A Universal Investigation of n-representations of n-quivers

We have two goals in this paper. First, we investigate and construct cofree coalgebras over n-representations of quivers, limits and colimits of n-representations of quivers, and limits and colimits of coalgebras in the monoidal categories of n-representations of quivers. Second, for any given quivers Q1,Q2,..., Qn, we construct a new quiver Q(Q1,Q2,...,Qn) , called an nquiver, and identify each category Repk(Qj) of representations of a quiver Qj as a full subcategory of the category Repk(Q(Q1,Q2,...,Qn)) of representations of Q (Q1,Q2,...,Qn) for every j ∈ {1, 2, . . . , n}.


Introduction
The notions of quiver and their representation can be traced back to 1972 when they were introduced by Gabriel [16].Since then, it has been studied as a vibrant subject with a strong linkage with many other mathematics areas.This comes from the modern approach that quiver representation theory suggests.Due to its inherent combinatorial flavor, this theory has re-cently been largely studied as extremely important theory with connections to many theories, such as associative algebra, combinatorics, algebraic topology, algebraic geometry, quantum groups, Hopf algebras, tensor categories.Further, it bridges the gap between combinatorics and category theory, and this simply comes from the well-known fact that there is a forgetful functor, which has a left adjoint, from the category of small categories to the category of quivers.It turns out that it gives "new techniques, both of combinatorial, geometrical and categorical nature" [11, p. ix].
To study a subject more extensively, one might need to generalize it in a certain way.The notion of n-representations of quivers can be introduced as a generalization of representations of quivers.We start with 2-representations of quivers and inductively define n-representations quivers.Then we mainly concentrate our study on 2-representations of quivers because they roughly give a complete description of n-representations of quivers which can be established analogously.We alternatively and preferably call 2-representations of quivers birepresentations of quivers.As the reader might notice, birepresentations of quivers are fundamentally different from representations of biquivers1 introduced by Sergeichuk in [27, p. 237].
This paper is mainly devoted to two parallel goals.The first one is to investigate and construct cofree coalgebras, limits and colimits of coalgebras in the categories of n-representations of quivers.
The other goal of this paper is to introduce a generalization for quivers and prove that this can be recast into n-representations of quivers.Accordingly, for any given quivers Q 1 , Q 2 , ..., Q n , one might build a new quiver Q (Q 1 ,Q 2 ,...,Qn) , called n-quiver, by which we are able to view each category Rep k (Q j ) of representations of a quiver Q j as a full subcategory of the category Rep k (Q (Q 1 ,Q 2 ,...,Qn) ) of representations of Q (Q 1 ,Q 2 ,...,Qn) for every j ∈ {1, 2, . . ., n}.It is worth mentioning that for dealing with finite dimensional representations, one could consider the corresponding map between the quiver coalgebra of Q 1 , Q 2 , ..., Q n and the quiver coalgebra of Q (Q 1 ,Q 2 ,...,Qn) [12].
