$\textstyle{Z}$$\textstyle{U}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{00}}$$\scriptstyle{g_{10}}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{00}}$$\scriptstyle{g_{01}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{00}}$$\scriptstyle{f_{10}}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{00}}$$\scriptstyle{f_{01}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Untangling this(!) but keeping the overall directed graph topology gives
precisely the ‘planar squares’ formalism and composition described above.
Thus, the rook squares formalism arises from taking the digraph representation
of matrices of partial injections and re-drawing it in a planar manner.
Similarly, the composition of IP is the natural ‘summing over paths’ operation
that – unlike matrix composition – preserves planarity111111An interesting
open question is how much of the $\bf Int$ construction (at least in the
‘particle-style’ setting) may be thought of as ‘imposing planarity’ on some
matrix calculus. We refer to [3] for further connections between planarity and
(particle-style) compact closure..

Readers familiar with the conventions of [1], as well as those of [40] may
wish to verify that Abramsky’s composition within $\bf GoI(pInj)$ may be drawn
in similar terms, and corresponds to vertical rather than horizontal pasting
of rook squares, as illustrated below :

$\textstyle{Y}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{b}$$\textstyle{Y}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a\cup\bigcup_{j=0}^{\infty}c(fd)^{j}fb}$$\scriptstyle{\bigcup_{j=0}^{\infty}g(df)^{j}b}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\scriptstyle{d}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{k}$$\textstyle{P}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k\cup\bigcup_{j=0}^{\infty}g(df)^{j}dh}$$\scriptstyle{\bigcup_{j=0}^{\infty}c(fd)^{j}h}$$\textstyle{P}$

Thus, at least at endomorphism monoids of self-dual objects (such as the
reflexive, or strictly reflexive objects we discuss), it is reasonable to
consider the compositions of [40] and [1] as distinct, but interacting,
operations on the same underlying set (see Section 17).

##  14 Self-similarity, and strict self-similarity in $\bf pInj$

We now exhibit self-similar objects of $({\bf pInj},\uplus)$, and demonstrate
the semi-monoidal strictification of this self-similarity, as a first step
towards exhibiting reflexive objects of $\bf Int(pInj)$, and strictifying this
reflexivity.

It is easy to find self-similar objects of $({\bf pInj},\uplus)$; any
countably infinite set $D$ will suffice, and appropriate bijections between
$D$ and $D\uplus D$ are well-illustrated by the Hilbert’s familiar parable of
the Grand Hotel. Following the conventions of [21, 22], we will take the
natural numbers as our canonical example, with self-similarity exhibited by
the usual (bijective) Cantor pairing.

######  Definition 72.

The Cantor pairing is the bijection
$\mbox{\scalebox{1.2}{$\vartriangleleft$}}:\mathbb{N}\uplus\mathbb{N}\rightarrow\mathbb{N}$
given by $\mbox{\scalebox{1.2}{$\vartriangleleft$}}(n,i)=2n+1$. We denote its
inverse by
$\mbox{\scalebox{1.2}{$\vartriangleright$}}:\mathbb{N}\rightarrow\mathbb{N}\uplus\mathbb{N}$;
explicitly,

$\mbox{\scalebox{1.2}{$\vartriangleright$}}(n)=\left\\{\begin{array}[]{lcr}\left(\frac{n}{2},0\right)&&n\mbox{
even,}\\\ &&\\\ \left(\frac{n-1}{2},1\right)&&n\mbox{ odd.}\end{array}\right.$

This is closely related to the dynamical algebra of [16, 21, 22], which is the
(inverse) submonoid of ${\bf pInj}({\mathbb{N}},{\mathbb{N}})$ generated by
the following partial injections

$p(n)=\left\\{\begin{array}[]{lcr}\frac{n}{2}&&n\ \mbox{ even }\\\ &&\\\
\mbox{undefined}&&n\ \mbox{ odd }\end{array}\right.\ \ \ \ ,\ \ \ \
q(n)=\left\\{\begin{array}[]{lcr}\mbox{undefined}&&n\ \mbox{ even }\\\ &&\\\
\frac{n-1}{2}&&n\ \mbox{ odd }\end{array}\right.$

together with their generalised inverses $p^{\ddagger}(n)=2n$ and
$q^{\ddagger}(n)=2n+1$. These satisfy the following key conditions :

  1. 1.

$pp^{\ddagger}=1=qq^{\ddagger}$

  2. 2.

$pq^{\ddagger}=0=qp^{\ddagger}$

  3. 3.

$p^{\ddagger}p\cup q^{\ddagger}q=1$.

######  Remark 73.

Conditions 1. and 2. above are the defining relations for the (two-generator)
polycyclic monoid of [54], and condition 3. is a natural condition on concrete
representations of polycyclic monoids. This observation was made in [26, 48]
and polycyclic monoids form a significant and active research area in inverse
semigroup theory generally.

######  Proposition 74.

The generators of the dynamical algebra, and their generalised inverses, arise
as composites of the Cantor pairing and the canonical projection / injection
arrows associated with the disjoint union, as

$p=\mbox{\scalebox{1.2}{$\vartriangleleft$}}\iota_{0}\ \ ,\ \
q=\mbox{\scalebox{1.2}{$\vartriangleleft$}}\iota_{1}\ \ ,\ \
p^{\ddagger}=\pi_{0}\mbox{\scalebox{1.2}{$\vartriangleright$}}\ \ ,\ \
q^{\ddagger}=\pi_{1}\mbox{\scalebox{1.2}{$\vartriangleright$}}$

###### Proof.

This may be verified almost instantly by direct calculation. It is also a
special case of a more general connection between polycyclic monoids and
categorical projections / injections described in [27]. ∎

######  Lemma 75.

The endomorphism monoids of ${\mathbb{N}}$ and
${\mathbb{N}}\uplus{\mathbb{N}}$ in $\bf pInj$ are isomorphic.

