How Kelly builds clubs from generators

PRELIMINARY DRAFT

Kelly does this in an extremely precise and careful way, so it seems best to quote him fairly exactly on this.

$\newcommand\calJ{{\mathcal J}} \newcommand\calT{{\mathcal T}}$

Actually, what we give is a considerable simplification of what Kelly gives in AAC Section 3.1 (the parenthesized numbers of the form (m.n)  are references to that document):

(1) It is only for single-sorted theories, not multi-sorted theories.

(2) It only describes algebraic theories that can describe structures borne by sets, not the more complex theories needed to describe structures borne by categories.

(3) It does not allow for permutation of variables.

Starting from 

(a graded set $\boxed{\leftcat\calB}$ of names of the basic generating operations), Kelly describes

(the free discrete club $\boxed\calT$ that it generates), (a monoid in the monoidal (under the wreath product $\circ$) category of graded sets). 

(The objects of $\calT$) and (their augmentations) are defined inductively by: 

(3.2) There is an element $\boxed\bfone$ (boldface 1) in $\calT$ of arity ${\bfone}\Gamma = 1 \in \N$. 

$\bfone$ is (the element in the theory $\calT$) 

which will be interpreted in (a model) as (the identity arrow (unary operation) for the model).

With $\bfone$ we can define the unit for the monoid $\calT$, $\boxed{ \leftadj\eta : \calJ \mathrel{\leftadj\to} \calT : 1 \mathrel{\leftadj\mapsto} \bfone }$.

(3.3) If ($\leftcat{B \in \calB}$) and ($T_1,\ldots,T_{\leftcat B \Gamma} \in \calT$), 

then there is an object (i.e. element) $\boxed{ \leftcat B \{T_1,\ldots,T_{\leftcat B\Gamma}\} } \in \calT$ wlth domain-type $(\leftcat B\Gamma)(T_1\Gamma,\ldots, T_{\leftcat B\Gamma}\Gamma) = T_1\Gamma + \cdots + T_{\leftcat B\Gamma}\Gamma$.

EXAMPLE

For a concrete, familiar example of this, consider $\leftcat{\calB = \{0,s\}}$, the names of the classic Peano successor algebra operators. 

Then $\calT$ and (its interpretation in the classic model in $\Set$, on $\N$) are:

\[  \boxed{  \begin{array}{l|llll| llll}    & &    \kern3em \leftcat\calT  &&&   {} \rlap{\kern3em \rightcat{ \text{interpretation in } \N }} &&& \kern0em  \\  \hline  \text{arity } 0   &   \leftcat{ 0\{\} }, &  \leftcat{  s\{0\{\}\} },  &  \leftcat{ s\{s\{0\{\}\} },  & \leftcat \dots  &     \rightcat 0, &  \rightcat{s0},  & \rightcat{ s^2 0},  & \rightcat \dots     \\  \text{arity } 1  &     \leftcat \bfone,  &   \leftcat{ s\{\bfone\} },  &   \leftcat{ s\{s\{\bfone\}\} },  &   \leftcat \ldots   &    \rightcat {  1_\N = \text{id}_\N },   & \rightcat { s:\N\to\N }, & \rightcat{  s^2 }, & \rightcat \ldots  \end{array}   }    \]

Using (3.2) and (3.3) we can define an embedding $\boxed{ \leftcat\calB \mathrel{\leftadj \hookrightarrow} \calT : \leftcat B \mathrel{\leftadj\mapsto} \leftcat B\{\bfone,\dots,\bfone\} }$.

We define the composition (or substitution) operation on $\calT$ 

\[  \boxed{ \begin{array}{} \calT \circ \calT &  \xrightarrow[\kern3em]{\textstyle \mu }  &  \calT   \\   T\leftcat[S_1,\dots,S_{T\Gamma}\leftcat]  &  \mapsto  &  T\rightcat(S_1,\dots,S_{T\Gamma}\rightcat) \end{array} }  \]  

inductively by setting 

(3.4)  $\bfone \leftcat[S\leftcat] \mapsto  \bfone \rightcat(S\rightcat) = S$,

(3.5) $\begin{equation} \big(\leftcat B\{T_1,\ldots,T_{\leftcat B \Gamma}\}\big)\leftcat[S_1,\ldots,S_m\leftcat]   \mapsto   \\   \big(\leftcat B \{T_1,\ldots,T_{\leftcat B\Gamma}\}\big) \rightcat(S_1,\ldots,S_m\rightcat) = \leftcat B \big\{T_1\rightcat(S_1,\ldots,S_{m_1}\rightcat),\ldots,T_{\leftcat B \Gamma}\rightcat( \ldots,S_m\rightcat)\big\}  \end{equation}$,

where $m_i = T_i\Gamma$ and $m$ is the sum of these. 

