Weighted Colimits of 2-Representations and Star Algebras

We apply the theory of weighted bicategorical colimits to study the problem of existence and computation of such colimits of birepresentations of finitary bicategories. The main application of our results is the complete classification of simple transitive birepresentations of a bicategory studied previously by Zimmermann. The classification confirms a conjecture he has made.


Introduction
Systematic study of finitary 2-representations of finitary 2-categories was initiated by the series of papers [23][24][25][26][27][28], after a number of successful instances and applications of categorification, and, in particular, categorical actions, in various areas of mathematics. These include the advances in knot theory following the introduction of Khovanov homology in [21], the proof of Broué's abelian defect group conjecture for symmetric groups in [4], the categorification of quantum groups developed in [18][19][20] as well as [7,34,36], and many more.
One of the main purposes of 2-representation theory is to gain a better abstract understanding of categorical actions of 2-categories (and, by extension, also of the 2-categories themselves), such as those in the above listed applications. Finitary 2-categories can be viewed as a 2-categorical counterpart to the classical notion of a finite-dimensional algebra, and from this point of view, 2-representation theory is analogous to classical representation theory of such algebras, which aims at a better understanding of the linear actions of the algebra.
One of the first problems to consider in the classical setting is the classification of simple modules. A 2-representation theoretic analogue of simple modules, known as simple transitive 2-representations, was introduced in [27], together with an associated weak Jordan-Hölder Communicated by Richard Garner. theory. Classification of such 2-representations has since become one of the central problems of the theory, with many complete classification results obtained, such as those in [27] and [31]. See also [22] for a slightly outdated overview.
A common feature of most of these results is the quasi-fiat structure of the 2-category considered. In the theory of tensor categories, this corresponds to rigidity of the tensor category. In the former case, one requires all 1-morphisms to have left and right adjoints; in the latter, one requires all objects to have left and right duals. Continuing the analogy with finite-dimensional algebras, a fiat 2-category can be viewed as analogous to a finitedimensional algebra with an involution.
The most notable exception is the main result of [32], which classifies simple transitive 2-representations of a large family of finitary 2-categories which need not be fiat. This is done by embedding the studied 2-category in a 2-category with additional adjunctions and lifting 2-representations to this bigger 2-category.
The present paper classifies the simple transitive 2-representations of a non-fiat 2-category B str n which does not have the crucial cell symmetry of the non-fiat 2-categories considered by [32]. The classification for B str n confirms [38,Conjecture 1], establishing a bijection between equivalence classes of simple transitive 2-representations and set partitions of {1, . . . , n}. In contrast to [32], neither the construction nor the classification employs an auxiliary fiat 2-category. Instead, we use weighted bicategorical colimits of prior known, not necessarily simple transitive, 2-representations of B str n , to construct new simple transitive 2-representations, and use the universal properties for the classification. The weighted colimits used can be thought of as a 2-representation theoretic categorification of quotient modules in classical representation theory. The prerequisite facts and a description of our application of such colimits is given in Sect. 3. To our best knowledge, this approach has not been considered previously in the study of 2-representations.
It was observed in [38] that for any simple transitive 2-representation M of B str n , there is a distinguished 2-transformation : C → M, i.e. a functor intertwining the B str nactions, from the cell 2-representation C. Further, [38] shows that sends indecomposable objects to indecomposable objects, possibly identifying certain isomorphism classes of such objects. The kind of potential identification observed there cannot be captured using the more familiar categorical constructions, such as orbit categories and skew group categories. Our construction uses the analogy with finite-dimensional algebras: if we instead considered a finite-dimensional algebra A and an A-module C, given elements x, y ∈ C we could universally construct a morphism ϕ of modules from C, satisfying ϕ(x) = ϕ(y), as the projection C C/ x − y , which is the coequalizer of the diagram A C 1 →x 1 →y . It is this latter realization we mimic in the bicategorical setting, replacing the regular module by the (representable) principal 2-representation P and, using Yoneda lemma, viewing indecomposable objects X , Y of C as parallel 2-transformations from P to C. We then study the bicategorical colimit which universally renders the 2-transformations isomorphic (rather than equal). This stands in stark contrast with the techniques employed in the theory of fiat 2-categories, where one heavily relies on 2-representations obtained from structures internal to the 2category whose 2-representations are studied, such as (co)algebra 1-morphisms and internal Hom studied in [29], whereas our approach uses the structure of the 2-category of 2representations. In fact, we need to embed the latter 2-category into the 2-category of all k-linear 2-functors to the 2-category Cat k of k-linear categories, which can be viewed as analogous to the embedding of the category A-mod of finitely generated modules over a k-algebra A, to the category A-Mod of all its modules.
This new approach, which in a sense categorifies quotient modules, can prove useful in similar problems concerning construction and classification of categorical actions. Indeed, a different classification problem for 2-representations of a non-finitary (although in a sense locally finitary) 2-category has been solved in [15] using the above approach, confirming [14,Conjecture 2] and generalizing it to the setting of [13]. In fact, the 2-representations constructed in [15] and in this document are the first non-trivial non-cell 2-representations constructed in the theory of simple transitive 2-representations of non-fiat 2-categories.
We now briefly explain the definition of the 2-category B str n . First, consider the double quiver R n on the star graph G n on n + 1 vertices, as depicted below for n = 5: The zigzag algebra A n on G n is a quotient of the path algebra of R n by the ideal generated by paths i → j → k with i = k, together with elements of the form In particular, A n is weakly symmetric, hence also self-injective. Under the complete set of pairwise orthogonal, primitive idempotents {e 0 , e 1 , . . . , e n }, induced by the above labelling of the quiver, we may consider the set L 0 = {A n } ∪ {A n e i ⊗ k e 0 A n | i = 0, 1, . . . , n} of A n -A n -bimodules. The additive, k-linear category B n := add L 0 is a monoidal subcategory of (A n -mod-A n , ⊗ A n ). Observe that B n is not symmetric or braided. We denote by B n the delooping bicategory of B n . The 2-category B str n is a strictification of B n . More precisely, it is obtained by delooping the strict monoidal category B str n of right exact endofunctors of A n -mod isomorphic to those given by the objects of B n .
We follow [30] in relaxing the 2-categorical setup to the bicategorical setup, which allows us to consider simple transitive birepresentations of B n rather than simple transitive 2representations of B str n . As observed in [30], the resulting two classification problems are equivalent. The bicategorical setup allows for greater flexibility in the computation of colimits.
We can now state our main result precisely: The paper is organized as follows. Section 2 contains the necessary preliminaries for the techniques of 2-representation theory we will employ, as well as a complete account of the notation we will use. In Sect. 3 we give an elementary account of weighted colimits, bicategorical cocompleteness of bicategories of k-linear pseudofunctors and preservation of weighted colimits by additive and Karoubi envelopes. Section 4 defines B n , summarizes and extends the results of [38], giving necessary properties of simple transitive birepresentations of B n without proving their existence. Section 5 constructs the simple transitive birepresentations and uses the results of Sect. 4 to obtain the classification.