###### Proof.

The isomorphisms between them are given by conjugation by the code / decode
arrows, as :

  * •

$\mbox{\scalebox{1.2}{$\vartriangleleft$}}(\\_)\mbox{\scalebox{1.2}{$\vartriangleright$}}:{\bf
pInj}({\mathbb{N}}\uplus{\mathbb{N}},{\mathbb{N}}\uplus{\mathbb{N}})\rightarrow{\bf
pInj}({\mathbb{N}}\uplus{\mathbb{N}},{\mathbb{N}}\uplus{\mathbb{N}})$

  * •

$\mbox{\scalebox{1.2}{$\vartriangleright$}}(\\_)\mbox{\scalebox{1.2}{$\vartriangleleft$}}:{\bf
pInj}({\mathbb{N}},{\mathbb{N}})\rightarrow{\bf
pInj}({\mathbb{N}}\uplus{\mathbb{N}},{\mathbb{N}}\uplus{\mathbb{N}})$

∎

The interpretation of the dynamical algebra as composites of the Cantor
pairing and projections / injections then allows us to make the link with
matrix representations of arrows.

######  Corollary 76.

Let $\small\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\in{\bf
pInj}({\mathbb{N}}\uplus{\mathbb{N}},{\mathbb{N}}\uplus{\mathbb{N}})$ be a
rook matrix. Then

$\mbox{\scalebox{1.2}{$\vartriangleleft$}}\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\mbox{\scalebox{1.2}{$\vartriangleright$}}=p^{\ddagger}ap\cup
p^{\ddagger}bq\cup q^{\ddagger}cp\cup q^{\ddagger}dq\in{\bf
pInj}({\mathbb{N}},{\mathbb{N}})$

Conversely, given $f\in{\bf pInj}({\mathbb{N}},{\mathbb{N}})$, then its
inverse image under the monoid isomorphism described above is the following
rook matrix:

$\mbox{\scalebox{1.2}{$\vartriangleright$}}f\mbox{\scalebox{1.2}{$\vartriangleleft$}}=\left(\begin{array}[]{cc}pfp^{\ddagger}&pfq^{\ddagger}\\\
qfp^{\ddagger}&qfq^{\ddagger}\end{array}\right)\in{\bf
pInj}({\mathbb{N}}\uplus{\mathbb{N}},{\mathbb{N}}\uplus{\mathbb{N}})$

######  Remark 77.

As well as the above monoid isomorphism between ${\bf
pInj}({\mathbb{N}},{\mathbb{N}})$ and ${\bf
pInj}({\mathbb{N}}\uplus{\mathbb{N}},{\mathbb{N}}\uplus{\mathbb{N}})$, there
is also, by construction, a bijection between the rook squares of ${\bf
Int(pInj)}(({\mathbb{N}},{\mathbb{N}}),({\mathbb{N}},{\mathbb{N}}))$, and the
rook matrices of ${\bf
pInj}({\mathbb{N}}\uplus{\mathbb{N}},{\mathbb{N}}\uplus{\mathbb{N}})$. Thus,
every arrow $f\in{\bf pInj}({\mathbb{N}},{\mathbb{N}})$ uniquely determines
and is determined by a rook square, with the correspondence given by
$f\mapsto$
$\textstyle{\mathbb{N}}$$\textstyle{{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pfp^{\ddagger}}$$\scriptstyle{qfp^{\ddagger}}$$\textstyle{{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pfq^{\ddagger}}$$\scriptstyle{qfq^{\ddagger}}$$\textstyle{\mathbb{N}}$
.

###  14.1 Strictifying self-similarity within $\bf pInj$

We first establish some notation & preliminary results :

######  Definition 78.

Let us denote by $({\bf pNat},\uplus,\emptyset)$ the full symmetric monoidal
subcategory of $\bf pInj$ generated by the natural numbers, together with
disjoint union. Note that $(\bf pNat,\uplus)$ is also traced, with trace given
by the trace of $({\bf pInj},\uplus)$.

We now give an explicit description of the semi-monoidal strictification of
self-similarity within $\bf pNat$. We take the code and decode arrows
$\mbox{\scalebox{1.2}{$\vartriangleleft$}}:{\mathbb{N}}\uplus{\mathbb{N}}\rightarrow{\mathbb{N}}$
and
$\mbox{\scalebox{1.2}{$\vartriangleright$}}:N\rightarrow{\mathbb{N}}\uplus{\mathbb{N}}$
to be as given in Definition 72, and similarly for the dynamical algebra.

Our starting point is a symmetric semi-monoidal tensor on the endomorphism
monoid of ${\mathbb{N}}$; this is well-known as Girard’s representation of
multiplicative conjunction within [21, 22].

######  Definition 79.

Given $f,g\in{\bf pNat}({\mathbb{N}},{\mathbb{N}})$, we define their tensor
$f\star g\in{\bf pNat}({\mathbb{N}},{\mathbb{N}})$ by $f\star g\ =\
\mbox{\scalebox{1.2}{$\vartriangleleft$}}(f\uplus
g)\mbox{\scalebox{1.2}{$\vartriangleright$}}$. This may alternatively and
equivalently be given in terms of the dynamical algebra, as $f\star
g=p^{\ddagger}fp\cup q^{\ddagger}gq$. Explicitly,

$(f\star g)(n)\ =\ \left\\{\begin{array}[]{lcr}2f\left(\frac{n}{2}\right)&&n\
\mbox{ even, and }\frac{n}{2}\in dom(f)\\\ &&\\\
2g\left(\frac{n-1}{2}\right)+1&&n\ \mbox{ odd, and }\ \frac{n-1}{2}\in
dom(g)\\\ &&\\\ \bot&&\mbox{otherwise}\end{array}\right.$

The symmetry and associativity isomorphisms $\sigma,\tau\in{\bf
pNat}({\mathbb{N}},{\mathbb{N}})$ for this tensor are given by $\sigma(n)\
=\left\\{\begin{array}[]{lcr}n+1&&n\mbox{ even}\\\ &&\\\ n-1&&n\mbox{
odd}\end{array}\right.$ and $\tau(n)\ =\ \left\\{\begin{array}[]{lcr}2n&&n\
(mod\ 2)=0\\\ &&\\\ n+1&&n\ (mod\ 4)=1\\\ &&\\\ \frac{n-1}{2}&&n\ (mod\
4)=3\\\ \end{array}\right.$

These also may be given explicitly, in terms of the dynamical algebra, as

$\sigma=p^{\ddagger}q\cup q^{\ddagger}p\ \ \mbox{ and }\ \
\tau=\left(p^{\ddagger}\right)^{2}p\cup p^{\ddagger}q^{\ddagger}pq\cup
q^{\ddagger}q^{2}$

######  Theorem 80.

The above operation $\\_\star\\_$ is indeed a symmetric semi-monoidal tensor
on the endomorphism monoid of ${\mathbb{N}}$, with associativity and symmetry
maps as given above.

