# The category of finite strings

Henning Krause  Fakultät für Mathematik
Universität Bielefeld
D-33501 Bielefeld
Germany<EMAIL_ADDRESS>
(Date: 13 September, 2021)

###### Abstract.

We introduce the category of finite strings and study its basic properties.
The category is closely related to the augmented simplex category, and it
models categories of linear representations. Each lattice of non-crossing
partitions arises naturally as a lattice of subobjects.

###### Key words and phrases:

String, simplex category, quiver representation, non-crossing partition, thick
subcategory

###### 2020 Mathematics Subject Classification:

05E10 (primary), 16G20, 18N50 (secondary)

##  1\. Introduction

Strings are considered to be one of the most basic combinatorial structures
arising in representation theory of associative algebras. In fact, many of the
interactions with neighbouring fields involve strings and their corresponding
representations, which are also known as string modules.

In this note we introduce a category of finite strings and establish some
connections. First of all, we notice that the category of connected strings is
equivalent to the augmented simplex category $\Delta$ (cf. [9, 17]), once the
initial and terminal objects in $\Delta$ are identified. Then we show that the
category of finite strings models categories of linear representations. More
precisely, we provide an equivalence between finite strings and certain
abelian categories (hereditary and uniserial length categories with only
finitely many simple objects and split over a fixed field, cf. [1]), where
morphisms between strings correspond to certain exact functors. In this
context it is appropriate to include cyclic strings which correspond to
abelian categories of infinite representation type. This is somewhat parallel
to the cyclic category of Connes and others [3, 4]; however we add new objects
(cyclic strings) while the cyclic category keeps the objects of $\Delta$ and
only morphisms are added.

Any morphism in the category of finite strings admits an epi-mono
factorisation. Thus it is of interest to study the subobjects of a given
object, at least for any connected string. We show for a linear string of
length $n$ that the lattice of subobjects is isomorphic to the lattice
$\operatorname{NC}(n+1)$ of non-crossing partitions [16], while the lattice
$\operatorname{NC}^{B}(n)$ of type $B$ non-crossing partitions [19] arises for
a cyclic string of length $n$.

The correspondence between strings and categories of linear representations
identifies subobjects of strings with thick subcategories of abelian
categories. In this way we recover the beautiful classification of thick
subcategories for quiver representations of type $A$ due to Ingalls and Thomas
[13], and we add a classification for nilpotent representations of cyclic
quivers which seems to be new. The cyclic case can be used to complete the
classification of all thick subcategories for representations of any tame
hereditary algebra, including the ones that are not generated by exceptional
sequences, and therefore complementing the work in [11, 12, 13]. We point out
that the category of strings can be extended to include all Dynkin types,
beyond the type $A$ in this work, and analogous to the categorification of
non-crossing partitions for all Dynkin types in [11].

Finally, let us mention the connection with some recent work which is
concerned with Iyama’s higher Auslander algebras of type $A$ [14]. These
algebras form a natural generalisation of the hereditary algebras of type $A$
arising in the present work. In [7] the authors point out the simplicial
structure of the representations for these higher Auslander algebras, using
some advanced categorical formalism. Wide subcategories of representations
generalise thick subcategories and these are studied for type $A$ higher
Auslander algebras in [10].

### Acknowledgements

It is a pleasure to thank Marc Stephan and Dieter Vossieck for several useful
comments on this work. In addition I wish to thank Christian Stump for
pointing me to the non-crossing partitions of type $B$.

##  2\. Connected strings

In this work we introduce the category of finite strings. For any natural
number $n$ the connected string of length $n$ is denoted by $\Sigma_{n}$. Each
string comes equipped with its set of (connected) substrings, together with a
multiplication on the set of substrings given by concatenation. The objects of
the category are finite coproducts of connected strings, and the morphisms are
maps that preserve substrings and their multiplication.

A _basic string_ is a pair $s=(s^{\prime},s^{\prime\prime})$ of integers
$s^{\prime}\leq s^{\prime\prime}$. We write
$\ell(s)=s^{\prime\prime}-s^{\prime}+1$ for the _length_ of $s$ and add a
_zero string_ $*$ satisfying $\ell(*)=0$. Strings of length one are called
_simple_ and we set $s_{i}:=(i,i)$, $i\in\mathbb{Z}$. For a string $s=(i,j)$
we call the simple strings $s_{i},s_{i+1},\ldots,s_{j}$ its _composition
factors_.

A multiplication of basic strings is given by concatenation. For $s,t$ set

$st:=\begin{cases}(s^{\prime},t^{\prime\prime})&\text{for
}s=(s^{\prime},s^{\prime\prime}),\,t=(t^{\prime},t^{\prime\prime}),\,s^{\prime\prime}+1=t^{\prime},\\\
t&\text{for }s=*,\\\ s&\text{for }t=*,\\\ *&\text{otherwise}.\end{cases}$

This multiplication is not associative. For instance, we have

$(s_{0}s_{1})s_{0}=(0,1)s_{0}=*\qquad\text{and}\qquad
s_{0}(s_{1}s_{0})=s_{0}*=s_{0}.$

For $n\in\mathbb{N}=\\{0,1,2,\ldots\\}$ the _connected string_ of length $n$
is by definition the set of basic strings

$\Sigma_{n}:=\\{s=(s^{\prime},s^{\prime\prime})\mid 0\leq s^{\prime}\leq
s^{\prime\prime}<n\\}\cup\\{*\\}.$

A morphism $\phi\colon\Sigma_{m}\to\Sigma_{n}$ is by definition a map such
that for all $s,t\in\Sigma_{m}$

(2.1) $\phi(st)=\phi(s)\phi(t)\qquad\text{and}\qquad
st\neq*=\phi(st)\implies\phi(s)=*=\phi(t).$

The string $*$ plays the role of a base point. In fact, a morphism is base
point preserving since

$\phi(*)=\phi(**)=\phi(*)\phi(*)=*.$

Any morphism is determined by the images of the simple strings but it need not
preserve the length of basic strings.

We define _standard morphisms_ as follows. Let $n\geq 1$. The morphism

$\delta^{i}_{n}\colon\Sigma_{n-1}\longrightarrow\Sigma_{n}\qquad(0\leq i\leq
n)$

is given by the unique injective map such that $s_{i-1}$ and $s_{i}$ are not
in its image. Note that $\delta_{1}^{0}=\delta_{1}^{1}$. The morphism

$\sigma^{i}_{n}\colon\Sigma_{n}\longrightarrow\Sigma_{n-1}\qquad(0\leq i<n)$

is given by the unique surjective map such that $s_{i}$ is sent to the zero
string.

The standard morphisms are analogues of the face and degeneracy maps for
simplices. In fact, they satisfy the following _simplicial identities_ [17,
VII.5].

######  Lemma 2.1.

The standard morphisms satisfy the following identities:

(2.2) $\displaystyle\delta^{i}_{n+1}\circ\delta^{j}_{n}$
$\displaystyle=\mathrlap{\delta^{j+1}_{n+1}\circ\delta^{i}_{n}}\hphantom{\begin{cases}\delta^{i}_{n}\circ\sigma^{j-1}_{n}&\\\
\operatorname{id}&\\\
\delta^{i-1}_{n}\circ\sigma^{j}_{n}&\end{cases}}\kern-1.2pti\leq j$
$\displaystyle\sigma^{j}_{n}\circ\sigma^{i}_{n+1}$
$\displaystyle=\mathrlap{\sigma^{i}_{n}\circ\sigma^{j+1}_{n+1}}\hphantom{\begin{cases}\delta^{i}_{n}\circ\sigma^{j-1}_{n}&\\\
\operatorname{id}&\\\
\delta^{i-1}_{n}\circ\sigma^{j}_{n}&\end{cases}}\kern-1.2pti\leq j$
$\displaystyle\sigma^{j}_{n}\circ\delta^{i}_{n}$
$\displaystyle=\begin{cases}\delta^{i}_{n}\circ\sigma^{j-1}_{n}&i<j\\\
\operatorname{id}&i=j\text{ or }i=j+1\\\
\delta^{i-1}_{n}\circ\sigma^{j}_{n}&i>j+1\end{cases}$

###### Proof.

This is easily checked, for instance by tracing the images of the simple
strings. We have

(2.3) $\delta_{n}^{i}(s_{j})=\begin{cases}s_{j}&j<i-1\\\ (j-1,j)&j=i-1\\\
s_{j+1}&j>i-1\end{cases}\qquad\text{and}\qquad\sigma_{n}^{i}(s_{j})=\begin{cases}s_{j}&j<i\\\
*&j=i\\\ s_{j-1}&j>i.\end{cases}$

It remains to note that any morphism $\phi$ is determined by the images
$\phi(s_{j})$. ∎

We denote by $\Sigma$ the _category of connected strings_ with objects given
by the strings $\Sigma_{n}$, $n\in\mathbb{N}$.

Any morphism can be written in some canonical form. First observe that there
is a canonical epi-mono factorisation.

######  Lemma 2.2.

Let $\phi\colon\Sigma_{m}\to\Sigma_{n}$ be a morphisms. Let $0\leq
j_{v}<\cdots<j_{0}<m$ be the indices $j$ such that $\phi(s_{j})=*$ and set
$\phi^{\prime}=\sigma_{m-v}^{j_{v}}\circ\cdots\circ\sigma_{m-1}^{j_{1}}\circ\sigma_{m}^{j_{0}}$.
Then there is a factorisation $\phi=\phi^{\prime\prime}\circ\phi^{\prime}$
such that $\phi^{\prime\prime}$ is injective.

