# An Infinitary Model of Diagrammatic Calculus in Unbiased Monoidal Categories

Vihaan Dheer

###### Abstract

Properties of morphisms represented by so-called ‘string diagrams’ of monoidal
categories (and their braided and symmetric derivatives), mainly their
resistance in value to isotopic deformation, have made the usage of graphical
calculi commonplace in category theory ever since the correspondence between
diagrams and tensor categories was rigorously established by Joyal and Street
in 1991. However, we find it important to make certain additions to the
existing theory of monoidal categories and their diagrams, with the goal of
extending to so-called ‘infinitary monoidal categories’. Most crucially, we
employ a structure inherently resistant to isotopic deformation, thus
replacing topological details with categorical ones. In the process, we
coherently introduce infinitary tensor product and transfinite composition
into the diagrammatic formalism.

###### keywords:

string diagram, tensor category, transfinite composition, infinitary monoidal
product, unbiased monoidal category, isotopy

###### Contents

  1. 1 Introduction
    1. 1.1 Motivations
    2. 1.2 Outline
  2. 2 Infinitary Monoidal Categories
    1. 2.1 Transfinite Composition
    2. 2.2 Unbiased Monoidal Categories
    3. 2.3 Colimit-expanded Categories
  3. 3 Monoidal Diagrams
    1. 3.1 Definitions
    2. 3.2 Unresolved Edges
    3. 3.3 Readouts & Validity
  4. 4 Conclusion
    1. 4.1 Summary
    2. 4.2 Future Work

##  1 Introduction

###  1.1 Motivations

The first successful design and application of the concept of string diagrams
similar to those employed by monoidal category theorists was likely due to
Roger Penrose in 1971, coining Penrose notation for tensors, in a series of
works beginning with Ref. 1, featuring various others including Ref. 2. In
1991, the celebrated work of Joyal and Street “The Geometry of Tensor
Calculus, I” main-source-js  formally introduced this concept into the scope
of monoidal category theory. Ever since MacLane’s definition of the monoidal
(tensor) category categories-smc , much effort had been placed in the study of
the coherence theorem nat-assoc , as well as the reformulation of the axioms
kelly-conditions . Upon Joyal and Street’s application of string diagrams to
the theory, addressing not only standard monoids but braided and symmetric
categories as well, attention was diverted towards studying and re-purposing
these diagrams to make use of in related categories. Twenty years from then,
it had become commonplace in monoidal category theory to such a degree as to
prompt expository works like Ref. 7.

More recently, as monoidal categories have begun to gain applicability to
external fields such as quantum information theory quantum-ct  and topological
field theory tqft-reference , and their diagrams as a result have become more
widely applied, the graphical theory may require modifications to stay in use.
In reference specifically to the proof for Theorem 1.2 of Ref. 3, we aim to
make a few major updates, from which a new diagrammatic theory supportive of
more general kinds of diagram will arise.

The mentioned theorem states that isotopic deformation of progressive plane
(string) diagrams does not change its ‘value’, i.e. the morphism which it
depicts. Clearly this makes checking validity of equations in these categories
much easier, but it is only half of the full picture: one may not only be
interested in isotopy-equivalence implying equality, but also whether equality
of morphisms implies that their diagrams can be isotopically deformed into one
another. Subtleties in this question, related to the various representations
of a single morphism, are valuable to consider to the goal of furthering the
theory.

Furthermore, Joyal and Street’s notion of progressive plane diagrams do not
allow for so-called ‘infinitary diagrams’, a concept which the author has yet
to see mentioned in the relevant literature, but feels it has potential for
later use. The reason is likely as follows: string diagrams have historically
been primarily for the purpose of human computation, thus the difficulties in
visualizing infinite diagrams (in either dimension) may have swayed most away
from the concept. However, the monoidal diagrams we introduce in this work
refer not just to the visual concept, but are a new type of mathematical
structure, which have use in their own right, not just as visual aids.

Our method for making these adjustments in some ways categorifies the original
proof; in other words, at least initially, diagrams are not topological
entities, and in fact are by design inherently resistant to isotopic
deformation, a concept which will be made formal in later sections. Instead,
they are mathematical structures which shall become members of a ‘category of
diagrams’ induced by some monoidal category; in the finitary case, the two
formulations are of course equivalent.

To support infinite arity of diagrams, we will rely heavily on Tom Leinster’s
unbiased-mc  unbiased definition of monoidal categories, and alter it to
define the notion of $\alpha$-ary monoidal categories, allowing us to work
with infinite monoidal products. Finally, we shall employ the standard notion
of transfinite composition in categories, effectively permitting categories to
be of infinite length in both the horizontal and vertical directions.

###  1.2 Outline

This paper will be structured as follows: in Sec. 2, we will review
transfinite composition, introduce our generalized definition of monoidal
categories, and discuss a novel construction called the ‘colimit expansion’.
Sec. 3 firstly defines monoidal diagrams, and then focuses on extracting from
such diagrams the morphisms in the underlying category that they encode.
Finally, we conclude with a discussion of applications and future work in Sec.
4.

##  2 Infinitary Monoidal Categories

This section will develop the categorical formalism necessary to extend
standard monoidal categories to their ‘infinitary counterparts’, in which
infinite products and compositions exist. This is important both to study the
monoidal diagrams that will be discussed in the next section, but also because
these categories are interesting in their own right. After reviewing
transfinite composition, we define these infinitary categories by converting
the axioms for a standard monoidal category into their unbiased form, and then
extending via ordinals. Finally, we conclude with a discussion of the so-
called “colimit-expansion”, which provides a systematic method to extend
standard monoidal categories to infinitary ones under certain conditions.

###  2.1 Transfinite Composition

Extending monoidal categories “horizontally” will involve considering
compositions of infinite sequences. Such ‘transfinite compositions’ appear all
over mathematics, perhaps most commonly in the study of model categories (see
e.g. Ref. 11). We briefly define transfinite composition and discuss some
basic results relevant here.

######  Definition 2.1 (transfinite composition).

In a category $\mathscr{C}$, for some ordinal $\alpha$ (taken as the
corresponding thin category), an $\alpha$-transfinite sequence is a functor

$X_{\bullet}\colon\alpha\to\mathscr{C}$ (2.1)

such that $X_{\beta}=\varinjlim_{\gamma<\beta}X_{\gamma}$ for each non-zero
limit ordinal $\beta<\alpha$. The transfinite composition of such a sequence
is the unique morphism

$X_{0}\longrightarrow\varinjlim_{\gamma<\alpha}X_{\gamma}$ (2.2)

induced by the colimit. The existence of this colimit is guaranteed by
transfinite induction.

The most intuitive example is given by successive inclusions in a concrete
category $(\mathscr{C},F\colon\mathscr{C}\to\mathbf{Set})$, for $F$ faithful.
A sequence

${X_{0}}$${X_{1}}$${X_{2}}$${\cdots}$ (2.3)

of inclusions under $F$ has a transfinite composition that is the obvious
inclusion $X_{0}\subseteq\varinjlim X_{\bullet}$ of underlying sets.

A basic but useful result is that two transfinite compositions are compatible
(i.e. rearrangeable) with each other:

######  Proposition 2.1.

Let $X_{\bullet},X^{\prime}_{\bullet}$ be $\alpha,\alpha^{\prime}$-transfinite
sequences respectively such that $X^{\prime}_{0}=\varinjlim X$. Let
$\beta,\beta^{\prime}$ be ordinals such that
$\alpha+\alpha^{\prime}=\beta+\beta^{\prime}$; then we may form
$\beta,\beta^{\prime}$-transfinite sequences $Y,Y^{\prime}$ respectively which
are segments/concatenations of $X,X^{\prime}$. Then $Y_{0}=X_{0}$,
$Y^{\prime}_{0}=\varinjlim Y$, and $\varinjlim Y^{\prime}=\varinjlim
X^{\prime}$.

Under certain conditions, a class of morphisms in a category may be closed
under transfinite composition; in other words the the composition of any
transfinite sequence with contained morphisms exists and is as well contained.
We call such a class saturated. As standard compositions are $2$-transfinite,
a saturated class is necessarily a subcategory. One example is that of
monomorphisms.

######  Proposition 2.2.

The wide subcategory of monomorphisms $\mathrm{Mono}(\mathscr{C})$ is
saturated.

Proof: We must show that for any transfinite sequence of monomorphisms

$X_{\bullet}\colon\alpha\hookrightarrow\mathscr{C},$ (2.4)

the canonical map $X_{0}\longrightarrow\varinjlim_{\gamma}X_{\gamma}$ is
monic. Suppose it weren’t; by transfinite induction, no leg of the limiting
cocone is monic. But this contradicts the uniqueness given by the universal
property of $\varinjlim_{\gamma}X_{\gamma}$.

In fact, this is a special case of a more general result stating that the
morphism class with the ‘left lifting property’ with respect to any class of
morphisms is saturated higher-cat-homotopical-alg . This of course has various
corollaries; for example, the classes of cofibrations and trivial cofibrations
in a model category are saturated.