###### Proof.

This is well-established [26, 27], and the interpretation as a semi-monoidal
strictification of self-similarity is given in [31]. Explicit elementary
arithmetic proofs of MacLane’s pentagon and hexagon conditions are also given
in [30]. ∎

Now consider the semi-monoidal category $({\bf pNat}_{-I},\uplus)$ given by
the ‘forgetting the unit’ functor of Definition 25. As well as the above
tensor, the strictification procedure of [31] gives, as described in Section
6, a semi-monoidal equivalence of categories between $({\bf
pNat}_{-I},\uplus)$ and $({\bf pNat({\mathbb{N}},{\mathbb{N}})},\star)$. This
proceeds as follows :

######  Definition 81.

For all objects $X\in Ob({\bf pNat}_{-I})$, we define mutually inverse
isomorphisms $C_{X}:X\rightarrow{\mathbb{N}}$ and
$D_{X}:{\mathbb{N}}\rightarrow X$ inductively, by
$C_{\mathbb{N}}=1_{\mathbb{N}}$, and for all $A,B\in Ob({\bf pNat}_{-I})$,

$C_{A\uplus B}=\mbox{\scalebox{1.2}{$\vartriangleleft$}}(C_{A}\uplus C_{B})\ \
,\ \ D_{A\uplus B}=(D_{A}\uplus
D_{B})\mbox{\scalebox{1.2}{$\vartriangleright$}}$

We then define a semi-monoidal functor $\Phi:({\bf
pNat}_{-I},\uplus)\rightarrow({\bf pNat({\mathbb{N}},{\mathbb{N}})},\star)$ by
:

Objects

$\Phi(X)={\mathbb{N}}$ for all $X\in Ob({\bf pNat})$.

Arrows

$\Phi(f)=C_{B}fD_{A}\in{\bf pNat({\mathbb{N}},{\mathbb{N}})}$, for all
$f\in{\bf pNat}(A,B)$

######  Theorem 82.

The functor $\Phi:({\bf pNat}_{-I},\uplus)\rightarrow({\bf
pNat({\mathbb{N}},{\mathbb{N}})},\star)$ is a semi-monoidal equivalence of
categories.

###### Proof.

This is immediate from the strictification procedure of [31] and heavily
prefigured (albeit without the interpretation as a semi-monoidal equivalence
of categories) in [26, 27]. ∎

We may now adjoint a strict unit to the above categories, as described in
Definition 25. This results in the the following categories :

  * •

$({\bf pNat({\mathbb{N}},{\mathbb{N}})},\star)_{+I}$, the above semi-monoidal
category with a strict unit adjoined.

  * •

$(\bf
pNat,\uplus)_{-{\tiny\begin{array}[]{cc}{\mathcal{E}}&\\!\\!{\mathcal{A}}\\\\[-1.5pt]
{\mathcal{F}}&\\!\\!{\mathcal{W}}\end{array}}}$, the de-elemented version of
$(\bf pNat,\uplus)$.

######  Corollary 83.

  1. 1.

$(\bf
pNat,\uplus)_{-{\tiny\begin{array}[]{cc}{\mathcal{E}}&\\!\\!{\mathcal{A}}\\\\[-1.5pt]
{\mathcal{F}}&\\!\\!{\mathcal{W}}\end{array}}}$ is monoidally equivalent to
$({\bf pNat({\mathbb{N}},{\mathbb{N}})},\star)_{+I}$.

  2. 2.

Both $(\bf
pNat,\uplus)_{-{\tiny\begin{array}[]{cc}{\mathcal{E}}&\\!\\!{\mathcal{A}}\\\\[-1.5pt]
{\mathcal{F}}&\\!\\!{\mathcal{W}}\end{array}}}$ and $({\bf
pNat({\mathbb{N}},{\mathbb{N}})},\star)_{+I}$ are traced.

###### Proof.

  1. 1.

This follows immediately from Corollary 31, and the monoidal equivalence is
given by a trivial extension of the $\Phi$ semi-monoidal functor of Definition
81 above to categories with strict units adjoined.

  2. 2.

This follows from Proposition 43. The trace on $(\bf
pNat,\uplus)_{-{\tiny\begin{array}[]{cc}{\mathcal{E}}&\\!\\!{\mathcal{A}}\\\\[-1.5pt]
{\mathcal{F}}&\\!\\!{\mathcal{W}}\end{array}}}$ is simply that of $({\bf
pNat},\uplus)$, restricted to non-element arrows. The trace of $({\bf
pNat}({\mathbb{N}},{\mathbb{N}}),\star)_{+I}$ is given by, for all $f\in{\bf
pNat}({\mathbb{N}},{\mathbb{N}})$,

     * •

$Tr_{I,I}^{I}(1_{I})=1_{I}$

     * •

$Tr_{{\mathbb{N}},{\mathbb{N}}}^{I}(f)=f$

     * •

$Tr_{{\mathbb{N}},{\mathbb{N}}}^{\mathbb{N}}(f)=f_{00}\cup\bigcup_{j=0}^{\infty}f_{01}f^{j}_{11}f_{10}$
where the components $f_{\\_,\\_}$ are as given in Corollary 76, as

$f_{00}=pfp^{\dagger}\ \ ,\ \ f_{01}=pfq^{\ddagger}\ \ ,\ \
f_{10}=qfp^{\ddagger}\ \ ,\ \ f_{11}=qfq^{\ddagger}$

∎

##  15 From self-similarity in $\bf pInj$ to (strict) reflexivity in $\bf
Int(pInj)$

As an immediate consequence of the self-similarity of the natural numbers, we
may give self-dual self-similar objects in a compact closed category :

######  Lemma 84.

The (strictly) self-dual object $(N,N)=(N,N)^{*}\in Ob({\bf Int(pInj)})$ is
self-similar.

