theoremequation theorem propequation prop lemmaequation lemma
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questionequation question defnequation defn exampleequation example
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# Composition closed premodel structures and the Kreweras lattice

Scott Balchin  Max Planck Institute for Mathematics<EMAIL_ADDRESS>,  Ethan MacBrough  Reed College<EMAIL_ADDRESS>and  Kyle Ormsby  Reed
College / University of Washington<EMAIL_ADDRESS>/<EMAIL_ADDRESS>
###### Abstract.

We investigate the rich combinatorial structure of premodel structures on
finite lattices whose weak equivalences are closed under composition. We prove
that there is a natural refinement of the inclusion order of weak
factorization systems so that the intervals detect these composition closed
premodel structures. In the case that the lattice in question is a finite
total order, this natural order retrieves the Kreweras lattice of noncrossing
partitions as a refinement of the Tamari lattice, and model structures can be
identified with stacked triangulations of a particular shape.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Composition closed premodel structures as lattice intervals
  4. 4 The case of finite total orders
  5. 5 Application: Model structures on $[n]$ as stacked triangulations

##  1\. Introduction

A weak factorization system on a category $\mathcal{C}$ consists of a pair of
morphisms $(\mathcal{L},\mathcal{R})$ satisfying lifting and factorization
properties, and they make up the backbone of categorical homotopy theory.
Given two such weak factorization systems $(\mathcal{L},\mathcal{R})$ and
$(\mathcal{L}^{\prime},\mathcal{R}^{\prime})$, there is an obvious ordering
given by inclusion of the right class (equivalently by reverse inclusion of
the left class). Such an ordered pair was called a _premodel structure_ by
Barton [Bar20]. For a premodel structure there is a natural notion of weak
equivalence, namely, we define
$\mathcal{W}=\mathcal{R}\circ\mathcal{L}^{\prime}$. If the class of morphisms
$\mathcal{W}$ are closed under the two-out-of-three property, then this is
exactly the data of a model structure on $\mathcal{C}$ in the sense of Quillen
[Qui67].

If the category $\mathcal{C}$ happens to be finite lattice $L$, then the
collection of weak factorization systems on $L$ is itself a finite lattice. As
such, the premodel structures can be described as intervals in this lattice,
in particular, the collection of premodel structures on a fixed $L$ form a
finite set. Among these intervals, one may hope to be able to identify the
model structures. It was this viewpoint that was taken by the first and third
authors along with Osorno and Roitzheim in [BOOR21], where this was done for
the finial total order $[n]=\\{0<1<\cdots<n\\}$. Such investigations were
dubbed as _homotopical combinatorics_ in loc. cit.

A key input to this was the identification of the weak factorization systems
for $[n]$, which was achieved by Balchin–Barnes–Roitzheim [BBR21] in tandem
with work of Franchere–Ormsby–Osorno–Qin–Waugh [FOO+22] under the guise of
_transfer systems_ via a surprising link to equivariant homotopy theory for
the group $C_{p^{n}}$. Namely, homotopy classes of $N_{\infty}$ operads for
$C_{p^{n}}$ in the sense of [BH15] are in bijection with weak factorization
systems on $\operatorname{Sub}(C_{p^{n}})\cong[n]$. From these results, it is
know that the lattice of weak factorization systems for $[n]$ is in bijection
with the Tamari lattice (of order $n+1$), and as such the premodel structures
are in bijection with intervals of the Tamari lattice. Among these intervals,
an enumeration was given for the number of model structures.

It was left as an open question in [BOOR21, §6] to explicitly identify the
model structures among the Tamari intervals. In this paper we provide an
answer to this question. To do so, we investigate a structure which lives in
between the premodel structures and model structures. Instead of asking for
the weak equivalences to satisfy the 2-out-of-3 property, we instead ask them
to only be closed under composition. It turns out that this collection of
_composition closed premodel structures_ enjoys many excellent structural
properties; in particular, we can identify them as intervals arising from a
refined ordering on the collection of weak factorization systems.

######  Theorem (Section 3).

Let $L$ be a finite lattice. Then there is a refinement of the ordering on
weak factorization systems on $L$ such that the intervals are exactly the
composition closed premodel structures. Moreover, the set of weak
factorization systems equipped with this ordering is a finite lattice.

This result suggests that the collection of composition closed premodel
structures may provide a sensible middle ground between premodel structures
and model structures. We highlight that even though there is a further
refinement of the ordering on weak factorization systems such that the
intervals are the model structures, the resulting poset is, not even in the
simplest non-trivial cases, a lattice (c.f., Section 3.1). As such, the
collection of model structures on a fixed lattice is poorly behaved.

In Section 4 we explore the implications of the above theorem for finite total
orders. Recall that we know that the ordering on weak factorization systems
retrieves the Tamari lattice. By a very explicit bijection we can consider
weak factorization systems on $[n]$ equivalently as noncrossing partitions on
$[n]$. The collection of noncrossing partitions under refinement of partitions
gives us the _Kreweras lattice_ , a well known, and ubiquitous, refinement of
the Tamari lattice. We are now in a position to state the main theorem of
Section 4.

######  Theorem (Section 4.1).

Let $n\geqslant 0$. Then the composition closed premodel structures on the
lattice $[n]$ are in bijection with intervals of the Kreweras lattice.

Equipped with an explicit description of the composition closed premodel
structures for $[n]$, in Section 5 we return to the question raised in
[BOOR21] regarding identifying the model structures among the Tamari
intervals. Building on work of Bernardi–Bonichon [BB09], we are able to
construct a bijection between _stacked triangulations_ and composition closed
premodel structures on $[n]$. We then further extend this bijection to certain
tricolored trees. We then provide an easy-to-check criterion for if a given
composition closed premodel structure is a model structure.

######  Theorem (Section 5).

There is an explicit bijection between certain tricolored trees and Kreweras
pairs under which a tricolored tree represents a model structure if and only
if it does not have a red branch descended from any non-red branch.

This theorem is intriguing for several reasons. Firstly, the Kreweras lattice
appears in many disparate areas of mathematics. Perhaps most interestingly for
this paper is the fact that the thick subcategories of the abelian category
$\mathrm{Rep}(A_{n})$ of all finite dimensional $k$-linear representations of
the quiver $A_{n}$ for a field $k$ under inclusion is the Kreweras lattice
[IT09, Kra21]. The investigations here suggest that there may be nontrivial
substance to understanding an explicit interpretation of the lattice intervals
in this setting, especially those that are in bijection to the model
structures.

Finally, returning to the setting of equivariant homotopy theory and
$N_{\infty}$ operads, Blumberg–Hill [BH22] introduce the notion of _compatible
pairs_ of $N_{\infty}$ operads, which encode _bi-incomplete Tambara functors_.
One can prove that the number of compatible pairs of $N_{\infty}$ operads for
$C_{p^{n}}$ is exactly the number of Kreweras intervals. In particular, this
suggests the existence of a (non-trivial) bijection between compatible pairs
of $N_{\infty}$ operads and composition closed premodel structures on the
subgroup lattice of $C_{p^{n}}$. We discuss this further in Section 4.1.

### Acknowledgements

The first author would like to thank the Max Planck Institute for Mathematics
for its hospitality, and was partially supported by the European Research
Council (ERC) under Horizon Europe (grant No. 101042990). The second author
thanks Coil Technologies for their generous donation to fund his tuition,
which enabled him to conduct this research. The third author’s work was
supported by the National Science Foundation under Grant No. DMS-2204365.

##  2\. Preliminaries

We begin by recalling the necessary preliminary results regarding weak
factorization systems and (pre)model structures. We refer the reader to
[BOOR21, §2] for further details.

######  Definition .

For any two morphisms $i\colon A\to B$ and $p\colon X\to Y$ in a category
$\mathcal{C}$, we say that $i$ _has the left lifting property (LLP) with
respect to $p$_, or $p$ _has the right lifting (RLP) property with respect to
$i$_, if for all commutative squares of the form

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there exists a lift $h\colon B\to X$ which makes the resulting diagram
commute. If $i$ lifts on the left of $p$ we write $i\boxslash p$. For any
class $\mathcal{S}$ of morphisms in $\mathcal{C}$ we write

$\displaystyle\mathcal{S}^{\boxslash}$
$\displaystyle=\\{g\in\operatorname{Mor}(\mathcal{C})\mid f\boxslash g\text{
for all }f\in\mathcal{S}\\},$ $\displaystyle{}^{\boxslash}\mathcal{S}$
$\displaystyle=\\{f\in\operatorname{Mor}(\mathcal{C})\mid f\boxslash g\text{
for all }g\in\mathcal{S}\\}.$

Note that $\mathcal{S}\subseteq{}^{\boxslash}\mathcal{T}$ if and only if
$\mathcal{T}\subseteq\mathcal{S}^{\boxslash}$. We write
$\mathcal{S}\boxslash\mathcal{T}$ when this holds.

######  Definition .

Let $\mathcal{L}$ and $\mathcal{R}$ be classes of morphisms in a category
$\mathcal{C}$. Then the pair $(\mathcal{L},\mathcal{R})$ is a _weak
factorization system_ if:

  1. (1)

every morphism $f$ in $\mathcal{C}$ can be factored as $f=pi$ with
$i\in\mathcal{L}$ and $p\in\mathcal{R}$,

  2. (2)

$\mathcal{L}\boxslash\mathcal{R}$,

  3. (3)

$\mathcal{L}$ and $\mathcal{R}$ are closed under retracts.

######  Definition .

A _premodel category_ is a category $\mathcal{C}$ with all finite limits and
colimits equipped with two weak factorization systems
$(\mathcal{L},\mathcal{R})$ and $(\mathcal{L}^{\prime},\mathcal{R}^{\prime})$
such that $\mathcal{R}\subseteq\mathcal{R}^{\prime}$ (equivalently
$\mathcal{L}\supseteq\mathcal{L}^{\prime}$). The _weak equivalences_ of a
premodel structure is the class of maps
$\mathcal{W}:=\mathcal{R}\circ\mathcal{L}^{\prime}$. For a fixed category
$\mathcal{C}$ we denote by $\mathsf{P}(\mathcal{C})$ the collection of
premodel structures of $\mathcal{C}$.

Premodel structures were introduced in [Bar20] as a generalization of model
structures (in the sense of Quillen) and are amenable to the theory of
homotopy limits and colimits. The model structures appear as the premodel
structures such that the weak equivalences $\mathcal{W}$ are closed under the
2-out-of-3 property. That is, if two of $f,g,f\circ g$ are weak equivalences,
then so is the third. We shall write $\mathsf{Q}(\mathcal{C})$ for the
collection of model structures on $\mathcal{C}$.

There is an inclusion
$\mathsf{P}(\mathcal{C})\subseteq\mathsf{Q}(\mathcal{C})$, which — even for
the most simple of $\mathcal{C}$ — is a strict inclusion. In this paper, we
wish to study a natural intermediate collection between these two which arises
as a weakening of the two-out-of-three property.