In the following discussion, when making categories compositionally
infinitary, we will use sequential cocompleteness to ensure a category has all
transfinite compositions. These saturated subcategories are especially well-
behaved because transfinite compositions in the subcategory coincide with
those in the category itself.

###  2.2 Unbiased Monoidal Categories

The contrast between biased and unbiased definitions is ubiquitous in
mathematics. In short, we are often given an operation intended to be of arity
$n$, that is presented solely in the nullary and binary cases. In this case
the definition is called biased, contrary to an unbiased definition which
constructs an $n$-ary operation at the outset.

In no way is the term meant to suggest some superiority of unbiased
definition: many cases, for example the monoidal categories at hand, require
much shorter and more efficient definitions, as well as auxiliary coherence
conditions. However, generalization with ordinal numbers is much easier with
an unbiased definition, which is the motivation for using one here.

We first formally define partitions of ordinal numbers, which greatly shrinks
the coherence conditions necessary for unbiased monoidal categories, which
shall be introduced thereafter. Finally, we will redevelop some of the
relevant theory of standard monoidal categories for their unbiased variants.

######  Definition 2.2 (ordinal partitions).

A $\gamma$-partition of an ordinal $\alpha$ is a $\gamma$-sequence which sums
to alpha. Explicitly, it is a function $p_{\bullet}\colon\gamma\to\alpha$ such
that

$\sum_{i<\gamma}{p_{i}}=\alpha.$ (2.5)

The set of all $\gamma$-partitions of $\alpha$ is written
$\alpha^{\gamma}_{\Sigma}$, while the set of all partitions in general of
$\alpha$ is $\alpha^{<}_{\Sigma}$. A $(\delta_{\bullet},p_{\bullet})$-double
partition of $\alpha$ is an indexed family
$p_{\bullet,\bullet}=\\{p_{i,\bullet}\in(p_{i})_{\Sigma}^{\delta_{i}}\,\colon
i<\gamma\\}$, for $\gamma$-partitions $\delta_{\bullet},p_{\bullet}$ of
ordinals $\delta,\alpha$ respectively. For each such double partition there is
an induced $\delta$-partition of $\alpha$
$(p_{\bullet,\bullet})^{\cup}_{\bullet}\colon\delta\to\alpha$, or simply
$(p^{\cup})_{\bullet}$, defined in the obvious fashion. When the domain and
range of a partition are clear, then an ordinal $\beta$ acts as a constant
partition $\beta\colon\gamma\to\delta$ such that $i\mapsto\beta$.

######  Definition 2.3 ($\alpha$-ary unbiased monoidal category).

An $\alpha$-ary unbiased monoidal category for some ordinal
$\alpha\nless\omega$ (where $\alpha=\mathbf{Ord}$ is permissable) is a tuple
$(\mathscr{C},\mathcal{T}_{\bullet},\iota,\eta)$, with an ordinary category
$\mathscr{C}$, a formal family of functors

$\mathcal{T}_{\bullet}\colon\alpha\to\bigcup_{\beta<\alpha}{\mathrm{Hom}{\left(\mathscr{C}^{\beta},\mathscr{C}\right)}}\quad\text{s.t.}\quad\mathcal{T}_{\beta}\colon\mathscr{C}^{\beta}\to\mathscr{C},$
(2.6)

a natural isomorphism
$\iota:\mathrm{id}_{\mathscr{C}}\stackrel{{\scriptstyle\textstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}\mathcal{T}_{1}$,
and a family of natural isomorphisms

$\eta=\big{\\{}\eta_{p_{\bullet}}\colon\mathcal{T}_{\gamma}\circ\prod_{i<\gamma}{\mathcal{T}_{p_{i}}}\stackrel{{\scriptstyle\textstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}\mathcal{T}_{\beta}\,\big{|}\,p_{\bullet}\in\beta^{\gamma}_{\Sigma}\,\,\,\forall\gamma,\beta<\alpha\big{\\}},$
(2.7)

such that for any $\gamma$-partition $p_{\bullet}$ of $\beta<\alpha$, and any
$(\delta_{\bullet},p_{\bullet})$-double partition, the following diagram
commutes,

${{\mathcal{T}_{\gamma}\circ\prod_{i<\gamma}{\left(\mathcal{T}_{\delta_{i}}\circ\prod_{j<\delta_{i}}{\mathcal{T}_{p_{i,j}}}\right)}}}$${\mathcal{T}_{\gamma}\circ\prod_{i<\gamma}{\mathcal{T}_{p_{i}}}}$${\mathcal{T}_{\delta}\circ\prod_{k<\delta}{\mathcal{T}_{(p^{\cup})_{k}}}}$${\mathcal{T}_{\beta}}$$\scriptstyle{\mathcal{T}_{\gamma}\left(\prod_{i<\gamma}{\eta({p_{i,\bullet}})}\right)}$$\scriptstyle{\eta(p_{\bullet})}$$\scriptstyle{\eta\big{(}(p^{\cup})_{\bullet}\big{)}}$
(2.8)

as does the following:

${\mathcal{T}_{\beta}}$${\mathcal{T}_{\beta}\circ\prod_{i<\gamma}{\mathcal{T}_{1}}}$${\mathcal{T}_{1}\circ\mathcal{T}_{\beta}}$${\mathcal{T}_{\beta}}$$\scriptstyle{\mathrm{id}_{\mathcal{T}_{\beta}}}$$\scriptstyle{\iota\mathcal{T}_{\beta}}$$\scriptstyle{\mathcal{T}_{\beta}\left(\prod_{i<\gamma}{\iota}\right)}$$\scriptstyle{\eta\left(1_{\bullet}\right)}$$\scriptstyle{\eta(\beta_{\bullet})}$
(2.9)

######  Definition 2.4 (infinitary, finitary, strict).

An $\alpha$-ary unbiased monoidal category is called finitary if
$\alpha=\omega$, and infinitary otherwise. If $\alpha=\mathbf{Ord}$, it is
fully infinitary. An $\alpha$-ary unbiased monoidal category is strict if each
component $\iota$ of each element of $\eta$ in Definition 2.3 is an identity.

From any unbiased monoidal category we can build a biased monoidal category by
extracting the associator/unitors from the family $\eta$ (see the proof of
Thm. 2.4). Likewise, given a biased monoidal category, we can build a finitary
unbiased monoidal category; the two notions are thus equivalent in the
finitary case.

######  Remark 1.

In the definition, we have made implicit reference to the categories
$\mathscr{C}^{\beta}$ for an ordinal $\beta$. To make this notion complete, we
wrap these objects in a power category of $\mathscr{C}$ which includes all
ordinal powers of $\mathscr{C}$ and the functors between them as morphisms.
This category becomes the prototypical example of an unbiased strict
(skeletal) monoidal category when endowed with the Cartesian product,
producing objects for general ordinal sequences. The strictness derives from
the fact that we do not distinguish between
$\mathscr{C}^{\beta}\times\mathscr{C}^{\gamma}$ and
$\mathscr{C}^{\beta+\gamma}$.

The most obvious example is given by generalizing (co)cartesian categories.
Any category $\mathscr{C}$ with all small (co)products will be fully
infinitary if each $\mathcal{T}_{\gamma}$ is taken to be the generalized
(co)product. A less trivial example is provided by the category of
endofunctors (that of which a monad is an object).

######  Example 2.1.

Suppose that, for a small category $\mathscr{C}$, the full subcategory
$\\{\mathscr{C}\\}\subseteq\mathbf{Cat}$ is cocomplete (or, in principle,
sequentially cocomplete). Then transfinite compositions of endofunctors in the
sense of Def. 2.1 form a fully infinitary product on the strict monoidal
category $(\mathrm{End}(\mathscr{C}),\circ,\mathrm{Id})$.

As is well-observed in the literature categories-smc ; main-source-js ;
quantum-ct , standard monoidal categories satisfy a ‘strictification theorem’
stating that each such category is monoidally equivalent to a strict monoidal
category. $\alpha$-ary unbiased monoidal categories follow a similar law; the
proofs are very similar (see e.g. Ref. 8), but it is necessary to define the
notions relevant to the theorem.