###### Proof.

From Corollary 40, as $\mathbb{N}\in Ob({\bf pInj})$ is self-similar, so is
$(\mathbb{N},\mathbb{N})\in Ob({\bf Int(pInj)})$, with the code / decode
arrows for $({\mathbb{N}},{\mathbb{N}})$ given by the following rook squares :

$\textstyle{{\mathbb{N}}\uplus{\mathbb{N}}}$$\textstyle{{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\vartriangleright$
$\scriptstyle{0}$$\textstyle{\mathbb{N}}$$\textstyle{{\mathbb{N}}\uplus{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\vartriangleleft$
$\scriptstyle{0}$$\textstyle{{\mathbb{N}}\uplus{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$
$\vartriangleleft$
$\textstyle{\mathbb{N}}$$\textstyle{{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$
$\vartriangleright$ $\textstyle{{\mathbb{N}}\uplus{\mathbb{N}}}$

∎

######  Corollary 85.

$({\mathbb{N}},{\mathbb{N}})\in Ob({\bf Int(pInj)})$ is an extensionally
reflexive object i.e. it is isomorphic to its own internal hom., so
$[({\mathbb{N}},{\mathbb{N}})\rightarrow({\mathbb{N}},{\mathbb{N}})]\cong({\mathbb{N}},{\mathbb{N}})$.

###### Proof.

This is immediate from the characterisation of extensionally reflexive objects
given in Lemma 9. ∎

Our stated aim is to provide concrete examples of how reflexivity may be
strictified in a compact closed category — how we may give a compact closed
subcategory containing the specified reflexive object, together with a
monoidal equivalence to another compact closed category in which this
reflexivity is exhibited by identity arrows. We do so by applying the abstract
procedures laid out in Section 9 to the above natural example of an
extensionally reflexive object in $\bf Int(pInj)$.

The following, although individually straightforward, will prove powerful :

######  Proposition 86.

  1. 1.

$\bf Int(pNat)$ is a compact closed category where all objects $(X,U)\in
Ob({\bf Int(pNat)})$ satisfying $X\ncong\emptyset\ncong U$ are isomorphic.

  2. 2.

Let us denote by $\bf IpN$ the full monoidal subcategory of $\bf Int(pNat)$
generated by the self-dual object $({\mathbb{N}},{\mathbb{N}})$. Then $\bf
IpN$ is a compact closed category where all non-unit objects are isomorphic.

###### Proof.

  1. 1.

By construction, arbitrary non-unit objects $X,Y,U,V\in Ob({\bf pNat})$ are
all isomorphic. Let us fix isomorphisms $\phi\in{\bf pNat}(X,Y)$ and
$\psi\in{\bf pNat}(U,V)$. Then the following rook squares give an isomorphism
in $\bf Int(pNat)((X,U),(Y,V))$, together with its inverse :

$\textstyle{Y}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{0}$$\textstyle{X}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{-1}}$$\scriptstyle{0}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\scriptstyle{\psi^{-1}}$$\textstyle{U}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\scriptstyle{\psi}$$\textstyle{V}$

  2. 2.

Note that point 1. above does not imply that all non-unit objects of $\bf
Int(pNat)$ are isomorphic; counterexamples are provided by $(X,I)$ and
$(I,X)$, where $X\neq\emptyset\in Ob({\bf pNat})$. However, by construction,
the full monoidal subcategory $\bf IpN$ generated by
$({\mathbb{N}},{\mathbb{N}})$ is closed under tensor and dual, and by point
3., all non-unit objects are isomorphic. As it is a full subcategory, it also
contains the relevant unit / co-unit arrows.

∎

Following the program laid out in above, we are of course moving towards a
monoidal equivalence between the category $\bf IpN$ of part 4. above, and a
compact closed category with a single non-unit object.

###  15.1 A monoidal strictification of extensional reflexivity

Corollary 83 above gives us precisely what we require for the strictification
of reflexivity at a reflexive object of a compact closed category. We have a
two-object traced monoidal category (i.e. $({\bf
pNat}({\mathbb{N}},{\mathbb{N}}),\star)_{+I}$) that is monoidally equivalent
to a small subcategory of a (de-elemented version of) $\bf pInj$, generated by
a reflexive object.

######  Remark 87.

The next step is to apply the $\bf Int$ construction to this two-object traced
monoidal category; this will result in a four-object compact closed category
that contains a strictly reflexive object, $({\mathbb{N}},{\mathbb{N}})$. We
then consider the two-object compact closed monoidal subcategory generated by
this distinguished object.

######  Definition 88.

As our notation is in danger of becoming unwieldy at this point, let us simply
denote by $\mathfrak{G}$ the two-object compact closed category resulting from
the steps of Remark 87 above.

Expanding out the definitions results in the following:

Objects

$Ob(\mathfrak{G})=\\{(I,I),({\mathbb{N}},{\mathbb{N}})\\}$

Hom-sets

By construction, all homsets (excluding the endomorphism monoid of the units)
are equal :

Scalars

$\mathfrak{G}((I,I),(I,I))=\\{1_{I}\\}$

Elements

$\mathfrak{G}((I,I),({\mathbb{N}},{\mathbb{N}}))={\bf
pNat}(I\star{\mathbb{N}},{\mathbb{N}}\star I)={\bf
pNat}({\mathbb{N}},{\mathbb{N}})$

Co-Elements

$\mathfrak{G}(({\mathbb{N}},{\mathbb{N}}),(I,I))={\bf pNat}({\mathbb{N}}\star
I,I\star{\mathbb{N}})={\bf pNat}({\mathbb{N}},{\mathbb{N}})$