We identify $\leftcat{B \in \calB}$ with $\leftcat B \{\bfone,\ldots,\bfone\} \in \calT$. 

Then $\leftcat B \{T_1,\ldots,T_{\leftcat B \Gamma}\}$ coincides with $\leftcat B \rightcat(T_1,\ldots,T_{\leftcat B \Gamma}\rightcat)$ by (3.4) and (3.5):

\[ \leftcat B \rightcat(T_1,\ldots,T_{\leftcat B \Gamma}\rightcat) \xlongequal{\leftcat\calB \hookrightarrow\calT}  \big(\leftcat B \{\bfone,\ldots,\bfone\}\big)\rightcat(T_1,\dots,T_{\leftcat B \Gamma}\rightcat) \xlongequal[\text{(3.5)}]{\mu} \leftcat B \big\{\bfone\rightcat(T_1\rightcat),\dots,\bfone\rightcat(T_{\leftcat B \Gamma}\rightcat)\big\} \xlongequal[\text{(3.4)}]{\mu} \leftcat B \{T_1,\dots,T_{\leftcat B \Gamma}\} \; , \]

and we can now drop (the curly-bracket notation) in favor of (round brackets). 

To see how (Kelly's various definitions) play together in (a simple and familiar case),

let us apply them to (the $\leftcat{ \calB = \{0,s\} }$ example above):

Identify ($\leftcat{ 0,s \in \calB }$) with ($\leftcat 0 \{\}, \leftcat s \{\bfone\} \in \calT$).

Then we have:

\[ \begin{array}{} \calT \circ \calT &  \xrightarrow[\kern3em]{\textstyle \mu }  &  &&  \calT   \\    \leftcat s \leftcat[0\leftcat]   &   \mapsto   &  \leftcat s \rightcat(0\rightcat)    &&&&  \leftcat s \{0\}    \\  \llap{\leftcat\calB \hookrightarrow\calT} \Vert   & \mapsto &  \llap{\calB\hookrightarrow\calT} \Vert    &&&& \Vert \rlap{\leftcat\calB \hookrightarrow\calT}  \\    \big(\leftcat s \{\bfone\}\big) \leftcat{\big[}0\{\}\leftcat{\big]} & \mapsto &  \big(\leftcat s \{\bfone\}\big)\rightcat{\big(}0\{\}\rightcat{\big)}  &  \xlongequal[\text{(3.5)}]{\mu}  &  \leftcat s \Big\{\bfone\rightcat{\big(}0\{\}\rightcat{\big)}\Big\}  &  \xlongequal[\text{(3.4)}]{\mu}  & \leftcat s \big\{0\{\}\big\}  \end{array} \]

Thus together we have a cospan (AKA opspan) in $(\Set\downarrow\N)$:

\[ \boxed{ \begin{array}{}   && 1 & \mapsto & \bfone   \\     &&&&    B\{\bfone,\ldots,\bfone\}   &  \leftarrow \kern-.2em\shortmid    &  B   \\    \hline  \leftadj\eta & : & \leftcat\calJ & \mathrel{\leftadj\to} & \calT & \leftadj\hookleftarrow & \leftcat\calB \\  &&&  \llap{ |\bfone|_\calA = \ulcorner 1_{\calA} \urcorner } \searrow  &  \Bigg\downarrow \rlap{|\,|_\calA} &   \swarrow    \\     &&&&   \{\calA,\calA\}   \ \end{array} } \]

Identity elements for monoidal categories

An identity element for a monoidal category depends of course on both the category and the monoidal structure.

Here we look at several examples.

(Just for linguistic variety, sometimes we will call an identity element a "neutral element".)

$\newcommand\calJ{{\mathcal J}}$

Most familiar probably is the category of sets.

For its coproduct structure, i.e. disjoint union, the identity is the empty set $\emptyset : X \cup \emptyset = X$ for all $X\in\Set$.

I.e. $X$ $+$ nothing is still $X$.