######  Definition 2.5 (unbiased monoidal functor).

Let $(\mathscr{C},\mathcal{T}_{\bullet},\iota,\eta)$,
$(\mathscr{C}^{\prime},\mathcal{T}_{\bullet}^{\prime},\iota^{\prime},\eta^{\prime})$
be unbiased monoidal categories of arity $\alpha$. An unbiased monoidal
functor between them is a pair $(F,\chi_{\bullet})$ consisting of a functor
$F\colon\mathscr{C}\to\mathscr{C}^{\prime}$, and a family of natural
isomorphisms

$\chi_{\bullet}\colon\alpha\to\bigcup_{\beta<\alpha}{\mathrm{mor}([\mathscr{C}^{\beta},\mathscr{C}^{\prime}])}\quad\text{s.t.}\quad\chi_{\beta}:\mathcal{T}^{\prime}_{\beta}\circ
F^{\beta}\stackrel{{\scriptstyle\textstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}F\circ\mathcal{T}_{\beta},$
(2.10)

such that the following diagram commutes for each ordinal partition
$p_{\bullet}\colon\gamma\to\alpha$,

${\mathcal{T}_{\gamma}^{\prime}\circ\prod_{i<\gamma}{\left(\mathcal{T}_{p_{i}}\circ
F^{p_{i}}\right)}}$${\mathcal{T}_{\gamma}^{\prime}\circ
F^{\gamma}\circ\prod_{i<\gamma}{\mathcal{T}_{p_{i}}}}$${\mathcal{T}_{\beta}^{\prime}\circ
F^{\beta}}$${F\circ\mathcal{T}_{\beta}}$${F\circ\mathcal{T}_{\gamma}\circ\prod_{i<\gamma}{\mathcal{T}_{p_{i}}}}$$\scriptstyle{\eta^{\prime}(p_{\bullet})F}$$\scriptstyle{\mathcal{T}_{\gamma}^{\prime}\prod_{i<\gamma}{\chi_{i}}}$$\scriptstyle{\chi_{\gamma}\prod_{i<\gamma}{\mathcal{T}_{p_{i}}}}$$\scriptstyle{\chi_{\beta}}$$\scriptstyle{F\eta(p_{\bullet})}$
(2.11)

called the pentagon equation, as does the following,

${F}$${\mathcal{T}_{1}^{\prime}\circ
F}$${F\circ\mathcal{T}}$$\scriptstyle{\iota^{\prime}F}$$\scriptstyle{F\iota}$$\scriptstyle{\chi_{1}}$
(2.12)

called the triangle equation.

As is usually done for monoidal categories, unbiased monoidal functors can be
used to define weak and strong equivalences. Similarly to the standard case,
we can promote the former to the latter using the axiom of choice categories-
smc .

######  Definition 2.6 (unbiased monoidal equivalence).

An unbiased monoidal functor $F:\mathscr{C}\to\mathscr{C}^{\prime}$ is a weak
equivalence if it is fully faithful and essentially surjective. It is a strong
equivalence if there is a specified unbiased monoidal functor
$G:\mathscr{C}^{\prime}\to\mathscr{C}$ such that $G\circ
F\simeq\mathrm{id}_{\mathscr{C}}$ and $F\circ
G\simeq\mathrm{id}_{\mathscr{C}^{\prime}}$.

######  Proposition 2.3.

A weak unbiased monoidal equivalence induces a strong equivalence given the
axiom of choice (AC).

Proof: Let $(F\colon\mathscr{C}\to\mathscr{C}^{\prime},\chi_{\bullet})$ be a
unbiased weak monoidal equivalence. For objects $A,B\in\mathscr{C}$, define
the equivalence relation $A\sim B\iff FA\simeq FB$. By AC there is a function
$f\colon\mathscr{C}/\\!\sim\,\to\mathscr{C}$ s.t. $F(f([A]))\simeq FA$;
essential surjectivity determines a function
$g\colon\mathscr{C}^{\prime}\to\mathscr{C}/\\!\sim$, and thus for each
$A^{\prime}\in\mathscr{C}^{\prime}$ let $GA^{\prime}=f(g(A^{\prime}))$. Let
$G$ take morphisms $A^{\prime}\to B^{\prime}$ to morphisms $GA^{\prime}\to
GB^{\prime}$ pre and post-composed by isomorphisms (chosen again by AC). Then
$FG$ and $GF$ are both naturally isomorphic to their respective identities
(say by maps $\zeta$ and $\xi$, respectively). It remains to show that
$G\colon\mathscr{C}^{\prime}\to\mathscr{C}$ is unbiased monoidal; consider
$\forall\beta<\alpha$ the natural isomorphism

$G\chi_{\beta}^{-1}G^{\beta}\colon GF\circ\mathcal{T}_{\beta}\circ
G^{\beta}\stackrel{{\scriptstyle\textstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}G\circ\mathcal{T}^{\prime}_{\beta}\circ
F^{\beta}G^{\beta}.$ (2.13)

Noting this we can define a natural isomorphism $\chi^{\prime}_{\beta}$ by the
commutativity of the following diagram,

${GF\circ\mathcal{T}_{\beta}\circ
G^{\beta}}$${G\circ\mathcal{T}^{\prime}_{\beta}\circ
F^{\beta}G^{\beta}}$${\mathcal{T}_{\beta}\circ
G^{\beta}}$${G\circ\mathcal{T}_{\beta}^{\prime}}$$\scriptstyle{G\chi_{\beta}^{-1}G^{\beta}}$$\scriptstyle{\xi(\mathcal{T}_{\beta}\circ
G^{\beta})}$$\scriptstyle{(G\circ\mathcal{T}_{\beta}^{\prime})\zeta^{\beta}}$$\scriptstyle{\chi^{\prime}_{\beta}}$
(2.14)

which clearly satisfies Eqs. 2.11 and 2.12 for all $\beta$.

######  Theorem 2.4 (Unbiased Strictification).

Every $\alpha$-ary unbiased monoidal category is monoidally equivalent to a
strict $\alpha$-ary unbiased monoidal category. With AC it is strongly
equivalent to one.

Proof: As discussed, the proof is almost identical the standard case given,
e.g., in Ref. 8. The key observation is that we may recover the unit object,
unitors, and associator from the unbiased axioms as follows: the unit is
$\mathcal{T}_{0}(*)$, the left unitor is the composite

${\mathcal{T}_{2}\circ(\mathcal{T}_{0}\times\mathrm{id})}$${\mathcal{T}_{2}\circ(\mathcal{T}_{0}\times\mathcal{T}_{1})}$${\mathcal{T}_{1}}$${\mathrm{id},}$$\scriptstyle{\mathcal{T}_{2}(\mathrm{id}\times\iota)}$$\scriptstyle{\eta_{(0,1)}}$$\scriptstyle{\iota^{-1}}$

and the associator is defined in the obvious fashion by noting that each way
of “bracketing” the two-fold product is naturally isomorphic by $\eta$ to
$\mathcal{T}_{3}$. The right unitor is obtained similarly to the left one with
the ordinal partition $(0,1)$.

Because of this result, studying $\alpha$-ary categories is no different from
studying their strict variants (at least regarding categorical properties that
respect the equivalence principle); we therefore choose the latter as to avoid
handling the $\eta$ morphisms.

This framework gives rise to an expected but interesting formulation with
infinitary Lawvere Theories (see e.g. Ref. 13). Let $\mathbf{Ord}$ be the
$\mathbf{Set}$ skeleton of von Neumann ordinals with inclusion functor
$\mathcal{I}_{\mathbf{Ord}}$. Let
$\mathcal{F}\colon\mathbf{Mon}\to\mathbf{Set}$ be the forgetful functor; this
admits a left adjoint $\mathcal{L}\dashv\mathcal{F}$ sending sets to their
corresponding free monoids. Since $\mathcal{L}\circ\mathcal{I}_{\mathbf{Ord}}$
is injective on objects, its image forms a full subcategory
$\mathscr{T}\subseteq\mathbf{Mon}$. Then, ignoring size issues, any functor

$\mathcal{M}\colon\mathscr{T}^{\mathrm{op}}\to\mathbf{Cat}$ (2.15)

that preserves small products is identified with a strict $\mathbf{Ord}$-ary
unbiased monoidal category. The $\alpha$-ary version can be recovered by
replacing $\mathbf{Ord}$ with the subcategory $\alpha\subseteq\mathbf{Set}$ in
the previous paragraph.

One can show with some difficulty that each free monoid $M$ in $\mathscr{T}$
can be written distinctly as the sequential limit of a constant functor

$M\cong\varinjlim{\left(\Delta_{M}\colon\beta\to\mathscr{T}\mathchar
24635\relax\;\gamma\mapsto 1\right)}$ (2.16)

for some $\beta<\alpha$ and the free monoid over a single element $1$, which
is just a $\beta$-fold product in $\mathscr{T}^{\mathrm{op}}$. Then
$\mathcal{M}(1)$ is an unbiased $\alpha$-ary monoidal category with product
functors

$\mathcal{T}_{\beta}=\mathcal{M}\left(1\to\beta\mathchar
24635\relax\;(0)\mapsto(012\dots\beta)\right)$ (2.17)

for all $\beta<\alpha$. As is intuitive, the image of $\mathcal{M}$ is the
full subcategory of $\mathbf{Cat}$ that is the aforementioned power category
of $\mathcal{M}(1)$.

A basic conclusion from the axioms of a standard monoidal category—yet the one
that gives rise to most of its diagrammatic theory—is the interchange law,
stating that, for properly composable morphisms,

$(g\circ f)\otimes(j\circ h)=(g\otimes j)\circ(f\otimes h).$ (2.18)

This is a direct consequence of bifunctoriality of $\otimes$. In the
infinitary case, however, while we can perform infinitary interchange
internally to the arguments of the tensor functors, we need additional
assumptions to do so on the outside.