Endomorphisms

$\mathfrak{G}(({\mathbb{N}},{\mathbb{N}}),({\mathbb{N}},{\mathbb{N}}))={\bf
pNat}({\mathbb{N}}\star{\mathbb{N}},{\mathbb{N}}\star{\mathbb{N}})={\bf
pNat}({\mathbb{N}},{\mathbb{N}})$

It will be convenient to describe arrows of homsets using the correspondence
between members of ${\bf pNat}({\mathbb{N}},{\mathbb{N}})$ and rook squares
over ${\bf pNat}({\mathbb{N}},{\mathbb{N}})$ of Remark 77. Given the same
endomorphism $f\in{\bf pNat}({\mathbb{N}},{\mathbb{N}})$, we draw it in
different ways as a rook square121212Note the use of a formal zero arrow, to
denote an empty homset, in the rook squares for elements / co-elements. This
is simply a notational convenience – we are not claiming the existence of a
zero arrow between the formal unit object and other objects in the same
category. Thus we have not quite arrived back in the category $\bf pInj$,
although the difference between an empty homset, and a homset containing the
nowhere-defined function on the empty set, is subtle! in different homsets of
$\mathfrak{G}$.

Elements

In $\mathfrak{G}((I,I),({\mathbb{N}},{\mathbb{N}}))$, we draw $f$ as
$\textstyle{\mathbb{N}}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\scriptstyle{1_{I}}$$\textstyle{{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{0}$$\textstyle{I}$

Co-Elements

In $\mathfrak{G}(({\mathbb{N}},{\mathbb{N}}),(I,I))$, we draw $f$ as
$\textstyle{I}$$\textstyle{{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\scriptstyle{f}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{I}}$$\scriptstyle{0}$$\textstyle{\mathbb{N}}$

Endomorphisms

In $\mathfrak{G}(({\mathbb{N}},{\mathbb{N}}),({\mathbb{N}},{\mathbb{N}}))$, we
draw $f$ as
$\textstyle{\mathbb{N}}$$\textstyle{{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pfp^{\ddagger}}$$\scriptstyle{qfp^{\ddagger}}$$\textstyle{{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pfq^{\ddagger}}$$\scriptstyle{qfq^{\ddagger}}$$\textstyle{\mathbb{N}}$
and refer to Remark 77 for the observation that this rook square uniquely
determines and is determined by $f\in{\bf pNat}({\mathbb{N}},{\mathbb{N}})$.

Composition

This is given by the ‘pasting rook squares and summing over paths’ described
in Definition 70.

The tensor

Tensors with the unit object are defined by strictness, so $(I,I)\Box\\_$ and
$\\_\Box(I,I)$ are identity functors. At the non-unit object, we have
$({\mathbb{N}},{\mathbb{N}})\Box({\mathbb{N}},{\mathbb{N}})=({\mathbb{N}},{\mathbb{N}})$,
and for arrows, the tensor is defined on rook square representations, as
$\vbox{ \vbox{ \small\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 6.37502pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{\mathbb{N}}$}}}}}}}\ignorespaces}}}}\ignorespaces} }\
{\large{\Box}}\ \vbox{ \vbox{ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 6.75002pt\raise 0.0pt\hbox{\hbox{\kern
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\ =\ \ \ \vbox{ \vbox{ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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The canonical isomorphisms

Associativity

From Lemma 36, the associator $T$ for the above tensor is
$\tau\star\tau^{-1}$, where $\star$ is the tensor of ${\bf
pNat}({\mathbb{N}},{\mathbb{N}})_{+I}$ and $\left(p^{\ddagger}\right)^{2}p\cup
p^{\ddagger}q^{\ddagger}pq\cup q^{\ddagger}q^{2}$ is the corresponding
associator. In rook square notation, this gives
$\textstyle{\mathbb{N}}$$\textstyle{{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{0}$$\textstyle{{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\scriptstyle{\tau^{-1}}$$\textstyle{\mathbb{N}}$

Symmetry

Also from Lemma 36, the commutativity isomorphism for $\\_\Box\\_$ is given by
$\sigma\star\sigma$, where $\sigma=q^{\dagger}p\cup p^{\dagger}q$ is the
commutativity isomorphism for $\\_\star\\_$. In rook square notation this is
simply
$\textstyle{\mathbb{N}}$$\textstyle{{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{0}$$\textstyle{{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\scriptstyle{\sigma}$$\textstyle{\mathbb{N}}$

The unit object

By construction, we have a strict unit object, $(I,I)$.

The compact closed structure

The dual

The dual of $\mathfrak{G}$ is a dagger, which we nevertheless write as $(\
)^{*}$, to avoid confusion with the generalised inverse of $\bf pInj$. Thus,
on objects, $\mathbb{N}^{*}=\mathbb{N}$, and on arrows it is given by
Definition 35. Explicitly, given an arrow in $\mathfrak{G}((X,U),(Y,V))$
represented as a rook square, its dual is given by  $\left(\vbox{ \vbox{
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\ {\large=}\ \ \vbox{ \vbox{ \small\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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The unit and co-unit

Recall that within $\mathfrak{G}$,

$({\mathbb{N}},{\mathbb{N}})\Box({\mathbb{N}},{\mathbb{N}})^{*}\ =\
({\mathbb{N}},{\mathbb{N}})\ =\
({\mathbb{N}},{\mathbb{N}})^{*}\Box({\mathbb{N}},{\mathbb{N}})$

We then have the two distinguished arrows for the compact closed structure,
$\eta:(I,I)\rightarrow({\mathbb{N}},{\mathbb{N}})$ and
$\epsilon:({\mathbb{N}},{\mathbb{N}})\rightarrow(I,I)$ given by, respectively
:

$\textstyle{\mathbb{N}}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\scriptstyle{0}$$\textstyle{{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{0}$$\textstyle{I}$
and
$\textstyle{I}$$\textstyle{{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\scriptstyle{\sigma}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\scriptstyle{0}$$\textstyle{\mathbb{N}}$

where $\sigma=p^{\ddagger}q\cup q^{\ddagger}p$ is the symmetry map for
$\\_\star\\_$, the tensor of the underlying traced monoidal category.

