Multisets of finite intervals and a universal category of poset representations

Abstract: For any finite totally ordered set, the multisets of intervals form an abelian category. Various classes of subcategories admit natural combinatorial descriptions, and counting them yields familiar integer sequences. Surprisingly, in some cases new integer sequences arise. The formulation of this counting problem leads to a universal construction which assigns to any poset a finitely cocomplete additive category; it is abelian when the poset is finite and does not depend on the choice of any ring of coefficients. For a general poset the universal category of representations is abelian if and only if for the lattice of ideals the meet of two compact elements is again compact.

• The paper examines finite intervals and representations, developing a universal category of poset representations highlighting combinatorial lattices for counting subcategories.
• Three specific classes of subcategories were shown to yield Catalan numbers: non-crossing partitions, Tamari lattices, and Dyck paths.
• The research introduced extension closed and colimit-closed categories, novel sequences without known formulas, characterizations of abelian representations and universality for poset construction.

Overview

This paper by Henning Krause and Balduin Stoye studies subcategories of representations of the linearly oriented quiver $A_n$ from a purely combinatorial viewpoint, and develops a universal construction that assigns to any poset $P$ a category $rep\,P$ of poset representations. The work has two intertwined aims: first, to give a systematic account of counting naturally defined classes of subcategories of the category of multisets of finite intervals, recovering several classical integer sequences (Catalan numbers, factorials) and producing sequences for which no closed formula is known; second, to isolate a lattice-theoretic condition — coherence of a poset — that characterizes exactly when the universal representation category is abelian, independently of any choice of coefficient ring.

The category of multisets of intervals

For $n \ge 1$, the authors define $\mathcal{I}_n$, the category of intervals of type $n$: objects are pairs $[a,b]$ with $1 \le a \le b \le n$, and there is a single non-zero morphism $[a,b] \to [c,d]$ precisely when $a \le c \le b \le d$. Composition is forced by transitivity and vanishes otherwise. Taking finite multisets of intervals with morphism spaces given by matrices over these Hom sets yields $P$0. The key structural fact is that $P$1 is abelian and equivalent to the category of finite-dimensional $P$2-linear contravariant representations of $P$3, via the assignment sending an interval $P$4 to the indecomposable representation supported on $P$5. Consequently every object has a finite composition series; an interval of length $P$6 has composition length $P$7, and the indecomposables are exactly the intervals. Since there are $P$8 intervals, the additive subcategories form a Boolean lattice of size $P$9.

The restriction to $rep\,P$0-linearity is inessential: the lattices of subcategories satisfying any of the closure conditions considered do not depend on the base field, so the counting problem is genuinely combinatorial.

Counting subcategories: known and new sequences

The paper counts full subcategories of $rep\,P$1 closed under finite direct sums and summands, subject to combinations of five closure operations: quotients (Q), subobjects (S), cokernels (C), kernels (K), and extensions (E). The results are summarized as follows:

Closure conditions	Interpretation	Sequence ($rep\,P$2)	Formula
Q, S, E	Serre subcategories ↔ sets of simples	2, 4, 8, 16, 32, 64	$rep\,P$3
C, K, E	Thick subcategories ↔ non-crossing partitions	2, 5, 14, 42, 132, 429	Catalan $rep\,P$4
Q, E	Torsion classes ↔ Tamari lattice	2, 5, 14, 42, 132, 429	same Catalan number
Q, S	Ideals of $rep\,P$5 ↔ Dyck paths	2, 5, 14, 42, 132, 429	same Catalan number
Q	Pretorsion classes ↔ sorting order on $rep\,P$6	2, 6, 24, 120, 720, 5040	$rep\,P$7
E	Extension closed	2, 7, 34, 199, 1308, 9300	no formula known
C, K	Exact abelian	2, 6, 22, 86, 345, 1411	no formula known
C	Finite colimit closed	2, 7, 37, 261, 2284, 23777	no formula known

The identifications with non-crossing partitions, the Tamari lattice, Dyck paths, and the sorting order each follow from existing structure theory (Ingalls–Thomas, Thomas, Oppermann–Reiten–Thomas, Armstrong). Notably, three distinct natural classes yield the same Catalan numbers while parameterizing different lattices, and the quotient-closed case grows factorially rather than exponentially in a Catalan-type way.

For the classes without known formulas, the paper provides effective combinatorial criteria that reduce the verification of closure properties to finitely many local checks:

• Extension closedness: it suffices to check extensions between indecomposables. There is a non-zero element of $rep\,P$8 for $rep\,P$9, $n \ge 1$0 if and only if $n \ge 1$1, $n \ge 1$2, and $n \ge 1$3; the middle term is then $n \ge 1$4.
• Cokernel closedness: it suffices to check cokernels of morphisms whose source is indecomposable and whose target is either indecomposable or a direct sum of two indecomposables. Explicitly, the cokernel of a morphism $n \ge 1$5 is $n \ge 1$6, with the convention $n \ge 1$7. The proof works directly in $n \ge 1$8 by constructing explicit bases of cokernels for arbitrary finite direct sum targets and reducing to the two-summand case.

These criteria make the enumeration a finite combinatorial search, though the growth of the colimit-closed sequence (already 23777 at $n \ge 1$9) suggests rapid divergence from Catalan-type behavior.