######  Definition 2.7 (smooth unbiased category).

An $\alpha$-ary unbiased monoidal category $\mathscr{C}$ is smooth if for each
object $X$ the functor

$\mathcal{T}_{2}(X,-)\colon\mathscr{C}\to\mathscr{C}$ (2.19)

is sequentially cocontinuous (i.e. preserves sequential colimits). It is
cosmooth if it preserves cosequentially cocontinuous (i.e. preserves colimits
of diagrams $\alpha^{\mathrm{op}}\to\mathscr{C}$).

It is not difficult to see that this condition implies that every tensor
functor is sequentially cocontinuous in the last argument. If the condition of
smoothness holds, we have the following obvious corollary.

######  Theorem 2.5 (Infinitary Interchange Law).

If $\mathscr{C}$ is a smooth $\alpha$-ary monoidal category with products
$\mathcal{T}_{\bullet}$, then the following two conditions hold:

  1. 1.

Consider two sequences of morphisms in $\mathscr{C}$, i.e. functors
$F,G\colon\beta\to\mathscr{C}$ for the thin category of $\beta<\alpha$ such
that both $F,G$ admit transfinite compositions $f\colon F_{0}\to\varinjlim F$
and $g\colon G_{0}\to\varinjlim G$ respectively. Then

$\mathcal{T}_{2}(f,g)=h\colon\mathcal{T}_{2}(F_{0},G_{0})\longrightarrow\varinjlim\big{(}\mathcal{T}_{2}\circ(F\times
G)\big{)}$ (2.20)

  2. 2.

If $(a_{0},a_{1}),(b_{0},b_{1}),\dots$ is a morphism sequence of length
$\gamma$ such that the composition of each pair exists, then

$\mathcal{T}_{\gamma}\big{(}a_{0}\circ a_{1},b_{0}\circ
b_{1},\dots\big{)}=\mathcal{T}_{\gamma}\big{(}a_{0},a_{1},\dots\big{)}\circ\mathcal{T}_{\gamma}\big{(}b_{0},b_{1},\dots\big{)}$
(2.21)

When such a smooth category is in addition cocomplete, it is a very well-
behaved setting for infinitary operations by virtue of the infinite
interchange law because all transfinite compositions exist and are compatible
with the product. If furthermore the category in question is a saturated
subcategory, then all possible infinitary manipulations of bracketings are
equalities. This condition, however, is too difficult to satisfy in practice
for any category, perhaps other than the thin category
$\alpha\subseteq\mathbf{Ord}$ itself.

###  2.3 Colimit-expanded Categories

Beyond the standard examples of categories with all small (co)products, it is
rare to naturally encounter infinitary monoidal categories. In addition, the
axioms are relatively difficult to show in practice. There is, however, a
general process by which one can construct a (smooth) fully infinitary
category from a standard one under certain conditions. This construction and
its derivatives will not only provide many examples, but also will also give
rather profound results about various existing constructions in category
theory.

The general idea is to take a monoidal category and use colimits of the finite
tensor products to define the functors $\mathcal{T}_{\beta}$. Although we will
not write it out explicitly, whenever we refer to sequential colimits (i.e.
those of diagrams $\alpha\to\mathscr{C}$ for an ordinal $\alpha$), we just as
well may use cosequential colimits taken over diagrams out of
$\alpha^{\mathrm{op}}$.

Now, let $(\mathscr{C},\otimes,I)$ be an ordinary monoidal category. Seemingly
the most general case depends on a choice of a cone from $I$ to the identity
$\mathrm{Id}\colon\mathscr{C}\to\mathscr{C}$, say
$\varepsilon\colon\Delta_{I}\to\mathrm{Id}$. We proceed with transfinite
recursion. Suppose we have a sequence
$(X_{\gamma})_{\gamma<\alpha}\in\mathscr{C}^{\alpha}$; to define the
infinitary tensor product $\mathcal{T}_{\alpha}$, we first recursively let
$F(0)=I$ and $F(\gamma+1)=F(\gamma)\otimes X_{\gamma}$. Under $F$ let
successor maps $\gamma\to\gamma+1$ go to the composite

$F(\gamma)\xrightarrow[\hskip 25.09373pt]{\rho^{-1}}F(\gamma)\otimes
I\xrightarrow[\hskip 25.09373pt]{1\otimes\varepsilon}F(\gamma)\otimes
X_{\gamma},$ (2.22)

and finally extend $F$ to a functor $\alpha\to\mathscr{C}$ by taking

$F(\beta)=\varinjlim_{\gamma<\beta}F_{\gamma}$ (2.23)

for any limit ordinal $\beta$. Because of the associator and unitors of the
underlying category $\mathscr{C}$, the axioms of a fully infinitary category
are almost satisfied, save for the fact that the coherence conditions cannot
be proven in the limit ordinal case. To resolve this, we must impose the
additional requirement that for any $F\colon\alpha\to\mathscr{C}$, there is an
is an isomomorphism

$X\otimes\varinjlim_{\gamma<\alpha}{F(\gamma)}\cong\varinjlim_{\gamma<\alpha}\big{(}X\otimes
F(\gamma)\big{)},$ (2.24)

i.e. $\mathscr{C}$ is smooth. Moreover, smoothness guarantees that each
colimit of the form in Eq. 2.23 exists. Notably, the above construction is
just an example of transfinite compositions of the maps $\varepsilon$, which
serve to ‘glue’ the full product together by partial products. This gives rise
to the following definition.

######  Definition 2.8 (colimit expansion).

Let $\mathscr{C}$ be a monoidal category equipped with a cone $\varepsilon$
from $I$ to $\mathrm{Id}$. $\mathscr{C}$ is colimit-expandable if for each
object $X$ of $\mathscr{C}$ the functor
$X\otimes-\colon\mathscr{C}\to\mathscr{C}$ is sequentially cocontinuous. The
resulting smooth, fully infinitary category as constructed above is the
colimit expansion of $\mathscr{C}$, and the pair $(\mathscr{C},\varepsilon)$
is termed a colimit-expanded category.

The obvious examples are $\mathbf{Set}$, $\mathbf{Rel},\mathbf{Cat}$, etc. all
equipped with their coproducts and initial objects as units. Recall that we
just as easily may use the product if instead a cone from $\mathrm{Id}$ to $I$
is employed. $\mathbf{Top}$ is an interesting non-example: sums of topological
spaces do not extend canonically to the infinitary case. As we will see later,
however, the “convenient category of topological spaces” is (dually) colimit-
expandable.

######  Proposition 2.6.

A colimit-expanded category $(\mathscr{C},\varepsilon)$ is necessarily
semicocartesian (i.e. $I$ is initial). Thus the legs of $\varepsilon$ are
trivially the unique maps out of $I$.

Proof: First notice that from the definition of a fully infinitary category,
in particular Eq. 2.9, it follows that for any $\alpha$ there is an
isomorphism $\mathcal{T}_{\alpha}(I,I,\dots)\cong I$ induced by $\eta$ and
$\iota$. Now suppose that the leg $\varepsilon_{I}\colon I\to I$ was not an
isomorphism and let $\alpha$ be a limit ordinal; since the product functor
$\mathcal{T}_{\alpha}$ is constructed via transfinite compositions of
$\epsilon$ which reflect isomorphisms, we must have
$I\ncong\mathcal{T}_{\alpha}(I,I,\dots)$, a contradiction.

Then since $\varepsilon_{I}\colon I\cong I$, the commutativity condition of
$\varepsilon$ guarantees that any two morphisms $f,g\colon I\to X$ are equal.
Of course each $I\to X$ exists by definition of $\varepsilon$, so $I$ is
initial in $\mathscr{C}$.

Because of this, there is not, in fact, freedom to choose various cones
$I\to\mathrm{Id}$, and thus we need not mention $\varepsilon$ hereafter. The
following statement, though a direct consequence of the limit preservation
properties of adjunctions, creates many of the rest of the examples of smooth
monoidal categories. We recall that a closed monoidal category is a symmetric
one in which, for each object $X$, the functor $-\otimes X$ has a right
adjoint $[X,-]\colon\mathscr{C}\to\mathscr{C}$ called the ‘internal hom’.

######  Theorem 2.7.

Any semicocartesian closed category is colimit-expandable. In particular, any
cocartesian closed category is colimit-expandable. By the same argument with
cosequential colimits, semicartesian closed categories, and thus cartesian
closed categories, are as well colimit-expandable.

Firstly, this gives an interesting result about closed monoidal categories; it
implies a rather unintuitive underlying relationship between the existence of
an internal hom and the possibility and uniqueness of an infinitary extension.
These semicocartesian closed categories have various special properties that
might give rise to such a relationship (e.g., by the adjunction formula,
$T\cong[X,I]$); the author believes that a study of these categories in their
own right would be worthwhile.

Secondly, it provides us with many obvious examples, circumventing the tedious
axioms. For instance, we can remedy the earlier non-example of $\mathbf{Top}$
by taking the cartesian closed subcategory $\mathbf{CGHaus}$ of compactly-
generated Hausdorff spaces categories-smc . In addition, we immediately have
that all topoi are colimit-expandable and thus uniquely smooth infinitary
categories, giving yet another useful characteristic of topoi that may have
future application.