###### Proof.

When $\phi(s_{i})=*$ for some simple string $s_{i}$, then
$\phi(s_{i}s)=\phi(s)=\phi(ss_{i})$ for all $s\in\Sigma_{m}$. This yields the
factorisation $\phi=\phi^{\prime\prime}\circ\phi^{\prime}$. Given strings
$s,t$ such that $\phi^{\prime\prime}(s)=\phi^{\prime\prime}(t)$, an induction
on their length shows that $s=t$. ∎

######  Lemma 2.3.

Every morphism $\phi\colon\Sigma_{m}\to\Sigma_{n}$ can be written uniquely as
composite

(2.4)
$\phi=\delta_{n}^{i_{u}}\circ\delta_{n-1}^{i_{u-1}}\circ\cdots\circ\delta_{n-u}^{i_{0}}\circ\sigma_{m-v}^{j_{v}}\circ\cdots\circ\sigma_{m-1}^{j_{1}}\circ\sigma_{m}^{j_{0}}$

with $0\leq i_{0}<\cdots<i_{u}\leq n$, $0\leq j_{v}<\cdots<j_{0}<m$, and
$n-u=m-v$.111We need to exclude $\delta_{1}^{0}$ as a factor and choose
instead $\delta_{1}^{1}$ in order to achieve uniqueness.

We call (2.4) the _canonical decomposition_ of $\phi$ in $\Sigma$.

###### Proof.

Let $0\leq j_{v}<\cdots<j_{0}<m$ be the indices $j$ such that $\phi(s_{j})=*$.
And let $0\leq i_{0}<\cdots<i_{u}\leq n$ be the indices such that for all
$(s^{\prime},s^{\prime\prime})\in\Sigma_{n}$ in the image of $\phi$ we have
$s^{\prime},s^{\prime\prime}+1\not\in\\{i_{0},\ldots,i_{u}\\}$. Then it is
clear that $\phi$ satisfies (2.4). Conversely, if $\phi$ is written as in
(2.4), then the indices $i_{0},\cdots,i_{u}$ and $j_{0},\cdots,j_{v}$ are
characterised as above. ∎

######  Lemma 2.4.

The category $\Sigma$ is generated by the objects $\Sigma_{n}$, the morphisms
$\delta_{n}^{i}$, $\sigma_{m}^{j}$, plus the relations
$\delta_{1}^{0}=\delta_{1}^{1}$ and (2.2).

###### Proof.

Let us denote by $\Sigma^{\prime}$ the category generated by the objects
$\Sigma_{n}$, the morphisms $\delta_{n}^{i}$, $\sigma_{m}^{j}$, plus the
relations $\delta_{1}^{0}=\delta_{1}^{1}$ and (2.2). Since the relations are
satisfied in $\Sigma$, there is a unique functor $\Sigma^{\prime}\to\Sigma$
which induces the identity on the objects and on the morphisms
$\delta_{n}^{i}$ and $\sigma_{m}^{j}$. Since every morphism in $\Sigma$ is a
composite of morphisms $\delta_{n}^{i}$ and $\sigma_{m}^{j}$, the induced map
$\operatorname{Hom}_{\Sigma^{\prime}}(\Sigma_{m},\Sigma_{n})\to\operatorname{Hom}_{\Sigma}(\Sigma_{m},\Sigma_{n})$
is surjective for all $m,n$. To show injectivity, choose $\phi,\psi$ in
$\operatorname{Hom}_{\Sigma^{\prime}}(\Sigma_{m},\Sigma_{n})$ with same image
in $\Sigma$. Since the relations $\delta_{1}^{0}=\delta_{1}^{1}$ and (2.2) in
$\Sigma^{\prime}$ are satisfied, there are decompositions (2.4) in
$\Sigma^{\prime}$ for $\phi$ and $\psi$. These decompositions coincide in
$\Sigma$ by Lemma 2.3, and therefore $\phi=\psi$. ∎

######  Example 2.5.

For any $n\geq 3$ we have the following pullback in $\Sigma$.

${\Sigma_{n}}$${\Sigma_{n-1}}$${\Sigma_{n-1}}$${\Sigma_{n-2}}$$\scriptstyle{\sigma_{n}^{0}}$$\scriptstyle{\sigma_{n}^{n-1}}$$\scriptstyle{\sigma_{n-1}^{n-2}}$$\scriptstyle{\sigma_{n-1}^{0}}$

##  3\. The simplicial category

Let $\Delta$ denote the _simplicial category_ , which is also known as
_augmented simplex category_ (terminology and notation follows [17, VII.5]).
The objects are given by the finite ordinals $[n]=\\{0,1,\ldots,n-1\\}$,
$n\in\mathbb{N}$, and the morphisms $\phi\colon[m]\to[n]$ are given by maps
satisfying $\phi(i)\leq\phi(j)$ for all $0\leq i\leq j<m$. For $n\geq 0$ there
is the _face map_

$\bar{\delta}^{i}_{n}\colon[n]\longrightarrow[n+1]\qquad(0\leq i\leq n)$

(the unique injective map not taking the value $i$) and for $n\geq 1$ the
_degeneracy map_

$\bar{\sigma}^{i}_{n}\colon[n+1]\longrightarrow[n]\qquad(0\leq i<n)$

(the unique surjective map taking twice the value $i$) which are known to
satisfy the simplicial identities (2.2). In fact, the category $\Delta$ is
generated by the objects $[n]$, the morphisms $\bar{\delta}_{n}^{i}$,
$\bar{\sigma}_{m}^{j}$, and the identities (2.2); see [9, II.2] or [17,
VII.5].

We write $\bar{\Delta}=\Delta[\alpha^{-1}]$ for the category which is obtained
by formally inverting the morphism $\alpha=\bar{\delta}_{0}^{0}$. This amounts
to identifying the initial and the terminal object in $\Delta$.

######  Proposition 3.1.

The assignments

$[n]\mapsto\begin{cases}\Sigma_{0}&n=0\\\
\Sigma_{n-1}&n>0,\end{cases}\qquad\bar{\delta}_{n}^{i}\mapsto\delta_{n}^{i},\qquad\bar{\delta}_{0}^{0}\mapsto\operatorname{id},\qquad\bar{\sigma}_{n}^{i}\mapsto\sigma_{n}^{i}$

provide a functor $p\colon\Delta\to\Sigma$ which induces an equivalence
$\bar{\Delta}\xrightarrow{\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\Sigma$.

###### Proof.

The functor $p$ is well defined since it maps generators to generators and the
simplicial identities are satisfied in both categories. The functor $p$
inverts $\bar{\delta}_{0}^{0}$ and induces therefore a functor
$\bar{\Delta}\to\Sigma$, which yields a bijection between the isomorphism
classes of objects. Also for the morphisms we obtain bijections since the
functor matches generators and relations. Note that
$\bar{\delta}_{1}^{0}=\bar{\delta}_{1}^{1}$ in $\bar{\Delta}$ since
$\bar{\delta}_{0}^{0}$ is invertible. ∎

The functor $p\colon\Delta\to\Sigma$ admits two sections $s_{0}$ and $s_{1}$
that are given by

$\Sigma_{n}\mapsto[n+1],\qquad\delta_{n}^{i}\mapsto\bar{\delta}_{n}^{i},\qquad\sigma_{n}^{i}\mapsto\bar{\sigma}_{n}^{i},$

except that $s_{i}(\delta_{1}^{0})=\bar{\delta}_{1}^{i}=s_{i}(\delta_{1}^{1})$
for $i=0,1$.

Let us summarise. We have for all $n\geq 1$ diagrams

$\leavevmode\hbox to91.06pt{\vbox
to36.01pt{\pgfpicture\makeatletter\hbox{\hskip 45.52792pt\lower-23.4837pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-45.52792pt}{-7.97859pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
11.93065pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.6251pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\Sigma_{n-1}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
11.93065pt\hfil&\hfil\hskip 23.99997pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 33.59727pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.29176pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\Sigma_{n}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
9.5973pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} {{}}{}{}{}{ {}{}{}}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{{}}{}{}{{{}{}}}{{}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-21.46663pt}{-11.9369pt}\pgfsys@lineto{25.73335pt}{-11.9369pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.93333pt}{-11.9369pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.42267pt}{-20.46982pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\delta_{n}^{i}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{}{}{ {}{}{}}{}{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{{}}{}{}{{{}{}}}{{}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-21.46663pt}{0.97972pt}\pgfsys@lineto{25.73335pt}{0.97972pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.93333pt}{0.97972pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.42267pt}{4.1936pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\delta_{n}^{i+1}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{26.13332pt}{-5.47859pt}\pgfsys@lineto{-21.06667pt}{-5.47859pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-21.26665pt}{-5.47859pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}
{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@rect{-3.01984pt}{-10.22838pt}{10.70636pt}{9.49957pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.86707pt}{-7.2145pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\sigma_{n}^{i}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad(0\leq i<n)$

satisfying the simplicial identities, but a difference from the usual simplex
category arises because of the extra identity $\delta_{1}^{0}=\delta_{1}^{1}$.

The equivalence in Proposition 3.1 can be explained in terms of linear
representations of posets. We refer to Theorem 4.9 and the appendix for
further details.

##  4\. Representations

The category of finite strings models certain categories of linear
representations. In the following we specify the relevant class of abelian
categories and the exact functors between them.

Let $P$ be a poset and $k$ a field. A $k$-linear representation of $P$ is by
definition a functor $P\to\operatorname{mod}k$ into the category of finite
dimensional $k$-spaces, where $P$ is viewed as a category (with objects the
elements in $P$ and a unique morphism $x\to y$ iff $x\leq y$). Morphisms
between representations are the natural transformations, and we denote by
$\operatorname{Rep}(P,k)$ the category of all finite dimensional $k$-linear
representations.

