat{1_{1_\Rightcat}}$.

reflection

The unit is an identity: $\leftadj\eta \mathrel{\leftadj=} \leftcat{1_{1_\Leftcat}}$.

coreflection

The counit is an identity: $\rightadj\epsilon \mathrel{\rightadj=} \rightcat{1_{1_\Rightcat}}$.

adjoint equivalence

The unit $\eta$ and the counit $\epsilon$ are both invertible 2-cells, i.e., are isomorphisms.

Partitions

A brief review of partitions. A partition $\pi$ of a set $X$ is a family of non-empty subsets of $X$ which are (pairwise) disjoint and such that their union is all of $X$. Thus each element $x\in X$ is a member of one and only one of the non-empty subsets which comprise $\pi$.
So we have a canonical surjection $X \twoheadrightarrow \pi$ : $x\in X \mapsto$ the member of $\pi$ which contains $x$. It is a surjection because each member of $\pi$ is non-empty.

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Partitions come in several varieties:

• Of a number or of a set? E.g., $3=2+1$ or $\{1,2,3\}=\{1,2\}+\{3\}$?
• Ordered or unordered? E.g., $2+1$ or $1+2$; $\{1,2,3\}=\{1,2\}+\{3\}$ or $\{1,2,3\}=\{3\}+\{1,2\}$?
• Weak or strong? I.e.: For number partitions, do we allow $0$? For set partitions, do we allow the empty set $\emptyset=\{\}$?

A note on terminology: Normal mathematical usage is that the word “partition”, without any modifier, is what the taxonomy above would call an “unordered strong partition” (number or set, as determined by the context). If an ordered partition is under discussion, it is called just that. Rather than the “weak/strong” distinction made above, weak partitions are sometimes (e.g., by Martin Aigner in Combinatorial Theory) called “generalized partitions”. Also, an ordered strong number partition is normally called a “composition”. So the above terminology is neither traditional nor necessary, but it does have the advantages, perhaps, of explicitness and orthogonality.

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We may summarize the types of partitions in the following two tables,
the first one for $\target r$‑partitions of an $n$-set,
the second one for $\target r$‑partitions of a number $n$.
The number in each cell of the table is
the number of partitions of that type with $\target r$ blocks (called $\target r$‑partitions)
for a set $X$ or number $n$.
The entries for the two strong unordered cases are the definitions of those symbols.

$r$‑partitions of an $\source n$-set $\source X$	ordered	unordered $- \target {{} / S_Y}$
weak	$\homst X\Set Y$ $\target r^{\source n}$	$\homst X\Set Y \target{{} / S_Y}$ $\displaystyle{\target{\sum\limits_{k=1}^r} {\source X \brace \target k}}$
strong	$\homst X\Surj Y$ $\displaystyle {\source X \brace \target r} \times \target{r!}$	$$\homst X\Surj Y \target{{}/ S_Y}$$ $\displaystyle {\source X \brace \target r}$

$\target r$‑partitions of a number $\source n$ $\source{S_X \backslash {}} -$	ordered	unordered $- \target {{} / S_Y}$
weak	$\source{S_X \backslash {}} \homst X\Set Y$ $\displaystyle {\source n+\target{r-1} \choose {\source n,\target{r-1}}}$	$\source{S_X \backslash {}} \homst X\Set Y \target{{} / S_Y}$ $\displaystyle {\target{\sum\limits_{k=1}^r} P_{\source n, \target k}}$
strong	$\source{S_X \backslash{}} \homst X\Surj Y$ $\displaystyle {\source{n-1} \choose \target{r-1}}$	$\source{S_X \backslash {}} \homst X\Surj Y \target{{} / S_Y}$ $\displaystyle {P_{\source n,\target r}}$

Here is a small but nontrivial example, taking
$X=[4]=\{0,1,2,3\}$, so $n=4$, and
$Y=\{a,b\}$, so $r=2$,
thus considering $2$-partitions of a $4$-set.

$2$‑partitions of the $4$-set $[4] = \{0,1,2,3\}$	ordered	unordered
weak	$|\hom {[4]}\Set {\{a,b\}}|$ $2^4 = 16$ The kernel-partitions of the 16 functions from [4] to {a,b} (click here to see a figure showing those).	$|\hom {[4]}\Set {\{a,b\}} / S_{\{a,b\}}|$ $\displaystyle{{[4] \brace 1} + {[4] \brace 2}} = 1+7 = 8$ The weak 2-partition $$0,1,2,3 \mid {}$$ together with the seven unordered strong 2-partitions of $[4]$ shown below.
strong	$|\hom {[4]}\Surj {\{a,b\}}|$ $\displaystyle {{[4] \brace 2} \times 2! = 7 \times 2 = 14}$ Take each of the two possible orderings of the seven unordered strong 2-partitions of $[4]$, e.g.,$$\begin{array}{c} 0,1,2 \mid 3 \\ 3 \mid 0,1,2 \end{array}$$	$|\hom {[4]}\Surj {\{a,b\}} / S_{\{a,b\}}|$ $\displaystyle {[4] \brace 2} = 7$ $$\left. \begin{array}{c} 0,1,2 \mid 3 \\ 0,1,3 \mid 2 \\ 0,2,3 \mid 1 \\ 1,2,3 \mid 0\end{array} \right\} (4)$$ $$\left. \begin{array}{c} 0,1 \mid 2,3 \\ 0,2 \mid 1,3 \\ 0,3 \mid 1,2 \end{array} \right\} (3)$$