Preliminaries
Throughout the text we always require the structure 2-morphisms of bicategorical structures to be invertible. The resulting setting of bicategories, pseudofunctors, strong transformations and modifications is what we will call the bicategorical setting, and we will give our results in this setup. In particular, we will study birepresentations of finitary bicategories, following [30].
Most of our results also hold in what we call the 2-categorical setting, where we require the structure 2-morphisms to be the identities, thus working with 2-categories, 2-functors, 2-transformations and modifications. We will comment on possible differences between the bicategorical and the 2-categorical results, when suitable.

Notation
Our notational conventions largely follow those of [30], with one difference and a few additions. Modifications are denoted by m, n and the like.
• Structure 2-morphisms of bicategorical structures are denoted by lower case fraktur. In particular, associators are denoted by a, left unitors are denoted by l, and right unitors are denoted by r. These 2-morphisms are denoted by α, v l , v r in [30]. • Categories are denoted by C⇔D, and the like; objects in a category C are denoted by capital letters, such as X ∈ Ob C. Morphisms are denoted by lower case letters, such as f ∈ C(X , Y ). • Given bicategories B , C , we denote the bicategory of pseudofunctors from B to C by [B , C ].
• Objects in a bicategory are denoted by i, j, and the like. 1-morphisms in a bicategory are denoted by F, G, and the like. 2-morphisms are denoted by lower case Greek letters, e.g. α, β. The identity 1-morphisms of objects will be denoted by i and the like, and the identity 2-morphisms of 1-morphisms will be denoted by id F , and the like. Similarly to, [30], we only indicate the 1-morphisms indexing the structure 2-morphisms of a bicategorical structure, while omitting the indexing objects from the notation, thus writing a H ,G,F rather than a i,j,k,l H ,G,F for the associator Note that our notational conventions for objects, 1-morphisms and 2-morphisms of a bicategory do not apply to the pseudofunctor bicategory [C , D ], where we prioritize our conventions for pseudofunctors, strong transformations and modifications. Similarly, our conventions do not apply to functor categories, or to the 2-category Cat, where we prioritize our separate conventions for categories and functors.

Finitary Bicategories and Their Birepresentations
Let k be an algebraically closed field of characteristic zero. We remark that, for the statements that do not involve finitary categories, this assumption can be relaxed to the assumption that k is a commutative ring.
Let Cat k denote the 2-category of small k-linear categories, k-linear functors and natural transformations. Under tensor product of k-linear categories, Cat k becomes a symmetric monoidal 2-category. For the general definition of a monoidal 2-category, we refer to [10,Definition 2.6].
We say that a bicategory B is k-linear if it is enriched in Cat k . Similarly for k-linear pseudofunctors, strong transformations and modifications. For the general definition and extensive treatment of bicategories enriched in a monoidal bicategory, we refer to [11]. In the case of Cat k , it follows that a k-linear bicategory is a bicategory B such that for any i, j ∈ Ob B , the category B(i, j) is k-linear and horizontal composition • h is k-bilinear.
Cat k itself is a k-linear 2-category: for any k-linear categories C, D, the category Cat k (C, D) is k-linear under pointwise formation of k-linear combinations of natural transformations of k-linear functors.
Given k-linear bicategories B, C , a k-linear pseudofunctor M : B → C is a pseudofunctor of underlying ordinary bicategories, such that, for any objects i, j of B, the local functor M i,j is k-linear. A k-linear strong transformation of k-linear pseudofunctors is a strong transformation of underlying pseudofunctors, with no additional requirements. Similarly, k-linear modifications are just modifications of said strong transformations. We will omit specifying k-linearity of bicategorical structures whenever it is a vacuous condition.
Whenever speaking of an ambient bicategory B, 2-category C or category C, we implicitly assume it to be essentially small. Our main aim is to prove the existence of certain k-linear pseudofunctors from a fixed, essentially small, bicategory B to Cat k . It will become clear that this result is invariant under k-linear biequivalence, so we may first construct such pseudofunctors from a biequivalent, small B and then pass under biequivalence. Thus, for our purposes, we may further assume that said essentially small structures are in fact small. Composition of two k-linear pseudofunctors is again a k-linear pseudofunctor, and so the collection of such pseudofunctors between the k-linear bicategories B and C , together with strong transformations and modifications, forms a bicategory which we denote by 2-natural in c, d, we say that (F, G) is 2-categorically adjoint. In particular, we avoid the common terminology calling bicategorical adjunctions biadjunctions, since 1-categorical ambidextrous adjunctions, abundant in 2-representation theory, are often referred to as biadjoint pairs. For an extensive account of bicategorical adjunctions, see [9].
Following the observation [16, 2.29], given a k-linear category A, the pair of 2-endofunctors of Cat k is 2-categorically adjoint, and hence the symmetric monoidal 2category Cat k is closed. As remarked in [11,Section 5], given k-linear bicategories B, C , D , we may form the k-linear bicategory B ⊗ k C , given by products on the level of objects and 1-morphisms and by tensor product over k on the level of 2-morphisms and structure 2morphisms, which yields the canonical k-linear biequivalences We may also form the k-linear bicategory B op , by reversing the direction of 1-morphisms in B.
We say that a k-linear category C is finitary if it is equivalent to the category A-proj of finite dimensional projective modules over a finite dimensional associative k-algebra A. A k-linear, abelian category C is said to be finite if it is equivalent to the category A-mod of finite dimensional modules over a finite dimensional associative k-algebra A. We let R k denote the 2-category of k-linear finite abelian categories, right exact functors and natural transformations. An abelian birepresentation of B is a k-linear pseudofunctor from B to R k . Given a finitary birepresentation M of B, its projective abelianization M is an abelian birepresentation such that M can be recovered from M by restricting to certain subcategories equivalent to M(i)-proj, for i ∈ Ob B. We refer the reader to [24] for details, and [29] for an improved construction. We will only use abelianization once in this document, and in that case the simpler construction of [24] can be used.
We say that a birepresentation of B is transitive if, for any i, j ∈ Ob B and any X ∈ Ob M(i), Y ∈ Ob M(j), there is a 1-morphism F ∈ B (i, j) such that Y is isomorphic to a direct summand of MF(X ).
A B-stable ideal I of a finitary birepresentation M of B is a tuple I(i) i∈Ob B of ideals of M(i) such that, for any X f − → Y ∈ I(i) and any F ∈ Ob B(i, j), we have MF( f ) ∈ I(j).
we may define the evaluation of α at f as We then have the commutative square where the upper horizontal arrow lies in I(j) by assumption, and hence the lower horizontal arrow also lies in I(j). This proves that MF( f ) ∈ ( −1 I)(j).

