A Universal Investigation of n-representations of n-quivers

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1 In press. A Universal Investigation of n-representations of n-quivers Adnan H. Abdulwahid Abstract. 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 Q,Q 2,..., Q n, we construct a new quiver Q (Q,Q 2, called an n-,...,qn) quiver, and identify each category Rep (Q j) of representations of a quiver Q j as a full subcategory of the category Rep (Q (Q,Q 2 ) of representations,...,qn) of Q (Q,Q 2 for every j {, 2,..., n}.,...,qn) Introduction The notions of quiver and their representation can be traced bac to 972 when they were introduced by Gabriel [6]. Since then, it has been studied as a vibrant subject with a strong linage 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- Keywords: Quiver, representation, birepresentation, n-representation, additive category, abelian category, -linear category. Mathematics Subject Classification [2]: 2G5, 47A67, 6B5, 6Gxx, 8D, 9D23, 8Axx, 8A35, 6T5. Received: 23 December 27, Accepted: 7 April 28 ISSN Print: Online: Shahid Beheshti University

2 2 A. H. Abdulwahid 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-nown 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" [, 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 biquivers introduced by Sergeichu 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, Q 2,..., Q n, one might build a new quiver Q (Q,Q 2, called n-quiver, by which we are able to view each category Rep (Q j ) of representations of a quiver Q j as a full subcategory of,...,qn) the category Rep (Q ) of representations of Q (Q,Q 2,...,Qn) (Q,Q 2 for every j {, 2,..., n}. It is worth mentioning that for dealing with finite,...,qn) dimensional representations, one could consider the corresponding map between the quiver coalgebra of Q, Q 2,..., Q n and the quiver coalgebra of Q [2]. (Q,Q 2,...,Qn) 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 A directed graph with usual and dashed arrows will be called a biquiver. Its representation is given by assigning to each vertex a complex vector space, to each usual arrow a linear mapping, and to each dashed arrow a semilinear mapping [27, p. 237].

3 A Universal Investigation of n-representations of n-quivers 3 is a category and U : A X is a faithful functor [2, p. 6]. Let (A, U) be a concrete category over X. Following [2, pp. 4-43], 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. 42]). 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 [7, pp ]. 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 ) Rep (Q,Q 2,...,Qn) (Q,Q 2 be the forgetful functor from,...,qn) the category of coalgebras in the category Rep (Q,Q 2 to the category,...,qn) Rep. The question is: does U (Q,Q 2,...,Qn) 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 Grothendiec and even hereditary. Our main focus is, however, to give explicit examples and constructions of several categorical elements such as limits, ernels or coernels. The sections of this paper can be summarized as follows. In Section 2, we give some detailed bacground 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

4 4 A. H. Abdulwahid of quivers. We also construct them in terms of limits of representations of quivers. In similar fashion, we investigate and construct colimits of n- representations 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 n- representations 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. 2 Preliminaries Throughout this paper is an algebraically closed field, n 2, and Q, Q, Q, Q 2,..., Q n are quivers. We also denote Q 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 [2, 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 ) = (ff, gg ). There is a projective functor P : A B A defined by P (A, B) = A and P (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.

5 A Universal Investigation of n-representations of n-quivers 5 Theorem 2.. (D-SAFT) [5, p. 48] 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. Proposition 2.2. Let (C,, I) be a monoidal category, CoMon(C) be the category of comonoids of C and U : CoMon(C) C be the forgetful functor. (i) If C is cocomplete, then CoMon(C) is cocomplete and U preserves colimits. (ii) If furthermore C is co-wellpowered, then so is CoM on(c). For the fundamental concepts of category theory, we refer to [8], [2], [5], [24], [2], [9], [5], [23], or [22]. Following [25], a quiver Q = (Q, Q, s, t) consists of Q Q a set of vertices, a set of arrows, s : Q Q a map from arrows to vertices, mapping an arrow to its starting point, t : Q Q a map from arrows to vertices, mapping an arrow to its terminal point. We will represent an element α Q 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,α Q of a quiver Q is a collection of -vector spaces M i one for each vertex i Q, and a collection of -linear maps ϕ α : M s(α) M t(α) one for each arrow α Q. A representation M is called finite-dimensional if each vector space M i is finite-dimensional. Let Q be a quiver and let M = (M i, ϕ α ), M = (M i, ϕ α) be two representations of Q. A morphism of representations f : M M is a collection (f i ) i Q of -linear maps f i : M i M i such that for each arrow s(α) α t(α)

6 6 A. H. Abdulwahid in Q the diagram M s(α) φ α M t(α) (2..) f s(α) M s(α) φ α f t(α) M t(α) commutes. 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 (Q) of -linear representations of Q. We denote by rep (Q) the full subcategory of Rep (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 [ ] φα M M = (M i M i, φ ) (2..2) α is the direct sum of M and M in Rep (Q) [3, p. 7]. 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 [4, p. 2]. Following [25, p. 4], the path algebra Q 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) =, otherwise. We will need the following propositions. Proposition 2.3. [3, p. 7] Let Q be a finite quiver. Then Rep (Q) and rep (Q) are -linear abelian categories. Proposition 2.4. [3, p. 74] Let Q be a finite, connected, and acyclic quiver. There exists an equivalence of categories Mod Q Rep (Q) that restricts to an equivalence Mod Q rep (Q), where Q is the path algebra of Q,