$2$‑partitions of the number $4$	ordered	unordered
weak	$|S_4 \backslash \hom {[4]}\Set {\{a,b\}}|$ $\displaystyle {{4+2-1 \choose {4,2-1}}} = {{5 \choose {4,1}}} = 5$ $$\begin{array}{c}4+0 \\ 3+1 \\ 2+2 \\ 1+3 \\ 0+4\end{array}$$	$|S_{[4]} \backslash \hom {[4]}\Set {\{a,b\}}/ S_{\{a,b\}}|$ $\displaystyle {P_{4,1} + P_{4,2}} = 1+2 = 3$ $$\begin{array}{c} 4+0 \\ 3+1 \\ 2+2\end{array}$$
strong	$|S_{[4]} \backslash \hom {[4]}\Surj {\{a,b\}}|$ $\displaystyle {4-1 \choose 2-1} = {3 \choose 1} = 3$ $$\begin{array}{c} 3+1 \\ 2+2 \\ 1+3 \end{array}$$	$|S_{[4]} \backslash \hom {[4]}\Surj {\{a,b\}}/ S_{\{a,b\}}|$ $\displaystyle {P_{4,2}} = 2$ $$\begin{array}{c} 3+1 \\ 2+2 \end{array}$$

Returning to the general case,
here is are some justifications for the entries in that table.

The entry for strong ordered set $k$‑partitions simply is
the number of strong unordered set $k$‑partitions times
the number of ways of ordering such a partition.

The entry for strong ordered number $k$‑partitions results from a simple representation.
Given a number $n$, write $n$ $1$'s in a row.
There are $n-1$ spaces between the $1$s.
Choose $k-1$ of those to insert a $+$ sign.
Add up the $1$s between $+$ signs.
Each ordered $k$‑partition of $n$ arises in one and only one such way.
E.g., there are $\binom{5}{2} = 10$ ordered $3$‑partitions of $6$:

\[ \begin{aligned} \{s,t\}\quad & & & & i+j+k\\ \{1,2\}\quad &11000 &.1.1.0.0.0.\quad &1{+}1{+}1111 &= 1+1+4\\ \{1,3\}\quad &10100 &.1.0.1.0.0.\quad &1\mathord+11\mathord+111 &= 1+2+3\\ \{1,4\}\quad &10010 &.1.0.0.1.0.\quad &1+111+11 &= 1+3+2\\ \{1,5\}\quad &10001 &.1.0.0.0.1.\quad &1{+}1111{+}1 &= 1+4+1\\ \{2,3\}\quad &01100 &.0.1.1.0.0.\quad &11+1+111 &= 2+1+3\\ \{2,4\}\quad &01010 &.0.1.0.1.0.\quad &11+11+11 &= 2+2+2\\ \{2,5\}\quad &01001 &.0.1.0.0.1.\quad &11+111+1 &= 2+3+1\\ \{3,4\}\quad &00110 &.0.0.1.1.0.\quad &111+1+11 &= 3+1+2\\ \{3,5\}\quad &00101 &.0.0.1.0.1.\quad &111+11+1 &= 3+2+1\\ \{4,5\}\quad &00011 &.0.0.0.1.1.\quad &1111+1+1 &= 4+1+1 \\ \end{aligned} \]

The above display may contain more information than is really necessary. The column labeled $\{s,t\}$ contains all $\binom{5}{2} = 10$ ways of choosing two numbers from the set $\{1,2,3,4,5\}$ (which we view as numbering the spaces between the six $1$s). For now, skip the second and third columns. The final two column implement the process described before the display.

On the other hand, there are only $3$ unordered $3$‑partitions of $6$: \[ \begin{aligned} 4+1+1 \\ 3+2+1\\ 2+2+2\\ \end{aligned} \] so $P_{6,3}=3$.

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\[ \begin{array}{ccc} \{(s,t)|1\le s\lt t\le 5\} & \substack {\longrightarrow \\ \longleftarrow} & \{(i,j,k)| 1\le i, 1\le j, 1\le k, i+j+k=6 \} \\ (s,t) & \mapsto & (s, t-s, 6-t) \\ (i,i+j) & \leftarrow & (i,j,k) \\ \end{array} \]

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Often we are interested in the ordered strong number partitions of a number $n$, not just of a given size $k$, but of all possible sizes, from $1$ to $n$. There are $2^{n-1}$ such, and they can easily be parametrized by the $2^{n-1}$ binary numbers of length $n-1$. The trick, as above, is that each $n-1$-digit binary number encodes where to place $+$ signs in the $n-1$ spaces between the $1$s in a string of $n$ $1$s. Here are tables showing the bijections, for $n$ from $1$ to $5$.