Definition 2.3 A finitary birepresentation M of B is simple transitive if it has no proper
B-stable ideals. In particular, a simple transitive birepresentation is transitive.

Cells
The left preorder ≤ L on the set of isomorphism classes of indecomposable 1-morphisms of B is defined by writing F ≤ L G if there is a 1-morphism H such that G is a direct summand of HF. We denote the resulting equivalence relation by ∼ L , and refer to the equivalence classes as left cells. Similarly one defines the right and two-sided preorders ≤ R , ≤ J , together with the right and two-sided equivalence relations and right and two-sided cells.

Biideals
A two-sided biideal I in a k-linear bicategory C consists of a collection of ideals (I j,k ) j,k∈Ob C of C (j, k), such that, for any 1-morphisms F of C (i, j) and G of C (k, l), and any 2-morphism γ ∈ I j,k , we have As a consequence, given any 2-morphisms α of C (i, j) and β of C (k, l), we have and similarly for α. Given a biideal I, the assignment (I 2 ) k,l := (I k,l ) 2 defines a biideal: In particular, where f , h are arbitrary morphisms of M(i), whereas the 1-morphisms F,G, the 2-morphism α and the morphism g are only required to be such that all the compositions and evaluations in the diagram are well-defined. It is clear that this collection is stable under vertical composition, since f , h are arbitrary. To see that it is also closed under horizontal composition, let H be such that the compositions HG, HF are defined. Applying MH on the defining diagram above, using the associators, we obtain:

M(HG)(g)
The Further, it shows that, if we modify the definition of ev M (I), so that instead of letting α be arbitrary, we require α = γ • h β as in (1), we again obtain an ideal of M, which we denote by ev M (I h,2 ), in view of the notation in (1). Inductively, we may define further ideals ev M (I h,k ), for k ∈ Z >0 . The above diagram also proves that ev M (I h,2 ) is the ideal generated by morphisms of the form MH( f ), for f ∈ ev M (I). If we express this by the suggestive notation then, setting I h,1 := I, inductively we also have Proof Since C (i, j) is finitary and I i,j contains no identity 2-morphisms, we have Given i, j ∈ Ob C , let n i,j be the nilpotency degree of Rad C (i, j). Since Ob C is finite, we may let n := max n i,j | i, j ∈ Ob C . Clearly then I n i,j = 0, for all i, j ∈ Ob C , and hence I n = 0.

Weighted Colimits of k-linear Pseudofunctors
We first recall the notion of a bicategorical weighted colimit. The simplified terminology we use here replaces the more precise terminology of pseudo-, bi-and lax colimits, which we will not need since we only use elementary bicategorical notions. Our choice of terminology here is the same as that in [1], and it is consistent with the notational conventions we made before.
giving rise to isomorphisms of categories It is then unique up to isomorphism in C , and if it always exists, we say that C is 2-categorically cocomplete.
Bicategorical and 2-categorical limits are obtained as bicategorical and 2-categorical colimits in B op and C op .

Proposition 3.2 The 2-category Cat k is complete and cocomplete, both bicategorically and 2-categorically.
Proof Cocompleteness can be shown by explicitly constructing certain colimits which can be used to obtain all colimits. A proof given by construction of coproducts, coinserters and coequifiers can be found in [1,Proposition 2.6]. For completeness, one easily verifies that the explicit constructions of products, cotensors, 2-equalizers and pseudoequalizers in Cat given in [2] apply also in the case of Cat k -this can be viewed as a consequence of preservation of limits by the forgetful 2-functor Cat k → Cat, which is 2-categorically right adjoint to the free k-linear category 2-functor. Completeness follows as a consequence of [37, 1.24].