7 A Universal Investigation of n-representations of n-quivers 7 M od Q denotes the category of right Q-modules, and M od Q denotes the full subcategory of Mod Q consisting of the finitely generated right Qmodules. This is a very brief review of the basic concepts involved with our wor. For the basic notions of quiver representations theory, we refer the reader to [3], [25], [4], [7], [4], [8], [3]. 3 n-representations of quivers: Basic concepts Let Q = (Q, Q, s, t), Q = (Q, Q, s, t ) be quivers. Definition 3.. A 2-representation of (Q, Q ) (or a birepresentation of (Q, Q )) is a triple M = ((M i, φ α ), (M i, φ ), (ψα )) i Q,i Q,α Q, where, Q (M i, φ α ), (M i, φ ) are representations of Q, Q respectively, and (ψ α) is a collection of -linear maps ψ α : M t(α) M s(), one for each pair of arrows (α, ) Q Q. 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 {,..., n}, let Q m = (Q (m), Q (m), s (m), t (m) ) be a quiver. An n-representation of (Q, Q 2,..., Q n ) is (2n )-tuple V = (V (), V (2),..., V (n), ψ, ψ 2,..., ψ n ), where for every m {, 2,..., n}, V (m) is a representation of Q m, and (ψ γ(m) mγ (m ) ) is a collection of -linear maps ψ γ(m) mγ (m ) : V (m ) t (m ) (γ (m ) ) V (m ) s (m) (γ (m) ), one for each pair of arrows (γ (m ), γ (m) ) Q (m ) Q (m) and m {2,..., n}. Remar 3.2. (i) When no confusion is possible, we simply write s and t instead of s and t, respectively, and for every m {, 2,..., n}, we write s and t instead of s (m) and t (m), respectively.

8 8 A. H. Abdulwahid (ii) It is clear that if (V (), V (2),..., V (n), ψ, ψ 2,..., ψ n ) is an n-representation of (Q, Q 2,..., Q n ), then (V (), V (2),..., V (n ), ψ, ψ 2,..., ψ n 2 ) is an (n )-representation of (Q, Q 2,..., Q n ) for every integer n 2. (iii) Part (ii) implies that for any integer n > 2, n-representations roughly inherit all the properties and universal constructions that (n )- 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 (3..) Q : 2 Q : and consider the following: M : M : 2 (3..2)

9 A Universal Investigation of n-representations of n-quivers 9 Then M and M are representations of Q and Q respectively [25]. following are birepresentations of (Q, Q ). The M 2 (3..3) N 2 (3..4) Definition 3.4. Let V = (V, V, ψ), W = (W, W, ψ ) be birepresentations of (Q, Q ). Write V = (V i, φ α ), V = (V i, µ ), W = (W i, φ α), W = (W i, µ ). A morphism of birepresentations f : V W is a pair f = (f, f ), where f = (f i ) : (V i, φ α ) (W i, φ α), f = (f i ) : (V i, µ ) (W i, µ ) are morphisms in Rep (Q), Rep (Q ) respectively such that the following

10 A. H. Abdulwahid diagram commutes. V s(α) φ α V t(α) f s(α) W s(α) φ α ψ α f t(α) (3..5) W t(α) V s() ψ α µ V t( f t() f s() W s() µ W t() The composition of two maps (f, f ) and (g, g ) can be depicted as the following diagram. V s(α) φ α V t(α) f s(α) W s(α) φ α f t(α) W t(α) g t(α) g s(α) ψ α U s(α) φ α U t(α) V s() f s() ψ α W s() µ µ V t( ψ α f t() W t() g s() U s() µ g t() U t() (3..6)

11 A Universal Investigation of n-representations of n-quivers In general, if V = (V (), V (2),..., V (n), ψ, ψ 2,..., ψ n ) and W = (W (), W (2),..., W (n), ψ, ψ 2,..., ψ n ) are n-representations of (Q, Q 2,..., Q n ), then a morphism of n-representations f : V W is n-tuple f = (f (), f (2),..., f (n ) ), where f (m) = (f (m) i (m) ) : (V i (m), φ i(m) γ (m) ) (W i (m), µ i(m) γ (m) ), is a morphism in Rep (Q m ) for any m {2,..., n}, and for each pair of arrows (γ (m ), γ (m) ) Q (m ) Q (m) the following diagram is commutative. V (m ) t (m ) (γ (m ) ) ψ γ(m) mγ (m ) V (m) s (m) (γ (m) ) (3..7) f (m ) t (m ) (γ (m ) ) f (m) s (m) (γ (m) ) W (m ) t (m ) (γ (m ) ) ψ γ (m) mγ (m ) W (m) s (m) (γ (m) ) for every m {2,..., n}. A morphism of n-representations can be depicted as: (3..8) Remar 3.5. (i) The above definition provides a category Rep (Q,Q ) of -linear birepresentations of (Q, Q ). We denote by rep (Q,Q ) the full subcategory of Rep (Q,Q ) consisting of the finite dimensional birepresentations. Similarly,