Before formulating our problem, we recall some categorical definitions.Let X be a category.A concrete category over X is a pair (A, U), where A is a category and U : A → X is a faithful functor [2, p. 61].Let (A, U) be a concrete category over X.Following [2, pp. 140-143], a free object over X-object X is an A-object A such that there exists a universal arrow (A, u) over X; that is, u : X → UA such that for every arrow f : X → UB, there exists a unique morphism f : A → B in A such that Uf u = f .We also say that (A, u) is the free object over X.A concrete category (A, U) over X is said to have free objects provided that for each X-object X, there exists a universal arrow over X.For example, the category V ect K of vector spaces over a field K has free objects.So do the category Top of topological spaces and the category of Grp of groups.However, some interesting categories do not have free objects [2, p. 142]).
Dually, co-universal arrows, cofree objects and categories that have cofree objects can be defined.For the basic concepts of concrete categories, free objects, and cofree objects, we refer the reader to [17, pp. 138-155].
It turns out that a concrete (A, U) over X has (co)free objects if and only if the functor that constructs (co)free object is a (right) left adjoint to the faithful functor U : A → X.Our problem can be formulated as follows.Let U n : CoAlg(Rep (Q 1 ,Q 2 ,...,Qn) ) → Rep (Q 1 ,Q 2 ,...,Qn) be the forgetful functor from the category of coalgebras in the category Rep (Q 1 ,Q 2 ,...,Qn) to the category Rep (Q 1 ,Q 2 ,...,Qn) .The question is: does U n have a right adjoint?
An expected strategy for the answer of this question is to use the dual of Special Adjoint Functor Theorem (D-SAFT).
We prove that the category of representations of n-quivers is equivalent to the category of usual quiver representations of a quiver (the n-quiver construction we introduce with respect to the n quivers).This shows that this category is abelian, has limits and colimits, is Grothendieck and even hereditary.Our main focus is, however, to give explicit examples and constructions of several categorical elements such as limits, kernels or cokernels.
The sections of this paper can be summarized as follows.
In Section 2, we give some detailed background on quiver representations and few categorical notions that we need for the next sections.
In Section 3, we introduce the notion of n-representations of quivers, and we explicitly give concrete examples of birepresentations of quivers.In addition, we establish the categories of n-representations of quivers.
In Section 4, we show that limits of birepresentations (2-representations of quivers) exist and inductively extend our results to limits of n-representations of quivers.We also construct them in terms of limits of representations of quivers.In similar fashion, we investigate and construct colimits of nrepresentations of quivers, and then we end the section by showing that the categories of n-representations of quivers are abelian.
In Section 5, we introduce the notion of 2-quivers and inductively define n-quivers.We explicitly give concrete examples of n-quivers and representations of n-quivers.By using the concept of n-quivers, we identify the categories of n-representations of quivers and the categories of representations of n-quivers as equivalent categories.This helps us to have an explicit description for the generators of the categories of n-representations of quivers and characterize some properties of n-representations of quivers.Finally, we investigate cofree coalgebras in the monoidal categories of nrepresentations of quivers.We also construct them in terms of colimits and generators and show that cofree coalgebras in these monoidal categories can be obtained from cofree coalgebras in the monoidal categories of quiver representations.