There are some examples of semi(co)cartesian closed categories that are not
cartesian closed; although most of these perhaps are slightly artificial, such
examples show that the colimit-expansion is versatile, and need not deal only
with the categorical product. The primary example, thought, that the author
has found in usage is that of nominal sets (see Andrew Pitts’ Nominal Sets
nominal-sets ; we follow his work here).

######  Example 2.2.

For any set $A$, by $\mathrm{Perm}(A)$ we denote the subgroup of
$\mathrm{Sym}(A)$ containing only those permutations that leave only finitely
many elements of $A$ changed. Fix a countably infinite $\mathbb{A}$, and let
$X$ be a $\mathrm{Perm}(\mathbb{A})$-set (i.e. an object in the
$\mathrm{Perm}(\mathbb{A})\mathbf{Set}$ topos). We say that a subset
$A\subseteq\mathbb{A}$ supports an $x\in X$ if
$\forall\pi\in\mathrm{Perm}(\mathbb{A})$, we have
$\left.\pi\right|_{A}=1_{A}\implies\pi\cdot x$. Any $x$ is finitely supported
if there is a finite subset of $\mathbb{A}$ which supports it. If it is so,
then we write the intersection of all finite supports as
$\mathrm{supp}_{X}{x}$ Then $\mathbf{Nom}$ is defined as the full subcategory
of $\mathrm{Perm}(\mathbb{A})\mathbf{Set}$ consisting of
$\mathrm{Perm}(\mathbb{A})$-sets in which each element is finitely supported
(Ref. 14, Ch. 2.1).

$\mathbf{Nom}$ inherits a cartesian closed structure from the topos
$\mathrm{Perm}(\mathbb{A})$ with the categorical product, so it is colimit-
expandable under thereunder. However it is also as such under the separated
product (Ref. 14, Ch. 3.4): we define the ‘freshness relation’
$\\#_{X,Y}\subseteq X\times Y$ by
$x\,\\#_{X,Y}\,y\iff\mathrm{supp}_{X}{x}\,\cap\,\mathrm{supp}_{Y}{y}=\varnothing$.
The relation is itself a nominal set, giving a bifunctor
$(X,Y)\mapsto\,\\#_{X,Y}$ called the separated product, under which
$\mathbf{Nom}$ is made monoidal with the terminal object as the unit. Finally
by Ref. 14, Thm. 3.12, it is semicartesian closed.

We conclude with a general demonstration of obtaining semi(co)carte- sian
closed categories, and thus colimit-expandable ones, from monoidally closed
ones via the (co)slice category.

######  Proposition 2.8.

Let $\mathscr{C}$ be closed closed monoidal with unit $I$. Both the slice
category $\mathscr{C}/I$ and the coslice category $I/\mathscr{C}$ are colimit-
expandable.

Proof: It is sufficient to consider the slice case. Observe that there is a
natural monoidal structure inherited from $\mathscr{C}$ on the slice category
given by

$(X,f)\otimes(Y,g)=(X\otimes Y,f\otimes g\colon X\otimes Y\to I\otimes I\cong
I),$ (2.25)

in which $(I,1_{I})$ is terminal, implying that $\mathscr{C}/I$ is
semicartesian. Next, since $\mathscr{C}$ is closed, it has a unique internal
hom

$[-,-]\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mathscr{C},$ (2.26)

so we may define a similar bifunctor for the slice category by

$(X,f),(Y,g)\mapsto\big{(}[X,Y],[f,g]\big{)}.$ (2.27)

We wish to show that the pair

$-\otimes\,(X,f)\dashv\big{[}(X,f),-\big{]}$ (2.28)

is an adjunction for each $(X,f)$ in $\mathscr{C}/I$. Quite simply, given a
morphism of $\mathscr{C}/I$ from $(X\otimes Y,f\otimes g)$ to some $(Z,h)$, we
get a unique corresponding map given by the adjunct. This is a valid morphism
in the slice category as it makes the diagram

${X}$${{[Y,Z]}}$${I}$${I\otimes
I}$$\scriptstyle{f}$$\scriptstyle{u}$$\scriptstyle{[g,h]}$$\scriptstyle{\lambda^{-1}}$
(2.29)

commute by the naturality of the relevant adjunction in $\mathscr{C}$, which
also gives the naturality of this bijection.

It is interesting that even though a general closed monoidal category is not
colimit-expandable, the corresponding (co)slice is. This suggests that ‘in
spirit’ all categories that are closed monoidal have the structure to be
infinitary monoidal, but rather it is a technicality of our definition that
they are not so. Future work, perhaps, could amend the axioms to resolve this.

##  3 Monoidal Diagrams

As discussed, the use of diagrams similar to those employed in monoidal
category theory dates back to the work of Penrose penrose-1 ; penrose-2 . When
applied specifically to monoidal categories, they become very useful because
isotopic deformations preserve represented morphisms.

In general, each diagram represents a morphism constructed by compositions and
tensor products of other morphisms; these morphisms are the nodes of the
diagram, while the edges are the objects. Composition takes place vertically
while the product acts horizontally. Constructing the diagram for two
morphisms and seeing if they are isotopic to one another provides an efficient
way to determine the equivalence of two morphisms, which is often non-trivial
given the monoidal product survey-graph-types .

In reality, the use of these ‘string diagrams’ in category theory is not
restricted to monoidal categories. In general, any weak 2-category operates in
a manner respectful of string diagrams string-diagrams  with 0-cells
represented by areas, 1-cells by lines, and 2-cells by points. Since monoidal
categories are weak 2-categories with one object, the formalism simplifies
considerably.

These string diagrams are usually topological entities which map to categories
main-source-js . While this framework is certainly the most natural and simple
to prove soundness/correctness within, it makes it difficult to generalize to
the strictly categorical notions we wish to consider, i.e. transfinite
composition and infinitary monoidal product. To remedy this, we shall use
define a diagram purely algebraically. This not only strengthens its
categorical properties, but permits for its use as a mathematical object in
its own right outside of calculational purposes.

The next few sections will look at, in detail, our upgraded definition for
monoidal diagrams. Sec. 3.1 focuses on defining diagrams on a monoidal
category, exemplifying this concept, and proving some preliminary results
therefor. Secs. 3.2 and 3.3 will both be devoted to understanding the morphism
which a given diagram represents; the ultimate goal will be to define a
category of diagrams, and then establish a functorial correspondence between
it and the underlying category.

###  3.1 Definitions

We shall construct monoidal diagrams as algebraic structures equipped with two
relations; we begin with the necessary definitions.

######  Definition 3.1 (diagrammatic relation).

A homogeneous relation $R\subseteq X\times X$ is diagrammatic if the following
conditions hold $\forall x,y\in X$:

  1. 1.

(weak totality) $\exists z\in X\,\big{(}xRz\lor zRx\big{)}$

  2. 2.

(antisymmetry) $xRy\implies\lnot yRx$

  3. 3.

(irreflexivity) $\lnot xRx$

Note that the last two conditions could be replaced with that of asymmetry; a
diagrammatic relation is equivalently an asymmetric relation where each
element is comparable to another.

######  Definition 3.2 (induced first-order relation).

We call a homogeneous relation $R$ on a set $X$ a first-order relation, a
homogeneous relation $S\subseteq R\times R$ on $R$ a second-order relation, ad
infinitum. For a second-order relation, we define the induced first-order
relation of $S$ as

$S^{*}=\bigcup_{i=1,2}{\pi^{2}_{i}[S]},$ (3.1)

where $\pi_{i}$ is the $i$th projection map of the Cartesian product $R\times
R$, $f^{2}\colon A\times A\to B\times B$ represents the Cartesian square of a
function, and the notation $f[X]$ for a function $f\colon A\to B$ refers to
the set $\\{f(x):x\in X\subseteq A\\}$.

Intuitively, if the relation $S$ in the above orders the edges of a digraph,
then $S^{*}$ orders the nodes based on their membership of an edge; this is
the case in which we will employ the definition.

######  Definition 3.3 (conditional construction).

Suppose that a second order-relation $S$ on a set $X$ is a homogeneous
relation on $R$. Then $S$ satisfies the axiom of conditional construction if

$\forall P\in R^{2}\,\left(\bigvee_{i,j\leq 2}{(\pi_{i}\times\pi_{j})(P)\in
S^{*}}\implies P\in S\right)$ (3.2)

holds; in this case, $S$ is conditionally constructed.

We will use the axiom of conditional construction simply to ensure the
coherence of a relation used to order the edges of a digraph as discussed
above. With this setup, we can introduce monoidal diagrams. For what follows,
fix a strict unbiased monoidal category $\mathscr{C}$.

######  Definition 3.4 (monoidal diagram).

A (monoidal $\mathscr{C}$-diagram $(\mathcal{D},E,H,\mu)$ consists of a set
$\mathcal{D}$, a diagrammatic relation $E$ on $\mathcal{D}$ called the
vertical comparator, a partial ordering $H$ on $E$ called the horizontal
comparator (or edge set), and a function
$\mu\colon\mathcal{D}\to\mathrm{mor}(\mathscr{C})$. These data are required to
satisfy the following conditions:

  1. 1.

$E,H$ are well-founded relations

  2. 2.