12 2 A. H. Abdulwahid it also creates a category Rep (Q,Q 2 of n-representations. We denote,...,qn) rep the full subcategory of Rep (Q,Q 2,...,Qn) (Q,Q 2 consisting of the finite,...,qn) dimensional n-representations. (ii) If Q is the empty quiver, then Rep (Q,Q ) is isomorphic to the category Rep (Q). Thus, representations of a quiver can be seen as a particular case of n-representations of quivers. (iii) For any m {,..., n}, let Q m = (Q (m), Q (m), s (m), t (m) ) be a quiver and fix j {,..., n}. Let Υ Rep (Q j ) be the subcategory of Rep (Q,Q 2,...,Qn) whose objects are (2n )-tuples X = (,,..., V (j),,...,, ψ, ψ 2,..., ψ n ), where V (j) is a representation of Q j, and ψ γ(m) = for every pair of mγ (m ) arrows (γ (m ), γ (m) ) Q (m ) Q (m) and m {2,..., n}. Then Υ Rep (Q j ) is clearly a full subcategory of Rep (Q,Q 2. Notably, we have an equivalence of categories Υ Rep (Q j ) Rep (Q j ), and thus by by Proposition,...,Qn) (2.4), we have Rep (Q j ) Υ Rep (Q j ) Mod Q j. It turns out that the category Rep (Q j ) and Mod Q j can be identified as full subcategories of Rep. (Q,Q 2,...,Qn) The category Υ Rep (Q j ) has a full subcategory Υ rep (Q j ) when we restrict the objects on the finite dimensional representations. Therefore, we also have rep (Q j ) Υ rep (Q j ) mod Q j. Remar 3.6. Let B be the class of all quivers. (i) One might consider the class B 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 -cells of B are simply the elements of B, for each Q, Q B, we have B(Q, Q ) = Rep (Q) Rep (Q ), whose objects are the -cells of B, and whose morphisms are the 2-cells of B, for each Q, Q, Q B, a composition functor F : Rep (Q ) Rep (Q ) Rep (Q) Rep (Q ) Rep (Q) Rep (Q )

13 A Universal Investigation of n-representations of n-quivers 3 defined by: F((N, N ), (M, M )) = (M, N ), F((g, g ), (f, f )) = (f, g ) on -cells (M, M ), (N, N ) and 2-cells (f, f ), (g, g ), for any Q B and for each (M, M ) B(Q, Q), we have F((M, M ), (M, M)) = (M, M ) and F((M, M ), (M, M )) = (M, M ). Also, for any 2-cell (f, f ), we have F((f, f ), (f, f )) = (f, f ) and F((f, f ), (f, f)) = (f, f ). Thus, the identity and the unit coherence axioms hold. The rest of bicategories axioms are obviously satisfied. For each Q, Q B, let ℶ (Q,Q ) be the full subcategory of Rep (Q,Q ) whose objects are the triples (X, X, Ψ), where (X, X ) Rep (Q) Rep (Q ) and Ψ α = for every pair of arrows (α, ) Q Q, and whose morphisms are usual morphisms of birepresentations between them. Clearly, ℶ (Q,Q ) = Rep (Q) Rep (Q ) for any Q, Q B. Thus, by considering the class B 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 Q, Q B, the product category Rep (Q) Rep (Q ) can be viewed as a full subcategory of Rep (Q,Q ). Further, it implies that the product category Rep (Q ) Rep (Q 2 )... Rep (Q n ) can be viewed as a full subcategory of Rep, (Q,Q 2,...,Qn) where Q, Q 2,..., Q n B and n 2. We also have the same analogue if we replace Rep, by rep, (Q,Q 2,...,Qn) (Q,Q 2,...,Qn) and Rep (Q ), Rep (Q 2 ),..., Rep (Q n ) by rep (Q ), rep (Q 2 ),..., rep (Q n ), respectively. For the basic notions of bicategories, we refer the reader to [9]. (ii) For any Q, Q, Q B, let H (Q,Q,Q ) be the full subcategory of Rep (Q,Q ) Rep (Q,Q ) 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 Rep (Q,Q ) Rep (Q,Q ) ) between them. Let Θ (Q,Q,Q ) : H (Q,Q,Q ) Rep (Q,Q ) be a map defined by: Θ (Q,Q,Q ) ((M, N, ψ), (N, L, ψ )) = (M, L, Ψ),

14 4 A. H. Abdulwahid Θ (Q,Q,Q ) ((f, f ), (g, g )) = (f, g ) : (M, L, Ψ) (M, L, Ψ ) for any objects ((M, N, ψ), (N, L, ψ )), ((M, N, ψ), (N, L, ψ )) in H (Q,Q,Q ), and for any morphism (f, f ) : (M, M, ψ) (N, N, ψ ), (g, g ) : (N, N, ψ ) (L, L, ψ ) where M = (M i, φ α ), M = (M i, φ α), N = (N i, µ ), N = (N i, µ ), L = (L i, ν γ ), L = (L i, ν γ), ψ = (ψ α ), ψ = (ψ α), ψ = ( ψ γ ), ψ = ( ψ γ), Ψ α γ = ψ γ µ ψ α and Ψ α γ = ψ γ µ ψ α. Note that Θ (Q,Q,Q ) : ((f, f ), (g, g )) = (f, g ) : (M, L, Ψ) (M, L, Ψ ) is a morphism in Rep (Q,Q ). To show this, consider the following diagram: M s(α) φ α M t(α) ψ α µ ψ N s() γ ν N t() γ L s(γ) L t(γ) f s(α) M s(α) f t(α) f g s() s() f g t() t() M φ t(α) N α ψ α s() N µ t() ψ γ For any α Q, Q, γ Q, we have: g s(γ) Ψα γ = g s(γ) ψ γ µ ψ α = = ψ γ g t() µ ψ α = g s(γ) ψ γ L s(γ) g s(γ) ν γ ψ γ µ ψ α f t() µ ψ α L t(γ) g t(γ) (3..9) = ψ γ µ f s() ψα = ψ γ µ ψ α f t() = Ψ α γ f t() Therefore, Θ (Q,Q,Q ) is a functor for any Q, Q, Q B. Indeed, for any Q, Q, Q B, Θ (Q,Q,Q ) has a left adjoint functor. Example 3.7. Let Q, Q be the quivers defined in Example 3.3 and consider the following: V : W : (3..)