Preliminaries
Throughout this paper k is an algebraically closed field, n ≥ 2, and are quivers.We also denote kQ the path algebra of Q. Unless otherwise specified, we will consider only finite, connected, and acyclic quivers.Let A be a (locally small) category and A, B objects in A. We denote by A(A, B) the set of all morphisms from A to B. Let A, B be categories.Following [21, p. 74], the product category A × B is the category whose objects are all pairs of the form (A, B), where A is an object of A and B an object of B. An arrow is a pair (f, g) : (A, B) → (A , B ), where f : A → A is an arrow of A and g : B → B is an arrow of B. The identity arrow for A × B is (id A , id B ) and composition is defined component-wise, so (f, g)(f , g ) = (f f , gg ).There is a projective functor P 1 : A × B → A defined by P 1 (A, B) = A and P 1 (f, g) = f .Similarly, we have a projective functor P 2 : A × B → B defined by P 2 (A, B) = B and P 2 (f, g) = g.
The following consequences are important for our investigation.(ii) If furthermore C is co-wellpowered, then so is CoM on(C).
Following [25], • Q 1 a set of arrows, • s : Q 1 → Q 0 a map from arrows to vertices, mapping an arrow to its starting point, • t : Q 1 → Q 0 a map from arrows to vertices, mapping an arrow to its terminal point.
We will represent an element α ∈ Q 1 by drawing an arrow from its starting point s(α) to its endpoint t(α) as: s(α) α − → t(α).A representation M = (M i , ϕ α ) i∈Q 0 ,α∈Q 1 of a quiver Q is a collection of k-vector spaces M i one for each vertex i ∈ Q 0 , and a collection of k-linear maps ϕ α : M s(α) → M t(α) one for each arrow α ∈ Q 1 .
A representation M is called finite-dimensional if each vector space M i is finite-dimensional.
Let Q be a quiver and let A morphism of representations f = (f i ) : M → M is an isomorphism if each f i is bijective.The class of all representations that are isomorphic to a given representation M is called the isoclass of M .
The above definition introduces a category Rep k (Q) of k-linear representations of Q.We denote by rep k (Q) the full subcategory of Rep k (Q) consisting of the finite dimensional representations.
Given two representations M = (M i , φ α ) and M = (M i , φ α ) of Q, the representation A nonzero representation of a quiver Q is said to be indecomposable if it is not isomorphic to a direct sum of two nonzero representations [14, p. 21].
Following [25, p. 114], the path algebra kQ of a quiver Q is the algebra with basis the set of all paths in the quiver Q and with multiplication defined on two basis elements c, c by c.c = cc , if s(c ) = t(c) 0, otherwise.
We will need the following propositions.There exists an equivalence of categories M od kQ Rep k (Q) that restricts to an equivalence M od kQ rep k (Q), where kQ is the path algebra of Q, M od kQ denotes the category of right kQ-modules, and M od kQ denotes the full subcategory of M od kQ consisting of the finitely generated right kQmodules.This is a very brief review of the basic concepts involved with our work.For the basic notions of quiver representations theory, we refer the reader to [3], [25], [4], [7], [14], [8], [30].
3 n-representations of quivers: Basic concepts , where Unless confusion is possible, we denote a birepresentation simply by M = (M, M , ψ).Next, we inductively define n-representations for any integer n ≥ 2.
For any m ∈ {1, . . ., n}, let where for every m ∈ {1, 2, . . ., n}, V (m) is a representation of Q m , and (i) When no confusion is possible, we simply write s and t instead of s and t , respectively, and for every m ∈ {1, 2, . . ., n}, we write s and t instead of s (m) and t (m) , respectively.
(ii) It is clear that if (V (1) , V (2) , ..., (iii) Part (ii) implies that for any integer n > 2, n-representations roughly inherit all the properties and universal constructions that (n − 1)representations have.Thus, we mostly focus on studying birepresentations, since they can be regarded as a mirror in which one can see a clear description of n-representations for any integer n > 2.
Example 3.3.Let Q and Q be the following quivers and consider the following: Then M and M are representations of Q and Q respectively [25].The following are birepresentations of (Q, Q ).
A morphism of n-representations can be depicted as: Thus, representations of a quiver can be seen as a particular case of n-representations of quivers.
(i) One might consider the class B 0 and full subcategories of the categories of birepresentations of quivers to build a bicategory.Indeed, there is a bicategory B consists of • the objects or the 0-cells of B are simply the elements of B 0 , objects are the 1-cells of B, and whose morphisms are the 2-cells of B, • for each Q, Q , Q ∈ B 0 , a composition functor defined by: • for any Q ∈ B 0 and for each (M, M ) ∈ B(Q, Q), we have Also, for any 2-cell (f, f ), we have Thus, the identity and the unit coherence axioms hold.
The rest of bicategories axioms are obviously satisfied.For each Q, Q ∈ B 0 , let (Q,Q ) be the full subcategory of Rep (Q,Q ) whose objects are the triples (X, X , Ψ), where (X, and whose morphisms are usual morphisms of birepresentations between them.Clearly, Thus, by considering the class B 0 and these full subcategories (described above) of the birepresentations categories of quivers, we can always build a bicategory as above.Obviously, the discussion above implies that for each We also have the same analogue if we replace For the basic notions of bicategories, we refer the reader to [19].
(ii) For any whose objects are pairs of triples of the type ((X, X , ψ), (X , X , ψ )), where (X, X , ψ) ∈ Rep (Q,Q ) and (X , X , ψ ) ∈ Rep (Q ,Q ) , and whose morphisms are usual morphisms (in ) be a map defined by: for any objects and for any morphism To show this, consider the following diagram: we have: Example 3.7.Let Q, Q be the quivers defined in Example 3.3 and consider the following: Then V and W (respectively, V and W ) are representations of Q (respectively, Q ) [25].It is also straightforward to verify that We refer the reader to [25] for more details.Consider the following.
(3.15) Then V and W are birepresentations of (Q, Q ), and V and W are birepresentations of (Q , Q).