######  Remark 89.

The above two-object compact closed category provides an example of the
strictification of extensional reflexivity described in abstract terms in the
first half of this paper; we have already seen that the following key
properties are satisfied :

  1. 1.

The compact closed category $(\mathfrak{G},\Box,(\ )^{*},(I,I))$ is monoidally
equivalent to the compact closed subcategory of $\bf Int(pInj)$ monoidally
generated by $({\mathbb{N}},{\mathbb{N}})\in Ob({\bf Int(pInj)})$.

  2. 2.

The object $({\mathbb{N}},{\mathbb{N}})\in Ob({\bf Int(pInj)})$ is
extensionally reflexive.

  3. 3.

The unique non-unit object $({\mathbb{N}},{\mathbb{N}})\in Ob(\mathfrak{G})$
is strictly extensionally reflexive.

##  16 Algebraic aspects

Having established concrete examples of the relevant abstract category theory,
we move on to considering the algebraic aspects – precisely, the embeddings of
Thompson’s $\mathcal{F}$ associated with coherence for associativity, and the
simple Abramsky-Heunen Frobenius monoid associated with strict reflexivity in
a compact closed category.

######  Definition 90.

Following Point 3 of Section 9.1, we observe that the endomorphism monoids
$\mathfrak{G}(({\mathbb{N}},{\mathbb{N}}),({\mathbb{N}},{\mathbb{N}}))$ and
$\bf pInj({\mathbb{N}},{\mathbb{N}})$ have the same underlying set. Let us
denote this by $\mathcal{H}$. It will be convenient to use rook matrix
notation for elements of ${\mathcal{H}}$, and write a partial injection $f$ as
$\left(\begin{array}[]{cc}pfp^{\ddagger}&pfq^{\ddagger}\\\
qfp^{\ddagger}&qfq^{\ddagger}\end{array}\right)$; entirely equivalently, we
say that $\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)$ is simply
shorthand for the partial injection $p^{\ddagger}ap\cup p^{\ddagger}bq\cup
q^{\ddagger}cp\cup q^{\ddagger}dq$. (As a notational device, we will also
denote composition within $\bf pInj$ itself simply by concatenation).

Let us denote the composition on ${\mathcal{H}}$ arising from $\bf pInj$ as
$\\_\cdot\\_:{\mathcal{H}}\times{\mathcal{H}}\rightarrow{\mathcal{H}}$, and
that arising from $\mathfrak{G}$ as
$\\_\circ\\_:{\mathcal{H}}\times{\mathcal{H}}\rightarrow{\mathcal{H}}$.
Explicitly, these are given by :

  * •

$\left(\begin{array}[]{cc}e&f\\\
g&h\end{array}\right)\cdot\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)=\left(\begin{array}[]{cc}ea\cup fc&eb\cup fd\\\ ga\cup
hc&gb\cup hd\end{array}\right)$.

  * •

$\left(\begin{array}[]{cc}e&f\\\
g&h\end{array}\right)\circ\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)=\left(\begin{array}[]{cc}\bigcup_{j=0}^{\infty}e\left(bg\right)^{j}a&f\cup\
\bigcup_{j=0}^{\infty}eb\left(gb^{j}\right)h\\\ c\cup\
\bigcup_{j=0}^{\infty}dg\left(bg^{j}\right)a&\bigcup_{j=0}^{\infty}d\left(fc\right)^{j}h\end{array}\right)$

By construction, we have two additional operations
$\\_\star\\_,\\_\Box\\_:{\mathcal{H}}\times{\mathcal{H}}\rightarrow{\mathcal{H}}$
that are symmetric semi-monoidal tensors for $({\mathcal{H}},\cdot)$ and
$({\mathcal{H}},\circ)$ respectively. These may be given explicitly, by :

  * •

$\left(\begin{array}[]{cc}e&f\\\
g&h\end{array}\right)\star\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)=\left(\begin{array}[]{cc}p^{\ddagger}ep\cup
p^{\ddagger}fq\cup q^{\ddagger}gp\cup q^{\ddagger}hq&0\\\ 0&p^{\ddagger}ap\cup
p^{\ddagger}bq\cup q^{\ddagger}cp\cup q^{\ddagger}dq\end{array}\right)$.

  * •

$\left(\begin{array}[]{cc}e&f\\\
g&h\end{array}\right)\Box\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)=\left(\begin{array}[]{cc}p^{\ddagger}ap\cup
q^{\ddagger}eq&p^{\ddagger}bq\cup q^{\ddagger}fp\\\ p^{\ddagger}cq\cup
q^{\ddagger}gp&p^{\ddagger}hp\cup q^{\ddagger}dq\end{array}\right)$.

The symmetry and associativity isomorphisms for the semi-monoidal monoid
$({\mathcal{H}},\cdot,\star)$ are given by, respectively,

$\sigma=\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)\ \ \mbox{ and }\
\ \tau=\left(\begin{array}[]{cc}p^{\ddagger}&q^{\ddagger}p\\\
0&q\end{array}\right)$

The inverse of $\tau$, with respect to the composition $\\_\cdot\\_$ is given
explicitly by $\tau^{\prime}=\left(\begin{array}[]{cc}p&0\\\
p^{\ddagger}q&q^{\ddagger}\end{array}\right)$, and $\sigma$ is self-inverse
w.r.t. the same composition.

Similarly, the associativity and symmetry isomorphisms for the semi-monoidal
monoid $({\mathcal{H}},\circ,\Box)$ are given by, respectively,
$S=\left(\begin{array}[]{cc}\sigma&0\\\ 0&\sigma\end{array}\right)$ and
$T=\left(\begin{array}[]{cc}\tau&0\\\ 0&\tau^{\prime}\end{array}\right)$.