15 A Universal Investigation of n-representations of n-quivers 5 V : 2 W : (3..) Then V and W (respectively, V and W ) are representations of Q (respectively, Q ) [25]. It is also straightforward to verify that Rep (Q)(V, W ) = and Rep (Q )(V, W ) = 2. We refer the reader to [25] for more details. Consider the following. V = (V, V, ψ) 2 (3..2) V = (V, V, ψ ) 2 [ ] [ ] [ ] (3..3)

16 6 A. H. Abdulwahid W = (W, W, ψ ) (3..4) W = (W, W, ψ ) (3..5) Then V and W are birepresentations of (Q, Q ), and V and W are birepresentations of (Q, Q). To compute Rep (Q,Q ) ( V, W ), consider the following diagram.

17 A Universal Investigation of n-representations of n-quivers 7 V 2 (3..6) [a] [c d] [e] [l] [m] [b] W The commuting squares give the relations a =, b =, c = a =, d = b =, e = c + d =, l = e =. (3..7) Hence, we obtain Rep (Q,Q ) ( V, W ) = {(,,,,,, m) : m } =. We leave it to the reader to compute Rep (Q,Q ) ( W, V ), Rep (Q,Q) ( V, W ) and Rep (Q,Q) ( W, V ). Definition 3.8. Let V = (V, V, ψ), W = (W, W, ψ ) be birepresentations of (Q, Q ). Write V = (V i, φ α ), V = (V i, φ ), W = ((W i, µ α ), W = (W i, µ ). Then [ ] [ ] [ ] V W φα φ = ((V i W i, ), (V µ i W i, α ψ α α µ ), α ψ α ), (3..8)

18 8 A. H. Abdulwahid [ ] [ ] φα φ where (V i W i, ), (V µ i W α i, α µ ) are the direct sums of α (V i, φ α ), (W i, µ α ) and (V i, φ ), (W i, µ ) in Rep (Q), Rep (Q ) respectively, is a birepresentation of (Q, Q ) called the direct sum of V, W (in Rep (Q,Q ) ). Similarly, direct sums in Rep (Q,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 (3..9) Definition 3.. A birepresentation V Rep (Q,Q ) is called indecomposable if M and M cannot be written as a direct sum of two nonzero birepresentations, that is, whenever M = L N with L, N Rep (Q,Q ), then L = or N =. Example 3.. Consider the birepresentations in Example 3.7. The birepresentation V is indecomposable, but the birepresentation W is not.

19 A Universal Investigation of n-representations of n-quivers 9 The above example also shows that if W = ((Wi, φ α ), (W i, φ ), (ψα )) is birepresentation of (Q, Q ) such that the representations (W i, φ α ) and (W i, φ ) are indecomposable in Rep (Q), Rep (Q ) respectively, then W need not be indecomposable. The proof of the following proposition is straightforward. Proposition 3.2. Let V = ((V i, φ α ), (V i, φ ), (ψα )) Rep be an indecomposable birepresentation, then the representations (W i, φ α ) and (W (Q,Q ) i, φ ) are indecomposable in Rep (Q) and Rep (Q ) respectively. 4 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 [5, p. 42]. Proposition 4.. The category Rep (Q,Q ) is complete and the forgetful functor U : Rep (Q,Q ) Rep (Q) Rep (Q ) is continuous. In addition, the limit of objects in Rep (Q,Q ) can be obtained by the corresponding construction for objects in Rep (Q) Rep (Q ). Proof. Let D be a small category, and let F : D Rep (Q,Q ) and consider the composition of the following functors. be a functor, D F Rep (Q,Q ) U Rep (Q) Rep (Q ) P Rep (Q) (4..) D F Rep (Q,Q ) U Rep (Q) Rep (Q ) P 2 Rep (Q ) (4..2) where U, U are the obvious forgetful functors, and P, P 2 are the projection functors. For all D D, let F D = V = (V, V, ψ D, ). Since Rep (Q), Rep (Q ) are