The universal construction

The second half of the paper constructs, for any poset $\mathcal{I}_n$0, a canonical functor $\mathcal{I}_n$1 into a finitely cocomplete additive category with a zero object. One adjoins a distinguished zero object $\mathcal{I}_n$2 to obtain the $\mathcal{I}_n$3-linear category $\mathcal{I}_n$4, takes the category $\mathcal{I}_n$5 of additive functors $\mathcal{I}_n$6, and corestricts the Yoneda embedding to finitely presented functors: $\mathcal{I}_n$7. This is the finite cocompletion of $\mathcal{I}_n$8, obtained by adjoining finite coproducts and cokernels. Although $\mathcal{I}_n$9 appears in the construction, the resulting category does not depend on a choice of coefficient ring, justifying the name universal category of poset representations.

The main structural result is that when $n$0 is finite, $n$1 is canonically equivalent to the module category $n$2 of the incidence algebra, hence is a length category. The universality is precise: for any finitely cocomplete category $n$3 with a zero object, restriction along $n$4 gives an equivalence between colimit-preserving functors $n$5 and all functors $n$6 preserving zero objects. Moreover, the assignment $n$7 is faithful up to isomorphism: two posets are isomorphic if and only if their universal categories are equivalent, since elements of $n$8 correspond to isoclasses of indecomposable projectives and the order is detected by non-vanishing Homs. For the chain $n$9, one recovers $[a,b]$0, with an interval $[a,b]$1 corresponding to the quotient $[a,b]$2.

Two caveats are stated explicitly. First, the universal functor lands in finitely cocomplete categories, but not in a universal property among additive finitely cocomplete categories: a functor preserving finite colimits and zero objects between additive categories need not be additive. Second, for infinite $[a,b]$3, subrepresentations of finitely presented objects need not be finitely presented, so the lattices of subobjects in $[a,b]$4 and $[a,b]$5 agree only in the noetherian case.

Coherent posets and the abelianness criterion

The final section characterizes when $[a,b]$6 is abelian. Using the bijection between ideals of a principal down-set $[a,b]$7 and additive subfunctors of the representable $[a,b]$8, chain conditions on ideals translate into chain conditions on subobjects: $[a,b]$9 is abelian with all objects of finite length if and only if every $1 \le a \le b \le n$0 is finite, and it is abelian with all objects noetherian if and only if ascending chains of ideals in each $1 \le a \le b \le n$1 stabilize. These characterizations are independent of the field $1 \le a \le b \le n$2.

The central definition is that of a coherent poset: for every diagram $1 \le a \le b \le n$3 there exist finitely many diagrams $1 \le a \le b \le n$4 such that every diagram $1 \le a \le b \le n$5 factors through some $1 \le a \le b \le n$6. The main theorem establishes four equivalent conditions:

1. $1 \le a \le b \le n$7 is coherent;
2. for each $1 \le a \le b \le n$8, the compact elements of the ideal lattice of $1 \le a \le b \le n$9 are closed under finite meets;
3. each representable $[a,b] \to [c,d]$0 is a coherent object of $[a,b] \to [c,d]$1;
4. $[a,b] \to [c,d]$2 is abelian.

Equivalently, $[a,b] \to [c,d]$3 is coherent if and only if in the lattice of ideals of $[a,b] \to [c,d]$4, the meet of two compact elements is again compact. The proof of (2) ⇒ (3) is the substantive part: given a morphism from a finite coproduct of representables to $[a,b] \to [c,d]$5, its kernel is exhibited as the image of an explicit map built from the finite presentations of the intersections $[a,b] \to [c,d]$6 supplied by coherence, using the tensor adjunction $[a,b] \to [c,d]$7.

This hierarchy mirrors ring theory — artinian implies noetherian implies coherent — with the poset conditions playing the role of the corresponding right-ideal conditions. It also connects to the recent result of Iyama and Marczinzik that a finite lattice $[a,b] \to [c,d]$8 is distributive if and only if $[a,b] \to [c,d]$9 is the module category of an Auslander regular algebra, indicating that properties of $a \le c \le b \le d$0 can detect order-theoretic properties of $a \le c \le b \le d$1.

Limitations and open questions

Three gaps remain open within the paper's own scope. First, no closed formulas are known for the numbers of extension-closed, exact abelian, or finite colimit-closed subcategories of $a \le c \le b \le d$2; only initial terms and local verification criteria are provided, and whether these sequences admit product formulas or asymptotics is unresolved. Second, the universality of $a \le c \le b \le d$3 is stated among finitely cocomplete categories without an additivity constraint, leaving open whether a refined universal property within the additive setting can be formulated. Third, the paper proposes studying properties of a poset through $a \le c \le b \le d$4 as a project but carries this out only for abelianness and the finite-length/noetherian dichotomy; the extent to which other categorical invariants of $a \le c \le b \le d$5 reflect order-theoretic properties of $a \le c \le b \le d$6 remains to be developed.

Conclusion

The paper accomplishes two things cleanly. It organizes the enumeration of naturally closed subcategories of interval representations into a uniform framework, recovering Catalan numbers, factorials, and powers of two for the classical classes while isolating three sequences that resist current methods. And it constructs a coefficient-free universal category of poset representations whose abelianness is characterized by a compactness condition on the lattice of ideals — the meet of compact elements being compact — providing a poset-level analogue of coherence for rings and a foundation for translating questions about posets into questions about their representation categories.