Definition 3.3
Let B be a k-linear bicategory. Let J be a small k-linear bicategory and let W be a k-linear pseudofunctor from J op to Cat k . Given a k-linear pseudofunctor F : J → B, a weighted k-linear bicategorical colimit W F is an object of B together with a representation of the k-linear pseudofunctor in i. In other words, there are k-linear equivalences of categories Similarly one obtains the notion of a 2-categorical k-linear colimit, of k-linear bicategorical and 2-categorical limits and the resulting notions of k-linear bicategorical and 2-categorical cocompleteness and completeness.
The next statement can be viewed as a direct corollary of [11,Proposition 11.2], however, since that paper is written in a much more general setting, we give an explanation of how one views our particular case as an instance of the setting of [11], and how one uses the results therein to obtain the statement.
Proof As we have observed before, we consider the case of enriching monoidal bicategory V = Cat k . As remarked in [11,Section 5], it is easy to verify that, given k-linear bicategories A, C , left A-modules in the sense of [11] are k-linear pseudofunctors from A to Cat k , and similarly, A-C -bimodules are k-linear pseudofunctors from A ⊗ k C op to Cat k . Given M, N ∈ [B , Cat k ] k , their internal hom, as studied in the general case in [11,Section 7], is given by the k-linear category [B , Cat k ] k (M, N), under the clear choice of the evaluation morphism ξ of [11,Section 7.3]. Combining this with the fact that B is assumed to be small, we conclude that the k-linear bicategory of moderate right B-modules studied in [11,Section 11] coincides with [B op , Cat k ] k . Its k-linear cocompleteness follows from [11,Proposition 11.2]. The colimit W F is given by W ⊗ JF , forF and ⊗ J as defined in [11]. Similarly, Taking all this into account, one sees that the result follows from [11, Corollary 6.10].
Using the categorification of [16, 3.8] in [11, 13.14], we conclude that ordinary bicategorical colimits in a k-linear bicategory B can be viewed as special cases of k-linear bicategorical colimits. We remark that in our case, the monoidal bicategorical adjunction obtained in [11, 13.14] is that given by the free k-linear category 2-functor Cat → Cat k and the forgetful 2-functor. As a consequence of the last two observations, together with Proposition 3.2, we find the following: For much more general results regarding weighted bicategorical limits, in a much more general setting of bicategories enriched in monoidal bicategories which are not necessarily symmetric or closed, see [11, Section 10, Section 11].

Additive and Karoubi Envelopes
Recall that a preadditive category is a category enriched in the category Ab of abelian groups. Let Cat Z denote the 2-category of preadditive categories.
The additive envelope is the universal solution to the problem of adding direct sums to a preadditive category. Similarly, the Karoubi envelope universally makes a category idempotent split. These constructions are used, for instance, to define the category Rep(S t )see [6]. A more detailed account is given in [5], and a very detailed account, which will be our main reference, is given in [35].
LetCat ⊕ Z , Cat K Z denote the 1, 2-full 2-subcategories of Cat Z , with the respective object sets being that of categories with finite direct sums and that of idempotent split categories, respectively. Let Cat ⊕ k , Cat K k be the k-linear variants thereof. Let Cat D k denote the 1, 2-full 2-subcategory of Cat k given by the categories which are both additive and idempotent split. We have the following results: C → C ⊕ and C : C → C K . Using the definitions of C , C , it is easy to verify that the following equations hold: for any k-linear functor F : C → D. Denote the respective inclusion 2-functors by (3) implies that we have 2transformations • [35, Proposition 75, Theorem 113]: Given C ∈ Ob Cat k and D ∈ Ob Cat ⊕ k , the functor is an equivalence. Similarly, given D ∈ Cat K k , the functor is an equivalence.
Let J D : Cat k → Cat D k denote the indicated inclusion 2-functor. Using the above listed facts about (−) ⊕ , (−) K , we make the following conclusion: Proof Given C ∈ Ob Cat k and D ∈ Ob Cat ⊕ k , the 2-naturality of the equivalence follows from Eq. (3). The 2-naturality in D follows from the associativity of composition of functors, since the equivalences are given by precomposition. The proof for Karoubi envelope is completely analogous.

Remark 3.7
Observe that the bicategorical adjunction above is almost 2-categorical, the only condition missing being that it is given by equivalences rather than isomorphisms of Homcategories.
Combining Proposition 3.5 with Proposition 3.6, we obtain the following: are bicategorically cocomplete.
Proof The first part of the statement is an immediate consequence of the preceding results. The second part follows since the bicategorical adjunctions give bicategorical adjunctions More precisely, to obtain the k-linear statement, one may argue directly on the level of cocones: given a colimiting cocone W F and a cocone H satisfying the one-dimensional universal property, one obtains 1-morphisms H → W F and W F → H. The respective one-dimensional universal properties then suffice to conclude that these 1-morphisms are mutually quasi-inverse equivalences.
Ideally, we would like to further restrict our treatment of colimits to the 2-subcategory A f k of Cat D k , consisting of finitary categories. However, the condition of finite dimensional hom-spaces between objects is not preserved under taking weighted colimits-in fact, it fails already for conical colimits. This is very different from the setting of classical representation theory: the category vec k of finite dimensional vector spaces over k is abelian, and hence, in particular, cocomplete. As a consequence, if C is a k-linear category, then the category C-mod, defined as Cat k (C, vec k ), also is k-linear and abelian. Our claim means that the categorification of this statement to finitary bicategories is false. We now give an example of this phenomenon. Let Cat f.d. k denote the 1, 2-full 2subcategory of Cat k whose objects are k-linear categories with finite dimensional homspaces.

Example 3.10
Let k be a field. Consider the free k-linear categories A 2 , A 3 on the quivers The coequalizer coeq(F b , F c ) in Cat k is given by the free k-linear category on the quiver Assume that Cat f.d. k admits a coequalizer of the above functors. The universal property of coeq(F b , F c ) gives a k-linear functor K : Applying the universal properties of the respective coequalizers, we obtain the following commutative diagram of k-linear functors: where T m , T f.d. m are obtained using the respective universal properties of their domains and hence are the unique k-linear functors making the left inner triangle and the outer triangle commute, respectively. The functor T m is determined by T m (x) = y. The right triangle part of the above diagram gives the following diagram of associative k-algebras: End coeq f.d.
Using image factorization, we replace End coeq f.d.