######  Definition .

Let $\mathcal{C}$ be a category equipped with a premodel structure. We say
that the premodel structure is a _composition closed premodel structure_ if
$\mathcal{W}$ is closed under composition. We write $\mathsf{C}(\mathcal{C})$
for the collection of composition closed premodel structures on $\mathcal{C}$.

######  Lemma .

Let $\mathcal{C}$ be a category. Then there are inclusions

$\mathsf{P}(\mathcal{C})\subseteq\mathsf{C}(\mathcal{C})\subseteq\mathsf{Q}(\mathcal{C}).$

Now that we have introduced the structures of interest in the general setting,
let us restrict to the case where the category $\mathcal{C}$ is a finite
lattice. In this case we have a concrete understanding of the collection of
weak factorization systems via the language of _transfer systems_ as we now
recall.

######  Definition .

Let $L=(\mathcal{P},\leqslant)$ be a lattice. A _transfer system_ on $L$
consists of a partial order $\mathcal{R}$ that refines $\leqslant$ and such
that for all $x,y,z\in L$, if $x\,\mathcal{R}\,y$ and $z\leqslant y$, then
$(x{\wedge}z)\,\mathcal{R}\,z$.

In the following result, we consider poset structures on the collections of
transfer systems and weak factorization systems. The order on transfer systems
is by refinement. That is, we say $\mathcal{R}\leqslant\mathcal{R}^{\prime}$
if and only if for all $x,y\in L$, if $x\,\mathcal{R}\,y$ then
$x\,\mathcal{R}^{\prime}\,y$. We will denote the collection of transfer
systems along with this ordering by $(\mathsf{Tr}(L),\leqslant)$. The order on
weak factorization systems is given, as usual, by inclusion of the right
class.

######  Proposition  ([FOO+22, Proposition 4.11, Theorem 4.13]).

Let $L$ be a finite lattice and $\mathcal{R}$ a transfer system on $L$. Then
there is a unique weak factorization system $(\mathcal{L},\mathcal{R})$ on
$L$. In particular the assignment

$\mathcal{R}\longleftrightarrow({}^{\boxslash}\mathcal{R},\mathcal{R})$

is a poset isomorphism between the poset of transfer systems and the poset of
weak factorization systems.

######  Remark .

In light of Section 2 we will henceforth use the terms weak factorization
system and transfer system interchangeably without any loss of generality.
Unless the left class is explicitly required, we will simply write
$\mathcal{R}$ for the unique transfer system $(\mathcal{L},\mathcal{R})$ that
it determines. When writing commutative diagrams, we will use the standard
convention of writing
$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{y}$
if $x\,\mathcal{R}\,y$ and
$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{y}$
if $x\,\mathcal{L}\,y$.

The collection $\mathsf{Tr}(L)$ (and consequently the collection of weak
factorization systems on $L$) has more structure than just being a poset. It
is in fact a complete lattice.

######  Lemma  ([FOO+22, Proposition 3.7]).

Let $L$ be a finite lattice. Then the poset $(\mathsf{Tr}(L),\leqslant)$ is
itself a finite lattice, and as such a complete lattice.

###### Proof.

The greatest element in the finite poset $(\mathsf{Tr}(L),\leqslant)$ is the
maximal transfer system $\mathcal{M}$ (i.e., $x\,\mathcal{M}\,y$ if and only
if $x\leqslant y$). The meet operation is given by intersection. Thus by
[Sta12, Proposition 3.3.1] the result follows, with the usual observation that
any finite lattice is complete. ∎

From Section 2, and the description of premodel structures, we obtain the
observation that premodel structures on a finite lattice can be realized by
intervals in the lattice of transfer systems. For $(P,\leqslant)$ a poset, we
denote by $\mathbb{I}(P)$ the poset of intervals of $L$. That is,
$\mathbb{I}(P)$ is the poset whose elements are pairs $(a\leqslant_{P}b)$ with
$a,b\in P$, and we say
$(a\leqslant_{P}b)\leqslant_{\mathbb{I}(P)}(a^{\prime}\leqslant_{P}b^{\prime})$
if and only if $a\leqslant_{P}a^{\prime}$ and $b\leqslant_{P}b^{\prime}$.

If $L$ is a complete lattice, then the collection of intervals is also a
complete lattice. Indeed, for a collection of intervals
$(a\leqslant_{P}b)_{i}$ we compute $\bigwedge(a\leqslant_{P}b)_{i}=(\bigwedge
a_{i})\leqslant_{P}(\bigwedge b_{i})$ using the fact that $L$ is a lattice
(and similarly for the join). As such, we obtain the result that for a fixed
lattice $L$ the collection $\mathsf{P}(L)$ is itself a complete lattice.

######  Lemma .

Let $L$ be a finite lattice. Then there is a bijection between the lattice of
premodel structures and the lattice of intervals
$\mathbb{I}(\mathsf{Tr}(L),\leqslant)$. In this lattice we have that for two
premodel structures $P_{1}$ and $P_{2}$, $P_{1}\leqslant P_{2}$ if and only if
the identity functor $\mathrm{id}\colon P_{1}\to P_{2}$ is right Quillen.

###### Proof.

The statement regarding the bijection between the lattice of premodel
structures and the lattice of intervals $\mathbb{I}(\mathsf{Tr}(L),\leqslant)$
is immediate from construction. If
$P_{1}=(\mathcal{R}_{1}\leqslant\mathcal{R}_{1}^{\prime})$ and
$P_{2}=(\mathcal{R}_{2}\leqslant\mathcal{R}_{2}^{\prime})$, then unravelling
the definition of the ordering on intervals, we have $P_{1}\leqslant P_{2}$ if
and only if $\mathcal{R}_{1}\leqslant\mathcal{R}_{2}$ and
$\mathcal{R}_{1}^{\prime}\leqslant\mathcal{R}_{2}^{\prime}$. Using terminology
more familiar to (pre)model structure, this happens if and only if the
identity functor preserves acyclic fibrations and fibrations. That is, the
identity functor is right Quillen. ∎

##  3\. Composition closed premodel structures as lattice intervals

In this section we will give conditions on a general premodel structure on a
finite lattice such that the weak equivalences are closed under composition.
We prove that there is an ordering which refines the natural inclusion
ordering on weak factorization systems such that the intervals realize the
composition closed premodel structures.

Following Section 2 we can succinctly record the data of a premodel structure
on a finite lattice as the pair $\mathcal{R}\leqslant\mathcal{R}^{\prime}$ of
weak factorization systems.

We begin with an observation regarding a characterization of when the weak
equivalences — which we recall are defined as
$\mathcal{R}\circ\mathcal{L}^{\prime}$ where
$\mathcal{L}^{\prime}={}^{\boxslash}\mathcal{R}$ — are closed under
composition.

######  Lemma .

Let $\mathcal{C}$ be a category. Suppose that
$\mathcal{A},\mathcal{B}\subseteq\mathrm{Mor}(\mathcal{C})$ are two sets of
morphisms containing all identity maps which are closed under composition.
Then $\mathcal{B}\circ\mathcal{A}$ is closed under composition if and only if
$\mathcal{A}\circ\mathcal{B}\subseteq\mathcal{B}\circ\mathcal{A}$.

###### Proof.

By definition, $\mathcal{B}\circ\mathcal{A}$ is closed under composition if
and only if
$(\mathcal{B}\circ\mathcal{A})\circ(\mathcal{B}\circ\mathcal{A})=\mathcal{B}\circ\mathcal{A}$.
Suppose that $\mathcal{B}\circ\mathcal{A}$ is closed under composition. Since
both classes of morphisms contain all identity maps,

$\mathcal{B}\circ\mathcal{A}\circ\mathcal{B}\circ\mathcal{A}\supseteq\mathcal{A}\circ\mathcal{B},$

and it follows that
$\mathcal{A}\circ\mathcal{B}\subseteq\mathcal{B}\circ\mathcal{A}$ as required.

Conversely, assume that
$\mathcal{A}\circ\mathcal{B}\subseteq\mathcal{B}\circ\mathcal{A}$. Since
$\mathcal{A}$ and $\mathcal{B}$ are both closed under composition we have that

$\mathcal{B}\circ(\mathcal{A}\circ\mathcal{B})\circ\mathcal{A}\subseteq\mathcal{B}\circ(\mathcal{B}\circ\mathcal{A})\circ\mathcal{A}=\mathcal{B}\circ\mathcal{A}.$

The result follows. ∎

We can now start moving towards our desired refined ordering on
$\mathsf{Tr}(L)$ whose intervals detect the composition closed premodel
structures. We highlight that the following results do not require any
finiteness assumptions on the underlying lattice.

######  Proposition .

Let
$\mathcal{R}\leqslant\mathcal{R}^{\prime}\leqslant\mathcal{R}^{\prime\prime}$
be weak factorization systems on an arbitrary lattice $L$ with corresponding
left classes
$\mathcal{L}\geqslant\mathcal{L}^{\prime}\geqslant\mathcal{L}^{\prime\prime}$.
Suppose further that $\mathcal{R}\circ\mathcal{L}^{\prime}$ and
$\mathcal{R}^{\prime}\circ\mathcal{L}^{\prime\prime}$ are closed under
composition. Then $\mathcal{R}\circ\mathcal{L}^{\prime\prime}$ is closed under
composition. In particular, being closed under composition is a transitive
relation.

###### Proof.

By Section 3 we have that $\mathcal{R}\circ\mathcal{L}^{\prime\prime}$ is
closed under composition if and only if
$\mathcal{L}^{\prime\prime}\circ\mathcal{R}\subseteq\mathcal{R}\circ\mathcal{L}^{\prime\prime}$.
Suppose $x\,\mathcal{R}\,y$ and $y\,\mathcal{L}^{\prime\prime}\,z$. Since
$\mathcal{L}^{\prime\prime}\subseteq\mathcal{L}^{\prime}$, we have
$y\,\mathcal{L}^{\prime}\,z$, and therefore by assumption there is some $w$
such that $x\,\mathcal{L}^{\prime}\,w\,\mathcal{R}\,z$. Similarly, since
$\mathcal{R}\subseteq\mathcal{R}^{\prime}$, there is some $w^{\prime}$ such
that $x\,\mathcal{L}^{\prime\prime}\,w^{\prime}\,\mathcal{R}^{\prime}\,z$. As
such, we have a square

$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{L}^{\prime}}$$\textstyle{w^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{w\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{z}$

and we conclude that $w\leqslant w^{\prime}$. However, $w\,\mathcal{R}\,z$ and
$w\leqslant w^{\prime}\leqslant z$ implies that $w\,\mathcal{R}\,w^{\prime}$,
and therefore $w\mathcal{R}^{\prime\prime}w^{\prime}$. This gives us another
square

$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{L}^{\prime\prime}}$$\textstyle{w\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime\prime}}$$\textstyle{w^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{z\hbox
to0.0pt{ .\hss}}$

It follows that $w=w^{\prime}$, and thus
$x\,\mathcal{L}^{\prime\prime}\,(w^{\prime}{=}w)\,\mathcal{R}\,z$ as required.
∎

######  Corollary .

There exists a partial order $\preccurlyeq$ on the set of weak factorization
systems of an arbitrary lattice which refines $\leqslant$ such that
$\mathcal{R}\preccurlyeq\mathcal{R}^{\prime}$ if and only if
$\mathcal{R}\circ\mathcal{L}^{\prime}$ is closed under composition.

