The category of finite strings

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.


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 ∆ (cf.[8,17]), once the initial and terminal objects in ∆ 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 ∆ 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 n-string that the lattice of subobjects is isomorphic to the lattice NC(n + 1) of non-crossing partitions [16], while the lattice NC B (n) of type B non-crossing partitions [19] arises for a cyclic n-string.
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 H.Krause 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. Also, I am grateful to an anonymous referee for pointing out an inaccuracy in an earlier version.

Connected strings
In this work we introduce the category of finite strings.For any natural number n the connected n-string is denoted by Σ 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 ′ , s ′′ ) of integers s ′ ⩽ s ′′ .We write ℓ(s) = s ′′ − s ′ + 1 for the length of s and add a zero string * satisfying ℓ( * ) = 0. Strings of length one are called simple and we set s i := (i, i), i ∈ Z.For a string s = (i, j) we call the simple strings s i , s i+1 , . . ., s j its composition factors.
A multiplication of basic strings is given by concatenation.For s, t set and the product st is undefined otherwise.This multiplication is associative.
For n ∈ N = {0, 1, 2, . ..} the connected n-string is by definition the set of basic strings The string * plays the role of a base point, and morphism are base point preserving.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 ⩾ 1.The morphism is given by the unique injective map such that s i−1 and s i are not in its image.Note that δ 0 is given by the unique surjective map that sends s i to the zero string.

The category of finite strings
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) Proof.This is easily checked, for instance by tracing the images of the simple strings.We have It remains to note that any morphism ϕ is determined by the images ϕ(s j ). □ Small examples are easily computed.We have Note that for any n ∈ N the maps δ 0 n and δ n n are special; they preserve the length of basic strings since We denote by Σ the category of connected strings with objects given by the strings Σ n , n ∈ N.
Any morphism can be written in some canonical form.First observe that there is a canonical epi-mono factorisation.
( 1) We need to exclude δ 0 1 as a factor and choose instead δ 1 1 in order to achieve uniqueness.

H. Krause
Proof.Let us denote by Σ ′ the category generated by the objects Σ n , the morphisms δ i n , σ j m , plus the relations δ 0 1 = δ 1 1 and (2).Since the relations are satisfied in Σ, there is a unique functor Σ ′ → Σ which induces the identity on the objects and on the morphisms δ i n and σ j m .Since every morphism in Σ is a composite of morphisms δ i n and σ j m , the induced map (2) in Σ ′ are satisfied, there are decompositions (4) in Σ ′ for ϕ and ψ.These decompositions coincide in Σ by Lemma 2.3, and therefore ϕ = ψ.□ Example 2.5.For any n ⩾ 3 we have the following pullback in Σ.

The simplicial category
Let ∆ 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, . . ., n − 1}, n ∈ N, and the morphisms ϕ unique injective map not taking the value i) and for n ⩾ 1 the degeneracy map (the unique surjective map taking twice the value i) which are known to satisfy the simplicial identities (2).In fact, the category ∆ is generated by the objects [n], the morphisms δi n , σj m , and the identities (2); see [8, §II.2] or [17, §VII.5].We write ∆ = ∆[α −1 ] for the category which is obtained by formally inverting the morphism α = δ0 0 .This amounts to identifying the initial and the terminal object in ∆.
Proposition 3.1.The assignments 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 δ0 0 and induces therefore a functor ∆ → Σ, 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 δ0 The functor p : ∆ → Σ admits two sections s 0 and s 1 that are given by ) for i = 0, 1.Let us summarise.We have for all n ⩾ 1 diagrams The category of finite strings satisfying the simplicial identities, but a difference from the usual simplex category arises because of the extra identity δ 0 1 = δ 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.

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 → 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 → y if and only if x ⩽ y).Morphisms between representations are the natural transformations, and we denote by Rep(P, k) the category of all finite dimensional k-linear representations.
For a k-linear abelian category A over a field k we consider the following conditions.(Ab1) A is connected, that is, 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.(Ab3) A is hereditary, that is, Ext 2 vanishes.(Ab4) A is uniserial, that is, every indecomposable object has a unique composition series.(Ab5) A is split, that is, End(S) ∼ = k for every simple object S. (Ab6) A is of finite type, that is, there are only finitely many isomorphism classes of indecomposable objects.Proof.See for example the description of uniserial categories in [1]. □ From now on we fix a field k and set of k-spaces.For any string s ∈ Σ n we define a representation M s ∈ A n as follows.Set M * = 0.For s = (s ′ , s ′′ ) let M s be the representation (2) Observe that the composition length of M s equals ℓ(s), and the composition factors of M s correspond bijectively to the composition factors of s. (1) The assignment s → M s induces a bijection between Σ n ∖ { * } and the isomorphism classes of indecomposable objects in 2) This is also known as string module in the terminology of [2].