\[\begin{array}{cccc} &n=1&\\ \text{binary} & \text{string} && \text{partition} \\ \emptyset & 1 &= &1 \\ \end{array}\]

\[\begin{array}{cccc} &n=2&\\\text{binary} & \text{string} && \text{partition} \\ 0 & 11 &= &2\\ 1 & 1+1 &= &1+1\\ \end{array}\]

\[\begin{array}{cccc} &n=3&\\ \text{binary} & \text{string} && \text{partition} \\ 00 & 111 &= &3\\ 01 & 11+1 &= &2+1\\ 10 & 1+11 &= &1+2\\ 11 & 1+1+1 &= &1+1+1\\ \end{array}\]

\[\begin{array}{cccc} &n=4&\\ \text{binary} & \text{string} && \text{partition} \\ 000 & 1111 &= &4\\ 001 & 111+1 &= &3+1\\ 010 & 11+11 &= &2+2\\ 011 & 11+1+1 &= &2+1+1\\ 100 & 1+111 &= &1+3\\ 101 & 1+11+1 &= &1+2+1\\ 110 & 1+1+11 &= &1+1+2\\ 111 & 1+1+1+1 &= &1+1+1+1\\ \end{array}\]

\[\begin{array}{cccc} &n=5&\\ \text{binary} & \text{string} && \text{partition} \\ 0000 & 11111 &= &5\\ 0001 & 1111+1 &= &4+1\\ 0010 & 111+11 &= &3+2\\ 0011 & 111+1+1 &= &3+1+1\\ 0100 & 11+111 &= &2+3\\ 0101 & 11+11+1 &= &2+2+1\\ 0110 & 11+1+11 &= &2+1+2\\ 0111 & 11+1+1+1 &= &2+1+1+1\\ 1000 & 1+1111 &= &1+4\\ 1001 & 1+111+1 &= &1+3+1\\ 1010 & 1+11+11 &= &1+2+2\\ 1011 & 1+11+1+1 &= &1+2+1+1\\ 1100 & 1+1+111 &= &1+1+3\\ 1101 & 1+1+11+1 &= &1+1+2+1\\ 1110 & 1+1+1+11 &= &1+1+1+2\\ 1111 & 1+1+1+1+1 &= &1+1+1+1+1\\ \end{array}\]

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Weak ordered number partitions

Weak ordered number partitions play an important role in mathematics.
Here is a look at some alternate representations for them,
one of which allows them to be counted easily.
We consider the case of weak ordered $k$-partitions of a number $n$.
There are in general ${n+k-1 \choose n} = {n+k-1 \choose k-1} = {n+k-1 \choose n,\,k-1}$ such partitions.
For example, here is a table showing three different ways of representing the ${4+3-1 \choose 4,\,3-1} = {6 \choose 4,\,2} = 15$ weak ordered $3$-partitions of $n=4$.

\[\begin{array}{ccl} \text{string} & \text{partition} & \text{monomial}\\[2ex] ++1111 & 0+0+4 & c^4 \\ +1+111 & 0+1+3 & bc^3 \\ +11+11 & 0+2+2 & b^2c^2 \\ +111+1 & 0+3+1 & b^3c \\ +1111+ & 0+4+0 & b^4 \\[1ex] 1++111 & 1+0+3 & ac^3 \\ 1+1+11 & 1+1+2 & abc^2 \\ 1+11+1 & 1+2+1 & ab^2c \\ 1+111+ & 1+3+0 & ab^3 \\[1ex] 11++11 & 2+0+2 & a^2c^2 \\ 11+1+1 & 2+1+1 & a^2bc \\ 11+11+ & 2+2+0 & a^2b^2 \\[1ex] 111++1 & 3+0+1 & a^3c \\ 111+1+ & 3+1+0 & a^3b \\[1ex] 1111++ & 4+0+0 & a^4 \\ \end{array}\]

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Strong unordered set partitions (i.e., just "partitions")

Rather than giving a closed form for these, such as we have for $n \choose k$, we give a recursive formula for them: $${n+1 \brace k} = {n \brace k-1} + k{n \brace k}$$ This is easily seen to be correct:
If a set with $n+1$ elements has a $k$-partition,
either the $n+1$st element stands alone, in which case the other $k-1$ blocks of the partition form a $k-1$-partition of the other $n$ elements,
or it is added to (an arbitrary) one of the $k$ blocks of a $k$-partition of the other $n$ elements.
Using that recursive formula,
we may form a triangular table for these partition numbers,
analogous to Pascal’s triangle for the binomial coefficients.
Perhaps a good name for this triangle is Stirling’s partition table
(vice Stirling’s cycle table). \[\bbox[navajowhite,10px,border:4px groove red]{\begin{array}{ccccccc} \displaystyle{n \brace k} & \displaystyle{\brace 0} & \displaystyle{\brace 1} & \displaystyle{\brace 2} & \displaystyle{\brace 3} & \displaystyle{\brace 4} & \displaystyle{\brace 5}\\ \displaystyle{0 \brace } & 1\\ \displaystyle{1 \brace } & 0 & 1\\ \displaystyle{2 \brace } & 0 & 1={2 \choose 2} & 1\\ \displaystyle{3 \brace } & 0 & 1 & 3={3 \choose 2}=2^2-1 & 1\\ \displaystyle{4 \brace } & 0 & 1 & 7=2^3-1 & 6 ={4 \choose 2} & 1\\ \displaystyle{5 \brace } & 0 & 1 & 15=2^4-1 & 25 & 10={5 \choose 2} & 1\\ \end{array}}\] The formulas ${n \brace 2} = {(2^n -2)\over 2} = 2^{n-1}-1$ and ${n \brace n-1} = {n \choose 2}$ are easily seen.