###### Proof.

Section 3 tells us that $\preccurlyeq$ is transitive. By the definition of a
weak factorization system the relation is clearly reflexive and antisymmetric.
∎

Now that we have determined that the desired partial ordering $\preccurlyeq$
exists, we work towards a more concrete understanding of what it entails for
$\mathcal{R}$ and $\mathcal{R}^{\prime}$.

######  Lemma .

Let $\mathcal{R}\leqslant\mathcal{R}^{\prime}$. Then
$\mathcal{R}\preccurlyeq\mathcal{R}^{\prime}$ if and only if any square of the
form

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0.0pt\lower-1.99997pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
5.6308pt\raise-35.27776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
12.54839pt\raise-41.22498pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.94722pt\hbox{$\scriptstyle{\mathcal{L}^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
29.85764pt\raise-35.27776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.85764pt\raise-35.27776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{w}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$
(3.1)

has a splitting of the form

$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{L}^{\prime}}$$\textstyle{w\hbox
to0.0pt{ .\hss}}$

###### Proof.

Suppose $\mathcal{R}\preccurlyeq\mathcal{R}^{\prime}$ and that we have a
square of the form (3.1). Since $x\,\mathcal{R}\,y\,\mathcal{L}^{\prime}\,w$
and $\mathcal{R}\preccurlyeq\mathcal{R}^{\prime}$ by assumption, we also have
$x\,\mathcal{L}^{\prime}\,z^{\prime}\,\mathcal{R}\,w$ for some $z^{\prime}$.
Thus we have a square

$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{L}^{\prime}}$$\textstyle{z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{z^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w}$

which implies that $x\leqslant z^{\prime}\leqslant z$. Since
$z^{\prime}\,\mathcal{R}\,w$, one obtains the desired splitting.

Conversely, suppose $\mathcal{R}\leqslant\mathcal{R}^{\prime}$ satisfies the
stated condition and suppose $x\,\mathcal{R}\,y\,\mathcal{L}^{\prime}\,w$.
Then we can factor $x\to w$ as
$x\,\mathcal{L}^{\prime}\,z\,\mathcal{R}^{\prime}\,w$, giving a square

$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{L}^{\prime}}$$\scriptstyle{\mathcal{R}}$$\textstyle{z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{L}^{\prime}}$$\textstyle{w\hbox
to0.0pt{ .\hss}}$

By assumption we can find a splitting

$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{L}^{\prime}}$$\textstyle{w\hbox
to0.0pt{ .\hss}}$

As $x\,\mathcal{L}^{\prime}\,z$ and $x\leqslant z^{\prime}\leqslant z$,
closure of the left class under pushouts gives us
$z^{\prime}\,\mathcal{L^{\prime}}\,z$. However we also have
$z^{\prime}\,\mathcal{R}\,w$ and $z^{\prime}\leqslant z\leqslant w$, and so by
closure of the right class under pullbacks we have
$z^{\prime}\,\mathcal{R}\,z$. As such it follows that $z^{\prime}=z$, and
hence $x\,\mathcal{L}^{\prime}\,z\,\mathcal{R}\,w$. ∎

We now give a second condition on when
$\mathcal{R}\preccurlyeq\mathcal{R}^{\prime}$ which we will see is more
amenable to computation.

######  Proposition .

Let $\mathcal{R}\leqslant\mathcal{R}^{\prime}$. Then
$\mathcal{R}\preccurlyeq\mathcal{R}^{\prime}$ if and only if any square

$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w}$

has a splitting of the form

$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{w}$

###### Proof.

Suppose $\mathcal{R}\leqslant\mathcal{R}^{\prime}$ satisfies the above
condition. Then for any square

$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{L}^{\prime}}$$\textstyle{w\hbox
to0.0pt{ ,\hss}}$

we obtain a splitting

$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{w\hbox
to0.0pt{ .\hss}}$

But then we have a diagram

$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{L}^{\prime}}$$\textstyle{w^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{w\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w}$

and hence $w^{\prime}=w$, so we’ve already obtained the desired splitting
required by Section 3, and as such
$\mathcal{R}\preccurlyeq\mathcal{R}^{\prime}$.

Conversely, assume that $\mathcal{R}\preccurlyeq\mathcal{R}^{\prime}$, and
suppose we have a square

$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w\hbox
to0.0pt{ .\hss}}$

We can factor $y\to w$ as
$y\,\mathcal{L}^{\prime}\,w^{\prime}\,\mathcal{R}^{\prime}\,w$, and we have
$\widetilde{z}:=(z{\wedge}w^{\prime})\,\mathcal{R}^{\prime}\,w^{\prime}$ by
closure under pullbacks. We obtain a square

$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{\widetilde{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{L}^{\prime}}$$\textstyle{w^{\prime}\hbox
to0.0pt{ ,\hss}}$

and hence by Section 3 we can find a splitting

$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{\widetilde{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{L}^{\prime}}$$\textstyle{w^{\prime}}$

which gives

$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{w}$

as required. ∎

The statement of Section 3 is particularly useful as it doesn’t make any
reference to cofibrations. In fact the structure of the constraint in Section
3 enables us to prove for any finite lattice $L$, that
$(\mathsf{Tr}(L),\preccurlyeq)$, is not just a poset but a lattice.

######  Corollary .

Let $L$ be a finite lattice and
$\mathcal{R}\preccurlyeq\mathcal{R}^{\prime},\mathcal{R}^{\prime\prime}$. Then
$\mathcal{R}\preccurlyeq\mathcal{R}^{\prime}\cap\mathcal{R}^{\prime\prime}$.

###### Proof.

Suppose we have a square

$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}\cap\mathcal{R}^{\prime\prime}}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w\hbox
to0.0pt{ ,\hss}}$

and let $\tilde{z}\,\mathcal{R}\,\tilde{w}$ be maximal among splittings

$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{\tilde{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}\cap\mathcal{R}^{\prime\prime}}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w\hbox
to0.0pt{ .\hss}}$

By Section 3, we have two splittings

$\textstyle{\tilde{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{\tilde{w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{w}$
$\textstyle{\tilde{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime\prime}}$$\textstyle{\tilde{w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime\prime}}$$\textstyle{w\hbox
to0.0pt{ .\hss}}$

By maximality of $\tilde{z}\to\tilde{w}$ we must have
$\tilde{z}=z^{\prime}=z^{\prime\prime}$ and
$\tilde{w}=w^{\prime}=w^{\prime\prime}$, and hence $\tilde{w}\to
w\in\mathcal{R}^{\prime}\cap\mathcal{R}^{\prime\prime}$. ∎

This observation brings us to the main result of this section.

######  Theorem .

Let $L$ be a finite lattice. Then $(\mathsf{Tr}(L),\preccurlyeq)$ is a finite
lattice. As such, the collection $\mathsf{C}(L)$ of composition closed
premodel structures on $L$ is a finite lattice.

###### Proof.

The poset $(\mathsf{Tr}(L),\preccurlyeq)$ has joins by Section 3, since we can
take $\mathcal{R}\vee\mathcal{R}^{\prime}$ to be the intersection of all
transfer systems $\mathcal{R}^{\prime\prime}$ with
$\mathcal{R}\preccurlyeq\mathcal{R}^{\prime\prime}$ and
$\mathcal{R}^{\prime}\preccurlyeq\mathcal{R}^{\prime\prime}$. Since it also
has a minimal element given by the trivial transfer system,
$(\mathsf{Tr}(L),\preccurlyeq)$ forms a lattice. The claim regarding
$\mathsf{C}(L)$ then follows again from the observation that the collection of
intervals on a finite lattice is once again a finite lattice. ∎

######  Remark .

The finiteness assumption in Section 3 is essential as we now demonstrate. Let
$L$ be the lattice $(\mathbb{N}\cup\\{\infty\\})\times[1]$ equipped with the
following transfer systems $\mathcal{R}_{1}\leqslant\mathcal{R}_{2}$.

$\textstyle{x_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}_{1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
x_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}_{1}}$$\textstyle{x_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}_{1}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
y_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{y_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{y_{\infty}}$
$\textstyle{x_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}_{2}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
x_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}_{2}}$$\textstyle{x_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}_{2}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}_{2}}$$\textstyle{y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
y_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{y_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{y_{\infty}}$

Suppose that $\mathcal{R}_{1}\preccurlyeq\mathcal{R}^{\prime}$ and
$\mathcal{R}_{2}\leqslant\mathcal{R}^{\prime}$ for some
$\mathcal{R}^{\prime}$. Suppose that there exists some $n$ such that for all
$m\geqslant n$, $y_{m}\,\cancel{\mathcal{R}^{\prime}}\,y_{\infty}$. Then we
have $y_{n}\,\mathcal{L}^{\prime}\,y_{\infty}$ and
$x_{n}\,\mathcal{R}_{1}\,y_{n}$, and since
$\mathcal{R}_{1}\preccurlyeq\mathcal{R}^{\prime}$, we must have
$x_{n}\,\mathcal{L}^{\prime}\,z\,\mathcal{R}_{1}\,y_{\infty}$. However, the
only $z$ with $z\,\mathcal{R}_{1}\,y_{\infty}$ is $z=y_{\infty}$. On the other
hand, since $\mathcal{R}_{2}\leqslant\mathcal{R}^{\prime}$, we have a square

$\textstyle{x_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{y_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{y_{\infty}}$

which clearly admits no lift, and hence
$x_{n}\,\cancel{\mathcal{L}^{\prime}}\,y_{\infty}$, which is a contradiction.

Conversely, let $(n)=(n_{1},n_{2},\dots)$ be any increasing infinite sequence,
and define $\mathcal{R}_{(n)}$ to be the smallest transfer system
$\mathcal{R}^{\prime}$ containing $\mathcal{R}_{2}$ such that
$y_{n_{k}}\,\mathcal{R}^{\prime}\,y_{\infty}$ for all $k$. Specifically, we
have $x_{m}\,\mathcal{R}_{(n)}\,y_{m}$ for all $m$ (including $m=\infty$), and
for all $k$ and all $m\geqslant n_{k}$ (including $m=\infty$) we have
$x_{n_{k}}\,\mathcal{R}_{(n)}\,x_{m}$, $x_{n_{k}}\,\mathcal{R}_{(n)}\,y_{m}$
and $y_{n_{k}}\,\mathcal{R}_{(n)}\,y_{m}$. These are the only relations in
$\mathcal{R}_{(n)}$.