H. Krause
Next we specify the class of exact functors which arises naturally in our context.For each exact functor F : A → B between abelian categories we denote by Ker F the full subcategory of A given by the objects X ∈ A such that F X = 0.This is a Serre subcategory and we denote by A/(Ker F ) the corresponding quotient, cf.[9].
We say that an exact functor F : A → B between abelian categories admits a homological factorisation if the induced functor A/A ′ → B with A ′ = Ker F induces for all objects X, Y ∈ A bijections 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 : A → B between hereditary abelian categories admits a homological factorisation if and only if A/(Ker F ) identifies with a thick subcategory of B.
Proof.Set A ′ = Ker F and suppose F identifies A/A ′ with a full subcategory B ′ ⊆ B. Clearly, B ′ is closed under kernels and cokernels of morphisms since F is exact.The subcategory B ′ is extension closed if and only if the induced map Ext Not all exact functors admit a homological factorisation.A simple example is for any field k the exact functor mod k → mod k given by X → X ⊗ k k 2 .
For m, n ∈ N we denote by Hom(A m , A n ) the set of k-linear exact functors A m → A n , up to natural isomorphism, that admit a homological factorisation.We define natural maps

canonical recollements of abelian categories
The

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The category of finite strings For n ⩾ 1 we set One checks that these functors satisfy the identities (2).Thus the assignment extends uniquely to maps β mn : Hom(Σ m , Σ n ) → Hom(A m , A n ) for all m, n ∈ N, using Lemma 2.4.Proof.The functor F identifies A m /(Ker F ) with a full subcategory of A n .The canonical functor A m → A m /(Ker F ) can be written as composite of functors of the form s i p : A p → A p−1 , which map indecomposables either to indecomposables or to zero.Thus for any s ∈ Σ m we have F (M s ) = M t for some t ∈ Σ n , using Lemma 4.2.This yields a morphism ϕ : Σ m → Σ n by setting ϕ(s) = t.□ The above lemma provides maps α mn : Proof.The category 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 ∈ N. Then β mn • α mn = id and α mn • β mn = id.
Proof.The identity α mn • β mn = 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 : A m → 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 α mn is injective and β mn • α mn = 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 Σ n → A n provides an equivalence between the category of connected strings and the category of k-linear abelian categories satisfying (Ab1)-(Ab6) (with morphisms given by k-linear exact functors admitting a homological factorisation).
We refer to the appendix for some further explanation of this result.In view of Proposition 3.1 we have the following consequence.

Finite coproducts
For a finite set of natural numbers n α ∈ N we define the coproduct α Σ nα of strings by taking from the product of the underlying sets all elements s = (s α ) such that H. Krause s α ̸ = * for at most one index α (that is, the coproduct of the pointed sets Σ nα ).For s = (s α ) and t = (t α ) in α Σ nα set st := (s α t α ).
For each index α and 0 ⩽ i < n α we denote by s α,i the simple string s given by s α = s i .
Each coproduct α Σ nα comes with canonical inclusions i α : Σ nα → α Σ nα and projections Morphisms α Σ mα → β Σ n β are by definition maps ϕ between the underlying sets such that the composite Lemma 5.1.There are canonical isomorphisms of pointed sets 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 Σ mα → α Σ mα .The second bijection uses the fact that each morphism We obtain the category of finite strings which has as objects the finite coproducts of connected strings. (3)