For a $k$-linear abelian category $\mathcal{A}$ over a field $k$ we consider
the following conditions.

  1. (Ab1)

$\mathcal{A}$ is _connected_ , that is,
$\mathcal{A}=\mathcal{A}_{1}\times\mathcal{A}_{2}$ implies $\mathcal{A}_{1}=0$
or $\mathcal{A}_{2}=0$.

  2. (Ab2)

$\mathcal{A}$ is a _length category_ , that is, every object has a finite
composition series, and there are only finitely many isomorphism classes of
simple objects.

  3. (Ab3)

$\mathcal{A}$ is _hereditary_ , that is, $\operatorname{Ext}^{2}$ vanishes.

  4. (Ab4)

$\mathcal{A}$ is _uniserial_ , that is, every indecomposable object has a
unique composition series.

  5. (Ab5)

$\mathcal{A}$ is _split_ , that is, $\operatorname{End}(S)\cong k$ for every
simple object $S$.

  6. (Ab6)

$\mathcal{A}$ is of _finite type_ , that is, there are only finitely many
isomorphism classes of indecomposable objects.

######  Lemma 4.1.

Let $\mathcal{A}$ be a $k$-linear abelian category satisfying _(Ab1)–(Ab6)_.
Then there is an equivalence
$\mathcal{A}\xrightarrow{\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\operatorname{Rep}([n]^{\mathrm{op}},k)$,
where $n$ equals the number of isomorphism classes of simple objects in
$\mathcal{A}$.

###### Proof.

See for example the description of uniserial categories in [1]. ∎

From now on we fix a field $k$ and set
$\mathcal{A}_{n}:=\operatorname{Rep}([n]^{\mathrm{op}},k)$ for
$n\in\mathbb{N}$. An object $M\in\mathcal{A}_{n}$ is a diagram

$M(n-1)\longrightarrow\cdots\longrightarrow M(1)\longrightarrow M(0)$

of $k$-spaces. For any string $s\in\Sigma_{n}$ we define a representation
$M_{s}\in\mathcal{A}_{n}$ as follows. Set $M_{*}=0$. For
$s=(s^{\prime},s^{\prime\prime})$ let $M_{s}$ be the representation222This is
also known as _string module_ in the terminology of [2].

$0\longrightarrow\cdots\longrightarrow 0\longrightarrow k\xrightarrow{\ 1\
}\cdots\xrightarrow{\ 1\ }k\longrightarrow
0\longrightarrow\cdots\longrightarrow 0$

such that $M_{s}(i)=k$ iff $s^{\prime}\leq i\leq s^{\prime\prime}$. Set
$M_{i}:=M_{s_{i}}$ for $0\leq i<n$. Observe that the composition length of
$M_{s}$ equals $\ell(s)$, and the composition factors of $M_{s}$ correspond
bijectively to the composition factors of $s$.

######  Lemma 4.2.

Let $n\in\mathbb{N}$.

  1. (1)

The assignment $s\mapsto M_{s}$ induces a bijection between
$\Sigma_{n}\setminus\\{*\\}$ and the isomorphism classes of indecomposable
objects in $\mathcal{A}_{n}$.

  2. (2)

For $s,t,u\in\Sigma_{n}$ there is an exact sequence $0\to M_{s}\to M_{t}\to
M_{u}\to 0$ if and only if $t=su$.

###### Proof.

Straightforward. ∎

Next we specify the class of exact functors which arises naturally in our
context. For each exact functor $F\colon\mathcal{A}\to\mathcal{B}$ between
abelian categories we denote by $\operatorname{Ker}F$ the full subcategory of
$\mathcal{A}$ given by the objects $X\in\mathcal{A}$ such that $FX=0$. This is
a Serre subcategory and we denote by $\mathcal{A}/(\operatorname{Ker}F)$ the
corresponding quotient, cf. [8].

We say that an exact functor $F\colon\mathcal{A}\to\mathcal{B}$ between
abelian categories admits a _homological factorisation_ if the induced functor
$\mathcal{A}/\mathcal{A}^{\prime}\to\mathcal{B}$ with
$\mathcal{A}^{\prime}=\operatorname{Ker}F$ induces for all objects
$X,Y\in\mathcal{A}$ bijections

$\operatorname{Ext}_{\mathcal{A}/\mathcal{A}^{\prime}}^{i}(X,Y)\longrightarrow\operatorname{Ext}_{\mathcal{B}}^{i}(FX,FY)\qquad(i\geq
0).$

A full subcategory of an abelian category is _thick_ if it is closed under
direct summands and the _two out of three property_ holds for any short exact
sequence (that is, if two terms belong to the subcategory, then also the
third).

######  Lemma 4.3.

An exact functor $F\colon\mathcal{A}\to\mathcal{B}$ between hereditary abelian
categories admits a _homological factorisation_ if and only if
$\mathcal{A}/(\operatorname{Ker}F)$ identifies with a thick subcategory of
$\mathcal{B}$.

###### Proof.

Set $\mathcal{A}^{\prime}=\operatorname{Ker}F$ and suppose $F$ identifies
$\mathcal{A}/\mathcal{A}^{\prime}$ with a full subcategory
$\mathcal{B}^{\prime}\subseteq\mathcal{B}$. Clearly, $\mathcal{B}^{\prime}$ is
closed under kernels and cokernels of morphisms since $F$ is exact. The
subcategory $\mathcal{B}^{\prime}$ is extension closed if and only if the
induced map
$\operatorname{Ext}_{\mathcal{A}/\mathcal{A}^{\prime}}^{1}(X,Y)\to\operatorname{Ext}_{\mathcal{B}}^{1}(FX,FY)$
is a bijection for all $X,Y\in\mathcal{A}$. ∎

Not all exact functors admit a homological factorisation. A simple example is
for any field $k$ the exact functor
$\operatorname{mod}k\to\operatorname{mod}k$ given by $X\mapsto
X\otimes_{k}k^{2}$.

For $m,n\in\mathbb{N}$ we denote by
$\operatorname{Hom}(\mathcal{A}_{m},\mathcal{A}_{n})$ the set of $k$-linear
exact functors $\mathcal{A}_{m}\to\mathcal{A}_{n}$, up to natural isomorphism,
that admit a homological factorisation. We define natural maps

$\operatorname{Hom}(\mathcal{A}_{m},\mathcal{A}_{n})\xrightarrow{\
\alpha_{mn}\
}\operatorname{Hom}(\Sigma_{m},\Sigma_{n})\quad\text{and}\quad\operatorname{Hom}(\Sigma_{m},\Sigma_{n})\xrightarrow{\
\beta_{mn}\ }\operatorname{Hom}(\mathcal{A}_{m},\mathcal{A}_{n})$

as follows.

Any morphism $\phi\colon[m]\to[n]$ induces an exact functor
$\phi^{*}\colon\mathcal{A}_{n}\to\mathcal{A}_{m}$ via precomposition. Let us
set $s_{n}^{i}:=(\bar{\delta}_{n-1}^{i})^{*}$ for $0\leq i<n$.

######  Lemma 4.4.

Let $n\geq 1$. There are canonical recollements of abelian categories

${\mathcal{A}_{1}}$${\mathcal{A}_{n}}$${\mathcal{A}_{n-1}}$${(0\leq
i<n)}$$\scriptstyle{s_{n}^{i}}$$\scriptstyle{d_{n}^{i}}$$\scriptstyle{d_{n}^{i+1}}$

such that

$\operatorname{Ker}s_{n}^{i}=\operatorname{add}M_{i},\qquad\operatorname{Im}d_{n}^{i}={M_{i}}^{\perp},\qquad\operatorname{Im}d_{n}^{i+1}={{}^{\perp}M_{i}}.$

The functors $s_{n}^{i}$, $d_{n}^{i}$, $d_{n}^{i+1}$ are exact. Moreover, they
send indecomposable objects to indecomposable objects or to zero.

###### Proof.

Let $\mathcal{C}=\operatorname{Ker}s_{n}^{i}$ denote the full subcategory of
objects in $\mathcal{A}_{n}$ that are annihilated by $s_{n}^{i}$. It is clear
that $M_{i}$ is the unique simple object in $\mathcal{C}$. Thus $\mathcal{C}$
equals the full subcategory given by the finite direct sums of copies of
$M_{i}$. Then the right adjoint of the quotient functor
$\mathcal{A}_{n}\to\mathcal{A}_{n}/\mathcal{C}$ identifies
$\mathcal{A}_{n}/\mathcal{C}$ with $\mathcal{C}^{\perp}$, while the left
adjoint identifies $\mathcal{A}_{n}/\mathcal{C}$ with ${}^{\perp}\mathcal{C}$.
Here, we consider the perpendicular categories defined with respect to
$\operatorname{Hom}$ and $\operatorname{Ext}^{1}$; cf. [8, III.2]. This yields
the descriptions of $d_{n}^{i}$ and $d_{n}^{i+1}$. The embedding of any
perpendicular category into $\mathcal{A}_{n}$ is exact since
$\operatorname{Ext}^{2}$ vanishes. The functor $s_{n}^{i}$ annihilates $M_{i}$
and sends all other indecomposables to indecomposable objects. ∎

For $n\geq 1$ we set

$\beta_{n,n-1}(\sigma_{n}^{i}):=s_{n}^{i}\qquad\text{and}\qquad\beta_{n-1,n}(\delta_{n}^{i}):=d_{n}^{i}.$

One checks that these functors satisfy the identities (2.2). Thus the
assignment extends uniquely to maps
$\beta_{mn}\colon\operatorname{Hom}(\Sigma_{m},\Sigma_{n})\to\operatorname{Hom}(\mathcal{A}_{m},\mathcal{A}_{n})$
for all $m,n\in\mathbb{N}$, using Lemma 2.4.