We claim that $\mathcal{R}_{1}\preccurlyeq\mathcal{R}_{(n)}$ and
$\mathcal{R}_{2}\preccurlyeq\mathcal{R}_{(n)}$ for any such
$\mathcal{R}_{(n)}$ constructed this way. Indeed, for any such
$\mathcal{R}_{(n)}$, one can explicitly determine $\mathcal{L}_{(n)}$: namely
we have $x_{n}\,\mathcal{L}_{(n)}\,x_{m}$ and
$y_{n}\,\mathcal{L}_{(n)}\,y_{m}$ if and only if $n_{k}<n\leqslant m\leqslant
y_{k+1}$ for some $k$, and no other arrows exist in $\mathcal{L}_{(n)}$. With
this explicit description, one is able to verify that
$\mathcal{R}_{1}\circ\mathcal{L}_{(n)}$ and
$\mathcal{R}_{2}\circ\mathcal{L}_{(n)}$ are both closed under composition.

Now we know that if $\mathcal{R}_{1}\preccurlyeq\mathcal{R}^{\prime}$ and
$\mathcal{R}_{2}\preccurlyeq\mathcal{R}^{\prime}$ then $\mathcal{R}^{\prime}$
must contain some $\mathcal{R}_{(n)}$, so in particular if the join
$\mathcal{R}_{1}\vee\mathcal{R}_{2}$ exists then it must be of the form
$\mathcal{R}_{(n)}$ for some sequence $(n)=(n_{1},n_{2},\dots)$ by minimality.
But then if we define $(n^{\prime})$ to be the same sequence with $n_{1}$
removed, we have $\mathcal{R}_{(n^{\prime})}<\mathcal{R}_{(n)}$ and
$\mathcal{R}_{1},\mathcal{R}_{2}\preccurlyeq\mathcal{R}_{(n^{\prime})}$,
contradicting minimality. Hence, the element
$\mathcal{R}_{1}\vee\mathcal{R}_{2}$ does not exist.

######  Remark .

Recall that in Section 2 we observed that the collection of premodel
structures and right Quillen functors between them formed a complete lattice.
In particular, any limit or colimit of premodel structures on a fixed finite
lattice $L$ along right Quillen functors exists.

We have proved in Section 3 that the collection of composition closed model
structures also forms a complete lattice. However, it does not follow that any
limit or colimit of composition closed premodel structures on a fixed finite
lattice $L$ along right Quillen functors exists. This is due to the fact that
for two composition closed premodel structures
$C_{1}=(\mathcal{R}_{1}\preccurlyeq\mathcal{R}_{1}^{\prime})$ and
$C_{2}=(\mathcal{R}_{2}\preccurlyeq\mathcal{R}_{2}^{\prime})$ we have
$C_{1}\leqslant_{\mathbb{I}(\mathsf{Tr}(L),\preccurlyeq)}C_{2}$ if and only if
$\mathcal{R}_{1}\preccurlyeq\mathcal{R}_{2}$ and
$\mathcal{R}_{1}^{\prime}\preccurlyeq\mathcal{R}_{2}^{\prime}$. In particular,
this is the data of a right Quillen functor satisfying an additional
condition. Let us locally call such a functor a _restricted right Quillen
functor_.

We have not been able to find a known description of what it means for a right
Quillen functor — even between model categories — to be restricted right
Quillen. Take for example the following model structures on the lattice $[2]$
and the right Quillen functor between them:

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Then this functor is not restricted right Quillen, even though it has the
strong property of preserving weak equivalences. We can see the failure of
this to be restricted right Quillen using Section 3. In particular, Setting
$x=z=0$, $y=1$, and $w=2$, we have the square

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{L}^{\prime}}$$\textstyle{2}$

which does not admit a splitting

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{L}^{\prime}}$$\textstyle{2}$

as $0$ is not fibrant in $C_{1}$.

###  3.1. A refined ordering for model structures

So far we have seen that premodel structures (resp., composition closed
premodel structures) on a finite lattice $L$ are in bijection with intervals
of the lattice $(\mathsf{Tr}(L),\leqslant)$ (resp., intervals of the lattice
$(\mathsf{Tr}(L),\preccurlyeq)$). One may hope for a similar result to detect
the model structures. Here we will show that such a refined ordering on
$\mathsf{Tr}(L)$ does exist, but the resulting poset is not a lattice. The
proof proceeds as in Section 3 where we show that the desired relation is
transitive.

######  Proposition .

Let
$\mathcal{R}\leqslant\mathcal{R}^{\prime}\leqslant\mathcal{R}^{\prime\prime}$
be transfer systems on an arbitrary lattice such that
$(\mathcal{R},\mathcal{R}^{\prime})$ and
$(\mathcal{R}^{\prime},\mathcal{R}^{\prime\prime})$ are model structures. Then
$(\mathcal{R},\mathcal{R}^{\prime\prime})$ is a model structure.

###### Proof.

We have by Section 3 that $\mathcal{R}\preccurlyeq\mathcal{R}^{\prime\prime}$,
so it suffices to show that if $f\circ
g\in\mathcal{R}\circ\mathcal{L}^{\prime\prime}$ then
$f\in\mathcal{R}\circ\mathcal{L}^{\prime\prime}$ if and only if
$g\in\mathcal{R}\circ\mathcal{L}^{\prime\prime}$. By duality it suffices to
show that $g\in\mathcal{R}\circ\mathcal{L}^{\prime\prime}$ implies that
$f\in\mathcal{R}\circ\mathcal{L}^{\prime\prime}$. To this end, suppose that
$x\to y\in\mathcal{R}\circ\mathcal{L}^{\prime\prime}$, and $x\to
z\in\mathcal{R}\circ\mathcal{L}^{\prime\prime}$, with $z\leqslant y$, we wish
to show that $z\to y\in\mathcal{R}\circ\mathcal{L}^{\prime\prime}$.

Since $\mathcal{L}^{\prime\prime}\subseteq\mathcal{L}^{\prime}$, we know that
$x\to y$ and $x\to z$ are in $\mathcal{L}^{\prime}\circ\mathcal{R}$, so since
$(\mathcal{R},\mathcal{R}^{\prime})$ forms a model structure, we have
$z\,\mathcal{L}^{\prime}\,u\,\mathcal{R}\,y$ for some $u$. Similarly using
that $\mathcal{R}\subseteq\mathcal{R}^{\prime}$ and
$(\mathcal{R}^{\prime},\mathcal{R}^{\prime\prime})$ forms a model structure,
we can find some $v$ such that
$z\,\mathcal{L}^{\prime\prime}\,v\,\mathcal{R}^{\prime}\,y$.

By pullback closure, we have $(u{\wedge}v)\,\mathcal{R}^{\prime}\,u$, and
since $z\,\mathcal{L}^{\prime}\,u$ and $z\leqslant u{\wedge}v$, we have a lift
$u\leqslant u{\wedge}v$, which implies that $u\leqslant v$. Similarly we have
$(u{\wedge}v)\,\mathcal{R}\,v$ and $z\,\mathcal{L}^{\prime\prime}\,v$, and
since $\mathcal{L}^{\prime\prime}\subseteq\mathcal{L}$, the same argument
forces $v\leqslant u$. Thus
$z\,\mathcal{L}^{\prime\prime}\,(v{=}u)\,\mathcal{R}\,y$ as required. ∎

######  Corollary .

There exists a partial ordering $\sqsubseteq$ on the set of transfer systems
of an arbitrary lattice which refines $\preccurlyeq$ such that
$\mathcal{R}\sqsubseteq\mathcal{R}^{\prime}$ if and only if the pair forms a
model structure.

######  Remark .

Although we now have a partial ordering $\sqsubseteq$ whose intervals detect
model structures, the resulting object is not a lattice. This should not be
surprising as we know that the collection of model structures even on a fixed
underlying category does not admit all homotopy limits and colimits.

Let us provide an example to show that $(\mathsf{Tr}(L),\sqsubseteq)$ fails to
be a lattice even in a relatively trivial case. Set $L=[2]$. Then
$(\mathsf{Tr}([2]),\sqsubseteq)$ takes on the form

---
$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

which is clearly not a lattice.

##  4\. The case of finite total orders

We will now restrict the theory from Section 3 to the case where $L=[n]$, a
finite total order. In [BOOR21], the premodel structures on $[n]$ were proved
to be in bijection with the intervals of the Tamari lattice. Here we will show
that the refined ordering $\preccurlyeq$ retrieves a well-known refinement of
the Tamari lattice, namely that of the _Kreweras ordering_ , which
demonstrates the rich structures that one can possibly obtain when considering
composition closed premodel structures.

###  4.1. Noncrossing partitions and the Kreweras ordering

######  Definition .

A partition of the set $\\{0,1,\dots n\\}$ is said to be _noncrossing_ if for
all $0\leqslant a<b<c<d\leqslant n$ such that $a$ and $c$ are in the same
block and $b$ and $d$ are in the same block, then $a,b,c$ and $d$ are all in
the same block.

There is a natural ordering on the collection $\mathsf{NC}_{n+1}$ of
noncrossing partitions on the set $\\{0,1,\dots,n\\}$ given by refinement of
partitions. This ordering is called the _Kreweras ordering_ which results in a
lattice (the _Kreweras lattice_) [Kre72]. The Tamari lattice is a strict
extension of the Kreweras lattice (i.e., the Kreweras lattice is a refinement
of the Tamari lattice).

######  Example .

Let us consider the case of $[2]$. Then the Tamari and Kreweras lattices take
the form

---
$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
---
$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

respectively. We note that there are 13 intervals in the Tamari lattice, while
there are only 12 in the Kreweras lattice.

We recall the bijection between noncrossing partitions and transfer systems
from [FOO+22].

######  Definition .

Let $\mathcal{R}$ be a transfer system on $[n]$. We can define a non-
decreasing function $\pi_{\mathcal{R}}\colon[n]\to[n]$ by setting
$\pi_{\mathcal{R}}(i)$ to be the maximal $j$ such that $i\,\mathcal{R}\,j$.

######  Proposition  ([FOO+22, Theorem 5.7]).

Let $\mathcal{R}$ be a transfer system on $[n]$, then the nonempty fibers of
$\pi_{\mathcal{R}}$ form a noncrossing partition of $[n]$. Moreover, this
provides a bijection between sets of noncrossing partitions and transfer
systems on $[n]$.

In [BOOR21] it was proved that the lattice $(\mathsf{Tr}([n]),\leqslant)$ is
isomorphic to the Tamari lattice. We can instead use the Kreweras ordering,
pulling back along the bijection above. In particular we have the following.

######  Lemma .

Let $\mathcal{R}$ and $\mathcal{R}^{\prime}$ be transfer systems on $[n]$.
Then $\mathcal{R}\leqslant\mathcal{R}^{\prime}$ in the Kreweras lattice if and
only if $\pi_{\mathcal{R}}(i)=\pi_{\mathcal{R}}(j)$ implies
$\pi_{\mathcal{R}^{\prime}}(i)=\pi_{\mathcal{R}^{\prime}}(j)$ for all
$i,j\in[n]$.