For its categorical product, the cartesian product, the neutral element is any one-element set: $1\times X \cong X$, where $1$ is any one-element set and $X$ is an arbitrary set.

<hr/>

More complex examples are given by (the three monoidal structures on $(\Set\downarrow\N) \simeq [\N,\Set]$), 

namely (the coproduct), (cartesian product), and (the $\circ$ binary operator defined by Kelly in his MVFC).

As in any functor category, the first two identity elements are just the functions (functors) $\N\to\Set$ which are constant respectively at $\emptyset\in\Set$ and $1\in\Set$, where $1$ can be any one-element set (often $1$ is chosen to be $\{\emptyset\} = \{0\}$).

The identity element for (the $\circ$ binary operator), given as an overset, has a single element over $1\in\N$, i.e. of arity or rank $1\in\N$ and nothing over the rest of $\N$.

There clearly is a choice for what to call that element; Kelly, with a view towards his intended application, denotes it $\bfone$ (bold face 1).

We should view it as (a formal symbol) which will be interpreted as the identity function (or arrow) in the semantics of structures.

As to the overset, the identity for $\circ$, Kelly in MVFC denotes it $\calJ$ (calligraphic J).

A concrete introduction to Kelly's clubs

Max Kelly introduced and discussed (a concept which he called "club") in a series of papers listed under the TOC entry for "Clubs" here: 

https://en.wikipedia.org/w/index.php?title=Max_Kelly&oldid=975556001

What is especially admirable, IMO, is Kelly's careful identification, description, and delineation, using the tools and ontology of elementary category theory, of the description of algebraic structures, a topic often approached using the tools of mathematical logic.

$\newcommand\calJ{{\mathcal J}}  \newcommand\calT{{\mathcal T}}$

 $\calI \calJ  \bfone $ 

Vi: Kelly introduced (a set of concepts) which use (the tools of category theory) to describe (structures that categories can bear).

This is a somewhat lengthy analysis.

Here we try to simplify the analysis by showing how (those same categ seeorical tools) can be used to describe (structures borne merely by sets, not categories).

Let us start  by introducing his notation in the most concrete example possible, 

(the cartesian product $X\times Y$ of two sets $X$ and $Y$).

To that end, regard (each element $x\in X$) as (an operator of arity $1$). 

Suppose (such an operator set $X$) operates on (a set $Y$).

Then Kelly would write the pair consisting of (the operator $x$) and (the operand $y$) as $\boxed{x[y]}$. Of course it is usual to write such a pair as merely an ordered pair $\langle x,y \rangle$, or some such notation, but Kelly uses (the square bracket notation) to distinguish operators from operands.

Further, he would write (the set of all such pairs) as $\boxed{X\circ Y}$ rather than $X\times Y$, indicating that he was regarding ($X$ as a set of operators) and ($Y$ as a set of operands).

Suppose now that $x$ operating on $y$ yields (a value in the set $Z$), 

a situation usually written in mathematics as 

$f : X\times Y \to Z : \langle x,y \rangle \mapsto f(x,y)$, where the function of two variables $f$  gives the action.

Kelly would, in the situation described above, and for a known function $f$, write it as 

$f : X\circ Y \to Z : x[y] \mapsto x(y)$, 

i.e. he uses going from (square brackets) to (round parentheses) to denote evaluation.

<hr />

Let us now advance to a less trivial subject, the theory of groups.

We must consider both the syntax and semantics for groups.

For defining and specifying (the structure of a group), we start with an abstract set that contains merely three elements:

${\mathcal A} = \{M,E,I\}$.

Here $M,E,I$ are considered merely abstract symbols, the <em>names</em> of the operations of the structure.

To each of those symbols we will associate a natural number, to be considered its "arity", via a function

$$\boxed{\Gamma : {\mathcal A} \to \N : M\mapsto2, E\mapsto0, I\mapsto1}$$

I.e., we will consider $M,E,I$ as abstract symbols which will be interpreted as the multiplication (or composition) operation, identity element, and inverse operation of a group.