Each $E$-triangle (set of two $E$-elements which take the same value on
$\pi_{1}$ or $\pi_{2}$) is $H$-comparable

  3. 3.

Any two $E$-minimal elements are $H^{\star}$-comparable

  4. 4.

$H$ is conditionally constructed

There is much to be said about this definition. The four conditions are non-
trivial, and it is not immediately clear that this definition, taken as an
axiomatic system, is consistent, complete, or independent, the first two being
quite important in this context. As we develop the theory further, however,
the three conditions will hold apparently.

Separately, Def. 3.4 is not entirely useful until we prescribe a method for
actually ‘viewing’ such a diagram, in the literal sense. To do so one begins
by laying out the nodes in $\mathcal{D}$ vertically by $E$, and then
connecting with a segment any $E$-related nodes. Next, these segments are
ordered horizontally by $H$; it is not difficult to see why this algebraic
structure is resistant to isotopic deformations.

######  Definition 3.5 (isomorphism of monoidal diagrams).

Let $(\mathcal{D},E,H,\mu)$,
$(\mathcal{D}^{\prime},E^{\prime},H^{\prime},\mu^{\prime})$ be monoidal
$\mathscr{C}$-diagrams. An isomorphism of diagrams is a bijection
$\delta\colon\mathcal{D}\to\mathcal{D}^{\prime}$ such that

$\delta\colon(\mathcal{D},E)\to(\mathcal{D}^{\prime},E^{\prime})$ (3.3)

is a digraph isomorphism, the induced bijection $\delta^{\prime}\colon E\to
E^{\prime}$ makes

$\delta^{\prime}\colon(E,H)\to(E^{\prime},H^{\prime})$ (3.4)

an order isomorphism, and finally $\mu^{\prime}\circ\delta=\mu$. These
conditions can be summarized by the commutative diagram below.

${H}$${{\mathcal{D}^{2}}^{2}}$${E}$${\mathcal{D}^{2}}$${\mathcal{D}}$${\mathrm{mor}(\mathscr{C})}$${H^{\prime}}$${{\mathcal{D}^{\prime
2}}^{2}}$${E^{\prime}}$${\mathcal{D}^{\prime
2}}$${\mathcal{D}^{\prime}}$$\scriptstyle{\delta}$$\scriptstyle{\mu}$$\scriptstyle{\mu^{\prime}}$

Here, the arrows $\rightharpoonup$, $\hookrightarrow$, $\dashrightarrow$
represent projections, inclusions, and induced bijections respectively.

We acknowledge that isomorphism of diagrams is an extremely strong condition:
diagrams are isomorphic if and only if they are exactly the same aside from
their nodes being labeled differently.

Diagrams may be adjoined to one another using the coproduct on the underlying
set and relations and the product functors in the underlying monoidal
category. Specifically, for an $\alpha$-sequence
$\\{(\mathcal{D}_{\gamma},E_{\gamma},H_{\gamma},\mu\gamma):\gamma<\alpha\\}$,
we set

$\coprod_{\gamma<\alpha}{(\mathcal{D}_{\gamma},E_{\gamma},H_{\gamma},\mu_{\gamma})}=\left(\bigsqcup_{\gamma<\alpha}\mathcal{D}_{\gamma},\coprod_{\gamma<\alpha}E_{\gamma},\coprod_{\gamma<\alpha}H_{\gamma},{\langle{\mu_{\gamma}}\rangle}_{\gamma<\alpha}\right),$
(3.5)

where the fourth component on the right-hand side is the unique map induced by
the coproduct. We call this process attachment. It is worth noting that the
attachment of diagrams coincides with the true coproduct in the category

$\mathbf{Set}\times\mathbf{Rel}\times\mathbf{Pos}\times(\mathbf{Set}/\mathrm{mor}(\mathscr{C})).$
(3.6)

This construction will be useful later in setting up a category of all
diagrams. Hereafter, we will turn our attention to understanding the internal
structure of monoidal diagrams. For what follows, fix a $\mathscr{C}$-monoidal
diagram $(\mathcal{D},E,H,\mu)$.

######  Proposition 3.1.

Given a well-founded diagrammatic relation $E$ on $D$, the reflexive
transitive closure $E^{+}_{r}$ is a well-founded partial order.

Proof: $E^{+}_{r}$ is trivially a partial order since $E$ is already
antisymmetric. Let $A\subseteq D$, and let $a\in A$ be a $E$-minimal element
(which exists by well-foundedness). We wish to show that $a$ is
$E^{+}_{r}$-minimal (and thus $E^{+}_{r}$ is well-founded); if it weren’t,
then $\exists b\neq a\in A$ s.t. $bE^{+}_{r}a$. Since $\lnot bEa$, $\exists
c\in A$ s.t. $bEc$ and $cEa$, but then $a$ is not $E$-minimal, a
contradiction.

######  Definition 3.6 (ordinal rank of a diagram).

Let $E^{+}_{r}$ be the reflexive transitive closure of $E$, and
$\mathfrak{C}_{m}(E^{+}_{r})$ be the set of $\subseteq$-maximal chains
therein. Each $C\in\mathfrak{C}_{m}(E^{+}_{r})$ is well-ordered, and thus
isomorphic to a unique ordinal, its ordinal rank, denoted $\mathrm{rank}(C)$.
The rank of $\mathcal{D}$ is then the largest such rank:

$\mathrm{rank}(\mathcal{D})=\max{\\{\mathrm{rank}(C):C\in\mathfrak{C}_{m}(E^{+}_{r})\\}}.$
(3.7)

######  Definition 3.7 (segmentation, layering).

Let $\alpha=\mathrm{rank}(\mathcal{D})$. The segmentation of $\mathcal{D}$ is
the function

$S_{\bullet}\colon\alpha\to\mathcal{P}(\mathcal{D})$ (3.8)

defined by transfinite recursion as follows: $S_{0}=\varnothing$, and
$\forall\beta<\alpha$

$S_{\beta}=\big{\\{}m\in\mathcal{D}:m\text{ is minimal in
}\mathcal{D}\setminus\Big{(}\bigcup_{\gamma<\beta}{S_{\gamma}}\Big{)}\big{\\}},$
(3.9)

which always exists by well-foundedness. We define the closely related concept
of the layering of $\mathcal{D}$, which is the map

$\begin{split}\mathcal{S}\colon\alpha&\to\mathcal{P}(\mathcal{D}).\\\
\beta&\mapsto\bigcup_{\gamma<\beta}{S_{\gamma}}\end{split}$ (3.10)

Then we may more simply write $S_{\beta}$ as the set of minimal elements of
the set difference $\mathcal{D}\setminus\mathcal{S}(\beta)$.

######  Proposition 3.2.

The segmentation of $\mathcal{D}$ partitions its nodes.

Proof: Suppose $S_{\bullet}$ wasn’t a disjoint family; then
$\exists\beta\neq\gamma$ such that $S_{\beta}\cap S_{\gamma}\neq\varnothing$,
and $\exists x\in S_{\beta}\cap S_{\gamma}$. Without loss of generality,
assume $\gamma<\beta$; $x\in S_{\gamma}\implies x\in\mathcal{S}(\beta)$, which
contradicts $x\in S_{\beta}$. We next must show that the extension
$\mathcal{S}(\alpha+1)=\mathcal{D}$. If this was false, then for $m\in
S_{\beta}$, $m$ would not be a minimal element, giving the required
contradiction.

######  Remark 2.

At present, ‘lone’ nodes (those not connected by an edge to any other node)
are not permissible in these diagrams, as they break the weak totality
condition on $E$. We do not consider this a large problem for multiple
reasons. Firstly, they could relatively easily be integrated into the theory,
however the formalism would lose a certain elegance it has, that of the actual
tuples storing only necessary information to reconstruct the original
morphisms. More importantly, this condition is imposed without loss of any
generality because, given a diagram that does not satisfy this weak totality
condition, we could add a layer below all of the others and affix identity
morphisms to the lone nodes, promoting it to a permissible monoidal diagram.

###  3.2 Unresolved Edges

The idea of ‘unresolved edges’ presents an initial problem in this
diagrammatic theory. When defining how to read a diagram, we shall break it
down into its segmentation. But, for morphisms to compose properly, we must
insert an identity edge to make uniform the width of each each layer. The
following construction will fix this issue; once again, fix a monoidal diagram
$(\mathcal{D},E,H,\mu)$.

######  Definition 3.8 ($\gamma$-unresolved edge).

An edge $e\in E$ is $\gamma$-unresolved if $\pi_{1}(e)\in\mathcal{S(\beta)}$
and $\pi_{2}(e)\in\mathcal{D}\setminus\mathcal{S}(\beta)$.

######  Definition 3.9 ($e$-incision, cohesion map).

Suppose $\mathcal{D}$ has an unresolved edge $e$ (at any $\gamma$). Let
$\mathcal{I}=\\{(\pi_{1}(e),1_{e}),(1_{e},\pi_{2},e)\\}$ be an $e$-incision,
and $E^{\prime}=E\setminus\\{e\\}\cup\mathcal{I}$ be the $e$-incision of $E$.
We then define the cohesion map $c\colon E^{\prime}\to E$ as the function such
that $c|_{E^{\prime}\setminus\mathcal{I}}$ is the identity, and
$c[\mathcal{I}]=\\{e\\}$.