For $\mathcal{R}$ and $\mathcal{R}^{\prime}$ as in the above lemma, we will,
for now, write $\mathcal{R}\leqslant_{K}\mathcal{R}^{\prime}$ for the order
relation in the Kreweras ordering. The main result of this section is to prove
that $\leqslant_{K}\,\equiv\,\preccurlyeq$, that is, composition closed model
structures on $[n]$ are in bijection with intervals in the Kreweras lattice.

######  Theorem .

Let $\mathcal{R}\leqslant\mathcal{R}^{\prime}$ be transfer systems on $[n]$.
Then $\mathcal{R}\preccurlyeq\mathcal{R}^{\prime}$ if and only if
$\pi_{\mathcal{R}}\leqslant\pi_{\mathcal{R}^{\prime}}$. That is, the ordering
$\preccurlyeq$ is exactly the Kreweras ordering.

###### Proof.

Observe first that $\pi_{\mathcal{R}}\leqslant\pi_{\mathcal{R}^{\prime}}$ if
and only if
$\pi_{\mathcal{R}^{\prime}}(\pi_{\mathcal{R}}(i))=\pi_{\mathcal{R}^{\prime}}(i)$
for all $i$. Indeed, if $\pi_{\mathcal{R}}\leqslant\pi_{\mathcal{R}^{\prime}}$
then by observing that we always have
$\pi_{\mathcal{R}}(i)=\pi_{\mathcal{R}}(\pi_{\mathcal{R}}(i))$ by transitivity
of $\mathcal{R}$, we obtain
$\pi_{\mathcal{R}^{\prime}}(\pi_{\mathcal{R}}(i))=\pi_{\mathcal{R}^{\prime}}(i)$.
Conversely it suffices to observe that
$\pi_{\mathcal{R}}(i)=\pi_{\mathcal{R}}(j)$ implies

$\pi_{\mathcal{R}^{\prime}}(i)=\pi_{\mathcal{R}^{\prime}}(\pi_{\mathcal{R}}(i))=\pi_{\mathcal{R}^{\prime}}(\pi_{\mathcal{R}}(j))=\pi_{\mathcal{R}^{\prime}}(j).$

Now if $\mathcal{R}\leqslant\mathcal{R}^{\prime}$ then for all $i$ we have a
diagram

$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{\pi_{\mathcal{R}}(i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{\mathcal{R}^{\prime}}(i).}$

If $\mathcal{R}\preccurlyeq\mathcal{R}^{\prime}$ then by Section 3 we obtain a
splitting

$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{\pi_{\mathcal{R}}(i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{j\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{\pi_{\mathcal{R}^{\prime}}(i)}$

but by maximality of $\pi_{\mathcal{R}}(i)$ we must have
$j=\pi_{\mathcal{R}}(i)$ and hence
$\pi_{\mathcal{R}^{\prime}}(\pi_{\mathcal{R}}(i))\geqslant\pi_{\mathcal{R}^{\prime}}(i)$.
But
$i\,\mathcal{R}\,\pi_{\mathcal{R}}(i)\,\mathcal{R}^{\prime}\,\pi_{\mathcal{R}^{\prime}}(\pi_{\mathcal{R}}(i))$,
so the reverse inequality follows immediately.

Conversely assume $\pi_{\mathcal{R}}\leqslant\pi_{\mathcal{R}^{\prime}}$ and
suppose we’re given a square

$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{j\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ell.}$

If $j\geqslant k$ then we obtain a trivial splitting

$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{j\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{j\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{j\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{\ell.}$

Otherwise since $[n]$ is totally-ordered we have $i\leqslant j<k$ and hence by
pullback-closure we have $i\,\mathcal{R}\,j\,\mathcal{R}^{\prime}\,\ell$ and
hence
$\pi_{\mathcal{R}^{\prime}}(\pi_{\mathcal{R}}(i))=\pi_{\mathcal{R}^{\prime}}(i)\geqslant\ell$.
If $\pi_{\mathcal{R}}(i)\geqslant\ell$ then we obtain another trivial
splitting

$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{j\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ell\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{\ell.}$

Else, we obtain a non-trivial splitting

$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}}$$\textstyle{j\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{\mathcal{R}}(i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}^{\prime}}$$\textstyle{\ell}$

and the result follows. ∎

######  Corollary .

Let $n\geqslant 0$. Then

$|\mathsf{C}([n])|=\dfrac{1}{2n+3}\binom{3n+3}{n+1}.$

###### Proof.

By Section 4.1 we have $|\mathsf{C}([n])|$ is the number of intervals in the
Kreweras lattice of noncrossing partitions on $[n]$. The result then follwos
from the computation of intervals from [Kre72]. ∎

######  Remark .

We return to the remark in the introduction regarding compatible pairs of
$N_{\infty}$ operads as introduced in [BH22]. In essence, a compatible pairs
of $N_{\infty}$ operads is a pair $(\mathcal{O}_{1},\mathcal{O}_{2})$ of
$N_{\infty}$ operads satisfying $\mathcal{O}_{1}\leqslant\mathcal{O}_{2}$ in
addition to a condition regarding coinduction of groups. In [Cha22] this
condition was passed through the equivalence to transfer systems to give a
more checkable condition which we now recall (in a purely categorical
fashion). Let $L$ be a finite lattice. A pair of transfer systems
$(\mathcal{R},\mathcal{R}^{\prime})$ on $L$ are _compatible_ if:

  1. (1)

$\mathcal{R}\leqslant\mathcal{R}^{\prime}$;

  2. (2)

For all $x\in L$ if $y,z\leqslant x$ are such that $y\,\mathcal{R}\,x$ and
$(y{\wedge}z)\,\mathcal{R}^{\prime}\,y$, then $z\mathcal{R}^{\prime}x$.

Let us write $\mathsf{Com}(L)$ for the collection of compatible pairs for a
fixed lattice $L$. One immediately sees that the pair
$(\mathcal{R},\mathcal{R})$ need not be compatible, and, in particular, the
collection of premodel structures and compatible pairs on a lattice cannot be
the same. However, in the case of $[n]$, M. Hill and collaborators111Private
communication – 2022. have computed

$|\mathsf{Com}([n])|=\dfrac{1}{2n+3}\binom{3n+3}{n+1}.$

That is, $|\mathsf{Com}([n])|=|\mathsf{C}([n])|$. This suggests the existence
of an explicit bijection between compostion closed premodel structures on
$[n]$ and the compatible pairs of transfer systems. We warn, however, that for
lattices other than $[n]$, computational evidence suggests that there are
always more compatible pairs than composition closed premodel structures.

###  4.2. Enumeration and asymptotics

Using Section 4.1 along with the results of [BOOR21] we can provide numerical
results regarding the density of composition closed premodel structures among
premodel structures and model structures among composition closed premodel
structures.

We remind the reader that we write $\mathsf{P}([n])$ (resp.,
$\mathsf{C}([n])$, $\mathsf{Q}([n])$) for the collection of premodel
structures (resp., composition closed premodel structures, model structures)
on $[n]$.

We recall from [BOOR21] that

$|\mathsf{P}([n])|=\dfrac{2}{(n+1)(n+2)}\binom{4n+5}{n}$

which follows from the count of intervals in the Tamari lattice achieved by
[Cha07], and

$|\mathsf{Q}([n])|=\binom{2n+1}{n}.$

In [BOOR21] it is further shown that

$\dfrac{|\mathsf{Q}([n])|}{|\mathsf{P}([n])|}\sim c2^{dn}n^{2}.$

where

$\displaystyle c=$ $\displaystyle\dfrac{243\sqrt{3/2}}{1024}\approx 0.290638,$
$\displaystyle d=$ $\displaystyle 3\log_{2}(3)-6\approx-1.24511.$

From Section 4.1 we have

$|\mathsf{C}([n])|=\dfrac{1}{2n+3}\binom{3n+3}{n+1}.$

By Stirling’s approximation we have

$\binom{n}{k}\sim\sqrt{\dfrac{n}{2\pi
k(n-k)}}\cdot\dfrac{n^{n}}{k^{k}(n-k)^{n-k}}$

for both $k$ and $n$ large. Applying this approximation to $\mathsf{Q}([n])$
and $\mathsf{C}([n])$ yields

$\displaystyle\dfrac{|\mathsf{Q}([n])|}{|\mathsf{C}([n])|}\sim$
$\displaystyle\dfrac{4\sqrt{\pi+\pi
n}\left(4\sqrt{2}n^{2}+8\sqrt{2}n+3\sqrt{2}\right)(2n+2)^{2n}(2n+1)^{2n}\sqrt{\dfrac{2n+1}{\pi
n^{2}+\pi n}}}{27\left(\sqrt{3}n+\sqrt{3}\right)(3n+3)^{3n}n^{n}}$
$\displaystyle\sim$ $\displaystyle\dfrac{32\sqrt{3}\sqrt{\pi
n}(2n)^{4n}n\sqrt{\dfrac{1}{\pi n}}}{81(3n)^{3n}n^{n}}$ $\displaystyle=$
$\displaystyle\dfrac{32\sqrt{3}}{81}\left(\dfrac{16}{27}\right)^{n}n.$

In other words we have

$\dfrac{|\mathsf{Q}([n])|}{|\mathsf{C}([n])|}\sim c^{\prime}2^{d^{\prime}n}n$

where

$\displaystyle c^{\prime}=$ $\displaystyle\dfrac{32\sqrt{3}}{81}\approx
0.6842670,$ $\displaystyle d^{\prime}=$ $\displaystyle
4-3\log_{2}(3)\approx-0.754888.$

As such, we see that $\frac{|\mathsf{Q}([n])|}{|\mathsf{C}([n])|}$ is
asymptotically exponential decay times a linear term, and therefore approaches
0 swiftly for large $n$.

We can now solve for

$\dfrac{|\mathsf{C}([n])|}{|\mathsf{P}([n])|}\sim
c^{\prime\prime}2^{d^{\prime\prime}n}n$

where

$\displaystyle c^{\prime\prime}=$
$\displaystyle\dfrac{c}{c^{\prime}}=\dfrac{19683}{65536}\sqrt{2}\approx
0.424743,$ $\displaystyle d^{\prime\prime}=$ $\displaystyle
d-d^{\prime}=6\log_{2}(3)-10\approx-0.490255.$

In conclusion, we see that composition closed premodel structures are rare
among premodel structures, and moreover model structures are rare among
composition closed premodel structures.

##  5\. Application: Model structures on $[n]$ as stacked triangulations

In the previous section we produced a bijection between composition closed
premodel structures on $[n]$ and Kreweras intervals. In this section we will
relate composition closed premodel structures to certain triangulations using
the work [BB09]. Following this we will identify which triangulations
correspond to model structures on $[n]$.