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What is the relation between ordered and unordered partitions of numbers and sets?
There is an extensive theory on that, of course, but a small example may provide a useful introduction.
The following table gives the (strong) partitions, ordered and unordered, of the number $4$ and, for the set $[4] \equiv \{1,2,3,4\}$, the number of elements of its stabilizer group and its orbit space for both unordered set partitions and unordered cycles.
The table’s bottom line gives the general form for the ``type'' of a partition, and the sizes of its corresponding stabilizers and orbits. \[\bbox[navajowhite,10px,border:6px groove red]{\begin{array}{ccccccccc} \text{type (ordered)} & \text{ordered} \# \text{partition} & \text{unordered} \# \text{partition} & \text{type (unordered)} & \text{template} & \text{partition stabilizer} & \text{partition orbit (partitions)} & \text{cycle stabilizer} & \text{cycle orbit (cycles)} \\[2ex] 000 & 4 & 4 & 4^1 & (\cdotp\cdotp\cdotp\cdotp) & (4!)^1 & 1 & 4^1 & 6 \\ 001 & 3+1 & 3+1 & 1^13^1 & (\cdotp)(\cdotp\cdotp\cdotp) & (3!)^1 & 4 & 3^1 & 8 \\ 010 & 2+2 & 2+2 & 2^2 & (\cdotp\cdotp)(\cdotp\cdotp) & 2!(2!)^2 & 3 & 2!2^2 & 3 \\ 011 & 2+1+1 & 2+1+1 & 1^22^1 & (\cdotp)(\cdotp)(\cdotp\cdotp) & 2!(2!)^1 & 6 & 2!2^1 & 6 \\ 100 & 1+3 \\ 101 & 1+2+1 \\ 110 & 1+1+2 \\ 111 & 1+1+1+1 & 1+1+1+1 & 1^4 & (\cdotp)(\cdotp)(\cdotp)(\cdotp) & 4! & 1 & 4! & 1 \\[3ex] &&& 1^{b_1}2^{b_2}\cdots i^{b_i}\cdots n^{b_n} && b_1!(1!)^{b_1} b_2!(2!)^{b_2} b_3!(3!)^{b_3} \cdots b_i!(i!)^{b_i} \cdots b_n!(n!)^{b_n} &n!/\text{(partition_stabilizer)} & b_1!1^{b_1} b_2!2^{b_2} b_3!3^{b_3} \cdots b_i!i^{b_i} \cdots b_n!n^{b_n} & n!/\text{(cycle_stabilizer)} \\ \end{array}}\] The appearances of $(1!)^{b_1}$ and $1^{b_1}$ are of course entirely unnecessary,
as each is equal to $1$;
they are there just to show the uniformity of the product of which they are a part.

Equivalence relations, partitions, and surjections

[This is a draft!]
In this post, we fix attention on a fixed set $X \in \bf\text{Set}$.
We consider, compare, and count three structures $X$ may possess:

1. Equivalence relations $E \rightrightarrows X$ on $X$;
2. Partitions $\pi \subset \mathcal P (X)$ of $X$;
3. Surjections (i.e., onto functions) $f: X \twoheadrightarrow Y$ out of $X$.

There are two fundamental relationships between those structures:

1. There is canonical bijection between equivalence relations on $X$ and partitions of $X$.
2. There is a coreflexive (pronounced co-reflexive) adjoint equivalence between the category of equivalence relations on $X$ and the category of surjections out of $X$.

The topics to be covered in this post are:

1. Equivalence relations
2. Partitions
3. Quotient sets, equivalence classes, and quotient maps
4. Kernels (sometimes known as “kernel pairs”)
5. Adjoint equivalences

[Everything below the horizontal line is just experimenting with formatting.]

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Equivalence relations

A brief review of equivalence relations. The standard definition of equivalence relations is that it is a relation which is reflexive, transitive, and symmetric.
Our definition of an equivalence relation is logically equivalent: that it is a thin groupoid, called by some a setoid.
A groupoid is just a category in which each arrow is invertible. Being a category gives the reflexive and transitive parts of the traditional definition, being “thin” (meaning that there is at most one arrow between any two objects) makes it a relation, and being each arrow being invertible makes the relation symmetric.

Partitions

A brief review of partitions. A partition $\pi$ of a set $X$ is a family of non-empty subsets of $X$ which are (pair-wise) disjoint and such that their union is all of $X$. Thus each element $x\in X$ is a member of one and only one of the non-empty subsets which comprise $\pi$.
So we have a canonical surjection $X \twoheadrightarrow \pi$ : $x\in X \mapsto$ the member of $\pi$ which contains $x$. It is a surjection because each member of $\pi$ is non-empty.