Non-crossing partitions
We wish to describe the subobjects of Σ n in the category of finite strings.This requires some preparations.
A set S ⊆ Σ n of non-zero strings is called non-crossing provided that s, t ∈ S and s ′ ⩽ t ′ ⩽ s ′′ ⩽ t ′′ implies s = t.Lemma 6.1.The assignment S → Thick(S) gives a bijection between the non-crossing subsets and the thick subsets of Σ n .
Proof.The inverse map takes a thick subset T ⊆ Σ n to the unique non-crossing subset S ⊆ T with Thick(S) = T .□ For non-crossing subsets S, S ′ of Σ n we set This yields the structure of a poset.In fact, the non-crossing subsets form a lattice since the thick subsets of Σ n are closed under intersections.We denote this lattice by NC(Σ n ).
Let n ∈ N. A partition P = (P α ) of [n] is given by pairwise disjoint non-empty subsets P α of [n] such that α P α = [n].Each partition is determined by the corresponding set of strings S(P ) ⊆ Σ n−1 , where by definition s = (s ′ , s ′′ ) ∈ S(P ) if for some α we have s ′ , s ′′ ∈ P α and i ̸ ∈ P α for all s ′ < i ⩽ s ′′ .This is clear since any part P α = {a 1 < a 2 < • • • < a r } is determined by the corresponding set of strings S α = {(a 1 , a 2 − 1), . . ., (a r−1 , a r − 1)}. (3)We may consider the category Hom(Σ op , Set * ) of functors Σ op → Set * into the category of pointed sets, which is the analogue of the category Hom(∆ op , Set) of simplicial sets.Then the category of finite strings identifies via the embedding X → Hom(−, X)| Σ with the full subcategory of finite coproducts of representable functors in Hom(Σ op , Set * ).

The category of finite strings
Call a subset S ⊆ Σ n−1 of non-zero strings partitioning when for any s, t ∈ S we have s ′ = t ′ if and only if s ′′ = t ′′ .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 Σ n−1 .
A partition P is non-crossing provided given elements i < j < i ′ < j ′ with i, i ′ in the same part and j, j ′ in the same part, then all elements belong to the same part.The partitions of [n] are partially ordered via refinement, so P ⩽ P ′ if any part of P is contained in a part of P ′ .The non-crossing partitions then form a lattice which is denoted by NC(n); cf.[16,22].Lemma 6.2.There is a lattice isomorphism NC(Σ n−1 ) ∼ − → NC(n) which is given by S → P (S).
Proof.It is clear that S ⊆ Σ n−1 is non-crossing if and only if P (S) is non-crossing.Let S ⩽ S ′ .This means any s ∈ S can be written as s = s 1 s 2 • • • s r with s 1 , . . ., s r in S ′ .On the other hand, P ⩽ P ′ means that for any part P α = {a 1 < a 2 < • • • < a u } of P and t = (a i , a i+1 − 1) ∈ S(P ), there is a part of P ′ containing a i , a i+1 and therefore t = t 1 t 2 • • • t r with t 1 , . . ., t r in S(P ′ ).Thus we have S ⩽ S ′ if and only if P (S) ⩽ P (S ′ ).□ We say that two monomorphisms X 1 ↣ X and X 2 ↣ X are equivalent if there exists an isomorphism X 1 → X 2 making the following diagram commutative.
An equivalence class of monomorphisms into X is called a subobject of X.Given subobjects X 1 ↣ X and X 2 ↣ X, we write X 1 ⩽ X 2 if there is a morphism X 1 → X 2 making the above diagram commutative; this yields a partial order.For a monomorphism ϕ : X → Σ n in the category of finite strings we set S(ϕ) := {ϕ(s) | s ∈ X simple}.
Proof.Consider the equivalence relation on S generated by s ∼ t when st ̸ = * .This yields a partition S = α S α and we set n α := card S α .Using the fact that S is noncrossing, there is a unique morphism ϕ : α Σ nα → Σ n which identifies the simple strings s α,i with the elements in S α .Thus S = S(ϕ).□ Lemma 6.4.A morphism ϕ : X → Σ 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 ϕ is a monomorphism and need to show that ϕ is given by an injective map.Let X = r i=1 Σ ni .The canonical decomposition of a morphism Σ ni → Σ 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 ϕ i : Σ ni → Σ n is given by an injective map by the first case.Then each subset Im ϕ i is thick, and Im ϕ 1 ∩ Im ϕ 2 = Thick(S) for some non-crossing S ⊆ Σ n .Let ψ : Y → Σ n be the corresponding morphism with S(ψ) = S which exists by Lemma 6.This yields the structure of a poset.In fact, the non-crossing subsets form a lattice since the thick subsets of Σn are closed under intersections.We denote this lattice by NC( Σn ).
Let n ∈ N. We consider the set where x is identified with x + n for 0 ⩽ x < n, and x := x.For a partition P = (P α ) of [2n] we require that P α is a part of P for each α.Each partition is determined by the corresponding set of strings S(P ) ⊆ Σn , where by definition s (n) ∈ Σn with 0 ⩽ s ′ < n belongs to S(P ) if for some α we have s ′ , s ′′ ∈ P α and i ̸ ∈ P α for all s ′ < i ⩽ s ′′ .The partitions of [2n] are partially ordered via refinement, and the non-crossing partitions then form a lattice which is denoted by NC B (n); cf.[19,22].
Proof.Adapt the proof of Lemma 6.2.□ Remark 8.3.Let P = (P α ) be a non-crossing partition of [2n] and denote by S = (S α ) the corresponding partition of S = S(P ).Then P has at most one part P α satisfying P α = P α .In fact, P α = P α holds if and only if Thick(S α ) is infinite.
As before, we write P (S) for the partition of [2n] corresponding to a non-crossing set S ⊆ Σn .For a monomorphism ϕ : X → Σn in the category of finite strings we set S(ϕ) := {ϕ(s) | s ∈ X simple}.
Theorem 8.4.Let n ∈ N. The subobjects of Σn in the enlarged category of finite strings form a lattice which is canonically isomorphic to the lattice of non-crossing partitions NC B (n).The isomorphism sends a monomorphism ϕ : X → Σn to P (S(ϕ)).
Proof.Adapt the proof of Theorem 6.6.□