######  Definition .

A _stacked triangulation_ is a triangulation formed by iteratively inserting a
vertex of degree 3 starting from the base triangulation of Figure 1.

Figure 1. The base triangulation for a stacked triangulation.

We can color the edges of a stacked triangulation iteratively using the
following rule:

$\star$ _color each added edge with whatever color the parent
vertex has in the direction of the edge._

The newly added node then gets colored so that within each adjacent triangle,
it takes the color of the opposite incoming edge. In this way one can define a
unique tricoloring of each stacked triangulation.

######  Example .

In Figure 2 we give an example of a stacked triangulation where we have
applied one step of iteration and colored the new edges and node
appropriately.

Figure 2. An example of a stacked triangulation after one iteration.

We now construct a colored tree on the internal nodes of the triangulation as
follows. Let the internal node of the base triangulation be the root of this
tree. Suppose some node $w$ subdivides the triangle $xyz$. Find whichever
among $x,y,z$ sits the highest on the tree (ignoring external nodes); there is
always a unique such choice. For concreteness suppose that this node is $x$.
We then add a branch going up from $x$ to $w$ colored with whatever color the
edge $xw$ is.

We canonically order the branches coming out of a node so that blue is to the
left of green and green is to the left of red. Clearly each node can have at
most one branch of each color coming out of it.

######  Example .

Figure 3 presents an example of the process of assigning a colored tree to a
stacked triangulation. Note that $d$ is attached to $c$ rather than $a$ since
$c$ sits higher on the tree than $a$.

$a$$b$$c$$d$$d$$c$$b$$a$ Figure 3. A tricolored tree associated to a stacked
triangulation.

Such trees are referred to as _realizers of triangulations which are both
minimal and maximal_ in [BB09], and are by construction unique. In particular
there is a clear bijection between such colored trees and stacked
triangulations. From now on when we say a colored tree, we mean a realization
of a triangulation which is both minimal and maximal.

######  Remark .

If for each missing branch color we add a leaf and then forget all the colors,
then we establish a bijection between stacked triangulations and (uncolored)
ternary trees.

The passage between Kreweras pairs of transfer systems
$\mathcal{R}\preccurlyeq\mathcal{R}^{\prime}$ and triangulations most
naturally passes through colored trees. As such, from now on we will restrict
to considering the connection between Kreweras pairs and colored trees. The
goal of this section is to prove the following simple characterization in
terms of colored trees of when a Kreweras pair defines a model structure.

######  Theorem .

There is a bijection between colored trees and Kreweras pairs (to be
explicitly constructed momentarily), under which a colored tree represents a
model structure if and only if it does not have a red branch descended from
any non-red branch.

We first explain the bijection between colored trees and Kreweras pairs
$\mathcal{R}\preccurlyeq\mathcal{R}^{\prime}$ from [BB09]. We start by sorting
the nodes of the tree left-to-right such that:

  1. (1)

if $x\to y$ is a blue branch, then everything above $y$ (including $y$ itself)
sits to the left of $x$;

  2. (2)

if $x\to y$ is a green branch, then everything above $y$ sits to the right of
$x$;

  3. (3)

if $x\to y$ is a red branch, then everything above $y$ sits to the right of
$x$ and also to the right of everything in the green branch of $x$.

We say that such a tree is _admissibly ordered_. Every tree has a unique
admissible ordering. We refer the reader to Section 5 for a tree ordered in
such a fashion.

Given some node $x$ in a tricolored tree $T$, the _red-green component_ of $x$
is defined to be the set of nodes $y$ for which the (unique) path connecting
$x$ to $y$ consists of only red and/or green edges. We write $x\sim_{T}y$ if
$y$ lies in the red-green component of $x$. This defines an equivalence
relation on the vertices of $T$. We write $x\to_{T}y$ if $y$ is descended from
$x$.

Given an admissibly ordered tricolored tree $T$, we define
$\pi_{\mathcal{R}^{\prime}}(x)$ to be the maximal $y\sim_{T}x$. We define
$\pi_{\mathcal{R}}(x)$ to be the maximal $y$ such that:

  1. (1)

$x\sim_{T}y$,

  2. (2)

$x\to_{T}y$, and

  3. (3)

either $x=y$ or the path from $x$ to $y$ starts with a green edge.

Equivalently, if $x$ has no green children then we define
$\pi_{\mathcal{R}}(x)=x$; otherwise if $w$ is the green child of $x$ then
$\pi_{\mathcal{R}}(x)$ is the right-most descendant of $w$.

######  Example .

We begin with the following tree that we have admissibly ordered:

012345678910111213

We then read off the values of $\pi_{\mathcal{R}^{\prime}}(x)$ to be:

$x$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\pi_{\mathcal{R}^{\prime}}(x)$ | 0 | 6 | 6 | 6 | 6 | 6 | 6 | 13 | 13 | 13 | 13 | 12 | 12 | 13

and the values of $\pi_{\mathcal{R}}(x)$ to be:

$x$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\pi_{\mathcal{R}}(x)$ | 0 | 2 | 2 | 5 | 5 | 5 | 6 | 10 | 9 | 9 | 10 | 12 | 12 | 13

######  Lemma .

Let $T$ be an admissibly ordered tricolored tree. If $x\to_{T}y$ and
$x\leqslant z\leqslant y$, then $x\to_{T}z$. Similarly if $x\to_{T}y$ and
$y\leqslant z\leqslant x$, then $x\to_{T}z$.

###### Proof.

We show the first statement; the proof of the second statement is analogous.
Since we already know by assumption that $z>x$, showing $x\to_{T}z$ is
equivalent to showing $z$ is descended from $x$. So let $x\wedge z$ be the
lowest node on the unique minimal path connecting $x$ and $z$, and suppose
$x\wedge z\neq x$. Let $B$ (resp., $G$, $R$) be the set of nodes descended
from $x\wedge z$ along a path starting with a blue (resp., green, red) edge.
Since $y$ is descended from $x$, $y$ and $x$ lie in the same set, say $G$ for
concreteness. Since $z\leqslant y$ we cannot have $z\in R$ since all nodes in
$R$ lie to the right of all nodes in $G$. Similarly since $z\geqslant x$ we
can’t have $z\in B\cup\\{x\wedge z\\}$ since all nodes in $B$ lie to the left
of $x\wedge z$ which lies to the left of all nodes in $G$. Thus $z\in G$; but
this contradicts the assumption that $x\wedge z$ lies on the minimal path
connecting $x$ and $z$. ∎

######  Theorem .

The above construction is a bijection from the set of tricolored trees with
$n+1$ nodes to the set of composition closed premodel structures on $[n]$.

###### Proof.

Since tricolored trees are in bijection with stacked triangulations, and from
[BB09], we know that stacked triangulations are in bijection with Kreweras
intervals, and by Section 4.1 we know Kreweras intervals are in bijection with
composition closed premodel structures, we at least know the cardinalities
match. Thus it suffices to show that

  1. (1)

the construction always outputs a composition closed premodel structure, and

  2. (2)

the construction is injective.

Let $\pi_{\mathcal{R}},\pi_{\mathcal{R}^{\prime}}$ be obtained from some
tricolored tree $T$. By construction $\pi_{\mathcal{R}}(x)$ and
$\pi_{\mathcal{R}^{\prime}}(x)$ are non-decreasing and idempotent.

Suppose $x\leqslant y\leqslant\pi_{\mathcal{R}^{\prime}}(x)$. We need to show
$\pi_{\mathcal{R}^{\prime}}(y)\leqslant\pi_{\mathcal{R}^{\prime}}(x)$. Let $z$
be the lowest node in the red-green path connecting $x$ to
$\pi_{\mathcal{R}^{\prime}}(x)$. Clearly $z\leqslant x$ and
$\pi_{\mathcal{R}^{\prime}}(z)=\pi_{\mathcal{R}^{\prime}}(x)$, so we might as
well assume $x=z$, so that $x\to_{T}\pi_{\mathcal{R}^{\prime}}(x)$. By the
lemma this implies $y$ is also descended from $x$. If the path connecting $y$
to $\pi_{\mathcal{R}^{\prime}}(y)$ passes through $x$ then we can restrict
this to give a red-green path from $x$ to $\pi_{\mathcal{R}^{\prime}}(y)$
which implies
$\pi_{\mathcal{R}^{\prime}}(y)\leqslant\pi_{\mathcal{R}^{\prime}}(x)$ by
maximality of $\pi_{\mathcal{R}^{\prime}}(x)$. Otherwise the path between $y$
and $\pi_{\mathcal{R}^{\prime}}(y)$ must remain entirely above $x$, which
implies $\pi_{\mathcal{R}^{\prime}}(y)$ is also descended from $x$. Let $w$ be
the right-most descendant of $x$, so that
$\pi_{\mathcal{R}^{\prime}}(y)\leqslant w$ by definition. By construction of
the admissible ordering, the path from $x$ to $w$ must be a red-green path,
and hence $w\leqslant\pi_{\mathcal{R}^{\prime}}(x)$.

Now suppose $x\leqslant y\leqslant\pi_{\mathcal{R}}(x)$. Then
$x\to_{T}\pi_{\mathcal{R}}(x)$ by definition, so by the lemma this implies $y$
is also descended from $x$. If $y=x$ there’s nothing to show. If $y$ were
descended along blue edge then $y<x$ is a contradiction, and if $y$ were
descended along a red edge then $y>\pi_{\mathcal{R}}(x)$ is a contradiction.
Thus $y$ is descended along a green edge. Let $x\to z$ be the green edge
descended from $x$, so that $y$ and $\pi_{\mathcal{R}}(x)$ are both descended
from $w$. Then by construction, $\pi_{\mathcal{R}}(x)$ is the rightmost
descendant of $w$, and hence in particular
$\pi_{\mathcal{R}}(x)\geqslant\pi_{\mathcal{R}}(y)$ since
$\pi_{\mathcal{R}}(y)$ is descended from $y$ which is descended from $w$.

Thus $\mathcal{R},\mathcal{R}^{\prime}$ are transfer systems. It remains to
show $\mathcal{R}\preccurlyeq\mathcal{R}^{\prime}$, i.e.,
$\pi_{\mathcal{R}}(x)=\pi_{\mathcal{R}}(y)$ implies
$\pi_{\mathcal{R}^{\prime}}(x)=\pi_{\mathcal{R}^{\prime}}(y)$. But
$\pi_{\mathcal{R}}(x)=\pi_{\mathcal{R}}(y)$ implies in particular that there
is a red-green path connecting $x$ and $y$, so this is immediate.

We now prove (2). Let $T$ be an admissibly-ordered tree on the vertex set
$[n]$, let $\mathcal{R}\preccurlyeq\mathcal{R}^{\prime}$ be the corresponding
composition closed pair, and let $x\in[n]$.