Quotient sets, equivalence classes, and quotient maps

A brief review of quotient sets, equivalence classes, and quotient maps. Given an equivalence relation $E$ on $X$, we define the equivalence class $[x]$ of $x\in X$ to be its orbit or connected component under the groupoid (!). That is equivalent to the customary definition.

For a simple example, we consider the set $X = \{1,2,3 \}$. $\{1,2,3 \}$ has five partitions, each of which is the quotient object in a short exact sequence. The exact sequences corresponding to three of the five partitions are: \[\begin{array}{cccccccl} E \ \text{as function} \ X\times X \to \{0,1\} & E \subseteq X\times X & \rightrightarrows & X & \twoheadrightarrow & X/E & \text{Partition type} & \text{comment} \\ \left ( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1\\ \end{array}\right ) & \{1,2,3\}\times\{1,2,3\} & \rightrightarrows & \{1,2,3\} & \twoheadrightarrow & \bigl\{\{1,2,3\}\bigr\} & 3 & \text{the coarsest, chaotic, case}\\ \left ( \begin{array}{ccc} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1\\ \end{array}\right ) & \{(1,1),(1,2),(2,1),(2,2),(3,3)\} & \rightrightarrows & \{1,2,3\} & \twoheadrightarrow & \bigl\{\{1,2\},\{3\}\bigr\} & 2+1 & \\ \left ( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\\ \end{array}\right ) & \{(1,1), (2,2), (3,3)\} & \rightrightarrows & \{1,2,3\} & \twoheadrightarrow & \bigl\{\{1\},\{2\},\{3\}\bigr\} & 1+1+1 & \text{the finest, discrete, case}\\ \end{array} \] The other two partitions are also of type $2+1$; the difference is that the block of the partition consisting of only one element contains respectively 1 and 2.

Consider the following diagram ("ker" is kernel): \[ \begin{array}{ccccc} \text{ker(}f\text{)} & \rightrightarrows & X & \overset{\text{class map}}{\twoheadrightarrow} & X/\text{ker(}f\text{)}\\ \Vert && \Vert && \downarrow \\ \text{ker(}f\text{)} & \rightrightarrows & X & \underset{f}{\twoheadrightarrow} & Y\\ \end{array} \] Here the surjection $f$ is at lower right;
the partition is $X/\text{ker(}f\text{)}$ at top right;
the equivalence relation is $\text{ker(}f\text{)}$ on the left.
An important adjoint equivalence (in combinatorics and elsewhere; $X \in \bf\text{Set}$) is \begin{equation} \mathrel{\mathcal{Equiv-relns}(X) \substack{\xrightarrow{X/-} \\\xleftarrow[\text{kernel}]{}} X\downarrow\mathcal{Surj}} \end{equation}

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\[\begin{array}{ccccc} \mathcal{Equiv-relns}(X) & \xrightarrow[\cong]{X/-} & \mathcal{Part}(X)\\ \text{kernel}\nwarrow &&\swarrow\text{form the class map} \\ &X\downarrow\mathcal{Surj}&\\ \end{array}\]

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\[\begin{array}{ccccc} \mathcal{Equiv-relns}(X) & \xrightarrow[\cong]{X/-} & \mathcal{Part}(X) & \xrightarrow{\text{form the class map}} & X\downarrow\mathcal{Surj}\\ \Vert & = && \Rightarrow & \Vert \\ \mathcal{Equiv-relns}(X) && \xleftarrow[\text{kernel}]{} && X\downarrow\mathcal{Surj}\\ \end{array}\]

The category structures on 2

• $\cattwo$ as a metacategory (CWM 1.1)
• $\cattwo$ as an internal category in $\Set$, i.e., $\cattwo\in\Cat(\Set)$ (CWM 1.2)
• $\cattwo$ as a category enriched in $\Set$, i.e., as a $\Set$-category, i.e., $\cattwo\in\Set$-$\Cat$ (CWM 1.8, 7.7)
• $\bftwo$ as a category enriched in itself, i.e., as a $\bftwo$-category, i.e., $\bftwo\in\bftwo$-$\Cat$ (BCECT 1.6.3)
• $\Set$ and $\cattwo$ as $\text{ccccc}$s - complete and cocomplete cartesian closed categories
• Relations between the categories

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$\cattwo$ as a metacategory (CWM 1.1)

(The metacategory $\cattwo$) has data consisting of:
two objects, $\bot$ and $\top$, and three arrows, $1_\bot:\bot\to\bot$, ${\lt}:\bot\to\top$, and $1_\top:\top\to\top$
(${\lt}:\bot\to\top$ is of course usually denoted $\bot\lt\top$).
(Its identity arrows) are evident, and (its composition operation) is defined by (the unit axiom).
Note that no mention of sets has been made.