######  Remark 4.5.

We have $d_{n}^{i}=(\bar{\sigma}_{n-1}^{i-1})^{*}$ for $0<i<n$. Thus
$d^{0}_{n}$ and $d^{n}_{n}$ are not obtained from morphisms $[n-1]\to[n]$.

######  Lemma 4.6.

Let $m,n\in\mathbb{N}$. An exact functor
$F\colon\mathcal{A}_{m}\to\mathcal{A}_{n}$ that admits a homological
factorisation induces a morphism $\phi\colon\Sigma_{m}\to\Sigma_{n}$ which is
given by $F(M_{s})=M_{\phi(s)}$.

###### Proof.

The functor $F$ identifies $\mathcal{A}_{m}/(\operatorname{Ker}F)$ with a full
subcategory of $\mathcal{A}_{n}$. The canonical functor
$\mathcal{A}_{m}\to\mathcal{A}_{m}/(\operatorname{Ker}F)$ can be written as
composite of functors of the form
$s_{p}^{i}\colon\mathcal{A}_{p}\to\mathcal{A}_{p-1}$, which map
indecomposables either to indecomposables or to zero. Thus for any
$s\in\Sigma_{m}$ we have $F(M_{s})=M_{t}$ for some $t\in\Sigma_{n}$, using
Lemma 4.2. This yields a morphism $\phi\colon\Sigma_{m}\to\Sigma_{n}$ by
setting $\phi(s)=t$. ∎

The above lemma provides maps
$\alpha_{mn}\colon\operatorname{Hom}(\mathcal{A}_{m},\mathcal{A}_{n})\to\operatorname{Hom}(\Sigma_{m},\Sigma_{n})$
satisfying $\alpha_{nn}(\operatorname{id})=\operatorname{id}$ and
$\alpha_{mp}(G\circ F)=\alpha_{np}(G)\circ\alpha_{mn}(F)$ for any pair of
composable functors $F,G$.

######  Lemma 4.7.

A $k$-linear equivalence
$\mathcal{A}_{n}\xrightarrow{\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\mathcal{A}_{n}$
is naturally isomorphic to the identity.

###### Proof.

The category $\mathcal{A}_{n}$ is _standard_ , that is, equivalent to the mesh
category given by its Auslander-Reiten quiver [20, 2.4]. Clearly, an
equivalence induces the identity on the Auslander-Reiten quiver and preserves
the mesh ideal. From this the assertion follows. ∎

######  Lemma 4.8.

Let $m,n\in\mathbb{N}$. Then $\beta_{mn}\circ\alpha_{mn}=\operatorname{id}$
and $\alpha_{mn}\circ\beta_{mn}=\operatorname{id}$.

###### Proof.

The identity $\alpha_{mn}\circ\beta_{mn}=\operatorname{id}$ is clear since
this can be checked on the standard morphisms, thanks to Lemma 2.4. We
consider only $k$-linear exact functors
$F\colon\mathcal{A}_{m}\to\mathcal{A}_{n}$ that admit a homological
factorisation. Such functors are determined, up to natural isomorphism, by the
values $F(M_{s})$ of the indecomposable objects; see Lemma 4.7. Thus
$\alpha_{mn}$ is injective and $\beta_{mn}\circ\alpha_{mn}=\operatorname{id}$
follows. ∎

Combining the above lemmas yields a combinatorial description of the abelian
categories that are specified in Lemma 4.1.

######  Theorem 4.9.

Let $k$ be a field. The assignment $\Sigma_{n}\mapsto\mathcal{A}_{n}$ provides
an equivalence between the category of connected strings and the category of
$k$-linear abelian categories satisfying _(Ab1)–(Ab6)_.∎

We refer to the appendix for some further explanation of this result.

##  5\. Finite coproducts

For a finite set of natural numbers $n_{\alpha}\in\mathbb{N}$ we define the
coproduct $\coprod_{\alpha}\Sigma_{n_{\alpha}}$ of strings by taking from the
product of the underlying sets all elements $s=(s_{\alpha})$ such that
$s_{\alpha}\neq*$ for at most one index $\alpha$ (that is, the coproduct of
the pointed sets $\Sigma_{n_{\alpha}}$). For $s=(s_{\alpha})$ and
$t=(t_{\alpha})$ in $\coprod_{\alpha}\Sigma_{n_{\alpha}}$ set

$st:=(s_{\alpha}t_{\alpha}).$

For each index $\alpha$ and $0\leq i<n_{\alpha}$ we denote by $s_{\alpha,i}$
the _simple string_ $s$ given by $s_{\alpha}=s_{i}$.

Each coproduct $\coprod_{\alpha}\Sigma_{n_{\alpha}}$ comes with canonical
inclusions
$i_{\alpha}\colon\Sigma_{n_{\alpha}}\to\coprod_{\alpha}\Sigma_{n_{\alpha}}$
and projections
$p_{\alpha}\colon\coprod_{\alpha}\Sigma_{n_{\alpha}}\to\Sigma_{n_{\alpha}}$
satisfying $p_{\alpha}\circ i_{\alpha}=\operatorname{id}$.

Morphisms
$\coprod_{\alpha}\Sigma_{m_{\alpha}}\to\coprod_{\beta}\Sigma_{n_{\beta}}$ are
by definition maps $\phi$ between the underlying sets such that the composite
$p_{\beta}\circ\phi\circ i_{\alpha}$ is a morphism
$\Sigma_{m_{\alpha}}\to\Sigma_{n_{\beta}}$ for all $\alpha,\beta$.

######  Lemma 5.1.

There are canonical isomorphisms of pointed sets

$\operatorname{Hom}\Big{(}\coprod_{\alpha}\Sigma_{m_{\alpha}},\coprod_{\beta}\Sigma_{n_{\beta}}\Big{)}\xrightarrow{\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\prod_{\alpha}\operatorname{Hom}\Big{(}\Sigma_{m_{\alpha}},\coprod_{\beta}\Sigma_{n_{\beta}}\Big{)}\xleftarrow{\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\prod_{\alpha}\coprod_{\beta}\operatorname{Hom}(\Sigma_{m_{\alpha}},\Sigma_{n_{\beta}}).$

###### Proof.

Isomorphisms of pointed sets are nothing but bijections, but it is important
to take (co)products of pointed sets. The first bijection is induced by the
canonical inclusions
$\Sigma_{m_{\alpha}}\to\coprod_{\alpha}\Sigma_{m_{\alpha}}$. The second
bijection uses the fact that each morphism
$\Sigma_{m_{\alpha}}\to\coprod_{\beta}\Sigma_{n_{\beta}}$ factors through the
inclusion $\Sigma_{n_{\beta}}\to\coprod_{\beta}\Sigma_{n_{\beta}}$ for one
index $\beta$. ∎

We obtain the _category of finite strings_ which has as objects the finite
coproducts of connected strings.333We may consider the category
$\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(\Sigma^{\mathrm{op}},\mathrm{Set}_{*})$
of functors $\Sigma^{\mathrm{op}}\to\mathrm{Set}_{*}$ into the category of
pointed sets, which is the analogue of the category
$\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(\Delta^{\mathrm{op}},\mathrm{Set})$
of simplicial sets. Then the category of finite strings identifies via the
embedding $X\mapsto\operatorname{Hom}(-,X)|_{\Sigma}$ with the full
subcategory of finite coproducts of representable functors in
$\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(\Sigma^{\mathrm{op}},\mathrm{Set}_{*})$.

##  6\. Non-crossing partitions

We wish to describe the subobjects of $\Sigma_{n}$ in the category of finite
strings. This requires some preparations.

Let $S\subseteq\Sigma_{n}$. We call $S$ _thick_ if $*\in S$ and for any pair
$s,t\in S$ of non-zero strings we have $st\in S$, and moreover
$(s^{\prime},t^{\prime}-1),(t^{\prime},s^{\prime\prime}),(s^{\prime\prime}+1,t^{\prime\prime})\in
S$ provided that $s^{\prime}\leq t^{\prime}\leq s^{\prime\prime}\leq
t^{\prime\prime}$. We denote by $\operatorname{Thick}(S)$ the smallest thick
subset of $\Sigma_{n}$ containing $S$.

A set $S\subseteq\Sigma_{n}$ of non-zero strings is called _non-crossing_
provided that $s,t\in S$ and $s^{\prime}\leq t^{\prime}\leq
s^{\prime\prime}\leq t^{\prime\prime}$ implies $s=t$.

######  Lemma 6.1.

The assignment $S\mapsto\operatorname{Thick}(S)$ gives a bijection between the
non-crossing subsets and the thick subsets of $\Sigma_{n}$.

###### Proof.

The inverse map takes a thick subset $T\subseteq\Sigma_{n}$ to the unique non-
crossing subset $S\subseteq T$ with $\operatorname{Thick}(S)=T$. ∎

For non-crossing subsets $S,S^{\prime}$ of $\Sigma_{n}$ we set

$S\leq
S^{\prime}\quad:\iff\quad\operatorname{Thick}(S)\subseteq\operatorname{Thick}(S^{\prime}).$

This yields the structure of a poset. In fact, the non-crossing subsets form a
lattice since the thick subsets of $\Sigma_{n}$ are closed under
intersections. We denote this lattice by $\operatorname{NC}(\Sigma_{n})$.