First observe that we can recover $\sim_{T}$, since
$\pi_{\mathcal{R}^{\prime}}(x)=\pi_{\mathcal{R}^{\prime}}(y)$ if and only if
$x$ and $y$ lie in the same red-green component. Let $y>x$ be minimal such
that $x\sim_{T}y$ and $x\,\mathcal{R}\,y$. If $x$ has no green child then by
definition of $\mathcal{R}$ we must have $x=y$. Otherwise letting $w$ be the
green child of $x$, by minimality of $y$ we have $x<y\leqslant w$. In the
latter case, by the lemma, $y$ must be descended from $x$ along a green edge,
and hence $y$ is descended from $w$. If $y<w$ then $y$ must be descended from
$w$ along a blue edge, but this contradicts the assumption that $x$ and $y$
lie in the same red-green component. Thus $y=w$ is the green child of $x$,
showing that we can recover all the green edges of $T$ from
$\mathcal{R},\mathcal{R}^{\prime}$.

Now let $y>x$ be minimal such that $x\sim_{T}y$ and
$x\,\cancel{\mathcal{R}}\,y$. Let $x\wedge y$ be the lowest point on the
minimal path between $x$ and $y$. We claim $y$ is the red child of $x\wedge
y$. Suppose first that $x\wedge y=x$. Then $y$ is descended from $x$ and
$x\sim_{T}y$, so if $y$ were descended along a green edge then we would have
$x\,\mathcal{R}\,y$, a contradiction. Thus $y$ is descended along a red edge,
and in particular $x$ has a red child $w$. By the same argument as before,
$y=w$ by minimality. In the general case, note that neither $x$ nor $y$ can be
descended from $x\wedge y$ along a blue edge since $x\sim_{T}y$. Since $x<y$
we thus must have $x$ descended from $x\wedge y$ along an initially green path
and $y$ descended from $x\wedge y$ along an initially red path. Another
minimality argument shows $y$ must be the red child of $x\wedge y$.

By induction we can assume that for all $x^{\prime}<x$ we’ve identified the
red child of $x^{\prime}$ using only $\mathcal{R}$ and $\mathcal{R}^{\prime}$.
If no $y$ exists as above then we know $x$ has no red children. So suppose
we’ve found some minimal $y$ as above. Then we know $y$ is the red child of
$x\wedge y$, and the path from $x\wedge y$ to $x$ is red-green. Since red-
green edges always point to the right this implies $x\wedge y\leqslant x$, so
either $y$ is the red child of $x$ or there exists some $x\wedge y<x$ such
that $y$ is the red child of $x\wedge y$. We can recognize the latter case by
the induction hypothesis, so this shows we can identify all the red edges in
$T$.

All that remains is to show how to recover the blue edges from $\mathcal{R}$
and $\mathcal{R}^{\prime}$. Let $T^{\prime}$ be the red-green forest that
we’ve constructed, and let $x$ be the root of some tree in $T^{\prime}$. We
claim either $x$ is the root of $T$ or $\pi_{\mathcal{R}^{\prime}}(x)+1$ is
the blue parent of $x$. Indeed this is forced by admissibility of the
ordering. Suppose $x$ is not the root of $T$ and let $y$ be the blue parent of
$x$. Since $x$ is not the red or green descendant of any node, the node
$\pi_{\mathcal{R}^{\prime}}(x)$ is the furthest-right descendant of $x$ and
hence $\pi_{\mathcal{R}^{\prime}}(x)<y$ by admissibility. But if
$y>\pi_{\mathcal{R}^{\prime}}(x)+1>x$ then by the lemma
$\pi_{\mathcal{R}^{\prime}}(x)+1$ must be descended from $y$ along a blue edge
which would imply $\pi_{\mathcal{R}^{\prime}}(x)+1$ is descended from $x$,
contradicting maximality of $\pi_{\mathcal{R}^{\prime}}(x)$. Thus
$y=\pi_{\mathcal{R}^{\prime}}(x)+1$ as claimed and we’re done. ∎

Now that we have proved that colored trees are in bijection with premodel
structures on $[n]$, we move towards isolating the collection of model
structures among them, and provide a proof of Section 5.

######  Proposition .

A colored tree has a red branch sitting somewhere above a blue branch (i.e.,
the source of the red branch is a descendant of the top of the blue branch) if
and only if there exists some $x$ such that
$\pi_{\mathcal{R}}(x)<\pi_{\mathcal{R}^{\prime}}(x)<n$.

###### Proof.

Suppose we have a red branch $x\to y$. Then $y$ must lie to the right of every
descendant of the (possibly non-existent) green branch coming out of $x$, and
hence we have $y>\pi_{\mathcal{R}}(x)$. But clearly
$\pi_{\mathcal{R}^{\prime}}(x)\geqslant y$, so this shows
$\pi_{\mathcal{R}}(x)<\pi_{\mathcal{R}^{\prime}}(x)$. As a partial converse,
if $\pi_{\mathcal{R}}(x)<\pi_{\mathcal{R}^{\prime}}(x)$ then either $x$ has a
red branch or there must be an upward-directed green path from some $z$ to $x$
such that $z$ has a red branch.

Now suppose $x$ sits somewhere above a blue branch $y\to z$ (i.e., $x$ is a
descendant of $z$ in the tree order). Then all descendants of $z$ must lie to
the left of $y$, and hence if there exists a red-green path from $x$ to some
$w$ then $w<y$. Thus in particular $\pi_{\mathcal{R}^{\prime}}(x)<n$.
Conversely, if all ancestor branches of $x$ are green or red then we have a
green-red path to the root, and clearly there is a red-green path from the
root $n$, so in this case $\pi_{\mathcal{R}^{\prime}}(x)=n$.

Combining these observations gives the desired equivalence. ∎

We shall write $x\,\mathcal{L}^{\prime}\to\mathcal{R}\,y$ to denote that there
exists some $z$ with $x\,\mathcal{L}^{\prime}\,z\,\mathcal{R}\,y$.

######  Proposition .

Let $\mathcal{R}\preccurlyeq\mathcal{R}^{\prime}$ be a Kreweras pair of
transfer systems on $[n]$. Then
$\pi_{\mathcal{R}}(x)<\pi_{\mathcal{R}^{\prime}}(x)$ implies
$\pi_{\mathcal{R}^{\prime}}(x)=n$ if and only if
$x\,\mathcal{L}^{\prime}\to\mathcal{R}\,y$,
$z\,\mathcal{L}^{\prime}\to\mathcal{R}\,y$, and $x\leqslant z$ implies
$x\,\mathcal{L}^{\prime}\to\mathcal{R}\,z$.

###### Proof.

Suppose $\pi_{\mathcal{R}}(x)<\pi_{\mathcal{R}^{\prime}}(x)<n$. Since
$\pi_{\mathcal{R}^{\prime}}(x)<n$, we know $x\,\mathcal{L}^{\prime}\,y$ where
$y=\pi_{\mathcal{R}^{\prime}}(x)+1$. Let $z=\pi_{\mathcal{R}^{\prime}}(x)$.
Then we know $z\,\mathcal{L}^{\prime}\,y$. But if
$x\,\mathcal{L}^{\prime}\,w\,\mathcal{R}\,z$, then $z<y$ implies $w=x$ and
hence $x\,\mathcal{R}\,z=\pi_{\mathcal{R}^{\prime}}(x)>\pi_{\mathcal{R}}(x)$,
a contradiction.

Conversely, suppose $x\,\mathcal{L}^{\prime}\,x^{\prime}\,\mathcal{R}\,y$ and
$z\,\mathcal{L}^{\prime}\,z^{\prime}\,\mathcal{R}\,y$ with $x\leqslant z$, but
there does not exist any $w$ with
$x\,\mathcal{L}^{\prime}\,w\,\mathcal{R}\,z$. We can suppose $z<x^{\prime}$
since otherwise $x^{\prime}\leqslant z\leqslant y$ and
$x^{\prime}\,\mathcal{R}\,y$ would imply
$x\,\mathcal{L}^{\prime}\,x^{\prime}\,\mathcal{R}\,z$. Now let $w\geqslant x$
be minimal such that $w\,\mathcal{R}\,z$ and suppose we have a square

${x}$${u}$${w}$${v}$$\scriptstyle{\mathcal{R}^{\prime}}$

If $u<w$ then this forces $u\,\mathcal{R}^{\prime}\,w$, but
$u\,\cancel{\mathcal{R}}\,w$ by transitivity of $\mathcal{R}$ and minimality
of $w$. Thus $\pi_{\mathcal{R}}(u)<\pi_{\mathcal{R}^{\prime}}(u)$. On the
other hand, $u<w\leqslant z<x^{\prime}$, so if
$\pi_{\mathcal{R}^{\prime}}(u)=n$ then we would have a diagram

${x}$${u}$${x^{\prime}}$${n}$$\scriptstyle{\mathcal{L}^{\prime}}$$\scriptstyle{\mathcal{R}^{\prime}}$

with no lift, a contradiction. ∎

Let $\theta_{\mathcal{R}}(z)=y$ where $y\leqslant z$ is minimal such that
$y\,\mathcal{L}\,z$. The following lemma records the properties satisfied by
this function that are dual to properties of $\pi$.

######  Lemma .

Let $\mathcal{R}$ be a transfer system on $[n]$. Then

  1. (1)

$\theta_{\mathcal{R}}$ is a non-increasing idempotent function, and
$\theta_{\mathcal{R}}(i)\leqslant j\leqslant i$ implies
$\theta_{\mathcal{R}}(j)\geqslant\theta_{\mathcal{R}}(i)$.

  2. (2)

$j\,\mathcal{L}\,i$ if and only if $\theta_{\mathcal{R}}(i)\leqslant
j\leqslant i$.

  3. (3)

For all $i\in[n]$, $\theta_{\mathcal{R}}(i)=\max\\{j<i\mid
j\,\mathcal{R}\,i\\}+1$.

One can observe that we have a red branch above a green branch somewhere if
and only if there exist $x<y<z$ such that $x\,\mathcal{R}\,z$ and
$y\,\mathcal{R}^{\prime}\,z$ but $w\,\cancel{\mathcal{R}}\,z$ for all
$y\leqslant w<z$. But $x\,\mathcal{R}\,z$ is equivalent to saying
$x<\theta_{\mathcal{R}}(z)$, and likewise $y\,\mathcal{R}^{\prime}\,z$ is
equivalent to $y<\theta_{\mathcal{R}^{\prime}}(z)$. Furthermore
$w\,\cancel{\mathcal{R}}\,z$ for all $y\leqslant w<z$ is equivalent to
$\theta_{\mathcal{R}}(z)\leqslant y$. Thus we have shown

######  Proposition .

A colored tree associated with a transfer system $\mathcal{R}$ has a red
branch sitting somewhere above a green branch if and only if there exists some
$z$ such that $0<\theta_{\mathcal{R}}(z)<\theta_{\mathcal{R}^{\prime}}(z)$.

Dualizing Section 5, we obtain the following result.