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$\cattwo$ as an internal category in $\Set$, i.e., $\cattwo\in\Cat(\Set)$ (CWM 1.2)

(The internal category $\cattwo$ in $\Set$) has data consisting of
a set of objects $\boxed{\cattwo_0 = \{\bot,\top\}}$ and
a set of arrows $\boxed{\cattwo_1 = \{1_\bot,{\lt},1_\top\}}$,
together with (source, target, identity, and composition functions) which implement (the operations defined for the metacategory $\cattwo$).
The only difference between the definition of $\cattwo$ (as a metacategory) and (as an internal category in $\Set$) is (the explicit mention of sets and functions).

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$\cattwo$ as a category enriched in $\Set$, i.e., as a $\Set$-category, i.e., $\cattwo\in\Set$-$\Cat$ (CWM 1.8, 7.7)

Again, we have the set of objects $\{\bot,\top\}$, now denoted by $\boxed{\ob\cattwo}$ (because the subscript $()_0$ will soon be given another meaning).
We also have a hom-function $\boxed{ \hom - \cattwo - : \ob\cattwo \times \ob\cattwo \to \Set}$,
taking each (pair of objects) to {the set of arrows from (the first, source, object) to (the second, target, object)}.
We represent the hom-function as a table:

target	$\top$		$\{\lt\}$	$\{1_\top\}$
	$\bot$		$\{1_\bot\} $	$\emptyset$
$\hom {\langle\text{source}\rangle} {\cattwo} {\langle\text{target}\rangle}$		$\bot$	$\top$
		source

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$\bftwo$ as a category enriched in itself, i.e., as a $\bftwo$-category, i.e., $\bftwo\in\bftwo$-$\Cat$ (BCECT 1.6.3)

(The internal hom for $\bftwo$) is shown in (the center of the display below), where it can be easily compared to (some related functions).

The $\Set$-valued hom-function (the external hom) for the $\Set$-category $\cattwo$ target $\top$ $\{ \bot \xrightarrow{\textstyle \lt} \top \}$ $\{ \top \xrightarrow{\textstyle 1_\top} \top \}$ $\bot$ $\{ \bot \xrightarrow{\textstyle 1_\bot} \bot\}$ $\emptyset$ $\hom {\langle\text{source}\rangle} {\cattwo} {\langle\text{target}\rangle}$ $\bot$ $\top$ source	The $\bftwo$-valued hom-function (the internal hom) for the $\bftwo$-category $\bftwo$ target $\top$ $\top$ $\top$ $\bot$ $\top$ $\bot$ $\hom {\langle\text{source}\rangle} \bftwo {\langle\text{target}\rangle}$ or $[\langle\text{source}\rangle,\langle\text{target}\rangle]$ $\bot$ $\top$ source	The truth-table for implication ($\Rightarrow$) consequent $\top$ $\top$ $\top$ $\bot$ $\top$ $\bot$ $\langle\text{ant.}\rangle \Rightarrow \langle\text{cons.}\rangle$ $\bot$ $\top$ antecedent
	The $\Set$-valued hom-function for the full subcategory $\cal T$ (for “tiny”) determined by $\emptyset$ and $\{\emptyset\}$ of the $\Set$-category $\Set$ target $\{\emptyset\}$ $\big\{\emptyset\to\{\emptyset\}\big\}$ $\big\{\{\emptyset\} \xrightarrow{\textstyle 1_{\{\emptyset\}} } \{\emptyset\}\big\}$ $\emptyset$ $\big\{\emptyset \xrightarrow{\textstyle 1_\emptyset } \emptyset\big\}$ $\emptyset$ $\hom {\langle\text{source}\rangle} \Set {\langle\text{target}\rangle}$ or $[\langle\text{source}\rangle,\langle\text{target}\rangle]$ $\emptyset$ $\{\emptyset\}$ source

We can compare (the two $\Set$-categories, $\cattwo$ and $\Set$) to (the $\bftwo$-category $\bftwo$) with (the following two diagrams).
The labels above the arrows are the names of the arrows in the $\Set$-categories
(for the last arrow on each line, there is no label above, indicating that there is no such arrow in the $\Set$-category; the $\Set$ arrow $\emptyset\to\{\emptyset\}$ exists, but is here unnamed);
labels below arrows indicate the value of $\big( \hom {\langle\text{source}\rangle} \bftwo {\langle\text{target}\rangle} = [\langle\text{source}\rangle,\langle\text{target}\rangle] \big)$ for the indicated source and target. \[\begin{array}{ccccccccccl} \bot & \xrightarrow[\textstyle \top]{\textstyle 1_\bot} & \bot & \xrightarrow[\textstyle \top]{\textstyle \lt} & \top & \xrightarrow[\textstyle \top]{\textstyle 1_\top} & \top & \xrightarrow[\textstyle \bot]{} & \bot & \mkern6em & \cattwo \text{ above, } \bftwo \text{ below} \\ \\ \emptyset & \xrightarrow{\textstyle 1_\emptyset} & \emptyset & \xrightarrow{} & \{\emptyset\} & \xrightarrow{\textstyle 1_{\{\emptyset\}}} & \{\emptyset\} & \xrightarrow{} & \emptyset && \Set \text{ above} \\ \end{array}\]