Let $n\in\mathbb{N}$. A _partition_ $P=(P_{\alpha})$ of $[n]$ is given by
pairwise disjoint non-empty subsets $P_{\alpha}$ of $[n]$ such that
$\bigcup_{\alpha}P_{\alpha}=[n]$. Each partition is determined by the
corresponding set of strings $S(P)\subseteq\Sigma_{n-1}$, where by definition
$s=(s^{\prime},s^{\prime\prime})\in S(P)$ if for some $\alpha$ we have
$s^{\prime},s^{\prime\prime}\in P_{\alpha}$ and $i\not\in P_{\alpha}$ for all
$s^{\prime}<i\leq s^{\prime\prime}$. This is clear since any part
$P_{\alpha}=\\{a_{1}<a_{2}<\cdots<a_{r}\\}$ is determined by the corresponding
set of strings $S_{\alpha}=\\{(a_{1},a_{2}-1),\ldots,(a_{r-1},a_{r}-1)\\}$.

Call a subset $S\subseteq\Sigma_{n-1}$ of non-zero strings _partitioning_ when
for any $s,t\in S$ we have $s^{\prime}=t^{\prime}$ iff
$s^{\prime\prime}=t^{\prime\prime}$. In that case there is a unique partition
$P=P(S)$ such that $S(P)=S$. This yields a bijective correspondence between
partitions of $[n]$ and partitioning sets of strings in $\Sigma_{n-1}$.

A partition $P$ is _non-crossing_ provided given elements
$i<j<i^{\prime}<j^{\prime}$ with $i,i^{\prime}$ in the same part and
$j,j^{\prime}$ in the same part, then all elements belong to the same part.
The partitions of $[n]$ are partially ordered via _refinement_ , so $P\leq
P^{\prime}$ if any part of $P$ is contained in a part of $P^{\prime}$. The
non-crossing partitions then form a lattice which is denoted by
$\operatorname{NC}(n)$; cf. [16, 21].

######  Lemma 6.2.

There is a lattice isomorphism
$\operatorname{NC}(\Sigma_{n-1})\xrightarrow{\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\operatorname{NC}(n)$
which is given by $S\mapsto P(S)$.

###### Proof.

It is clear that $S\subseteq\Sigma_{n-1}$ is non-crossing if and only if
$P(S)$ is non-crossing. Let $S\leq S^{\prime}$. This means any $s\in S$ can be
written as $s=s_{1}s_{2}\cdots s_{r}$ with $s_{1},\ldots,s_{r}$ in
$S^{\prime}$. On the other hand, $P\leq P^{\prime}$ means that for any part
$P_{\alpha}=\\{a_{1}<a_{2}<\cdots<a_{u}\\}$ of $P$ and $t=(a_{i},a_{i+1}-1)\in
S(P)$, there is a part of $P^{\prime}$ containing $a_{i},a_{i+1}$ and
therefore $t=t_{1}t_{2}\cdots t_{r}$ with $t_{1},\ldots,t_{r}$ in
$S(P^{\prime})$. Thus $S\leq S^{\prime}$ if and only if $P(S)\leq
P(S^{\prime})$. ∎

We say that two monomorphisms $X_{1}\rightarrowtail X$ and
$X_{2}\rightarrowtail X$ are _equivalent_ if there exists an isomorphism
$X_{1}\to X_{2}$ making the following diagram commutative.

${X_{1}}$${X_{2}}$${X}$

An equivalence class of monomorphisms into $X$ is called a _subobject_ of $X$.
Given subobjects $X_{1}\rightarrowtail X$ and $X_{2}\rightarrowtail X$, we
write $X_{1}\leq X_{2}$ if there is a morphism $X_{1}\to X_{2}$ making the
above diagram commutative; this yields a partial order.

For a monomorphism $\phi\colon X\to\Sigma_{n}$ in the category of finite
strings we set

$S(\phi):=\\{\phi(s)\mid s\in X\text{ simple}\\}.$

######  Lemma 6.3.

Let $S\subseteq\Sigma_{n}$ be non-crossing. Then there exists a monomorphism
$\phi\colon X\to\Sigma_{n}$ such that $S(\phi)=S$.

###### Proof.

Consider the equivalence relation on $S$ generated by $s\sim t$ when
$st\neq\ast$. This yields a partition $S=\bigcup_{\alpha}S_{\alpha}$ and we
set $n_{\alpha}:=\operatorname{card}S_{\alpha}$. Using the fact that $S$ is
non-crossing, there is a unique morphism
$\phi\colon\coprod_{\alpha}\Sigma_{n_{\alpha}}\to\Sigma_{n}$ which identifies
the simple strings $s_{\alpha,i}$ with the elements in $S_{\alpha}$. Thus
$S=S(\phi)$. ∎

######  Lemma 6.4.

A morphism $\phi\colon X\to\Sigma_{n}$ is a monomorphism if and only if it is
given by an injective map.

###### Proof.

Clearly, any injective map yields a monomorphism. Thus we suppose that $\phi$
is a monomorphism and need to show that $\phi$ is given by an injective map.

Let $X=\coprod_{i=1}^{r}\Sigma_{n_{i}}$. The canonical decomposition of a
morphism $\Sigma_{n_{i}}\to\Sigma_{n}$ from Lemma 2.3 yields the case $r=1$.
For the general case we may assume that $r=2$. For each index $i$ the
restricted morphism $\phi_{i}\colon\Sigma_{n_{i}}\to\Sigma_{n}$ is given by an
injective map by the first case. Then each subset $\operatorname{Im}\phi_{i}$
is thick, and
$\operatorname{Im}\phi_{1}\cap\operatorname{Im}\phi_{2}=\operatorname{Thick}(S)$
for some non-crossing $S\subseteq\Sigma_{n}$. Let $\psi\colon Y\to\Sigma_{n}$
be the corresponding morphism with $S(\psi)=S$ which exists by Lemma 6.3.
Clearly, $\psi$ factors through each $\phi_{i}$ via a morphism $\psi_{i}\colon
Y\to\Sigma_{n_{i}}$. We obtain a diagram

${Y}$${X}$${\Sigma_{n}}$$\scriptstyle{\psi_{1}}$$\scriptstyle{\psi_{2}}$$\scriptstyle{\phi}$

where both composites equal $\psi$. Thus $\psi_{1}=\psi_{2}$ and therefore
$\operatorname{Im}\phi_{1}\cap\operatorname{Im}\phi_{2}=\\{*\\}$. We conclude
that $\phi$ is given by an injective map. ∎

######  Lemma 6.5.

Let $\phi\colon X\to\Sigma_{n}$ be a monomorphism. Then the set $S(\phi)$ is
non-crossing and we have
$\operatorname{Thick}(S(\phi))=\operatorname{Im}\phi$. Moreover, $\phi$
factors through a monomorphism $\phi^{\prime}\colon X^{\prime}\to\Sigma_{n}$
if and only if $S(\phi)\leq S(\phi^{\prime})$.

###### Proof.

Let $X=\coprod_{i=1}^{r}\Sigma_{n_{i}}$ and $m\in\mathbb{N}$. The set $S$ of
simple strings in $\Sigma_{m}$ is non-crossing and we have
$\operatorname{Thick}(S)=\Sigma_{m}$. This property is preserved under a
monomorphism $\Sigma_{m}\to\Sigma_{n}$ and yields the case $r=1$. The general
case follows since the restrictions
$\phi_{i}\colon\Sigma_{n_{i}}\to\Sigma_{n}$ satisfy
$\operatorname{Im}\phi_{i}\cap\operatorname{Im}\phi_{j}=\\{*\\}$ for $i\neq
j$, by Lemma 6.4.

For a monomorphism $\phi^{\prime}\colon X^{\prime}\to\Sigma_{n}$ we have

$\displaystyle S(\phi)\leq S(\phi^{\prime})$
$\displaystyle\iff\operatorname{Thick}(S(\phi))\subseteq\operatorname{Thick}(S(\phi^{\prime}))$
$\displaystyle\iff\operatorname{Im}\phi\subseteq\operatorname{Im}\phi^{\prime}$
$\displaystyle\iff\phi\text{ factors through }\phi^{\prime}.\qed$

######  Theorem 6.6.

Let $n\in\mathbb{N}$. The subobjects of $\Sigma_{n}$ in the category of finite
strings form a lattice which is canonically isomorphic to the lattice of non-
crossing partitions $\operatorname{NC}(n+1)$. The isomorphism sends a
monomorphism $\phi\colon X\to\Sigma_{n}$ to $P(S(\phi))$.

This result could be deduced from [11, 13], using the correspondence between
strings and representations from Theorem 4.9, which identifies thick subsets
of $\Sigma_{n}$ with thick subcategories of $\mathcal{A}_{n}$. We refer to
[22, §4] for a detailed exposition. The following is a direct proof.