######  Proposition 3.3.

Given the cohesion map $c\colon E^{\prime}\to E$ as in Def. 3.9, there is a
unique partial ordering $H^{\prime}$ such that the extension
$c\colon(E^{\prime},H_{st}^{\prime})\to(E,H_{st})$ is order-reflecting (where
$A_{st}$ is the corresponding strict partial order of $A$).

Proof: This immediately follows from the definition of an order-reflecting
map. Uniqueness is also preserved by the strictness, since the map $A\mapsto
A_{st}$ is bijective on partial orders.

######  Proposition 3.4.

Proceeding with the language of Def. 3.9 and Prop. 3.3, let
$\mathcal{D}^{\prime}=\mathcal{D}\cup\\{1_{e}\\}$ and
$\mu^{\prime}:\mathcal{D}^{\prime}\to\mathrm{mor}(\mathscr{C})$ such that
$\mu^{\prime}|_{\mathcal{D}}=\mu$ and
$\mu^{\prime}(1_{e})=(\mathrm{cod}\circ\mu\circ\pi_{1})(e)$. Then
$(\mathcal{D}^{\prime},E^{\prime},H^{\prime},\mu^{\prime})$ forms a monoidal
diagram.

Proof: This is an obvious consequence of the fact that $(\mathcal{D},E,H,\mu)$
is itself a monoidal diagram.

######  Definition 3.10 ($e$-resolution, local resolver).

Let $\mathfrak{D}$ be the class of all $\mathscr{C}$-diagrams, and for each
diagram $\mathcal{D}$ let $\mathcal{U}^{\gamma}_{\mathcal{D}}$ be the set of
$\gamma$-unresolved edges in $\mathcal{D}$, and let
$\mathcal{U}_{\mathcal{D}}=\bigcup_{\gamma<\mathrm{rank}(\mathcal{D})}{\mathcal{U}^{\gamma}_{\mathcal{D}}}$
be the set of all unresolved edges in $\mathcal{D}$. Following Prop. 3.4, we
call the diagram
$\mathrm{res}(\mathcal{D},e)=(\mathcal{D}^{\prime},E^{\prime},H^{\prime},\mu^{\prime})$
the $e$-resolution of $\mathcal{D}$. We thus have a well-defined class
function

$\mathrm{res}\colon\bigsqcup_{\mathcal{D}\in\mathfrak{D}}{\mathcal{U}_{\mathcal{D}}}\to\mathfrak{D}$
(3.11)

called the local resolver of $\mathscr{C}$-diagrams.

######  Proposition 3.5.

For a monoidal diagram $\mathcal{D}$, the set $\mathcal{U}_{\mathcal{D}}$ can
be well-ordered without invoking AC.

Proof: We construct a well-ordering on $\mathcal{U}_{\mathcal{D}}$ as follows:
take the lexicographic ordering $E\times E$ on $\mathcal{D}^{2}$; since
$\mathcal{U}_{\mathcal{D}}\subseteq E\subseteq\mathcal{D}^{2}$, there is a
well-founded poset uniquely defined through $E\times E$ which exists and
satisfies the preceding condition on $\mathcal{U}_{\mathcal{D}}$.

Because of the existence of this well-ordering, we can well-define the
following piecewise function:

$\begin{split}\mathrm{res}_{*}\colon\mathfrak{D}&\to\mathfrak{D}\\\
\mathcal{D}&\mapsto\begin{cases}\mathrm{res}(\mathcal{D},\min{\left(\mathcal{U}_{\mathcal{D}}\right)})&\mathcal{U}_{\mathcal{D}}\neq\varnothing\\\
\mathcal{D}&\mathcal{U}_{\mathcal{D}}=\varnothing\\\ \end{cases}\end{split}$
(3.12)

This map allows us to, edge-by-edge, resolve all unresolved edges, until it
becomes idempotent (i.e. the entire diagram has been resolved).

######  Definition 3.11 (resistivity, global resolver, resolution).

The resistivity, $\mathfrak{r}\colon\mathfrak{D}\to\mathbf{Ord}$, of a diagram
$\mathfrak{r}(\mathcal{D})$ is the smallest ordinal $\alpha$ such that

$\mathrm{res}^{(\alpha+1)}_{*}(\mathcal{D})=\mathrm{res}^{(\alpha)}_{*}(\mathcal{D}).$
(3.13)

We define the global resolver as

$\begin{split}\mathrm{res}\colon\mathfrak{D}&\to\mathfrak{D}\\\
\mathcal{D}&\mapsto\mathrm{res}_{*}^{(\mathfrak{r}(\mathcal{D}))}(\mathcal{D}).\end{split}$
(3.14)

We call $\mathrm{res}(\mathcal{D})$ the resolution of the diagram
$\mathcal{D}$, and define the subclass $\overline{\mathfrak{D}}$ as the image
of $\mathrm{res}$, elements of which are resolved diagrams.

Having resolved these edges, each diagram can be deconstructed into layers
properly; this will be demonstrated in the following section.

###  3.3 Readouts & Validity

We may move towards extracting the morphism encoded by a diagram, which will
be termed its readout. Before doing so, however, there is an important
subtlety to note: as we have not strongly restricted the map $\mu$, it is
possible that the morphisms dictated to compose by the structure of a diagram
are simply unable to do so due to their domains and codomains; we would like
to think of such diagrams as invalid. To rigorously address this, we will
first determine the potential readout of a diagram, and then formalize its
validity. To do so, the following result will be useful.

######  Proposition 3.6.

Let $(\mathcal{D},E,H,\mu)$ be a monoidal diagram such that
$\mathcal{D}\in\bar{\mathfrak{D}}$. For any
$\beta<\mathrm{rank}(\mathcal{D})$, let

$H^{*}_{\beta}=H^{*}\cap\big{(}S_{\beta}\times S_{\beta}\big{)}.$ (3.15)

Then $(S_{\beta},H^{*}_{\beta})$ is a well-ordered set.

Proof: Suppose $H^{*}$ was not well-founded; then there is a non-empty subset
$U\subseteq\mathcal{D}$ with no minimal element. Let $U^{\prime}=\\{e\in
E:\bigvee_{i=1,2}{\pi_{i}(e)\in U}\\}$. By condition 1 of monoidal diagrams,
$(E,H)$ is a well-founded set, and thus $U^{\prime}$ has a minimal element,
say $(c,d)$. Choose $x\in\\{c,d\\}\cap U$; since $U$ has no minimal element,
$\exists$ distinct $y\in U$ such that $yH^{*}x$. By weak totality, $y$ must be
part of at least one pair in $U^{\prime}$, say $(a,b)\in U^{\prime}$ that is
unique from $(c,d)$. But since $H$ is well-constructed as a second-order
relation, this implies that $(a,b)H(c,d)$, which contradicts $(c,d)$ being a
minimal element, showing the well-foundedness of $H^{*}$.

Since $H^{*}_{\beta}\subseteq H^{*}$ and is therefore well-founded itself, it
remains to show that $S_{\beta}$ is totally ordered. We will proceed with
transfinite induction on $\beta$; thus first assume
$(S_{\beta},H_{\beta}^{*})$ is totally ordered. For the successor ordinal
case, let $x,y\in S_{\beta+1}$. Note that there must exist $a\in S_{\beta}$
such that $aEx$; if not, then $x$ would have been minimal in
$\mathcal{D}\setminus\mathcal{S}(\beta)$ and thus a member of $S_{\beta}$
which it cannot be since $S_{\bullet}$ partitions $\mathcal{D}$. Similarly,
there is a $b\in S_{\beta}$ such that $bEy$. By supposition either
$aH^{*}_{\beta}b$ or $bH^{*}_{\beta}a$, or identically $aH^{*}b$ or $bH^{*}a$.
But well-constructedness then implies either that $(a,x)H(b,y)$ or
$(b,y)H(a,x)$, which by definition means that $x$ and $y$ are $H^{*}$
comparable and thus obviously $H^{*}_{\beta+1}$ comparable.

For the limit ordinal case, suppose $x,y\in\beta$ for some limit ordinal
$\beta$, and that $(S_{\delta},H^{*}_{\delta})$ is totally ordered for all
$\delta<\beta$. By the same argument as above, $\exists
a,b\in\mathcal{S}(\beta)$ such that $aEx$ and $bEy$. Furthermore, $a\in
S_{\delta_{1}}$ and $b\in S_{\delta_{2}}$ for $\delta_{1},\delta_{2}<\beta$;
if $\delta_{1}\neq\delta_{2}$, say without loss of generality that
$\delta_{1}>\delta_{2}$, then the diagram would contain a
$\delta_{1}$-unresolved edge, which is a contradiction since
$\mathcal{D}\in\bar{\mathfrak{D}}$. Thus $\delta_{1}=\delta_{2}=\delta$, and
so $x$ and $y$ are $H^{*}_{\beta}$ comparable by the same logic as above.
Finally, the base case $\beta=1$ is trivial since $S_{1}$ consists of all
$E$-minimal elements, but by condition three of a monoidal diagram, these must
be $H^{*}$ comparable (and thus $H^{*}_{\beta}$ comparable as well).
$S_{\beta}$ is therefore well-ordered.