There is a lot of information packed into those hom-tables. For example:
[That (the edge corresponding to ($\langle\text{source}\rangle=\bot$)) is all $\top$] is equivalent to ($\bot$ being initial in $\bftwo_0$).
[That (the edge corresponding to ($\langle\text{target}\rangle=\top$)) is all $\top$] is equivalent to ($\top$ being terminal in $\bftwo_0$).
[That (the edge corresponding to ($\langle\text{source}\rangle=\top$)) is identical to its input] is equivalent to ($[\top,-] \xlongequal{\textstyle i} 1_\bftwo$). (The letter $i$ comes from a generalization in the paper [CC].)
[That (the edge corresponding to ($\langle\text{target}\rangle=\bot$)) interchanges truth values] is equivalent to ($[-,\bot] = \neg$ (negation)).

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The subscript $()_0$ changes meaning when going between (internal category theory) and (enriched category theory)

On the one hand, (an internal category $\catA$ in a category $\calE$) can be viewed as (a truncated simplicial object in $\calE$),
with $\catA_0$ = the object of objects, $\catA_1$ = the object of arrows, $\catA_2$ = the object of composable pairs of arrows, and $\catA_3$ = the object of composable triples of arrows.
That is a useful point of view, and it sets the meaning in internal category theory for $\catA_0$.

On the other hand, in (enriched category theory), there is a long tradition, starting with the Eilenberg-Kelly paper “Closed Categories” and continuing through Kelly’s basic text, Basic Concepts of Enriched Category Theory, where (the $()_0$ subscript) has a different meaning when applied to (an enriched category).
If $\catA$ is (an enriched category), $\catA_0$ is NOT ($\catA$’s set of objects) (Kelly denotes that by $\ob\catA$, while some other authors use $|\catA|$),
but rather ($\catA$’s underlying ordinary ($\Set$-enriched) category), as defined in section 1.3 of [BCECT].
We wish to emphasize (the enriched categorical point of view), so we adopt its notation.
Thus henceforth $\boxed{\bftwo_0}$ is not the mere set $\{\bot,\top\}$, but rather an isomorph of the ordinary category, and linear order, $\cattwo$.

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$\bftwo$ and $\Set$ as $\text{ccccc}$s - complete and cocomplete cartesian closed categories

We assume as already known the fact that $\bftwo$ and $\Set$ are complete and cocomplete cartesian closed categories, whose operations include:

$\bftwo$ and $\Set$ as $\text{ccccc}$s (complete and cocomplete cartesian closed categories)

		completeness				cocompleteness				internal hom
arity:		nullary	binary	arbitrary		nullary	binary	arbitrary		binary
$\bftwo$		$\top$	$\wedge$	$\forall$ or $\bigwedge$		$\bot$	$\vee$	$\exists$ or $\bigvee$		$\Rightarrow$
$\Set$		$1$	$\times$	$\displaystyle\prod$		$\emptyset$	$+$	$\sum$ or $\displaystyle\coprod$		$[,]$

We denote the internal hom in $\Set$ (the set of functions from, say, set $\setX$ to set $\setY$), often denoted by exponentiation $\setY^\setX$, by $[\setX,\setY]$.
The other operations necessary for completeness and cocompleteness in $\Set$ are equalizers and coequalizers.
For $\Set$ equalizers are easy, while coequalizers involve completing a parallel pair of functions into an equivalence relation, then taking the quotient of that relation.

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References

CWM, Mac Lane, Saunders, Categories for the Working Mathematician (1971,1998)
BCECT, Kelly, Max, Basic Concepts of Enriched Category Theory (1982/2005)
CC, Eilenberg and Kelly, “Closed Categories” (1966)
NAMC, Janelidze and Kelly, “Note on Actions of a Monoidal Category” (2001)

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Draft stuff:

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Relations between the categories

This section is a rough draft:

From (the total order $\cattwo$) to (the $\bftwo$-category $\bftwo$):
The total order $(\cattwo,{\le})$ has
binary products = binary meets = conjunction = $\land$,
and, for each $\objB\in\cattwo$, the order-preserving function $\boxed{-\land\objB : \cattwo\to\cattwo}$
has a right adjoint, denoted $\boxed{\objB\Rightarrow -}$ or $\boxed{[\objB,-]}$ or, combining the previous two notations, $\boxed{[\objB\Rightarrow-]}$ , thus we have:
\[ \pi: \big( \objA\land\objB \le \objC \big) \text{ iff } \big( \objA \le [\objB\Rightarrow\objC] \big), \mkern6em \text{a special case of} \mkern6em \hom {\objectA\tensor\objectB} {(\cal V_0)} {\objectC} \mathop\cong\limits^{\textstyle\pi}_{\text{II, $\S$3}} \hom \objectA {(\cal V_0)} {[\objectB,\objectC]} \mkern6em \text{for $\calV_0 = \cattwo$} \;. \]