F b ,Fc
(K (X )) with its subalgebra generated by K X ,X (x) =: z. Since End coeq f.d.

F b ,Fc
(K (X )) is finite dimensional by assumption, the aforementioned subalgebra is a finite dimensional quotient of k[x], hence isomorphic to k[w]/ w k , for some k. Since z is mapped to y under ( T f.d. m ) K (X ),K (X ) , we see that z must be a radical element with nilpotency degree greater than or equal to m. This implies k ≥ m. But, while we may vary m, the coequalizer, and thus also the integer k, remain unchanged, so the existence of a coequalizer in Cat f.d.
k would imply the existence of k such that k ≥ m for all m > 0, which is a contradiction.
In view of the Krull-Schmidt theorem for finitary categories, we see from Corollary 3.8 that colimits of finitary birepresentations, although themselves not necessarily finitary, may be computed starting from the indecomposable objects of the underlying categories. Formally, we have the following: Proposition 3.11 Let C be a finitary category. Let C-indec be the full subcategory of indecomposable objects of C. Then C (C-indec) ⊕ .
Let X be a set of objects of C, and let X be the full subcategory of C satisfying Ob X = X. Then add X X D .
Proof Since C is additive, by [35,Proposition 99], the functor C : C → C ⊕ is an equivalence. Let I be the inclusion functor C-indec → C. Since I is full and faithful, by [35, Lemma 101], so is I ⊕ : (C-indec) ⊕ → C ⊕ . The Krull-Schmidt theorem implies that all the isomorphism classes of objects of C are given by those of finite direct sums of objects in C-indec. Thus, I ⊕ is essentially surjective, and hence an equivalence.
Similarly, since add X is additive and idempotent split, we have (add X ) D add X , the full and faithful functor X → add X gives rise to X D → (add X ) D , which also is full and faithful, and further also essentially surjective, since any object of add X is isomorphic to a direct summand of a direct sum of objects in X .

The Bicategories B and B
Given a, b ∈ Z with a ≤ b, we denote by [a, b] the set {a, a + 1, . . . , b}. Let S n denote the star graph on n + 1 vertices. Label the unique internal node of S n by 0, and the leaves by 1, 2, . . . , n. Let n denote the zigzag algebra on S n , i.e. the quotient of the path algebra of 0 1 · · · n a 1 a n b 1 b n by the ideal given by the sum of the third power of the arrow ideal and the ideal given by relations a j b i = 0, for i = j, and b 1 a 1 = · · · = b n a n =: c. Let e 0 , . . . , e n denote the complete set of pairwise orthogonal, primitive idempotents induced by the labelling on S n . For k ∈ [1, n], denote by c k the element given by the 2-cycle a k b k . For a more extensive study of zigzag algebras, see for example [8].
Consider the k-linear monoidal subcategory B n of ( n -modn , ⊗ n ) given by the additive closure add n n n , n e k ⊗ k e 0 n | k ∈ [0, n] . Viewed as a bicategory with a unique object, B n is a finitary bicategory. We denote its unique object by i. In particular, B n is biequivalent to its essential image in Cat k ( n -mod, n -mod), under the pseudofunctor sending a bimodule M to the functor M ⊗ n −. Simple transitive 2-representations of the 2-category B str n given by this essential image were studied in [38]. In this section, we use the partial results of [38] to give a description of the underlying categories and action matrices for simple transitive birepresentations of B n . As observed in [30, 2.3], studying the simple transitive birepresentations of B n is equivalent to studying the simple transitive 2-representations of B str n . To simplify the notation, we now fix n ∈ Z >0 and denote n by , denote B str n by B str , and finally write B for B n . We will introduce the subscripts again whenever there is a risk of ambiguity, in particular when different values of n need to be considered simultaneously.
Using the canonical isomorphism End -mod ( ) op , we identify morphisms of indecomposable projective modules with elements of . For example, given i ∈ [1, n], this yields Hom -proj ( e i , e 0 ) = k {a i }. This applies also to the indecomposable bimodules e k ⊗ k e 0 , where we identify morphisms with the images of the generators e k ⊗ e 0 . We obtain Hom -mod-( e j ⊗ k e 0 , e k ⊗ k e 0 ) = e j e k ⊗ k e 0 e 0 .
Similarly, we identify morphisms from with the images of the generator 1. An easy but tedious calculation yields Hom -mod-( e j ⊗ k e 0 , ) = e j e 0 ; Proof It suffices to show that indecomposable non-identity 1-morphisms send the 2morphisms given by I back to I under horizontal composition. Since, for any v ∈ e 0 and any j ∈ [1, n], we have vc j = 0, it follows that id e j ⊗ k e 0 ⊗α = 0, for α ∈ I.
To see that I is also a right biideal, consider the case of the morphism Identifying isomorphic bimodules and identifying morphisms between decomposable bimodules with matrices of those given in (5), we may write from which it is clear that the resulting morphism remains in I. The remaining cases are similar.
Since I consists of radical morphisms, Lemma 2.5 implies that it is nilpotent. Consider the unique cell birepresentation C of B, and the unique cell 2-representation C str of B str . From the definition, we know that C(i) is the quotient of the category add { e k ⊗ k e 0 | k ∈ [1, n]} by the unique maximal nilpotent ideal stable under the left B-action given by tensor products over . Under the identification in (4), it corresponds to the maximal ideal I of ⊗ k e 0 e 0 such that e 0 ⊗ I belongs to ⊗ k Rad(e 0 e 0 ). A simple calculation yields which implies that for the ideal J = k {c i | i ∈ [1, n]}, we have Let A := /J . Remembering the index n, we obtain an algebra A n , for every n ∈ Z >0 . It follows that Since C is equivalent to C str as a birepresentation of B, we obtain Let B denote the bicategory given by Consider its strictification B str , defined similarly to B str . Remembering the index, we obtain the bicategories B n and the 2-categories B str n . From (8), we see that the assignments give a 2-functor Passing under biequivalences, this gives a pseudofunctor Q : B → B. As described earlier, Q maps indecomposable 1-morphisms to the corresponding indecomposable 1-morphisms, and hence is essentially surjective. Abusing notation, we identify elements of and their images of the projection from onto A, whenever such images are non-zero. A simple but tedious calculation yields Proof I annihilates C str by Lemma 2.4, so, by construction, Q sends I to zero. There is thus an induced pseudofunctor Q : B /I → B . It is essentially surjective, since Q is such.
Further, as a consequence of the calculations following Lemma 4.1, the cell birepresentation C is equivalent to the pullback birepresentation Q A-proj, where A-proj has the structure of the defining birepresentation of B. The latter is a faithful birepresentation (it is given by a locally faithful pseudofunctor) and so Ker Q = I. Indeed, if the kernel properly contained I, its image under Q would give a non-zero biideal of B annihilating A-proj. Thus, Q is locally faithful.
To see that it is also locally full, observe that, for bimodules M, N ∈ B(i, i), using  I(M, N ).
The statement follows from the fact that an injective map of equidimensional finite dimensional vector spaces is an isomorphism.