###### Proof.

The assignment $\phi\mapsto P(S(\phi))$ gives a well defined map from the
poset of subobjects of $\Sigma_{n}$ to $\operatorname{NC}(n+1)$ by Lemmas 6.2
and 6.5. In fact, the map is injective and $\phi$ factors through a
monomorphism $\phi^{\prime}$ if and only if $P(S(\phi))\leq
P(S(\phi^{\prime}))$. Thus it remains to show surjectivity. Let
$P\in\operatorname{NC}(n+1)$ and set $S=S(P)$. Then there is a morphism
$\phi\colon\coprod_{\alpha}\Sigma_{n_{\alpha}}\to\Sigma_{n}$ satisfying
$S=S(\phi)$ by Lemma 6.3. Thus $P=P(S(\phi))$. ∎

##  7\. Cyclic strings

We enlarge the category of finite strings and add cyclic strings as follows.
Let $\Sigma_{\mathbb{Z}}$ denote the set of all basic strings. There is a
natural action of the group of integers given by

$*^{z}:=*\qquad\text{and}\qquad
s^{z}:=(s^{\prime}+z,s^{\prime\prime}+z)\qquad\text{for
}s=(s^{\prime},s^{\prime\prime}),\,z\in\mathbb{Z}.$

For $n>0$ the _cyclic string_ of length $n$ is the set of orbits with respect
to the action of the subgroup $(n)=n\mathbb{Z}$. Thus

$\tilde{\Sigma}_{n}:=\\{s^{(n)}\mid s\in\Sigma_{\mathbb{Z}}\\}=\\{s^{ni}\mid
s\in\Sigma_{\mathbb{Z}},\,i\in\mathbb{Z}\\}$

with multiplication given by

$s^{(n)}t^{(n)}:=u^{(n)}$

where $u=*$ except when there is a pair of integers $i,j$ such that
$s^{ni}t^{nj}=u\neq*$. We set $\tilde{\Sigma}_{0}:=\\{*\\}$.

A morphism $\phi\colon\tilde{\Sigma}_{m}\to\tilde{\Sigma}_{n}$ is by
definition a map satisfying (2.1). We define _standard morphisms_ which are
given by their values on simple strings as in (2.3). Let $n\geq 1$. Then the
morphism

$\tilde{\delta}^{i}_{n}\colon\tilde{\Sigma}_{n-1}\longrightarrow\tilde{\Sigma}_{n}\qquad(0\leq
i<n)$

is given by the injective map such that $s^{(n)}_{i-1}$ and $s^{(n)}_{i}$ are
not in its image, and the morphism

$\tilde{\sigma}^{i}_{n}\colon\tilde{\Sigma}_{n}\longrightarrow\tilde{\Sigma}_{n-1}\qquad(0\leq
i<n)$

is given by the surjective map such that $s^{(n)}_{i}$ is sent to the zero
string. The _cyclic permutation_

$\tau^{i}_{n}\colon\tilde{\Sigma}_{n}\longrightarrow\tilde{\Sigma}_{n}\qquad(0\leq
i<n)$

is given by $s^{(n)}\mapsto(s^{i})^{(n)}$.

######  Lemma 7.1.

The standard morphisms satisfy the identities (2.2), and every morphism
$\phi\colon\tilde{\Sigma}_{m}\to\tilde{\Sigma}_{n}$ admits a unique
decomposition

$\phi=\delta_{n}^{i_{u}}\circ\delta_{n-1}^{i_{u-1}}\circ\cdots\circ\delta_{n-u}^{i_{0}}\circ\sigma_{m-v}^{j_{v}}\circ\cdots\circ\sigma_{m-1}^{j_{1}}\circ\sigma_{m}^{j_{0}}\circ\tau_{m}^{k}$

with $0\leq i_{0}<\cdots<i_{u}<n$, $0\leq j_{v}<\cdots<j_{0}<m$, $0\leq k<m$,
and $n-u=m-v$.

###### Proof.

Adapt the proof of Lemmas 2.1 and 2.3. The only difference arises from cyclic
permutations. ∎

Next we consider morphisms $\Sigma_{m}\to\tilde{\Sigma}_{n}$ and
$\tilde{\Sigma}_{m}\to\Sigma_{n}$, which are by definition maps satisfying
(2.1). Let $n\geq 1$. The standard morphism

$\varepsilon^{i}_{n}\colon\Sigma_{n-1}\longrightarrow\tilde{\Sigma}_{n}\qquad(0\leq
i<n)$

is given by the unique injective map such that $s^{(n)}_{i}$ is not in its
image. We consider the morphism

${\tilde{\Sigma}_{n}}$${\tilde{\Sigma}_{n}}$${\tilde{\Sigma}_{n-1}}$${\cdots}$${\tilde{\Sigma}_{1}}$$\scriptstyle{\tau_{n}^{-i}}$$\scriptstyle{\tilde{\sigma}_{n}^{n-1}}$$\scriptstyle{\tilde{\sigma}_{n-1}^{n-2}}$$\scriptstyle{\tilde{\sigma}_{2}^{1}}$

and note that its _kernel_ (that is, the set of elements sent to the zero
string) equals the image of $\varepsilon_{n}^{i}$.

######  Lemma 7.2.

Let $m,n\in\mathbb{N}$. Every morphism $\Sigma_{m}\to\tilde{\Sigma}_{n}$
factors through $\varepsilon^{i}_{n}$ for some $0\leq i<n$, and every morphism
$\tilde{\Sigma}_{m}\to\Sigma_{n}$ factors through
$\tilde{\Sigma}_{0}=\Sigma_{0}$.

###### Proof.

First consider a morphism $\phi\colon\Sigma_{m}\to\tilde{\Sigma}_{n}$. It is
easily checked that the longest string $s^{(n)}$ in the image of $\phi$ has
length at most $n-1$, because the image of $\phi$ is finite. Choose an index
$i$ such that the simple $s_{i}$ does not arise as a composition factor of
$s$. It follows that $\phi$ factors through $\varepsilon_{n}^{i}$, since all
composition factors of strings in the image of $\phi$ are composition factors
of $s$.

Now consider for $\psi\colon\tilde{\Sigma}_{m}\to\Sigma_{n}$ its epi-mono
factorisation $\psi=\psi^{\prime\prime}\circ\psi^{\prime}$. The image is of
the form $\tilde{\Sigma}_{p}$ for some $p\leq m$. Because
$\psi^{\prime\prime}$ is injective and $\tilde{\Sigma}_{p}$ is infinite for
$p>0$ we conclude that $p=0$. ∎

Let us consider the category of all connected strings (linear and cyclic). The
objects are of the form $\Sigma_{n}$ or $\tilde{\Sigma}_{n}$ with
$n\in\mathbb{N}$. As before, we add finite coproducts and obtain the _enlarged
category of finite strings_. The objects are of the form

$\Big{(}\coprod_{\alpha}\Sigma_{m_{\alpha}}\Big{)}\amalg\Big{(}\coprod_{\beta}\tilde{\Sigma}_{n_{\beta}}\Big{)}$

given by a finite set of natural numbers $m_{\alpha}$ and $n_{\beta}$.

Let $k$ be a field. For the quiver

${n-1}$${n-2}$${\cdots}$${1}$${0}$${(n\geq 1)}$

we denote by $\tilde{\mathcal{A}}_{n}$ the category of all finite dimensional
and nilpotent $k$-linear representations. Then we have the following analogue
of Lemma 4.1.

######  Lemma 7.3.

Let $k$ be a field and $\mathcal{A}$ a $k$-linear abelian category. Suppose
that $\mathcal{A}$ satisfies _(Ab1)–(Ab5)_ but not _(Ab6)_. Then there is an
equivalence
$\mathcal{A}\xrightarrow{\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\tilde{\mathcal{A}}_{n}$,
where $n$ equals the number of isomorphism classes of simple objects in
$\mathcal{A}$.

###### Proof.

See for example the description of uniserial categories in [1]. ∎

We continue with analogues of Lemmas 4.2 and 4.4. Let $n\geq 1$. The
indecomposable objects of $\tilde{\mathcal{A}}_{n}$ are parameterised by the
elements of $\tilde{\Sigma}_{n}\setminus\\{*\\}$. There are canonical
recollements of abelian categories

${\mathcal{A}_{1}}$${\tilde{\mathcal{A}}_{n}}$${\tilde{\mathcal{A}}_{n-1}}$${(0\leq
i<n)}$$\scriptstyle{\tilde{s}_{n}^{i}}$$\scriptstyle{\tilde{d}_{n}^{i}}$$\scriptstyle{\tilde{d}_{n}^{i+1}}$

(with $\tilde{d}_{n}^{0}=\tilde{d}_{n}^{n}$) and composing them yields a
recollement

${\mathcal{A}_{n-1}}$${\tilde{\mathcal{A}}_{n}}$${\tilde{\mathcal{A}}_{1}.}$$\scriptstyle{e_{n}^{0}}$$\scriptstyle{\tilde{s}_{2}^{1}\cdots\tilde{s}_{n}^{n-1}}$

Furthermore, there are equivalences

$t_{n}^{i}\colon\tilde{\mathcal{A}}_{n}\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\tilde{\mathcal{A}}_{n}\qquad(0\leq i<n)$

which are given by a cyclic permutation $S_{j}\mapsto S_{j+i}$ of the simple
representations. We obtain a correspondence between standard morphisms in
$\tilde{\Sigma}_{n}$ and exact functors:

$\tilde{\delta}_{n}^{i}\longleftrightarrow\tilde{d}_{n}^{i}\qquad\tilde{\sigma}_{n}^{i}\longleftrightarrow\tilde{s}_{n}^{i}\qquad\varepsilon_{n}^{i}\longleftrightarrow
e_{n}^{i}\qquad\tau_{n}^{i}\longleftrightarrow t_{n}^{i}\qquad(0\leq i<n).$

The following result generalises Theorem 4.9. As before, we consider
$k$-linear abelian categories together with $k$-linear exact functors, up to
natural isomorphism, that admit a homological factorisation.

######  Theorem 7.4.

Let $k$ be a field. The assignments $\Sigma_{n}\mapsto\mathcal{A}_{n}$ and
$\tilde{\Sigma}_{n}\mapsto\tilde{\mathcal{A}}_{n}$ provide an equivalence
between the enlarged category of finite strings and the category of $k$-linear
abelian categories satisfying _(Ab2)–(Ab5)_.