20 2 A. H. Abdulwahid complete categories, the functors P UF, P 2 UF have limits. Let (L, (η D ) D D ), (L, (η D) D D ) be limits of P UF, P 2 UF respectively, where η D, η D are the morphisms L ηd V D, L η D V D, (4..3) for every D D. Write L = (L i, φ α), L = (L i, µ ), V D = (Vi D, φ D α ), V D = (Vi D, φ D α ) V D = (V i D, µ D ), V D = (V i D, µ D ). Let h : D D be a morphism in D and consider the following diagram η D s() L s(α) φ α η D s(α) Ξ s(α) η D s(α) η D t(α) L t(α) Ξ t(α) L s(α) φ α η D s(α) η D s(α) V D s(α) φ D α V D t(α) (F h) s(α) V D s(α) φ D α ψ D,α V s() D η D s() (F h ) s() V µ D s() D V D t() η D t() η (F h) t(α) D t(α) (F h ) t() ηt(α) D L t(α) Vt(α) D ψ α η t(α) D µ D V D t() η D s() L s() µ L t() η D t() Ξ s() η D t() η D s() Ξ t() η D t() L s() µ L t() ψ α (4..4) Clearly, (L s(), (η D s() ) D D ) can be viewed as a limit of a functor ρ : D V ect. Also, for any D, D D, we have (F h ) s() ψ D,α ηt(α) D = ψ D,α (F h) t(α) ηt(α) D (since F h = (F h, F h ) is a morphism in Rep (Q,Q ) ) = ψ D,α ηt(α) D (since (L, (η D ) D D ) is a limit of P UF ) Thus, (L t(α), (ψ D, ηt(α) D ) ) is a cone on ρ. Since V ec D D is complete, there exists a unique -linear map ψ α : L t(α) L s() with η D s() ψ α = ψd,α ηt(α) D for every D D and for each pair of arrows (α, ) Q Q. Hence, ηd is a morphism in Rep (Q,Q ) for any D D. Let L = (L, L, ψ ), η D = (η D, η D ). We claim that ( L, ( η D ) D D ) is ψ D,α

21 A Universal Investigation of n-representations of n-quivers 2 a limit of F. Then obviously all we need is to show that for any cone ( L, ( η D ) D D ), there exists a unique morphism Ξ = (Ξ, Ξ ) in Rep (Q,Q ) with η D Ξ = η D for every D D. Let ( L, ( η D ) D D ) be a cone on F and write L = ( L, L, ψ ). Since Rep (Q), Rep (Q ) are complete categories, there exist unique morphisms Ξ, Ξ in Rep (Q), Rep (Q ) respectively such that η D Ξ = η D, η D Ξ = η D for every D D. It remains to show that Ξ = (Ξ, Ξ ) is a morphism in Rep (Q,Q ). For any D D, we have η D s() Ξ s() ψ α = η D s() ψ α (since η D Ξ = η D ) = ψ D,α η t(α) D (since η D is a morphism in Rep (Q,Q ) ) = ψ D,α ηt(α) D Ξ t(α) (since η D Ξ = η D ) = η s() D ψ α Ξ t(α) (since η D is a morphism in Rep (Q,Q ) ) Notably, ( L t(α), ( η s() D ψ ) D D ) can be viewed as a limit of a functor ρ : D V ect. In addition, ( L t(α), ( η s() D ψ α ) ) is a cone on D D ρ since (F h ) s() η s() D ψ α = F h s() η D s() Ξ s() ψ α (since η D = η D Ξ) = η s() D Ξ s() ψ α (since F h η D = η D ) = η s() D ψ α. Since V ec is complete, it follows from the universal property of the limit that Ξ s() ψ α = ψ α Ξ t(α). Thus, Ξ is a morphism in Rep (Q,Q ). Thus, ( L, ( η D ) D D ) is a limit of F, as desired. From Proposition 4. and Remar 3.2, we obtain. Proposition 4.2. The category Rep (Q,Q 2,...,Qn) functor is complete and the forgetful U n : Rep Rep (Q,Q 2,...,Qn) (Q ) Rep (Q 2 )... Rep (Q n ) is continuous. Further, the limit of objects in Rep (Q,Q 2 can be obtained,...,qn) by the corresponding construction for objects in Rep (Q ) Rep (Q 2 )...

22 22 A. H. Abdulwahid Rep (Q n ). 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. Corollary 4.3. The category Rep (Q,Q 2,...,Qn) has ernels. Proposition 4.4. The category Rep (Q,Q ) functor is cocomplete and the forgetful U : Rep (Q,Q ) Rep (Q) Rep (Q ) is cocontinuous. Moreover, the colimit of objects in Rep (Q,Q ) can be obtained by the corresponding construction for objects in Rep (Q) Rep (Q ). Proof. Let D be a small category, and let F : D Rep (Q,Q ) and consider the composition of the following functors. be a functor, D F Rep (Q,Q ) U Rep (Q) Rep (Q ) P Rep (Q) (4..5) D F Rep (Q,Q ) U Rep (Q) Rep (Q ) P 2 Rep (Q ) (4..6) where U, U are the obvious forgetful functors, and P, P 2 are the projection functors. For all D D, let F D = V = (V, V, ψ D, ). Since Rep (Q), Rep (Q ) are cocomplete categories, the functors P UF, P 2 UF have colimits. Let (C, (ζ D ) D D ), (C, (ζ D) D D ) be colimits of P UF, P 2 UF respectively, where ζ D, ζ D are the morphisms V D ζd C, V D ζ D C, (4..7) for every D D. Write C = (C i, φ α), C = (C i, µ ), V D = (Vi D, φ D α ), V D = (Vi D, φ D α ) V D = (V i D, µ D ), V D = (V i D, µ D ). Let h : D D be a morphism in D and consider the following diagram.