Let us illustrate how this notation works in the most familiar example possible, the additive group of the integers $(\Z, +, 0, -)$:

\[ \begin{array}{}  {\mathcal A}  \circ \Z  &  \to  & \Z  \\   M[s,t]  &  \mapsto  & M(s,t) \equiv s+t \\  E[]  &  \mapsto  & E() \equiv 0  \\  I[s]  &  \mapsto  & I(s) \equiv -s   \end{array} \]

Now, starting from $({\mathcal A}, \Gamma)$, we can proceed alternatively in two directions, which ultimately converge (to be filled in):

\[ \boxed{  \begin{array}{c|c|c} & \text{no composition} & \text{composition}  \\   \hline  \text{syntax} & \Big( ({\mathcal A} = \{M,E,I\}) \; , \; (\Gamma : {\mathcal A} \to \N) \Big) & \Big( {\mathcal T} \, , \, (\Gamma : {\mathcal T} \to \N) \Big)  \\  \hline { \textstyle \text{semantics;} \atop \textstyle \text{algebras} }    &  \begin{array}{}  {\mathcal A}  \circ \Z  & \to  & \Z   \\ M[s,t] & \mapsto & M(s,t) \equiv s+t \\ E[] & \mapsto & E() \equiv 0 \\ I[s] & \mapsto & I(s) \equiv -s    \\   \hline  {\mathcal A}  & \buildrel {||} \over \to & \{\Z,\Z\}  \\  M & \mapsto & |M|={+} : (\Z^2\cong \Z\times\Z) \to \Z  \\   E & \mapsto & |E|=0 : (\Z^0\cong 1) \to \Z  \\ I  & \mapsto & |I|={-} : (\Z^1\cong\Z) \to \Z   \end{array}   \\  \end{array}  } \]

<hr />

A segue into semantics:

For two arbitrary sets $Y$ and $Z$ we want to consider all $n$-ary functions from $Y$ to $Z$, i.e., all $Y^n \to Z$, for arbitrary natural numbers $n \in \N$. 

For $n=0$ these are the functions $(Y^0 \cong 1) \to Z$, which are in bijection with the elements of $Z$. 

For $n=1$ these are (the functions $Y^1 \to Z$). 

Since $Y^1 \cong Y$, these are in bijection with (the set of functions $Y\to Z$), which is usually denoted either $[Y,Z]$ or $Z^Y$.

For $n=2$ these may be viewed as all possible binary operations (or connectives) from $Y$ to $Z$.

Thus, (given two sets $Y$ and $Z$),

we define a graded set $\boxed{ \{Y,Z\} }$, i.e. an object of $(\Set\downarrow\N)$, by

\[ \boxed{ \{Y,Z\} = \{(n,T) : n\in \N \text{ and } T:Y^n \to Z \} } \kern4em \text{and} \kern4em  \boxed{ \Gamma : \{Y,Z\} \to \N : (n,T) \mapsto (n,T)\Gamma = n } \; ,\]

i.e.,  an element of $\{Y,Z\}$ is (an $n\in \N$) together with (an $n$-ary operation on $Y$ with values in $Z$); the arity $(n,T)\Gamma$ is of course $n$.

As just seen, $\{Y,Z\}$ contains as subsets sets in bijection with $Z$ (the $0$-ary, i.e. nullary, operations) and with the ordinary set of functions from $Y$ to $Z$, usually denoted $Z^Y$ or $[Y,Z]$ (the unary operations).

The two constructs, binary operations actually, $\circ$ and $\{,\}$, may be easily extended to define two bifunctors 

\[ \boxed{ \circ : (\Set\downarrow\N) \times \Set \to \Set} \kern4em \text{and} \kern4em \boxed{ \{,\} : \Set^\op \times \Set \to (\Set\downarrow\N) } \]

such that, for all (graded sets $\calA\in (\Set\downarrow\N)$) and (mere sets $Y,Z \in \Set$), there is a natural bijection of sets 

\[ \boxed{ \hom {(\calA\circ Y)} \Set Z   \cong    \hom \calA {(\Set\downarrow\N)} {\{Y,Z\}} } \; , \]

the bijection being exemplified in a box above.

The two constructs, $\circ$ and $\{,\}$, may be easily extended further to define two bifunctors 

\[ \boxed{ \circ : (\Set\downarrow\N) \times (\Set\downarrow\N) \to (\Set\downarrow\N) } \kern4em \text{and} \kern4em \boxed{ \{,\} : (\Set\downarrow\N)^\op \times (\Set\downarrow\N) \to (\Set\downarrow\N) } \; , \]

making $(\Set\downarrow\N)$ a <em>closed monoidal category</em>.