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 A be a k-linear abelian category satisfying (Ab2)-(Ab5).Then every thick subcategory of A satisfies again (Ab2)-(Ab5).
Proof.Let C ⊆ A be a thick subcategory.Then C is closed under images of morphisms in C because A is hereditary.It follows that the category C is abelian and again hereditary.Also, C is necessarily a length category.If X ∈ C is simple, then End(X) is isomorphic to k[t]/(t p ) for some p ⩾ 1, since X is indecomposable in A. Schur's lemma then implies p = 1.It remains to show that C is a uniserial category with finitely many simple objects.We may assume that either A = A n or A = Ãn for Algebraic Combinatorics, Vol. 6 #3 (2023)

Lemma 4 . 1 .
Let A be a k-linear abelian category satisfying (Ab1)-(Ab6).Then there is an equivalence A ∼ − → Rep([n] op , k), where n equals the number of isomorphism classes of simple objects in A.

Remark 4 . 5 .Lemma 4 . 6 .
We have d i n = (σ i−1 n−1 ) * for 0 < i < n.Thus d 0 n and d n n are not obtained from morphisms [n − 1] → [n].Let m, n ∈ N.An exact functor F : A m → A n that admits a homological factorisation induces a morphism ϕ : Σ m → Σ n which is given by F (M s ) = M ϕ(s) .

ϕLemma 8 . 1 .
3. Clearly, ψ factors through each ϕ i via a morphism ψ i : Y → Σ ni .We obtain a diagram Algebraic Combinatorics, Vol. 6 #3 (2023)The category of finite strings The assignment S → Thick(S) gives a bijection between the non-crossing subsets and the thick subsets of Σn .Proof.The inverse map takes a thick subset T ⊆ Σn to the unique non-crossing subset S ⊆ T with Thick(S) = T .□For non-crossing subsets S, S ′ of Σn we set S ⩽ S ′ : ⇐⇒ Thick(S) ⊆ Thick(S ′ ).
[9, C = Ker s i n denote the full subcategory of objects in A n that are annihilated by s i n .It is clear that M i is the unique simple object in C. Thus C equals the full subcategory given by the finite direct sums of copies of M i .Then the right adjoint of the quotient functor A n → A n /C identifies A n /C with C ⊥ , while the left adjoint identifies A n /C with ⊥ C. Here, we consider the perpendicular categories defined with respect to Hom and Ext 1 ; cf.[9,  §III.2].This yields the descriptions of d i n and d i+1 n .The embedding of any perpendicular category into A n is exact since Ext 2 vanishes.
The functor s i n annihilates M i and sends all other indecomposables to indecomposable objects.□ Algebraic Combinatorics, Vol. 6 #3 (2023)