Prior Results
Since I is nilpotent, Lemma 2.4 shows that simple transitive birepresentations of B are the same as the simple transitive birepresentations of B. Hence, all the analysis given in [38] applies also if we make this replacement. On the level of the underlying k-algebras, we replace by A. That the results of [38] apply is even clearer if one observes the following: • For j ∈ [0, n], we have e 0 Ae j e 0 e j . Since A is a quotient of by a nilpotent ideal, this implies that the multisemigroup given by composition of indecomposable 1-morphisms is the same for B and B.
• The module Ae 0 is projective-injective, and so the bimodule Ae 0 ⊗ k e 0 A gives a selfadjoint endofunctor of A-proj. Let P be a set partition of a set X . We will associate a function to P, which, abusing notation, we denote by P : X → 2 X . This function sends an element x ∈ X to the subset P(x) of X it belongs to in the partition. For the remainder of this section, let M be a simple transitive birepresentation of B, let be the above described strong transformation, and let P M partition [0, n] into r + 1 subsets; recall that P(0) = {0}. We now give a very short summary of the results of [38] we will use.

Proposition 4.3
Given k, k ∈ [0, n], let X, Y be indecomposable objects of C(i) belonging to the isomorphism classes of A n e k , A n e k in A n -proj under the identification C(i) A n -proj given in (7).

• The objects (X ), (Y ) are indecomposable, with (X ) (Y ) if and only if P M (k) = P M (k ).
• If k = k or P(k) = P(k ), then X ,Y gives a bijection of Hom-spaces.
• As a consequence, if U = {u 1 , . . . , u r } ⊆ [0, n] is a transversal of P M , then the restriction of to add {A n e k | k ∈ U } is an equivalence of categories. In particular, M(i) A r -proj.
We obtain an induced complete set of representatives of isomorphism classes of indecomposable objects of M(i), which we denote by A M r e P(u 1 ) , . . . , A M r e P(u r ) . Using M(i) A r -proj and identifying bimodules and endofunctors, we have Proof For n = 1, all the claims follow from the proof of [38,Theorem 4.1]. The exact same proof works for n > 1 after a modification of indices, since all the claims follow from the self-adjointness of A n e 0 ⊗ k e 0 A n , which is independent of n, and the action matrices, which are completely determined by P M .
Since action matrices are an invariant of finitary birepresentations, the following statement can be concluded already from their characterization in [38, Theorem 5.1], but it is even clearer in view of Proposition 4.3:

Equifying Modifications
Since B admits a unique object i, we denote the principal birepresentation P i = B(i, −) simply by P. It admits a transitive subbirepresentation N, given by the subcategory For any birepresentation M of B, Yoneda lemma gives

B-afmod(P, M) M(i).
Given an object X ∈ M(i), we denote by X : P → M the strong transformation that sends a 1-morphism F to the object MF(X ). Denote the strong transformation N → P, given by inclusion, by . Let j := A n e j ⊗ k e 0 A n be the indicated strong transformation from P to C. The identification C(i) A-proj of (7) sends the object Ae j ⊗ k e 0 A of the quotient category to Ae j . We may thus identify j with the functor − ⊗ A n A n e j .

Lemma 4.5 Given j, k ∈ [1, n], there is an invertible modification
Proof Under the identifications above, a modification m from j • to k • is given by a natural transformation m : − ⊗ A Ae j → − ⊗ A Ae k of functors from N(i) = add {Ae k ⊗ k e 0 A | k ∈ [0, n]} to A-proj, such that, for any M, N ∈ N(i), s From the earlier statements, it follows that the diagram commutes, which proves that s satisfies the axiom (10). Further, s N is natural in N , since it is defined in terms of the right A-action on N , which commutes with left A-module morphisms. Proof From the description of in Proposition 4.3, we have The result follows by Yoneda lemma applied on P.
Similarly to the argument preceding (7), under the identification (4), the unique maximal B -stable ideal of N, corresponds to A ⊗ k Rad(e 0 Ae 0 ). Using the identification of i with − ⊗ A Ae i , we see that i sends this ideal to zero, which implies that i • factors canonically through the projection N → C. Let i • : C → C denote the resulting transformation. Since C is simple transitive, the strong transformation • i • is faithful. Since i sends the isomorphism class represented by Ae j ⊗ k e 0 A to that represented by Ae j , we see that • i • sends indecomposables to indecomposables, as prescribed by P M . Thus, the underlying functor of • i • is determined by a faithful, k-linear functor from A n -proj to A r -proj, which maps indecomposable objects to indecomposable objects, as prescribed by P M . Let T denote the set of isomorphism classes of such functors. Recall that Rad End A n -mod (A n e 0 ) = k {c}, independently of n (abusing notation by identifying c ∈ A n for varying n). From the above description it follows that the restriction of a functor F in T to that subspace corresponds to an endomorphism of k {c}, hence a scalar, which we denote by χ F .