Thus, for all graded sets $\calA,\calB,\calC\in (\Set\downarrow\N)$, there is a natural bijection of sets 

\[ \boxed{ \pi : \hom {(\calA\circ\calB)} {(\Set\downarrow\N)} \calC \cong \hom \calA {(\Set\downarrow\N)} {\{\calB,\calC\}} } \;  \tag{BCECT 1.23; MVFC 2.33} \]

which enriches to a natural isomorphism in $(\Set\downarrow\N)$:

\[ \boxed{ p : \big\{ {\calA\circ\calB}, \calC \big\} \cong \big\{ \calA, {\{\calB,\calC\}} \big\} } \; .  \tag{BCECT 1.27} \]

The extensions are as follows:

If $(\calA,\calB) \in (\Set\downarrow\N) \times (\Set\downarrow\N)$, then $\boxed{\calA\circ\calB}$ consists of pairs $A\in \calA$ and $B : A\Gamma \to \calB$, written as $\boxed{ A[B] }$..

If $A\Gamma = n$, we may write such a pair as $\boxed{A[B_1, \ldots, B_n]}$, listing the valued of the sequence $B$. 

(This is the format that was used in our earlier examples $M[s,t]$ etc.)

If $A\Gamma=0$ then of course it is written as $\boxed{ A[] }$.

We then define the arity of such a pair by

$(A[B_1, \ldots, B_n])\Gamma = B_1\Gamma + \ldots + B_n\Gamma$, i.e., the sum of the arities of the "inputs".

We have given two versions of $\circ$, 

\[ \boxed{ \circ : (\Set\downarrow\N) \times \Set \to \Set} \kern4em \text{and} \kern4em \boxed{ \circ : (\Set\downarrow\N) \times (\Set\downarrow\N) \to (\Set\downarrow\N) } \: ,  \]

the latter extending the former.

In fact, (the category of ordinary sets $\Set$) embeds fully and faithfully in (the category of graded sets $(\Set\downarrow\N)$) as (the full subcategory of graded sets concentrated over $0$).

\[ \Set \to (\Set\downarrow\N) : X \mapsto \Bigg(X \xrightarrow[\kern2em]{\textstyle !_X} 1 \xrightarrow[\kern2em]{\textstyle \ulcorner 0 \urcorner} \N \Bigg) , f \mapsto f \]

If $\calB\in (\Set\downarrow\N)$ is in that subcategory, i.e. is a graded set concentrated over $0$, 

then so to is $\calA\circ\calB$ for all $\calA\in (\Set\downarrow\N)$, 

i.e., that subcategory is a left ideal.

$\circ$ is pseudo-associative: $\boxed{ (\calA\circ\calB)\circ\calC \cong \calA\circ(\calB\circ\calC) }$ via

$\big(A[B_1,\dots,B_n]\big)[C_1,\ldots,C_m] \cong A\Big[ B_1[C_1,\ldots,C_{m_1}], B_2[C_{m_1+1},\ldots,C_{m_1+m_2}], \ldots, B_n[C_{m-m_n+1},\ldots,C_m] \Big]$

where $B_i\Gamma = m_i$ and $m = m_1+\cdots+m_n$.

$\circ$ has a pseudo-identity $\boxed{\calJ}$, the graded set with one element $\boxed{\bfone\in\calJ}$ whose arity is $\boxed {1\in\N}$, 

for $\calJ \circ \calA \cong \calA \cong \calA\circ\calJ$.

Picking a name for the unique element of $\calJ$ is an issue.

Kelly names it $\bfone$, and I will go with that despite the many other uses of $1$.

Again, crucial is that (the element $\bfone\in\calJ$) has (arity $1\in\N$).

Anyhow, with that naming convention, the identity isomorphisms are:

\[ \begin{array}{}  \calJ\circ\calA & \cong & \calA & \kern 6em & \calA\circ\calJ & \cong & \calA    \\    \bfone[A] & \leftrightarrow & A   &&   A[\bfone,\ldots,\bfone] & \leftrightarrow & A  \\  \end{array}  \]

<hr />

References

All are to papers by Kelly, whose titles are abbreviated as 

MVFC

AAC

C&D

See (the list of Kelly's papers on clubs cited above) for (the full bibliographical information).

Much more excellent information is at 

https://ncatlab.org/nlab/show/club

<hr />

---------

Extra junk:

Note that in the right column, the symbols $+,0,-$ to the left of the equals sign are symbols in the abstract theory club $\calK$, while to the right of the equals sign they are the familiar actual operations on integers.