23 A Universal Investigation of n-representations of n-quivers 23 ζ D s() ζ D s(α) ζ D s(α) V D s(α) φ D α C s(α) C s(α) ζs(α) D ζ s(α) D φ α Λ s(α) ζ D t(α) φ α ζ D t(α) V D t(α) (F h) s(α) ζs(α) D (F h) t(α) V D s(α) φ D α ψ D,α V s() D ζ D s() (F h ) s() V µ D s() D V D t() ζ D t() µ D ζ D s() C s() µ C t() Λ s() ζ D t() ζ D s() Λ t() C s() µ C t() C t(α) Λ t(α) C t(α) ζ D t(α) ζ D t(α) V D t(α) (F h ) t() ψ α V D t() ζ D t() ζ D t() ψ α (4..8) It is clear that (C t(α), (ζ D t(α) ) ) can be viewed as a colimit of a functor D D υ : D V ect. Moreover, we have for any D, D D, s() (F h) s() ψ D,α ζ D s() ψd,α (F h ) t(α) = ζ D ψ D,α (since F h = (F h, F h ) is a morphism in Rep (Q,Q ) ) = ζ D s() ψ D,α (since (C, (ζ D ) D D ) is a colimit of P 2 UF ). Therefore, (C s(), (ζ D s() ψ D, ) D D ) is a cocone on υ. Since V ec is cocomplete, there exists a unique -linear map ψ α : C t(α) C s() with ψ α ζd t(α) = ζ D s() ψ D,α for every D D and for each pair of arrows (α, ) Q Q. It turns out that ζ D is a morphism in Rep (Q,Q ) for any D D. Let C = (C, C, ψ ), ζ D = (ζ D, ζ D ). We claim that ( C, ( ζ D ) D D ) is a colimit of F. To substantiate this claim, we need to show that for any cocone ( C, ( ζd) D D ), there exists a unique morphism Λ = (Λ, Λ ) in Rep (Q,Q ) with Λ ζ D = ζd for every D D. Let ( C, ( ζd) D D ) be a cocone on F and write C = ( C, C, ψ ). Since Rep (Q), Rep (Q ) are cocomplete categories, exists unique morphisms Λ, Λ in Rep (Q), Rep (Q ), respectively, such that Λ ζ D = ζ D, Λ ζ D = ζ D for every D D. It remains to show that Λ = (Λ, Λ ) is a morphism in Rep (Q,Q ). For any D D, we have

24 24 A. H. Abdulwahid ψ α Λ t(α) ζ D t(α) = ψ α ζ D t(α) (since Λ ζ D = ζd ) = ζ D s() ψd,α (since ζd is a morphism in Rep (Q,Q ) ) = Λ s() ζ D s() ψd,α (since Λ ζ D = ζd ) = Λ s() ψ α ζd t(α) (since ζ D is a morphism in Rep (Q,Q ) ) Notably, (C t(α), (ζt(α) D ) ) can be viewed as a colimit of a functor D D υ : D V ect. Moreover, ( C s(), ( ψ α ζ t(α) D ) ) is a cocone on D D υ since for any D, D D, we have ψ α D ζ t(α) (F h) t(α) = ψ α Λ t(α) ζt(α) D (F h) t(α) (since Λ ζ D = ζd ) = ψ α Λ t(α) ζt(α) D (since (C, (ζ D ) D D ) is a colimit of P UF ) = ψ α ζ D s() (since Λ ζ D = ζd ) Since V ec is cocomplete, it follows from the universal property of the colimit that Λ s() ψ α = ψ α Λ t(α). Consequently, Λ is a morphism in Rep (Q,Q ), which completes the proof. From Proposition 4.4 and Remar 3.2, we obtain. is cocomplete and the forget- Proposition 4.5. The category Rep (Q,Q 2,...,Qn) ful functor U n : Rep (Q,Q 2,...,Qn) Rep (Q ) Rep (Q 2 )... Rep (Q n ) is cocontinuous. Furthermore, the colimit of objects in Rep can (Q,Q 2,...,Qn) be obtained by the corresponding construction for objects in Rep (Q ) Rep (Q 2 )... Rep (Q n ). Since a category is cocomplete if and only if it has coproducts and coequalizers [2], we have the following immediate consequence. Corollary 4.6. The category Rep (Q,Q 2,...,Qn) has coernels.

25 A Universal Investigation of n-representations of n-quivers 25 Next, we aim to show that the Categories of n-representations are abelian. Following [3, 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. (ii) There exists a zero object C such that C(, ) =. (iii) (Existence of direct sums.) For any objects X, X C, the direct sum X X C. Let be a field. An additive category C is said to be -linear if for any objects X, Y C, C(X, Y ) is equipped with a structure of a vector space over, such that composition of morphisms is -linear. An abelian category is an additive category C in which for every morphism f : X Y there exists a sequence K X ι I j Y c C (4..9) with the following properties: (i) ji = f, (ii) (K, ) = Ker(f), (C, c) = Coer(f), (iii) (I, ι) = Coer(), (I, j) = Ker(c). A sequence (4..) is called a canonical decomposition of f. Let f = (f, f ) : X Ȳ be a morphism in Rep (Q,Q ). It follows that f : X Y, f : X Y are morphisms in Rep (Q), Rep (Q ) respectively. From Proposition 2.3, f : X Y, f : X Y have the canonical decompositions K X ι I j Y c C (4..) K X ι I j Y c C