To compute Rep
The commuting squares give the relations Hence, we obtain We leave it to the reader to compute (3.18)where Similarly, direct sums in Rep (Q 1 ,Q 2 ,...,Qn) can be defined.
Example 3.9.Consider the birepresentations in Example 3.7.Then the direct sum V ⊕ W is the birepresentation and M cannot be written as a direct sum of two nonzero birepresentations, that is, whenever Example 3.11.Consider the birepresentations in Example 3.7.The birepresentation V is indecomposable, but the birepresentation W is not.
The above example also shows that if W = ((W i , φ α ), (W i , φ β ), (ψ α β )) is birepresentation of (Q, Q ) such that the representations (W i , φ α ) and W need not be indecomposable.The proof of the following proposition is straightforward.

Completeness, cocompleteness and canonical decomposition of morphisms in the categories of n-representations
Recall that a category C is cocomplete when every functor F : D → C, with D a small category has a colimit [9].For the basic notions of cocomplete categories and examples, we refer to [2], [9], or [26].A functor is cocontinuous if it preserves all small colimits [15, p. 142].
Proposition 4.1.The category Rep (Q,Q ) is complete and the forgetful functor is continuous.In addition, the limit of objects in Rep (Q,Q ) can be obtained by the corresponding construction for objects in Proof.Let D be a small category, and let F : D → Rep (Q,Q ) be a functor, and consider the composition of the following functors.
where U, U are the obvious forgetful functors, and P 1 , P 2 are the projection functors.For all D ∈ D, let F D = V = (V, V , ψ D, ).Since Rep k (Q), Rep k (Q ) are complete categories, the functors P 1 UF , P 2 UF have limits.Let (L, (η D ) D∈D ), (L , (η D ) D∈D ) be limits of P 1 UF , P 2 UF respectively, where η D , η D are the morphisms for every . Let h : D → D be a morphism in D and consider the following diagram 4.4. Ls(α) Clearly, (L s(β) , (η D s(β) ) D∈D ) can be viewed as a limit of a functor ρ : D → V ect k .Also, for any D, D ∈ D, we have for every D ∈ D and for each pair of arrows (α, Let L = (L, L , ψ ), ηD = (η D , η D ).We claim that ( L, (η D ) D∈D ) is a limit of F .Then obviously all we need is to show that for any cone ( L, ( ηD ) D∈D ), there exists a unique morphism Ξ = (Ξ, Ξ ) in Rep (Q,Q ) with ηD Ξ = ηD for every D ∈ D. Let ( L, ( ηD ) D∈D ) be a cone on F and write Notably, ( Lt(α) , (η D s(β) ψ ) D∈D ) can be viewed as a limit of a functor ρ : Since V ec k is complete, it follows from the universal property of the limit that Ξ s(β) Thus, ( L, (η D ) D∈D ) is a limit of F , as desired.
From Proposition 4.1 and Remark 3.2, we obtain.
Proposition 4.2.The category Rep (Q 1 ,Q 2 ,...,Qn) is complete and the forgetful functor is continuous.Further, the limit of objects in Rep (Q 1 ,Q 2 ,...,Qn) can be obtained by the corresponding construction for objects in Since a category is complete if and only if it has products and equalizers [2], the following is an immediate consequence of Proposition 4.2.
Proof.Let D be a small category, and let F : D → Rep (Q,Q ) be a functor, and consider the composition of the following functors.
where U, U are the obvious forgetful functors, and P 1 , P 2 are the projection functors.For all D ∈ D, let for every . Let h : D → D be a morphism in D and consider the following diagram.
x x Ct(α) for every D ∈ D and for each pair of arrows (α, We claim that ( C, ( ζD ) D∈D ) is a colimit of F .To substantiate this claim, we need to show that for any cocone ) D∈D ) can be viewed as a colimit of a functor υ : (since Λ ζD = ζD ) Since V ec k is cocomplete, it follows from the universal property of the colimit that Λ s(β) . Consequently, Λ is a morphism in Rep (Q,Q ) , which completes the proof.
From Proposition 4.4 and Remark 3.2, we obtain.Proposition 4.5.The category Rep (Q 1 ,Q 2 ,...,Qn) is cocomplete and the forgetful functor Since a category is cocomplete if and only if it has coproducts and coequalizers [2], we have the following immediate consequence.
Next, we aim to show that the Categories of n-representations are abelian.Following [13, p. 2], an additive category is a category C satisfying the following axioms: (i) Every set C(X, Y ) is equipped with a structure of an abelian group (written additively) such that composition of morphisms is biadditive with respect to this structure.
(iii) (Existence of direct sums.)For any objects X, X ∈ C, the direct sum Let k be a field.An additive category C is said to be k-linear if for any objects X, Y ∈ C, C(X, Y ) is equipped with a structure of a vector space over k, such that composition of morphisms is k-linear.
An abelian category is an additive category C in which for every morphism f : X → Y there exists a sequence with the following properties: From the above results, kernels and cokernels exist in Rep (Q,Q ) .It turns out that f has a canonical decomposition in Rep (Q,Q ) , and this decomposition can explicitly be seen in the following commutative diagram.
X t(α) s s e e (4.12)This implies that any morphism f : V → W of n-representations has a canonical decomposition in ) are equipped with a structure of an abelian group such that composition of morphisms is biadditive with respect to this structure [3, p. 70].Since f , ḡ are morphisms in Rep (Q,Q ) and since the category V ec k is abelian, we have the following commutative diagram: Thus, the set Rep (Q,Q ) ( V , W ) is equipped with a structure of an abelian group such that composition of morphisms is biadditive with respect to the above structure.
We end this section with the following result.