###### Proof.

We adapt the proof of Theorem 4.9. Any $k$-linear abelian category satisfying
(Ab2)–(Ab5) decomposes into a finite coproduct of connected abelian
categories, which are (up to an equivalence) of the form $\mathcal{A}_{n}$ or
$\tilde{\mathcal{A}}_{n}$, respectively, by Lemmas 4.1 and 7.3. Thus we obtain
a bijection between the isomorphism classes of objects. It remains to consider
the morphisms, and we may restrict ourselves to connected categories. An exact
functor $F\colon\mathcal{A}\to\mathcal{B}$ which admits a homological
factorisation sends indecomposable objects either to indecomposables or to
zero; see Lemma 4.6. This yields a morphism
$\phi\colon\Sigma_{\mathcal{A}}\to\Sigma_{\mathcal{B}}$ between the
corresponding strings, given by $M_{\phi(s)}=F(M_{s})$ for each
$s\in\Sigma_{\mathcal{A}}$. The assignment $F\mapsto\phi$ is injective since
$k$-linear exact functors which admit a homological factorisation are
naturally isomorphic when they coincide on indecomposable objects; see Lemma
4.7. The assignment is surjective, by Lemmas 2.3 and 7.1, in combination with
Lemma 7.2. ∎

##  8\. Non-crossing partitions of type $B$

We wish to describe the subobjects of $\tilde{\Sigma}_{n}$ in the enlarged
category of finite strings. This description is parallel to that for
$\Sigma_{n}$ and involves the non-crossing partitions of type $B$.

Let $S\subseteq\Sigma_{\mathbb{Z}}$. We call $S$ _thick_ if $*\in S$ and for
any pair $s,t\in S$ of non-zero strings we have $st\in S$, and moreover
$(s^{\prime},t^{\prime}-1),(t^{\prime},s^{\prime\prime}),(s^{\prime\prime}+1,t^{\prime\prime})\in
S$ provided that $s^{\prime}\leq t^{\prime}\leq s^{\prime\prime}\leq
t^{\prime\prime}$. We denote by $\operatorname{Thick}(S)$ the smallest thick
subset of $\Sigma_{\mathbb{Z}}$ containing $S$.

A set $S\subseteq\Sigma_{\mathbb{Z}}$ of non-zero strings is called _non-
crossing_ provided that $s,t\in S$ and $s^{\prime}\leq t^{\prime}\leq
s^{\prime\prime}\leq t^{\prime\prime}$ implies $s=t$.

For $n>0$ consider the canonical projection
$p\colon\Sigma_{\mathbb{Z}}\to\tilde{\Sigma}_{n}$. Then a subset
$S\subseteq\tilde{\Sigma}_{n}$ is _thick_ if $p^{-1}(S)$ is thick, and $S$ is
_non-crossing_ if $p^{-1}(S)$ is non-crossing. Note that $\ell(s)\leq n$ for
any $s^{(n)}$ when $S$ is non-crossing.

######  Lemma 8.1.

The assignment $S\mapsto\operatorname{Thick}(S)$ gives a bijection between the
non-crossing subsets and the thick subsets of $\tilde{\Sigma}_{n}$.

###### Proof.

The inverse map takes a thick subset $T\subseteq\tilde{\Sigma}_{n}$ to the
unique non-crossing subset $S\subseteq T$ with $\operatorname{Thick}(S)=T$. ∎

For non-crossing subsets $S,S^{\prime}$ of $\tilde{\Sigma}_{n}$ we set

$S\leq
S^{\prime}\quad:\iff\quad\operatorname{Thick}(S)\subseteq\operatorname{Thick}(S^{\prime}).$

This yields the structure of a poset. In fact, the non-crossing subsets form a
lattice since the thick subsets of $\tilde{\Sigma}_{n}$ are closed under
intersections. We denote this lattice by
$\operatorname{NC}(\tilde{\Sigma}_{n})$.

Let $n\in\mathbb{N}$. We consider the set

$[2n]=\\{0,1,\cdots,n-1,\bar{0},\bar{1},\cdots,\overline{n-1}\\},$

where $\bar{x}$ is identified with $x+n$ for $0\leq x<n$, and
$\bar{\bar{x}}:=x$. For a partition $P=(P_{\alpha})$ of $[2n]$ we require that
$\overline{P_{\alpha}}$ is a part of $P$ for each $\alpha$. Each partition is
determined by the corresponding set of strings
$S(P)\subseteq\tilde{\Sigma}_{n}$, where by definition
$s^{(n)}\in\tilde{\Sigma}_{n}$ with $0\leq s^{\prime}<n$ belongs to $S(P)$ if
for some $\alpha$ we have $s^{\prime},s^{\prime\prime}\in P_{\alpha}$ and
$i\not\in P_{\alpha}$ for all $s^{\prime}<i\leq s^{\prime\prime}$. The
partitions of $[2n]$ are partially ordered via refinement, and the non-
crossing partitions then form a lattice which is denoted by
$\operatorname{NC}^{B}(n)$; cf. [19, 21].

######  Lemma 8.2.

There is a lattice isomorphism
$\operatorname{NC}(\tilde{\Sigma}_{n})\xrightarrow{\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\operatorname{NC}^{B}(n)$
which is given by $S\mapsto P(S)$.

###### Proof.

Adapt the proof of Lemma 6.2. ∎

######  Remark 8.3.

Let $P=(P_{\alpha})$ be a non-crossing partition of $[2n]$ and denote by
$S=(S_{\alpha})$ the corresponding partition of $S=S(P)$. Then $P$ has at most
one part $P_{\alpha}$ satisfying $\overline{P_{\alpha}}=P_{\alpha}$. In fact,
$\overline{P_{\alpha}}=P_{\alpha}$ holds if and only if
$\operatorname{Thick}(S_{\alpha})$ is infinite.

As before, we write $P(S)$ for the partition of $[2n]$ corresponding to a non-
crossing set $S\subseteq\tilde{\Sigma}_{n}$. For a monomorphism $\phi\colon
X\to\tilde{\Sigma}_{n}$ in the category of finite strings we set

$S(\phi):=\\{\phi(s)\mid s\in X\text{ simple}\\}.$

######  Theorem 8.4.

Let $n\in\mathbb{N}$. The subobjects of $\tilde{\Sigma}_{n}$ in the enlarged
category of finite strings form a lattice which is canonically isomorphic to
the lattice of non-crossing partitions $\operatorname{NC}^{B}(n)$. The
isomorphism sends a monomorphism $\phi\colon X\to\tilde{\Sigma}_{n}$ to
$P(S(\phi))$.

###### Proof.

Adapt the proof of Theorem 6.6. ∎

##  9\. Thick subcategories

Results about subobjects in categories of strings correspond to statements
about thick subcategories of abelian categories, because of the correspondence
from Theorem 7.4.

Recall that a full subcategory of an abelian category is _thick_ if it is
closed under direct summands and the two out of three property holds for any
short exact sequence.

######  Lemma 9.1.

Let $k$ be a field and let $\mathcal{A}$ be a $k$-linear abelian category
satisfying _(Ab2)–(Ab5)_. Then every thick subcategory of $\mathcal{A}$
satisfies again _(Ab2)–(Ab5)_.

###### Proof.

Let $\mathcal{C}\subseteq\mathcal{A}$ be a thick subcategory. Then
$\mathcal{C}$ is closed under images of morphisms in $\mathcal{C}$ because
$\mathcal{A}$ is hereditary. It follows that the category $\mathcal{C}$ is
abelian and again hereditary. Also, $\mathcal{C}$ is necessarily a length
category. If $X\in\mathcal{C}$ is simple, then $\operatorname{End}(X)$ is
isomorphic to $k[t]/(t^{p})$ for some $p\geq 1$, since $X$ is indecomposable
in $\mathcal{A}$. Schur’s lemma then implies $p=1$. It remains to show that
$\mathcal{C}$ is a uniserial category with finitely many simple objects. We
may assume that either $\mathcal{A}=\mathcal{A}_{n}$ or
$\mathcal{A}=\tilde{\mathcal{A}}_{n}$ for some $n\in\mathbb{N}$. Then a
representative set of simple objects in $\mathcal{C}$ identifies with a non-
crossing subset $S$ in $\Sigma_{n}$ or $\tilde{\Sigma}_{n}$. The set $S$ is
finite since the length of any string in $S$ is bounded by $n$. Let
$M_{s},M_{t}$ be simple objects in $\mathcal{C}$ corresponding to strings
$s,t\in S$. Then $\operatorname{Ext}^{1}(M_{t},M_{s})\neq 0$ iff $st\neq*$. It
is clear that for each $s\in S$ there is at most one $t\in S$ with $st\neq*$,
and dually there is at most one $r\in S$ with $rs\neq*$. Then a criterion from
[1] implies that $\mathcal{C}$ is uniserial. ∎

Now we can deduce classifications of thick subcategories from Theorems 6.6 and
8.4. The first part is due to Ingalls and Thomas [13] and only included for
completeness; the second part seems to be new.

######  Corollary 9.2.

Let $k$ be a field and $n\in\mathbb{N}$.

  1. (1)

There is a canonical isomorphism between the lattice of thick subcategories of
$\mathcal{A}_{n}$ and the lattice $\operatorname{NC}(n+1)$.

  2. (2)

There is a canonical isomorphism between the lattice of thick subcategories of
$\tilde{\mathcal{A}}_{n}$ and the lattice $\operatorname{NC}^{B}(n)$.