26 26 A. H. Abdulwahid in Rep (Q), Rep (Q ) respectively. From the above results, ernels and coernels exist in Rep (Q,Q ). It turns out that f has a canonical decomposition K X ῑ Ī j Ȳ c C (4..) in Rep (Q,Q ), and this decomposition can explicitly be seen in the following commutative diagram. φα K s(α) K t(α) s(α) t(α) X φ s(α) α X t(α) C s(α) c s(α) Y s(α) ι ψ α s(α) j s(α) ι t(α) I s(α) φ α I t(α) C t(α) ϕ α c t(α) Y t(α) ϕ α j t(α) K s() ψ α µ K t( c C t() s() Y t() Ψ α s() X s() ψ α t() µ X t() ι t() Ψ α ν C s() ν ι s() Y s() I s() µ I t() c t() j t() j t() This implies that any morphism f : V canonical decomposition in Rep. (Q,Q 2,...,Qn) (4..2) W of n-representations has a Remar 4.7. Let f, ḡ : V W be morphisms in Rep(Q,Q ). Write f = (f, f ), ḡ = (g, g ), f = (f i ), g = (g i ), f = (f i ), g = (g i ), V = (V, V, ψ), W = (W, W, ψ ). Define f + ḡ = (f + g, f + g ) = ((f i + g i ), (f i + g i )). Since Rep (Q) and Rep (Q ) are abelian, the sets Rep (Q)(V, W ), Rep (Q)(V, W ) are equipped with a structure of an abelian group such that composition of morphisms is biadditive with respect to this structure [3, p.

27 A Universal Investigation of n-representations of n-quivers 27 7]. Since f, ḡ are morphisms in Rep(Q,Q ) and since the category V ec is abelian, we have the following commutative diagram: V s(α) φ α V t(α) f t(α) +g t(α) (4..3) f s(α) +g s(α) W s(α) φ α ψ α W t(α) V s() ψ α µ V t( f t() +g t() f s() +g s() W s() µ W t() 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. Theorem 4.8. The category Rep (Q,Q ) is a -linear abelian category. More generally, the category Rep (Q,Q 2 is a -linear abelian category for any,...,qn) integer n 2. 5 n-quivers and n-representations Let V = (V, V, ψ), W = (W, W, ψ ) be birepresentations of (Q, Q ). For simplicity, we suppress from the tensor product and use instead. Define V W = (V W, V W, ψ ψ ). (5..) Then V W is clearly a birepresentation of (Q, Q ). Let f : V W, ḡ : M N be morphisms in Rep(Q,Q ) and write V = (V, V, ψ), W = (W, W, ψ ), M = (M, M, Ψ), N = (N, N, Ψ ). Define f ḡ : V M W N. (5..2)

28 28 A. H. Abdulwahid Then it is clear that f ḡ is a morphism in Rep (Q,Q ), and hence the following diagram is commutative. f s(α) g s(α) φ α ϕ α V s(α) M s(α) V t(α) M t(α) W s(α) N s(α) φ α ϕ α W t(α) N t(α) f t(α) g t(α) ψ α Ψα ψ α Ψ α f s() g s() V s(α) M s(α) µ ν V t(α) M t(α) W s(α) N s(α) µ ν W t(α) N t(α) f t() g t() (5..3) Thus, the category Rep (Q,Q ) is a monoidal category, and hence by Remar 3.2, Rep (Q,Q 2 is a monoidal category for any n 2. For the,..., Qn) basic notions of monoidal categories, we refer the reader to [3], [28], [6], and [, Chapter 6]. Definition 5.. Let Q = (Q, Q, s, t), Q = (Q, Q, s, t ) be quivers, and let {ϱ α : α Q, Q } be a collection of of arrows t(α) ϱα s () one for each pair of arrows (α, ) Q Q. A 2-quiver induced by (Q, Q ) is a quiver Q (Q,Q ) = ( Q, Q, s, t ), where Q = Q Q, Q = Q Q {ϱα : α Q, Q }, s : Q Q a map from arrows to vertices, mapping an arrow to its starting point,

29 A Universal Investigation of n-representations of n-quivers 29 t : Q Q a map from arrows to vertices, mapping an arrow to its terminal point. The notation above denotes the disjoint union. The above definition turns out that one can inductively define n-quivers for any integer n 2. For any m {,..., n}, let Q m = (Q (m), Q (m), s (m), t (m) ) be a quiver, and let {ϱ γ(m) : γ (m) Q (m), m {2,..., n}} be a collection of of arrows mγ (m ) t (m ) (m) ϱγ mγ (m ) s (m) one for each pair of arrows (γ (m ), γ (m) ) Q (m ) Q (m). An n-quiver induced by (Q, Q 2,..., Q n ) is a quiver Q (Q,Q 2,...,Qn) = ( ˆQ, ˆQ, ŝ, ˆt), where ˆQ = n m= Q(m), ˆQ = n m= Q(m) {ϱ γ(m) mγ (m ) : γ (m) Q (m), m {2,..., n}}, ŝ : ˆQ ˆQ a map from arrows to vertices, mapping an arrow to its starting point, ˆt : ˆQ ˆQ a map from arrows to vertices, mapping an arrow to its terminal point.

30 3 A. H. Abdulwahid Example 5.2. (5..4) Q : 2 Q : Q : Q (Q,Q ) : (5..5) 5 Q (Q,Q,Q ) : (5..6)

31 A Universal Investigation of n-representations of n-quivers [ ] [ ] 3 We have (Q) =, (Q ) =, (Q (Q,Q ) ) 3 9 = 3, and (Q (Q,Q,Q ) ) = Consider the following: (5..7) V : V : V : 2 Then V, V, V are clearly representations of Q, Q, Q respectively.