Expanding out these definitions in terms of the dynamical algebra, we get
$S=\left(\begin{array}[]{cc}p^{\ddagger}q\cup q^{\ddagger}p&0\\\
0&p^{\ddagger}p\cup q^{\ddagger}p\end{array}\right)$ and

$T=\left(\begin{array}[]{cc}\left(p^{\ddagger}\right)^{2}p\cup
p^{\ddagger}q^{\ddagger}pq\cup q^{\ddagger}q^{2}&0\\\ 0&p^{\ddagger}p^{2}\cup
q^{\ddagger}p^{\ddagger}qp\cup\left(q^{\ddagger}\right)^{2}q\end{array}\right)$

$S$ is then its own inverse, w.r.t. both compositions on ${\mathcal{H}}$. The
inverse of $T$ (again, w.r.t. both compositions) is given by

$T^{\prime}=\left(\begin{array}[]{cc}p^{\ddagger}p^{2}\cup
q^{\ddagger}p^{\ddagger}qp\cup\left(q^{\ddagger}\right)^{2}q&0\\\
&\left(p^{\ddagger}\right)^{2}p\cup p^{\ddagger}q^{\ddagger}pq\cup
q^{\ddagger}q^{2}\end{array}\right)$

The following identities are then immediate from both the algebraic
description, and the details of the $\bf Int$ construction.

######  Lemma 91.

The above distinct associativity and symmetry elements are related as follows
:

  * •

$S=\sigma\star\sigma=\sigma\cdot S\cdot\sigma$

  * •

$T=\tau\star\tau^{\prime}$

  * •

$T^{\prime}=\sigma\cdot T\cdot\sigma$

###### Proof.

These follow, simply by construction. ∎

The above structure, consisting of two distinct monoid compositions and two
distinct semi-monoidal structures, along with non-trivial interactions between
them, then provides examples of the algebraic structures discussed in Sections
5.4 and 10.2.

######  Theorem 92.

  1. 1.

The submonoid of $({\mathcal{H}},\cdot)$ generated by
$\\{\tau,\tau^{\prime},1\star\tau,1\star\tau^{\prime}\\}$ is a group
isomorphic to Thompson’s group $\mathcal{F}$, and is also generated by the
closure of $\\{\tau,\tau^{\prime}\\}$ under the composition $\cdot$ and the
tensor $\star$.

  2. 2.

The submonoid of $({\mathcal{H}},\circ)$ generated by $\\{T,T^{\prime},1\Box
T,1\Box T^{\prime}\\}$ is again a subgroup isomorphic to Thompson’s group
$\mathcal{F}$, and is also generated by the closure of $\\{T,T^{\prime}\\}$
under the composition $\circ$ and the tensor $\Box$.

  3. 3.

The elements

     * •

$\Delta=\tau\cdot(1\star\sigma)$

     * •

$\nabla=(1\star\sigma)\cdot\tau^{\prime}$

satisfy $\Delta\cdot\nabla=1=\nabla\cdot\Delta$, but

$\nabla\circ\Delta=1\ \ \mbox{ and }\ \ \Delta\circ\nabla\neq 1$

Hence $\\{\Delta,\nabla\\}$ generates a copy of the bicyclic monoid within
$({\mathcal{H}},\circ)$.

  4. 4.

The above elements satisfy

The Frobenius condition

$(1\Box\nabla)\circ T^{\prime}\circ(\Delta\Box
1)=\Delta\circ\nabla=(\nabla\Box 1)\circ T(1\Box\Delta)$

Associativity

$\nabla\circ(1\Box\nabla)=\nabla\circ(\nabla\Box 1)\circ T$

Co-Associativity

$(1\Box\Delta)\circ\Delta=((\Delta\Box 1)\circ\Delta)\circ T$.

and hence generate a copy of the simple Abramsky-Heunen Frobenius algebra
within the semi-monoidal monoid $({\mathcal{H}},\circ,\Box)$.

###### Proof.

  1. 1.

As $({\mathcal{H}},\cdot,\star)$ is a semi-monoidal monoid, this follows
directly from Theorem 23.

  2. 2.

Similarly, $({\mathcal{H}},\circ,\Box)$ is a semi-monoidal monoid; this again
follows directly from Theorem 23.

  3. 3.

Although this is a corollary of the categorical reasoning of Part 3. of
Theorem 60, it is also perhaps the last point at which a purely algebraic
proof is readily accessible, so we prove this directly, as a check that our
categorical reasoning is indeed correct.

Expanding out the definition gives that

$\Delta=\tau\cdot(1\star\sigma)=\left(\begin{array}[]{cc}p^{\ddagger}&q^{\ddagger}p\\\
0&q\end{array}\right)\left(\begin{array}[]{cc}1&0\\\ 0&q^{\ddagger}p\cup
p^{\ddagger}q\end{array}\right)=\left(\begin{array}[]{cc}p^{\ddagger}&q^{\ddagger}q\\\
0&p\end{array}\right)$

Similarly,

$\nabla=(1\star\sigma)\cdot\tau^{\prime}=\left(\begin{array}[]{cc}1&0\\\
0&q^{\ddagger}p\cup
p^{\ddagger}q\end{array}\right)\left(\begin{array}[]{cc}p&0\\\
p^{\ddagger}q&q^{\ddagger}p\end{array}\right)=\left(\begin{array}[]{cc}p&0\\\
q^{\ddagger}q&p^{\ddagger}\end{array}\right)$

Direct matrix composition, together with the defining relations of the
dynamical algebra, then show that
$\Delta\cdot\nabla=\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)=\nabla\cdot\Delta$. For the composite derived from the
compact closed structure, a straightforward route to calculating the
composites $\nabla\circ\Delta$ and $\Delta\circ\nabla$ is given by moving to
the ‘rook squares’ formalism, and summing over paths within the following two
diagrams :

$\scriptstyle{p}$$\scriptstyle{q^{\ddagger}q}$$\scriptstyle{p^{\ddagger}}$$\scriptstyle{0}$$\scriptstyle{0}$$\scriptstyle{p^{\ddagger}}$$\scriptstyle{p}$$\scriptstyle{q^{\ddagger}q}$
$\scriptstyle{p^{\ddagger}}$$\scriptstyle{0}$$\scriptstyle{p}$$\scriptstyle{q^{\ddagger}q}$$\scriptstyle{q^{\ddagger}q}$$\scriptstyle{p}$$\scriptstyle{p^{\ddagger}}$$\scriptstyle{0}$

Relying on the key identities $pq^{\ddagger}=0=qp^{\ddagger}$ simplifies this
considerably, and gives the composites as :

$\nabla\circ\Delta=\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right)\ \
\mbox{ and }\ \
\Delta\circ\nabla=\left(\begin{array}[]{cc}p^{\ddagger}p&q^{\ddagger}q\\\
q^{\ddagger}q&p^{\ddagger}p\end{array}\right)$

so $\nabla\circ\Delta=1$ and $\Delta\circ\nabla\neq 1$ as required. (It is
worth observing the curiosity that
$(\Delta\circ\nabla)\cdot(\Delta\circ\nabla)=1$. The categorical significance
of this is currently unknown).