Lemma 4.7
The map T → k\{0}, sending F to χ F , is a bijection. An automorphism τ of F in T is determined uniquely by τ A n e 0 .
Proof As a consequence of Proposition 3.11, an additive functor of finitary categories is determined, up to natural isomorphism, by its restriction to the full subcategory of indecomposable objects. Further, a natural transformation between such functors is uniquely determined by its components indexed by indecomposable objects. Since we do not consider or assume any monoidal or strict monoidal structure on our categories and functors, we may simplify further by replacing the domain and codomain categories by equivalent, skeletal categories N, R. We write Ob N = {A n e 0 , . . . , A n e n } and Ob R = {A r e 0 , . . . , A r e r }. Choose F, F ∈ T . Identifying c ∈ A n with c ∈ A r , the map F A n e 0 ,A n e 0 corresponds to an algebra endomorphism of k[c]/ c 2 . Since F is faithful, it sends c to χ F c with χ F = 0. Assume χ F = χ F , and let σ : F → F be a natural transformation. We have F(A n e 0 ) = F (A n e 0 ) = A r e 0 . We write σ 0 (e 0 ) = σ 0 e 0 + σ c c. Naturality implies the commutativity of

A r e P(i)
A r e 0 A r e P(i) A r e 0 e P(i) →λ i a P(i) e P(i) →σ i e P(i) e P(i) →σ 0 e 0 +σ c c e P(i) →λ i a P(i) and so, for a fixed σ 0 , we may set σ i = σ 0 λ i λ i . Given i, i ∈ [1, n] such that i = i , the Hom-space Hom A n -proj (A n e i , A n e i ) is zero, so the commutativity of the above diagram, together with the commutativity of does define a natural isomorphism. In particular, we see that σ is completely determined by its component indexed by A n e 0 . This concludes the proof. Proof Clearly, if the statement holds for a functor F, it also holds for any functor F isomorphic to F. We may thus set F = Be i ⊗ k e j B ⊗ B − for some primitive idempotents e i , e j ∈ B.
, and X Be i . For any f ∈ End B (M), the morphism F f maps k e i ⊗ v | v ∈ e j M to itself, and so F f cannot be a radical morphism. If F f |X , F f |X are linearly independent automorphisms of X , then, since the top of X is simple, there is a linear combination which is a contradiction. There is λ ∈ k\{0} such that m = λm.