n-quivers and n-representations
Let V = (V, V , ψ), W = (W, W , ψ ) be birepresentations of (Q, Q ).For simplicity, we suppress k from the tensor product ⊗ k and use ⊗ instead.
Then it is clear that f ⊗ ḡ is a morphism in Rep (Q,Q ) , and hence the following diagram is commutative.
• s : Q1 → Q0 a map from arrows to vertices, mapping an arrow to its starting point, • t : Q1 → Q0 a map from arrows to vertices, mapping an arrow to its terminal point.
The notation above denotes the disjoint union. (5.5) Consider the following: Then V, V , V are clearly representations of Q, Q , Q respectively.Now, consider the following: where kΩ is the subspace of kQ (Q 1 ,Q 2 ,...,Qn) generated by the set Ω n .To avoid confusion, we identify kQ (Q 1 ) as kQ 1 .
(ii) We note that V in the previous example can be identified as a birepresentations of (Q, Q ).Indeed, every representation of Q (Q,Q ) can be identified as a birepresentation of (Q, Q ).Conversely, every birepresentation of (Q, Q ) can be viewed as a representation of Q (Q,Q ) .Similarly, V can be identified as a 3-representation of Q (Q ,Q,Q ) .This can be explicitly stated as the following.

Proposition 5.4. There exists an equivalence of categories Rep
Proof.This follows from the construction of Q (Q,Q ) in Definition 5.1.In fact, there is a combining functor F : Using Induction and Remark 3.2, we have the following.Proposition 5.5.For any n ≥ 2, there exists an equivalence of categories The following is an immediate consequence of Proposition 5.4 and Proposition 2.4.Proposition 5.6.There exists an equivalence of categories M od kQ By Remark 3.2 and Propositions 5.5, 2.4, we have the following.Proposition 5.7.For any n ≥ 2, there exists an equivalence of categories M od kQ We recall the definitions of a co-wellpowered category and a generating set for a category.Let E be a class of all epimorphisms of a category A. Then A is called co-wellpowered provided that no A-object has a proper class of pairwise non-isomorphic quotients [2, p. 125].In other words, for every object the quotients form a set [26, p. 92, 95].We refer the reader to [2] basics on quotients and co-wellpowered categories.
Following [?, p. 127], a set G of objects of the category C is said to generate C when any parallel pair f, g : X → Y of arrows of C , f = g implies that there is an G ∈ G and an arrow α : G → X in C with f α = gα (the term "generates" is well established but poorly chosen; "separates" would have been better).For the basic concepts of generating sets, we refer to [?], [2], or [15].
The following proposition immediately follows from Remark 3.2, Proposition 5.7 and the fact that the categories of modules are co-wellpowered with generating sets.Proposition 5.8.For any n ≥ 2, the category Rep (Q 1 ,Q 2 ,...,Qn) is co-well powered.
Let CoAlg(Rep (Q,Q ) ) be the category of coalgebras in Rep (Q,Q ) .By [1, p. 30], the left kQ (Q,Q ) -module coalgebras which are finitely generated as left kQ (Q,Q ) -modules form a system of generators for CoAlg(M od kQ (Q,Q ) ).Thus, from Proposition 2.2 and Theorem 2.1, we have the following.Ḡ.

Theorem 2 . 1 .Proposition 2 . 2 .
(D-SAFT)[15, p. 148]  If A is cocomplete, co-wellpowered and with a generating set, then every cocontinuous functor from A to a locally small category has a right adjoint.Let (C, ⊗, I) be a monoidal category, CoM on(C) be the category of comonoids of C and U : CoM on(C) → C be the forgetful functor.(i)If C is cocomplete, then CoM on(C) is cocomplete and U preserves colimits.

Theorem 4 . 8 .
The category Rep (Q,Q ) is a k-linear abelian category.More generally, the category Rep (Q 1 ,Q 2 ,...,Qn) is a k-linear abelian category for any integer n ≥ 2.

For
any m ∈ {1, 2, . . ., n}, let U m : CoAlg(Rep k (Q m )) → Rep k (Q m ) be the obvious forgetful functor with a right adjoint V m .Proposition 5.14 and Remark 3.2 immediately imply the following consequence.Corollary 5.15.The product functor