######  Remark 3.

Actually, for the above result, we need not restrict ourselves to the subclass
$\bar{\mathfrak{D}}$ of resolved diagrams; doing so does greatly simplify the
proof however, especially since we will only need this well-ordering for
resolved diagrams. The more general form can be proved by noting that any two
nodes in a segment either trace back to an $E$-triangle at some point in the
diagram by chain, or they lead to $E$-minimal elements; thus their
comparability recurses back up through the diagram by means of the axiom of
conditional construction.

As discussed, the segments of this diagram indicate where to apply composition
and monoidal product. The segmentation is already constructed to be well-
ordered, in part, by the vertical comparator. Given now that the horizontal
comparator well-orders each segment, we are aware in what order to carry out
monoidal product when constructing the readout.

######  Definition 3.12 (reading).

Let $(\mathcal{D},E,H,\mu)$ be a monoidal diagram such that
$\mathcal{D}\in\bar{\mathfrak{D}}$. Let
$\ell_{\beta}=\mathrm{rank}(S_{\beta})$ for all $\beta$. Let

$S^{\bullet}_{\beta}\colon\ell_{\beta}\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}S_{\beta}$
(3.16)

be the unique order-isomorphism between the well-ordering and its ordinal
rank, by means of Prop. 3.6. Using the notation of Sec. 2.2, the reading of
$\mathcal{D}$ is the ordinal sequence

$\begin{split}\mathrm{read}(\mathcal{D})_{\bullet}\colon\mathrm{rank}(\mathcal{D})&\to\mathrm{mor}(\mathscr{C})\\\
\beta&\mapsto\mathcal{T}_{\ell_{\beta}}\circ\prod_{\gamma<\ell_{\beta}}{\mu\big{(}S_{\beta}^{\gamma}\big{)}}.\end{split}$
(3.17)

Since we now have a sequence of morphisms in the category which are to be
composed, we must impose the validity requirement before doing so.

######  Definition 3.13 (valid diagram).

Let $(\mathcal{D},E,H,\mu)$ be a monoidal diagram such that
$\mathcal{D}\in\bar{\mathfrak{D}}$. Then let

$\mathrm{inv}(\mathcal{D})=\\{\gamma<\mathrm{rank}(\mathcal{D}):\mathrm{dom}(\mathrm{read}(\mathcal{D})_{\gamma+1})\neq\mathrm{cod}(\mathrm{read}(\mathcal{D})_{\gamma})\\}$
(3.18)

We shall write $\mathrm{val}(\mathfrak{D})$ for the preimage of $\varnothing$
under
$\mathrm{inv}\circ\mathrm{res}\colon\mathfrak{D}\to\mathcal{P}(\mathbf{Ord})$.
An element of $\mathrm{val}(\mathfrak{D})$ is a valid diagram.

It is not in general true that
$\mathrm{val}(\mathfrak{D})\subseteq\bar{\mathfrak{D}}$; the $\mathrm{res}$
map allows us to consider diagrams valid even if they contain unresolved
edges. With this, we intuitively would like to define the readout map as the
transfinite composition (as in Def. 2.1) of the reading. To do so, consider
for the diagram $\mathcal{D}\in\mathrm{val}(\mathfrak{D})$ a functor

$\mathcal{D}_{\bullet}\colon\mathrm{rank}({\mathcal{D}})\to\mathscr{C},$
(3.19)

such that
$\mathcal{D}_{\bullet}=\mathrm{dom}\circ\mathrm{read}(\mathcal{D})_{\bullet}$
on objects. On the morphism $\beta\to\beta+1$, let $\mathcal{D}_{\bullet}$
take the value $\mathrm{read}(\mathcal{D})_{\beta}$. The composition law is
satisfied since $\mathcal{D}$ is a valid diagram.

######  Definition 3.14 (readout).

The readout is the function
$\mathcal{R}\colon\mathrm{val}(\mathfrak{D})\to\mathrm{mor}(\mathcal{C})$
which maps a diagram $\mathcal{D}$ to the unique morphism induced by the
colimit:

${\mathcal{D}_{0}}$${\varinjlim{\mathcal{D}_{\bullet}}}$$\scriptstyle{\mathcal{R}(\mathcal{D})}$
(3.20)

Under this formalism, we can define an unbiased monoidal category of diagrams
with a product structure given by attachment. The readout map will get
promoted to a functor, and compositions of diagrams are given by gluing of
transfinite compositions.

######  Definition 3.15 (category of diagrams).

For any $\alpha$-ary unbiased monoidal category $\mathscr{C}$, there is a
category of diagrams $\mathrm{dgm}(\mathscr{C})$ that shares the same objects
as $\mathscr{C}$, and has morphisms that are equivalence classes of valid
diagrams under the equivalence relation of isomorphism of diagrams.
$\mathrm{dgm}(\mathscr{C})$ is given a strict unbiased monoidal structure via
unbiased attachment of diagrams, written $\coprod$.

Since isomorphisms of diagrams preserve the readout, we can treat it as an
identity-on-objects functor

$\mathcal{R}\colon\mathrm{dgm}(\mathscr{C})\to\mathscr{C}\qquad[\mathcal{D}]_{\simeq}\mapsto\mathcal{R}(\mathcal{D}).$
(3.21)

Moreover, it is easy to see that $\mathcal{R}$ is an unbiased monoidal
functor, since

$\mathcal{R}(\coprod_{\gamma<\beta}\mathcal{D}_{\gamma})=\mathcal{T}_{\beta}\big{(}\mathcal{R}(\mathcal{D}_{\beta})_{\gamma<\beta}\big{)}$
(3.22)

and thus the data required therefor are just identities. This concludes our
study of monoidal diagrams.

##  4 Conclusion

###  4.1 Summary

Overall, we have provided a coherent theory in which infinitary monoidal
diagrams can be evaluated, in the process fully constructing and exploring the
notion of unbiased monoidal categories.

In Sec. 2, we introduce the latter, starting by defining unbiased monoidal
categories similarly to Ref. 10, and then extending the definition to any
ordinal arity. In addition, we discuss transfinite composition and prove that
it commutes properly with our notion of infinitary product. Finally, we define
the colimit expansion, which both provides various examples of infinitary
monoidal categories and gives rather unintuitive results regarding them.

Sec. 3 defines the monoidal diagram, a novel algebraic structure which
resembles a graph structure over a directed graph. After fixing the issue of
$\gamma$-unresolved edges in these diagrams, we construct the readout functor
which evaluates monoidal diagrams identifying them uniquely with morphisms in
the underlying infinitary monoidal category. This theory is presented in such
a way to respect the infinitary structure of the category; for this reason
infinitary monoidal diagrams satisfy much of the same properties one derives
from isotopic deformations. We conclude by promoting the class of diagrams to
a category itself, making the readout an unbiased monoidal identity-on-objects
functor.

###  4.2 Future Work

There are many directions in which to build upon our theory. Firstly, as has
been noted in the relevant literature unbiased-mc , there is no canonical one
extension from finitary to infinitary monoidal categories. Thus, diagrams
presented here could potentially have different evaluations and satisfy
different properties if a different extension is chosen. The author believes,
however, that the use of unbiased monoidal categories is most natural due to
the infinitary Lawvere Theory formulation discussed at the end of Sec. 2.2.

Another important choice made here was that between an algebraic theory of
diagrams and a graphical theory (as it is, after all, a graphical calculus).
As we chose an algebraic theory, the resultant diagrams were naturally
resistant to isotopic deformation, and the difficulty in proving coherence was
shunted to category theory. In contrast, working early on in a topological
formalism simplify the details of the proof. Thus, one way to generalize could
be by making Joyal and Street’s topological progressive plane diagrams
infinitary. An important and interesting step would be proving the direct
equivalence of the topological and categorical methods, thus demonstrating the
true ‘naturality’ of string diagrams for monoidal categories of unrestricted
arity.

A study of the colimit-expansion presented here, the author believes, would be
very fruitful. It is possible, and very likely, that many categories beyond
just the semicocartesian closed ones discussed are colimit-expandable.
Specifically, here we only considered satisfying the smoothness necessary for
the expansion with right adjoint. But, of course, sequential cocontinuity is a
much weaker condition than general cocontinuity—therefore, classifying this
would provide much insight into the colimit expansion and infinitary monoidal
categories.

Finally, and most apparently, one could generalize our notion of unbiased
monoidal categories to the braided (and even symmetric) variants, and
thereafter the diagrams that stem from it. This presents an additional
challenge because of the lack of commutativity in ordinal addition.
Considering our Lawvere theory approach, the isomorphism established by the
braiding is mapped to by non-commutative ordinals, make it difficult to
define. Furthermore, even $\mathbf{Set}$ with products would not form an
example. Improving the formalism to support such notions would be an important
advancement, though, and will therefore be the future focus of the author.

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