Simple Transitive Birepresentations of B n
We will use the theory of bicategorical weighted colimits to construct and classify simple transitive birepresentations of B . The bicategories in which we will consider colimits are Cat k and the pseudofunctor bicategory [B , Cat k ] k . Observe that both these bicategories are, in fact, 2-categories. This simplifies the notions below.
Let C be a 2-category. Let J 1 be the 2-category given by • • . In particular, J 1 has no non-identity 2-morphisms. Let F be the 2-functor J 1 → C given by i denotes the terminal category with a unique object and only its identity morphism, and Iso is the walking isomorphism category, with two objects P 1 , P 2 and, as its only non-identity morphisms, two mutually inverse morphisms between the two objects.
The bicategorical coisoinserter of F, G above is the bicategorical colimit W F. One may verify that it is given by an object w of C together with a 1-morphism W : j → w and an commutes. Further, the pair ( H, γ ) is unique up to an invertible 2-morphism compatible with γ . This is the one-dimensional aspect of the universal property of (w, W, ζ ). In our applications we will not need the two-dimensional aspect, and hence we omit describing it.
where Arr is the walking arrow category, with two objects Q 1 , Q 2 and a unique morphism Q 1 → Q 2 . In particular, the two 2-cells in the last diagram coincide.
The bicategorical coequifier of α and β above is the colimit W F. One may verify that it is given by an object r together with a 1-morphism R : j → r such that there are equivalences strongly natural in k. Here, C (j, k) eqf denotes the full subcategory of C (j, k) given by where Iso(P) is the category presented by We now return to the setting of the bicategory B n studied in the previous sections. F W (P) be the 2-functor from J n to [B n , Cat k ] k given by the diagram The colimit W(P) F W (P) can be obtained by iterating coisoinserters, and is given by a k-linear pseudofunctor C W (P), a strong transformation W (P) and a set of invertible modifications x m l : Proof The statement follows from the one-dimensional aspect of the universal property of (C WR (P), WR (P), x m l ) obtained by combining the universal properties of the iterated coisoinserter C W (P) and the coequifier C R (P). Note that the statement of the lemma does not exhaust the one-dimensional universal property.
We now describe the underlying category for C W (P). Since it is constructed pointwise, it suffices to find the coisoinserter of the underlying diagram in Cat k . We will only determine the underlying category up to equivalence, and thus, using Proposition 3.6, Corollary 3.8, and Proposition 3.11 we may restrict the domain category P(i) to its full subcategory P(i)-indec of indecomposable objects. We may also restrict the codomain to the common essential image of the underlying functors of i , for i ∈ [1, n], which is exactly C(i)-indec. This is clear under the identification of C(i) with A n -proj and of i with − ⊗ A n A n e i . We have thus reduced the problem of finding the underlying coisoinserter to that of finding W(P) D, where D is the diagram
−⊗ An A n e i 1 1 −⊗ An A n e i r kr (12) Remark 5. 2 We recommend the reader to first consider the case n = 2 and P 0 given by [0, 2] = {0} {1, 2}. In fact, the only aspect of the remaining arguments that does not carry over verbatim from that case to the general case is taken care of by the explicit description of the weight W(P) above. We will see that, in the case of P 0 , one only needs to, in a sense, adjoin a single isomorphism to the category. In the general case, the form of the weight W(P) tells us to add as few isomorphisms as possible to still obtain the sought identifications of isomorphism classes. This considerably facilitates the next step (using coequifiers), which is to ensure that the result is finitary.
Let P S be the skeletal subcategory of P(i)-indec with and let I P S be its inclusion functor. Further, consider the full subcategories C A , C S of A-indec. proj given by Using e 0 A ⊗ A Ae j e 0 Ae j = k b j for j ∈ [1, n], let F A,S : C A → C S be the k-linear functor induced by A-module isomorphisms In particular, F A,S ((Ae k ⊗ k e 0 A) ⊗ A Ae j ) = Ae k , F A,S (A ⊗ A Ae j ) = Ae j , and F A,S is an equivalence. The full images of (− ⊗ A Ae j ) • I P S , for j ∈ [1, n], are contained in C A , and so we may consider the diagram . .
in Cat k . Since the inclusions of the subcategories C S , P S are both equivalences of categories, Diagram 12 and Diagram 13 are equivalent as 2-functors to Cat k . We may thus compute the colimit of this diagram in order to obtain an underlying category equivalent to the colimit of diagram 12, and, similarly, the universal cones will be compatible under such equivalence. This again reduces the problem of finding the colimit C W (P). To simplify the notation, let It is easy to describe F i explicitly. For example, we have and the image is exactly the same if we omit s i m l ,i m l+1 from the diagram. We conclude that s m l (Ae j ⊗ e 0 A) = id Ae j . Observe that, after substituting ξ m l (k) = id Ae k in 14, the equations still hold, so the functor described in the lemma is well-defined, and the assignment ζ m l (Ae k ⊗ k e 0 A) := id Ae k , (ζ m l )(A) = ξ m l (A) gives an isomorphism coinciding with s m l on Ae j ⊗ k e 0 A, for j ∈ [0, n].
We conclude that a k-linear functor G : C W → D coequifies s m l and x m l if and only if it sends ξ m l (k) to id G(Ae k ) . It is now clear that there is a unique functor G : C WR → D such that G • R = G. If we view C WR as a subcategory of C W , under the clear embedding, we may describe G as the corresponding restriction of G. That C WR is the coequifier now follows from Proposition 3.9.
To see that (C WR ) D A r -proj, note that C WR is obtained from the category of indecomposable A n -projectives by adjoining isomorphisms that do not increase the dimensions of the Hom-spaces, due to the relations given in the lemma and the fact that we only adjoin isomorphisms along the linear orders [i m 1 , i m k m ], so no new automorphisms can be obtained by composing the adjoined isomorphisms. • Given indecomposable objects X , Y ∈ C(i), the map WR X ,Y (P) is a bijection unless X Y and WR (P)(X ) WR (P)(Y ). • If U is a transversal of P, then the restriction of WR (P) to add {Ae k | k ∈ U } is an equivalence of categories.
Proof As observed earlier, applying (−) D to the diagram gives a diagram in Cat D k equivalent to that underlying Since (−) D sends faithful functors to faithful functors, and the remaining statements concern indecomposable objects and are preserved under equivalences of categories, the result follows from the explicit description of Diagram (17).

Proposition 5.8 The birepresentation C WR (P) is simple transitive.
Proof Let I be a B -stable ideal of C WR (P). Consider the B -stable ideal WR (P) −1 I of C, as defined in Lemma 2.2. Since C is simple transitive, this latter ideal is either zero, or coincides with all of C. In the former case, it is immediate that I = 0. Since Let I = C. Since WR restricted to add {Ae k | k ∈ U }, where U is a transversal of P, is an equivalence of categories, it follows that the functor WR is surjective on morphismsits essential image is all of C WR (P)(i). It follows that WR (P) −1 I = C implies I = C WR (P). We conclude that I = 0 or I = C WR (P), which proves that C WR (P) is simple transitive. . Let U be a transversal for P. Denote by |U the restriction of to add {Ae k | k ∈ U }. Using similar notation for • WR (P), we obtain But Propositions 4.3 and 5.7 prove that |U and WR (P) |U are equivalences of categories. It follows that is an equivalence of categories, and thus also an equivalence of birepresentations.
is a bijection.
Proof From Proposition 5.8 we have that, for any P, the birepresentation C WR (P) is simple transitive. From Corollary 4.4 it follows that for different set partitions P, P , we have C WR (P) C WR (P ). Finally, Theorem 5.9 shows that any simple transitive birepresentation M is equivalent to C WR (P), for P = P M .

Proposition 5.11
Let P be a set partition, let P be a refinement of P and let P be a refinement of P . There is a strong transformation (P, P ) : C WR (P) → C WR (P ) such that WR (P ) (P, P ) • WR (P).

Further, we have
(P, P ) (P , P ) • (P, P ) Proof Recall the universal property of C WR (P) described in Lemma 5.1. Applying it on the strong transformation WR (P ), we obtain (P, P ) satisfying WR (P ) (P, P ) • WR (P).
Further, we conclude that a strong transformation satisfying the above property is unique up to invertible modification. Similarly, we have a strong transformation (P, P ), unique up to invertible modification, such that (P, P ) • WR (P) WR (P ).
We have − → f 0 B f 0 is a k-linear isomorphism. The quiver automorphism swapping 1, 2 and fixing the remaining vertices gives an automorphism ψ (12) of B . The resulting linear isomorphism These properties are completely analogous to those of star algebras, and using them one may verify that there is a finitary birepresentation of (add B B B , B f k ⊗ k f 0 B , ⊗ B ) given by an analogous (multiple) coequifier of (multiple) coisoinserter.
However, for such examples, the analysis of the action matrices and Cartan matrices such as that for star algebras in [38] has not been conducted, and so we cannot easily generalize our classification of simple transitive birepresentations to these cases.