Note that in (the right column), (the symbols $M,E,I$ to the left of the equivalence symbol) are (symbols in the abstract theory ${\mathcal B}$), while (to the right of the equivalence symbol) they are interpreted as (the appropriate familiar actual operations on integers).

Now, let us consider the.application of the concept to a very simple, concrete, and well understood situation, the theory of groups.

To denote the binary group operation we will use the symbol $'\cdot'$.

This should be distinguished from two other binary connectives $'\circ'$ and $'\times'$ which will also play prominent roles.

<hr/>

Draft of remarks on Prop. 23.2 in Kelly TC

The passage across the long horizontal line is the $({-} \circ \calB) \dashv \{\calB,{-}\}$ adjunction bijection

including using its natutality at $\eta$.

\[ \boxed {   \begin{array}{}     \calJ \circ \{\calB,\calA\} & {} \rlap{ \kern-5em\xrightarrow[\textstyle \kern17em ?? \kern17em]{\textstyle l_{\{\calB,\calA\}} } } &&&&& \{\calB,\calA\} \\  1 [S]  & \mapsto &  \bfone [S]  && \mapsto &&  S     \\     \calJ \circ \{\calB,\calA\}   &  \xrightarrow[\textstyle \eta \circ \{\calB,\calA\}] {}  &  \calT \circ \{\calB,\calA\}   &  {} \rlap{ \kern-3em\xrightarrow[\textstyle a_\calB]{\kern26em}  } &&& \{\calB,\calA\}   \\  \hline  \\   [1][S][B_1,\ldots,B_{\Gamma S}] & \mapsto & [\bfone][S][B_1,\ldots,B_{\Gamma S}] & \mapsto & [\bfone]\big[S(B_1,\ldots,B_{\Gamma S})\big] & \mapsto & S(B_1,\ldots,B_{\Gamma S})     \\    \calJ \circ \{\calB,\calA\} \circ \calB  &  \xrightarrow{\textstyle \eta \circ \{\calB,\calA\} \circ \calB } & \calT \circ \{\calB,\calA\} \circ \calB  &   \xrightarrow{\textstyle \calT \circ e }  &  \calT \circ \calA  &  \xrightarrow[\kern3em]{\textstyle a}  & \calA    \\    & \llap{\calJ \circ e} \searrow  &  \eta\circ e   &  \nearrow \rlap{\eta\circ\calA}    & √ &    \nearrow \rlap{l_\calA}     \\    &&   \calJ\circ\calA     \\      &  \llap{ l_{\{\calB,\calA\} \circ \calB} = l_{\{\calB,\calA\}} \circ \calB } \searrow  &  1 \big[S(B_1,\ldots,B_{\Gamma S})\big]  &&&  \nearrow \rlap e  \\    && \{\calB,\calA\} \circ \calB    \\   &&  S[B_1,\ldots,B_{\Gamma S}]   \end{array}  }   \]

We have ($ \bar f \in T{-}\Alg$) iff ($f \in T{-}\Alg$).

Because: 

The commuting of (the outer square in the diagram below) 

is the definition of ($f$ being a $T$-algebra map).

Now apply one aspect of the $({-}\tensor B) \dashv [B,{-}]$ adjunction, namely: 

(the arrow $e$ at lower right below) is 

(a universal arrow from $(-\tensor B)$ to $A$), i.e.,

(a terminal object in $({-}\tensor B)\downarrow A$).

\[ \boxed {   \begin{array}{}   & {} \rlap{ \kern-4em \xrightarrow[\kern28em]{\textstyle T \tensor f} }   \\    T \tensor P \tensor B  &  \xrightarrow{\textstyle T \tensor \bar f \tensor B}  &  T \tensor [B,A] \tensor B  &  \xrightarrow{\textstyle T \tensor e}  &  T \tensor A      \\    \llap{ p \tensor B }  \Bigg\downarrow  & \big( \bar f \in T{-}\Alg \big) \tensor B  &  \Bigg\downarrow \rlap{ a_B \tensor B }  & \text{defn }a_B&  \Bigg\downarrow \rlap{ a }  \\ P \tensor B  &   \xrightarrow[\textstyle \bar f  \tensor  B]{}  &  [B,A] \tensor B  &  \xrightarrow[\textstyle e]{}  &  A   \\   & {} \rlap{ \kern-4em \xrightarrow[\textstyle f]{\kern28em} }  \\   \end{array}  }   \]