  4. 4.

For this, we must simply appeal to the abstract category theory already
developed, and claim it as a Corollary of Theorem 56.

∎

##  17 Future directions

I would like to thank Samson Abramsky for the advice that I should never give
a comprehensive account of a subject, but rather leave some ‘low-hanging
fruit’ so that other authors have the opportunity to reference me. Although
this was undoubtedly firmly tongue-in-cheek, there are nevertheless many
directions that could be pursued.

Categorically

It is hard to avoid the conclusion that this paper needs a good dose of the
scalars. The strictification of reflexivity described in Section 9 relies on
the underlying traced monoidal category having trivial scalars, and the
interaction of Frobenius algebras, Thompson’s $\mathcal{F}$ and the bicyclic
monoid is also predicated on the monoid of scalars being trivial. It is
presumably possible to get rid of this requirement in both cases, and give a
somewhat more sophisticated procedure that can also deal with a non-trival
monoid of scalars. Although this work remains to be carried out, the key to it
is undoubtedly the methods of [2] of adjoining a non-trivial commutative
monoid (with involution) of scalars to a traced or compact closed category.

Logically

This paper has concentrated on categorical & algebraic aspects of the Geomery
of Interaction system, rather than logical interpretations. A logical puzzle
arises nevertheless; in the system of [21, 22], there is unavoidably a simple
A-H Frobenius monoid, as studied in Section 10.2, and given explicitly in
Section 16. The split / merge distinguished arrows of a Frobenius algebra have
the interpretation with quantum-mechanical systems as ‘fan-out’ – a restricted
form of copying that does not violate the no-cloning theorem [59], but does
provide a great deal of computational power to quantum computational systems
[61]. The question is then simply, ‘to which features of (resource-sensitive)
linear logic do these arrows correspond?’. Possibly relevant is the fact that
we have not, so far, given a treatment of how copying is treated within linear
logic generally, or within the systems of [21, 22, 6] in particular.
Logically, this is via the $!(\\_)$ modality, which – as shown in [26] –
appears in [21, 22] as a fixed-point functor on a semi-monoidal monoid
$(M,\star)$, defined by $f\star!(f)=!(f)$, for all $f\in M$.

Algebraically

Canonical coherence arrows of semi-monoidal monoids have a habit of appearing
as interesting or well-known purely algebraic structures – the case of
Thompson’s $\mathcal{F}$ is a prime example. Similarly, the logicians’
dynamical algebra has long been identified as not only Nivat and Perot’s
polycyclic monoids [26, 48], but also the algebra of projections / injections
for a semi-monoidal monoid [26, 27]. When we consider symmetry as well as
associativity arrows, we instead have to deal with Thompson’s group
$\mathcal{V}$ [49]. This immediately raises the natural question of the
standard or simple A-H Frobenius monoid(s). It is entirely reasonable to
expect these, considered simply as monoids rather than semi-monoidal monoids,
to be well-known & well-studied for their algebraic properties.

Theorem 60 provides some justification for the intuition that the simple A-H F
monoid is a combination of Thompson’s $\mathcal{F}$ and the bicyclic monoid,
with interactions between the two determined by the Frobenius condition – thus
uniting three different iconic structures from different branches of algebra.

What is, however, missing from the above account is some subset of the A-H F
monoid that generates it by closure under composition only (i.e. not closure
under composition and tensor). This would undoubtedly make it more accessible
to the algebra community, which is a natural setting in which it should also
be studied. Their identification as well-known algebra must surely be close at
hand.

Notation and diagrammatics

The power of a good formalism for both making a subject accessible, and for
developing new theory, can hardly be overestimated. This is shown by the
utility of the string diagrams formalism for compact closed categories
generally, and its application in the categorical quantum mechanics program in
particular.

Unfortunately, it is also singularly inappropriate for working with strict
reflexivity generally, and A-H Frobenius monoids in particular, due to its
reliance on strict monoidal tensors. Recall from [31] that one cannot
simultaneously have strict associativity and strict self-similarity (& hence
strict reflexivity), except in the trivial case where everything collapses to
the unit object. As the key building blocks of the A-H Frobenius monoids are
(necessarily non-strict) associators, such a diagrammatic formalism is simply
inapplicable.

A suitable formalism that illustrates the underlying concepts – which in many
cases are actually quite simple – without becoming bogged down in syntax, is
sorely needed.

## Acknowledgements

This paper owes a large debt to Chris Heunen (Edinburgh), both for the
concrete results indicated in the text, and many stimulating discussions on
the topics listed of this chapter. He has also been responsible for correcting
several wrong turns I had taken, which is greatly appreciated. Thanks are due
to Mark Lawson for numerous references and results on the inverse semigroup
theoretic side, as well as for neat constructions that significantly simplify
otherwise complex results. Phil Scott (Ottawa) has been very helpful, with
discussions about linear logic and the Geometry of Interaction in general, how
untyped logical and computational systems should be modeled, and in particular
the rôle of units in untyped systems. I am also very grateful to Noson
Yanofsky, for discussions on the structure and properties of Thompson’s group
$\mathcal{F}$ and related structures, and their interaction with categorical
coherence generally, and Hilbert Hotel style operations in particular.

Finally, thanks are due to Samson Abramsky for uncountably many reasons;
attempting to list them all would be futile.

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