###### Proof.

We apply Lemma 9.1. From the homological factorisation of an exact functor it
follows that each subobject of $\mathcal{A}_{n}$ or $\tilde{\mathcal{A}}_{n}$
is given by the inclusion of a thick subcategory. On the other hand, all thick
subcategories arise in this way. Theorem 7.4 provides the correspondence with
subobjects of $\Sigma_{n}$ and $\tilde{\Sigma}_{n}$, respectively. Then the
assertion follows for $\mathcal{A}_{n}$ from Theorem 6.6 and for
$\tilde{\mathcal{A}}_{n}$ from Theorem 8.4. ∎

######  Remark 9.3.

(1) The classification of thick subcategories for abelian categories of the
form $\mathcal{A}_{n}$ or $\tilde{\mathcal{A}}_{n}$ given by a field and
$n\in\mathbb{N}$ generalises to any connected hereditary and uniserial length
category with finitely many isomorphism classes of simple objects. The proof
is essentially the same, because indecomposable objects can be identified with
strings which encode their composition series. Then thick subcategories
correspond bijectively to non-crossing sets of strings.

(2) The category of regular modules over a tame hereditary algebra is an
example of an hereditary and uniserial length category [6]. For the module
category of a tame hereditary algebra one can show that any thick subcategory
is contained in the thick subcategory of regular modules, provided it is not
generated by an exceptional sequence [5]. This yields a classification of all
thick subcategories, complementing the work in [11, 12, 13].

(3) For an hereditary abelian category $\mathcal{A}$, thick subcategories of
the bounded derived category $\mathbf{D}^{b}(\mathcal{A})$ correspond
bijectively to thick subcategories of $\mathcal{A}$ via

$\mathbf{D}^{b}(\mathcal{A})\supseteq\mathcal{C}\longmapsto\\{H^{0}(X)\in\mathcal{A}\mid
X\in\mathcal{C}\\}\subseteq\mathcal{A},$

see [15, Proposition 4.4.17].

##  Appendix A Basic strings

The properties of basic strings and the connection with linear representations
become more transparent if we consider analogues of face and degeneracy maps
for the poset of integers. We view this poset as a category, and this means
that morphisms $\mathbb{Z}\to\mathbb{Z}$ are viewed as functors. For
$i\in\mathbb{Z}$ we define morphisms

$\delta^{i}\colon\mathbb{Z}\longrightarrow\mathbb{Z},\qquad
j\mapsto\begin{cases}j&j<i\\\ j+1&j\geq i\end{cases}$

and

$\sigma^{i}\colon\mathbb{Z}\longrightarrow\mathbb{Z},\qquad
j\mapsto\begin{cases}j&j\leq i\\\ j-1&j>i.\end{cases}$

These satisfy the simplicial identities (2.2). Moreover, they are related via
adjunctions:

$\cdots\dashv\delta^{i+1}\dashv\sigma^{i}\dashv\delta^{i}\dashv\sigma^{i-1}\dashv\cdots$

Let $k$ be a field. Then for each $i\in\mathbb{Z}$ precomposition with
$\delta^{i}$ and $\sigma^{i}$ yields exact functors

${\operatorname{Rep}(\mathbb{Z}^{\mathrm{op}},k)}$${\operatorname{Rep}(\mathbb{Z}^{\mathrm{op}},k).}$$\scriptstyle{(\delta^{i})^{*}}$$\scriptstyle{(\sigma^{i})^{*}}$

This assignment is contravariant and therefore reverses the directions of
functors. Thus the dual simplicial identities but the same adjunctions

$\cdots\dashv(\delta^{i+1})^{*}\dashv(\sigma^{i})^{*}\dashv(\delta^{i})^{*}\dashv(\sigma^{i-1})^{*}\dashv\cdots$

are satisfied.

For $i\in\mathbb{Z}$ let $S_{i}$ denote the simple representation concentrated
in $i$, that is, $S_{i}(j)=0$ for all $j\neq i$. Let $n\in\mathbb{N}$. Then
precomposition with the inclusion $[n]\to\mathbb{Z}$ yields an exact functor

$\operatorname{Rep}(\mathbb{Z}^{\mathrm{op}},k)\longrightarrow\operatorname{Rep}([n]^{\mathrm{op}},k)=\mathcal{A}_{n}$

which becomes an equivalence when restricted to the full subcategory of
objects in $\operatorname{Rep}(\mathbb{Z}^{\mathrm{op}},k)$ with composition
factors in $\\{S_{0},\ldots,S_{n-1}\\}$. Viewing this as an identification,
the functors $(\delta^{i})^{*}$ and $(\sigma^{i-1})^{*}$ restrict to exact
functors

$\mathcal{A}_{n}\xrightarrow{\ (\delta^{i})^{*}\
}\mathcal{A}_{n-1}\qquad(0\leq i<n)$

and

$\mathcal{A}_{n-1}\xrightarrow{\ (\sigma^{i-1})^{*}\
}\mathcal{A}_{n}\qquad(0\leq i\leq n).$

Recall that $\Sigma_{\mathbb{Z}}$ denotes the set of basic strings. Then
$\Sigma_{\mathbb{Z}}\setminus\\{*\\}$ identifies with the indecomposable
objects of finite length in $\operatorname{Rep}(\mathbb{Z}^{\mathrm{op}},k)$
via $s\mapsto M_{s}$, as in Lemma 4.2. This identification yields maps
$\Sigma_{\mathbb{Z}}\to\Sigma_{\mathbb{Z}}$ which are induced by
$(\delta^{i})^{*}$ and $(\sigma^{i})^{*}$, respectively. Restricting these
maps for any $n\in\mathbb{N}$ to the set $\Sigma_{n}$ of basic strings with
composition factors in $\\{s_{0},\ldots,s_{n-1}\\}$ gives

$\sigma_{n}^{i}=(\delta^{i})^{*}|_{\Sigma_{n}}\qquad\text{and}\qquad\delta_{n}^{i}=(\sigma^{i-1})^{*}|_{\Sigma_{n-1}}.$

Then the following elementary observation (reflecting a duality for the
simplicial category $\Delta$, cf. [18, VIII.7]) explains the simplicial
relations (and any further properties) for
$\delta^{i}_{n}\colon\Sigma_{n-1}\to\Sigma_{n}$ and
$\sigma^{i}_{n}\colon\Sigma_{n}\to\Sigma_{n-1}$.

######  Lemma A.1.

Consider symbols $(\delta^{i},\sigma^{i})$ and $(d^{i},s^{i})$ for some
integers $i\in\mathbb{Z}$. After substituting $\delta^{i}\mapsto s^{i}$ and
$\sigma^{i}\mapsto d^{i+1}$ and reversing the order of composition, the
identities (2.2) hold for $(d^{i},s^{i})$ if and only if they hold for
$(\delta^{i},\sigma^{i})$.∎

## References

  * [1] I. Kr. Amdal and F. Ringdal, Catégories unisérielles, C. R. Acad. Sci. Paris Sér. A-B 267 (1968), A85–A87 and A247–A249.
  * [2] M. C. R. Butler and C. M. Ringel, Auslander–Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra 15 (1987), no. 1-2, 145–179.
  * [3] P. Cartier, Homologie cyclique: rapport sur des travaux récents de Connes, Karoubi, Loday, Quillen$\ldots$, Astérisque No. 121-122 (1985), 123–146.
  * [4] A. Connes, Cohomologie cyclique et foncteurs ${\rm Ext}^{n}$, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 23, 953–958.
  * [5] N. D. Dichev, Thick subcategories for quiver representations, PhD thesis, Universität Paderborn, 2009.
  * [6] V. Dlab and C. M. Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. 6 (1976), no. 173, v+57 pp.
  * [7] T. Dyckerhoff, G. Jasso and T. Walde, Simplicial structures in higher Auslander-Reiten theory, Adv. Math. 355 (2019), 106762, 73 pp.
  * [8] P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448.
  * [9] P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Springer-Verlag New York, Inc., New York, 1967.
  * [10] M. Herschend and P. Jørgensen, Classification of higher wide subcategories for higher Auslander algebras of type $A$, J. Pure Appl. Algebra 225 (2021), no. 5, Paper No. 106583, 22 pp.
  * [11] A. Hubery and H. Krause, A categorification of non-crossing partitions, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 10, 2273–2313.
  * [12] K. Igusa and R. Schiffler, Exceptional sequences and clusters, J. Algebra 323 (2010), no. 8, 2183–2202.
  * [13] C. Ingalls and H. Thomas, Noncrossing partitions and representations of quivers, Compos. Math. 145 (2009), no. 6, 1533–1562.
  * [14] O. Iyama, Cluster tilting for higher Auslander algebras, Adv. Math. 226 (2011), no. 1, 1–61.
  * [15] H. Krause, Homological Theory of Representations, Cambridge Studies in Advanced Mathematics, 195, Cambridge University Press, Cambridge, 2021.
  * [16] G. Kreweras, Sur les partitions non croisées d’un cycle, Discrete Math. 1 (1972), no. 4, 333–350.
  * [17] S. Mac Lane, Categories for the working mathematician, second edition, Graduate Texts in Mathematics, 5, Springer-Verlag, New York, 1998.
  * [18] S. Mac Lane and I. Moerdijk, Sheaves in geometry and logic, corrected reprint of the 1992 edition, Universitext, Springer-Verlag, New York, 1994.
  * [19] V. Reiner, Non-crossing partitions for classical reflection groups, Discrete Math. 177 (1997), no. 1-3, 195–222.
  * [20] C. M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, 1099, Springer-Verlag, Berlin, 1984.
  * [21] R. Simion, Noncrossing partitions, Discrete Math. 217 (2000), no. 1-3, 367–409.
  * [22] C. M. Ringel, The Catalan combinatorics of the hereditary Artin algebras, in Recent developments in representation theory, 51–177, Contemp. Math., 673, Amer. Math. Soc., Providence, RI, 2016.