######  Proposition .

Let $\mathcal{R}\preccurlyeq\mathcal{R}^{\prime}$ be a Kreweras pair. Then
$\theta_{\mathcal{R}}(x)<\theta_{\mathcal{R}^{\prime}}(x)$ implies
$\theta_{\mathcal{R}}(x)=0$ if and only if
$x\,\mathcal{L}^{\prime}\to\mathcal{R}\,y$,
$x\,\mathcal{L}^{\prime}\to\mathcal{R}\,z$, and $z\leqslant y$ implies
$z\,\mathcal{L}^{\prime}\to\mathcal{R}\,y$.

Combining the above results leads immediately to a proof of Section 5 as
required. ∎

######  Remark .

Via Section 5 and the bijection between colored trees and stacked
triangulations from [BB09], we also have a bijection between model structures
on finite total orders and stacked triangulations constructed in the following
fashion:

  1. (1)

Iteratively subdivide the triangle furthest to the bottom right some finite
number of times.

  2. (2)

Iteratively subdivide the resulting diagram only in top and bottom left
triangles some finite number of times.

We call triangulations formed in this manner _model triangulations_. The
following diagrams exhibit a model triangulation with $6+1$ internal vertices
(Figure 5) along with its associated admissible tree (Figure 6), Kreweras
interval of noncrossing partitions on $[6]$ (Figure 7), transfer systems
$\mathcal{R}\leqslant\mathcal{R}^{\prime}$ (Figure 8). The corresponding model
structure on $[6]$ is as follows (Figure 4):

0123456 Figure 4. The model structure on $[6]$ encoded by Figure 5.

Figure 5. A stacked triangulation with $6+1$ internal nodes. 0123456 Figure 6.
The admissible tree associated to Figure 5.

$x$ | 0 | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---|---
$\pi_{\mathcal{R}}(x)$ | 0 | 3 | 2 | 3 | 4 | 5 | 6

$x$ | 0 | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---|---
$\pi_{\mathcal{R}^{\prime}}(x)$ | 0 | 6 | 2 | 6 | 6 | 5 | 6

0123456$\leqslant_{K}$0123456

Figure 7. noncrossing partitions on $[6]$ associated with Figure 5
01234560123456$\preccurlyeq$ Figure 8. The transfer systems associated to
Figure 5.

###  5.1. Involutions

The characterization of Section 5 is — by construction — invariant under
swapping blue and green,222On triangulations, this corresponds to reflection
across the vertical axis of symmetry. and hence this operation induces an
involution on the collection of model structures. Presently, we characterize
this involution explicitly. We begin with an intermediate lemma, referring the
reader to [BOOR21, Definition 3.1] for the definition of interval partitions.

######  Lemma .

Let $(\mathcal{R},\mathcal{R}^{\prime})$ be a model structure. Then $x$ and
$y$ are weakly equivalent if and only if they lie in the same blue-green
component of the colored tree corresponding to
$(\mathcal{R},\mathcal{R}^{\prime})$.

###### Proof.

By Section 5, we know the colored tree corresponding to
$(\mathcal{R},\mathcal{R}^{\prime})$ has a unique maximal red path $y_{0}\to
y_{1}\to\cdots\to y_{n}$ where $y_{0}$ is the root, and such that no other
branches are red. Given the method for which we order the tree nodes, one
observes that the blue-green components partition $[n]$ into an interval
partition $[n]=P_{0}\amalg P_{1}\amalg\cdots\amalg P_{n}$ where $y_{i}\in
P_{i}$ and $x<y$ for all $x\in P_{i}$, $y\in P_{j}$, $i<j$. The claim is then
that the $P_{i}$ are the weak equivalence classes of
$(\mathcal{R},\mathcal{R}^{\prime})$.

Let $x$ be the minimal element of $P_{i}$ and $y$ the maximal element. Then by
the ordering that we use, it follows that the path from $y_{i}$ to $x$ is
entirely blue and the path from $y_{i}$ to $y$ is entirely green. Thus
$x\,\mathcal{L}^{\prime}\,y_{i}\,\mathcal{R}\,y$. By the decomposition
property of weak equivalences for model structures on a poset (c.f.,
[BOOR21]), this implies that all elements of $P_{i}$ are weakly equivalent.

Now suppose $x\in P_{i}$ and $y\in P_{j}$ for some $i<j$. Then the path from
$x$ to $y$ passes through the red path $y_{i}\to y_{j}$. Suppose
$x\,\mathcal{L}^{\prime}\,z\,\mathcal{R}\,y$. Then $y$ must be a descendent of
$z$ along a green branch. But the first ancestors of $y$ are $y_{0}\to
y_{1}\to\cdots\to y_{j}$, and since none of these branches are green we must
have $z$ descended from $y_{j}$, which in particular forces $z\in P_{j}$. On
the other hand, if $\tilde{x}$ is maximal in $P_{i}$ and $\tilde{y}$ is
maximal in $P_{j}$ then one observes that
$\tilde{x}\,\mathcal{R}^{\prime}\,\tilde{y}$, so if
$x\,\mathcal{L}^{\prime}\,z$ then $z\leqslant\tilde{x}\in P_{i}$, a
contradiction. ∎

######  Remark .

From the above lemma we can retrieve the enumeration result of [BOOR21,
Theorem 4.10] regarding model structures on $[n]$. Indeed, observe that a
colored tree corresponding to a model structure is necessarily made up from a
collection of blue-green trees growing up from a red field. Since blue-green
trees correspond to binary trees, it follows that such trees correspond to
ordered collections of binary trees whose node counts sum to $n$. Since binary
trees are counted by Catalan numbers, this exactly recovers the enumeration of
model structures by partitioning $[n-1]$ into weak equivalence classes and
counting transfer systems on each part as in [BOOR21].

We are now ready to give an explicit description of the blue–green involution
on colored trees with respect to model structures.

######  Proposition .

Let $(\mathcal{R},\mathcal{R}^{\prime})$ be a model structure on $[n]$, and
let $P$ be the interval partition generated by the weak equivalence classes.
Then $(\mathcal{R},\mathcal{R}^{\prime})$ restricts to a model structure
$(\mathcal{R}_{S},\mathcal{R}^{\prime}_{S})$ on each $S\in P$. The model
structure constructed by swapping blue and green in the colored tree
corresponding to $(\mathcal{R},\mathcal{R}^{\prime})$ is the unique model
structure $(\tilde{\mathcal{R}},\tilde{\mathcal{R}}^{\prime})$ with the same
weak equivalences as $(\mathcal{R},\mathcal{R}^{\prime})$ but for which
$\tilde{\mathcal{R}}^{\prime}_{S}=\mathcal{L}_{S}^{\mathsf{op}}$ and
$\tilde{\mathcal{R}}_{S}=(\mathcal{L}^{\prime}_{S})^{\mathsf{op}}$ for all
$S\in P$.

###### Proof.

By Section 5.1 and the way we sort things, clearly the operation of swapping
blue and green preserves each weak equivalence class and reverses the order of
all the nodes in each class. In what follows whenever we use $<$ or
$\leqslant$ without clarification we refer to the order relevant for
$\tilde{\mathcal{R}}$ and $\tilde{\mathcal{R}}^{\prime}$, i.e., the order
post-reversal. Thus when we refer to the “opposite order” we really mean the
original order.

Suppose $x<y$ lie in some blue-green component $P_{i}$. Then
$x\,\tilde{\mathcal{R}}\,y$ if and only if $y$ is descended from $x$ along a
path that begins with a blue branch (in the original colored tree
corresponding to $(\mathcal{R},\mathcal{R}^{\prime})$). But this means in the
opposite order that for all $y\leqslant z<x$, $z$ is also descended from $x$
along a blue branch and hence $z\,\cancel{\mathcal{R}}^{\prime}\,x$ for all
such $z$, which implies $y\,\mathcal{L}^{\prime}\,x$. Conversely, if $y$ is
not descended from $x$ along a path that begins with a blue branch then since
$x<y$ there must be some $z$ such that $y$ is descended from $z$ along a blue
branch and $x$ is descended from $z$ along a green branch; but then it’s easy
to see in the opposite order that $y<z<x$ and $z\,\mathcal{R}^{\prime}\,x$,
and hence $y\,\cancel{\mathcal{L}}^{\prime}\,x$. Thus
$x\,\tilde{\mathcal{R}}\,y$ if and only if $y\,\mathcal{L}^{\prime}\,x$ as
claimed. Since $P_{i}$ is a weak equivalence class, we have
$\tilde{\mathcal{R}}=\tilde{\mathcal{R}}^{\prime}$ on $P_{i}$, so by [BOOR21,
Theorem 3.10] we’re done. ∎

## References

  * [Bar20] R. W. Barton.  A model 2-category of enriched combinatorial premodel categories.  arXiv:2004.12937v1, 2020.
  * [BB09] O. Bernardi and N. Bonichon.  Intervals in Catalan lattices and realizers of triangulations.  J. Combin. Theory Ser. A, 116(1):55–75, 2009.
  * [BBR21] S. Balchin, D. Barnes, and C. Roitzheim.  $N_{\infty}$-operads and associahedra.  Pacific J. Math., 315(2):285–304, 2021.
  * [BH15] A. J. Blumberg and M. A. Hill.  Operadic multiplications in equivariant spectra, norms, and transfers.  Adv. Math., 285:658–708, 2015.
  * [BH22] A. J. Blumberg and M. A. Hill.  Bi-incomplete Tambara functors.  In Equivariant topology and derived algebra, volume 474 of London Math. Soc. Lecture Note Ser., pages 276–313. Cambridge Univ. Press, Cambridge, 2022.
  * [BOOR21] S. Balchin, K. Ormsby, A. M. Osorno, and C. Roitzheim.  Model structures on finite total orders.  arXiv:2109.07803v1, 2021.
  * [Cha07] F. Chapoton.  Sur le nombre d’intervalles dans les treillis de Tamari.  Sém. Lothar. Combin., 55:Art. B55f, 18, 2005/07.
  * [Cha22] D. Chan.  Bi-incomplete Tambara functors as $\mathcal{O}$-commutative monoids.  arXiv:2208.05555v1, 2022.
  * [FOO+22] E. E. Franchere, K. Ormsby, A. M. Osorno, W. Qin, and R. Waugh.  Self-duality of the lattice of transfer systems via weak factorization systems.  Homology, Homotopy and Applications, 24(2):115–134, 2022.
  * [IT09] C. Ingalls and H. Thomas.  Noncrossing partitions and representations of quivers.  Compos. Math., 145(6):1533–1562, 2009.
  * [Kra21] H. Krause.  The category of finite strings.  arXiv:2109.05745v1, 2021.
  * [Kre72] G. Kreweras.  Sur les partitions non croisées d’un cycle.  Discrete Math., 1(4):333–350, 1972.
  * [Qui67] D. G. Quillen.  Homotopical algebra.  Lecture Notes in Mathematics, No. 43. Springer-Verlag, Berlin-New York, 1967.
  * [Sta12] R. P. Stanley.  Enumerative combinatorics. Volume 1, volume 49 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, second edition, 2012.