From (the $\bftwo$-category $\bftwo$) to (its underlying total order $\bftwo_0$):
Given the internal hom $[,]$, i.e., $\Rightarrow$, define \[ v: \big( \objA\le\objB \big) \text{ iff } \big( \top = [\objA\Rightarrow\objB] \big), \mkern6em \text{which is almost a special case of} \mkern6em \hom \objectA {(\cal V_0)} {\objectB} \mathop\cong\limits^{\textstyle v}_{\text{II (3.12)}} \hom \objectI {(\cal V_0)} {[\objectA,\objectB]} \;. \] (The general cases indicated above) are standard situations in (the theory of closed categories; see section II.3 of [CC]).
The references below the $\cong$ symbols are references to that paper.
The “almost” qualifier is required because (the situation in which it appears) is (a definition of $\le$ in $\cal V_0$), not (an isomorphism between already existing objects).
(The order relation thus defined on $\{\bot,\top\}$) is exactly the same as (that given by fiat on $\{\bot,\top\}$ in the definition of $\cattwo$ as a metacategory, etc.).
Thus $\bftwo_0$ and $\cattwo$ are isomorphic total orders.

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\[\begin{array}{} \boxed{\begin{array}{} \textstyle \ar\cattwo \atop {\Rule{2em}{1px}{1px}} \\ 1_\bot \\ \lt \\ 1_\top \\ \end{array}} & \cong & \boxed{\begin{array}{} {\textstyle \le_2} \atop {\Rule{3em}{1px}{1px}} \\ \langle\bot,\bot\rangle \\ \langle\bot,\top\rangle \\ \langle\top,\top\rangle \\ \end{array}} & \cong & \mkern2em \boxed{\begin{array}{} \textstyle \ar(\bftwo_0) \atop {\Rule{4em}{1px}{1px}} \\ \top=\hom \bot\bftwo\bot \\ \top=\hom \bot\bftwo\top \\ \top=\hom \top\bftwo\top \\ \end{array}} & \to & 1 \\ & \searrow & \big\downarrow & \swarrow && \lrcorner & \big\downarrow \rlap\top \\ 1 & \xrightarrow{\textstyle \langle\objectA,\objectB\rangle} & \boxed{\begin{array}{} \{\bot,\top\} \times \{\bot,\top\} \\ \langle\bot,\bot\rangle \\ \langle\bot,\top\rangle \\ \langle\top,\top\rangle \\ \langle\top,\bot\rangle \\ \end{array}} & {}\rlap{\xrightarrow[\boxed{\begin{array}{} \textstyle \hom - \bftwo - \\ [,]_2 \\ \Rightarrow_2 \\ \le_2 \\ \end{array}} ]{\mkern13em}} &&& \boxed{\begin{array}{} \{\bot,\top\} \\ \bot \\ \top \\ \end{array}} \\ && \llap{\text{projections from cartesian product}} \bigg\downarrow \bigg\downarrow \\ && \boxed{\begin{array}{} \{\bot,\top\} \\ \bot \\ \top \\ \end{array}} \\ \end{array}\]

The cartesian closed category 2

\[\begin{array}{l} & \objA \Longrightarrow [ \objB \Rightarrow \objC ] \\ \neg\big( \objA \wedge \neg[ \objB \Rightarrow \objC ] \big) && \neg\objA \vee [ \objB \Rightarrow \objC ] \\ \neg\big( \objA \wedge \neg\neg( \objB \wedge \neg\objC ) \big) \\ \neg\big( \objA \wedge ( \objB \wedge \neg\objC ) \big) && \neg\objA \vee \neg( \objB \wedge \neg \objC ) \\ \neg\big( \objA \wedge \objB \wedge \neg\objC \big) && \neg\objA \vee \neg\objB \vee \objC \\ \neg\big( (\objA \wedge \objB) \wedge \neg\objC \big) && \neg(\objA \wedge \objB) \vee \objC \\ & \objA \wedge \objB \Longrightarrow \objC \\ \end{array}\]

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\[\begin{array}{} &&& \radjtop & \unicode{0x2500} & \radjtop \\ && \unicode{0x2571} && \unicode{0x2571} \\ & \radjtop & \unicode{0x2500} & \radjtop \\ \\ && \radjtop & \unicode{0x2500} & \ladjbot \\ & \unicode{0x2571} && \unicode{0x2571} \\ \radjtop & \unicode{0x2500} & \radjtop \\ \end{array}\]

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		$\objB$
		$\Big\uparrow$	$\objB$
$\top$			$\Rule{100px}{2px}{0px}$
$\neg\objA$		$\Rule{2px}{50px}{50px}$		$\Rule{2px}{50px}{50px}$	$\objA$
$\bot$			$\Rule{100px}{2px}{0px}$		$\longrightarrow \objA$
		$\bot$	$\neg\objB$	$\top$

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$\top$		$\top$	$\Rule{100px}{2px}{0px}$	$\top$
		$\Rule{2px}{50px}{50px}$		$\Rule{2px}{50px}{50px}$
$\bot$		$\top$	$\Rule{100px}{2px}{0px}$	$\bot$
		$\bot$		$\top$

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\[\begin{array}{} \bbox[100px, border:1px black solid]{AA} \end{array}\]