32 32 A. H. Abdulwahid Now, consider the following: (5..8) V : [ ] V [ ] 2 [ ] (5..9) Then V is a representations of Q (Q,Q ) and V is a representation of Q (Q,Q,Q ). Remar 5.3. (i) In general, if Ω n = {ϱ γ(n) we have or nγ (n ) Q (Q,Q 2,...,Qn) = : γ (n ) Q (n ), γ (n) Q (n) }, then Q (Q,Q 2,...,Q n ) Ω n Q n Q (Q,Q 2,...,Q n ) Q (Q,Q 2,...,Qn) = Ω n Q n where Ω is the subspace of Q (Q,Q 2,...,Qn) generated by the set Ω n. To avoid confusion, we identify Q (Q ) as Q. (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

33 A Universal Investigation of n-representations of n-quivers 33 (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 (Q,Q ) Rep (Q (Q,Q ) ) that restricts to an equivalence rep (Q,Q ) rep (Q (Q,Q ) ). Proof. This follows from the construction of Q (Q,Q ) in Definition 5.. In fact, there is a combining functor F : Rep (Q,Q ) Rep (Q (Q,Q ) ) that views any birepresentation of (Q, Q ) as one piece, a representation of Q (Q,Q ). Its inverse is the decomposing functor G : Rep (Q (Q,Q ) ) Rep (Q,Q ), which breas each representation of Q (Q,Q ) into three parts (two representations of Q, Q and a collection of compatibility maps) to form a birepresentation of (Q, Q ). Using Induction and Remar 3.2, we have the following. Proposition 5.5. For any n 2, there exists an equivalence of categories Rep Rep (Q,Q 2,...,Qn) (Q (Q, Q 2 ) that restricts to an equivalence,...,qn) rep rep (Q,Q 2,..., Qn) (Q ). (Q,Q 2,...,Qn) The following is an immediate consequence of Proposition 5.4 and Proposition 2.4. Proposition 5.6. There exists an equivalence of categories Mod Q (Q,Q ) Rep (Q,Q ) that restricts to an equivalence mod Q (Q,Q ) rep (Q,Q ). By Remar 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 Mod Q Rep (Q,Q 2,...,Qn) (Q,Q 2 that restricts to an equivalence,...,qn) mod Q rep. (Q,Q 2,...,Qn) (Q,Q 2,...,Qn) 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. 25]. 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.

34 34 A. H. Abdulwahid Following [2, p. 27], 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 fα 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], [2], or [5]. The following proposition immediately follows from Remar 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,Q 2 is co-wellpowered.,...,qn) has a gener- Proposition 5.9. For any n 2, the category Rep (Q,Q 2,...,Qn) ating set. Using Theorem (2.), we have the following. Proposition 5.. For any n 2, the obvious forgetful functor U n : Rep (Q,Q 2,...,Qn) Rep (Q ) Rep (Q 2 )... Rep (Q n ) has a right adjoint. Equivalently, for any n 2, the concrete category (Rep (Q,Q 2,...,Qn), U n) has cofree objects. Let CoAlg(Rep (Q,Q ) ) be the category of coalgebras in Rep (Q,Q ). By [, p. 3], the left Q (Q,Q ) -module coalgebras which are finitely generated as left Q (Q,Q ) -modules form a system of generators for CoAlg(Mod Q (Q,Q ) ). Thus, from Proposition 2.2 and Theorem 2., we have the following. Proposition 5.. The forgetful functor U : CoAlg(Rep (Q,Q ) ) Rep (Q,Q ) has a right adjoint. Equivalently, the concrete category (Rep (Q,Q ), U ) has cofree objects. Remar (3.2) implies the following consequence. Proposition 5.2. For any n 2, the forgetful functor U n : CoAlg(Rep ) Rep (Q,Q 2,...,Qn) (Q,Q 2,...,Qn) has a right adjoint. Equivalently, for any n 2, the concrete category (Rep, U (Q,Q 2,...,Qn) n) has cofree objects.

35 A Universal Investigation of n-representations of n-quivers 35 Remar 5.3. If the forgetful functor U : CoAlg(Rep (Q,Q ) ) Rep (Q,Q ) has a right adjoint V and M Rep (Q,Q ), then the cofree coalgebra over M can be given by V ( M) = lim [ f:ḡ M] Rep (Q,Q ), Ḡ CoAlg(Rep (Q,Q ) ), Ḡ rep (Q,Q ) Thus, the concrete category (CoAlg(Rep (Q,Q ) ), U ) has cofree objects given in terms of colimits and generators. It follows that, for any n 2, if the forgetful functor Ḡ. U n : CoAlg(Rep (Q,Q 2,...,Qn) ) Rep (Q,Q 2,...,Qn) has a right adjoint V n and over M can be given by M Rep (Q,Q 2, then the cofree coalgebra,...,qn) V n ( M) = lim [ f:ḡ M] Rep (Q,Q 2,...,Qn), Ḡ CoAlg(Rep (Q,Q 2,...,Qn) ), Ḡ rep (Q,Q 2,...,Qn) The following proposition explicitly describes colimits and cofree objects in the product categories. Proposition 5.4. For any m {, 2, 3,..., n}, let U m : A m X m be a forgetful functor with a right adjoint V m. Then the product functor V V 2... V n : X X 2... X n A A 2... A n is a right adjoint of the obvious forgetful functor U U 2... U n : A A 2... A n X X 2... X n. Furthermore, for any (X, X 2,..., X n ) X X 2... X n, the cofree object over (X, X 2,..., X n ) is exactly (V (X ) V 2 (X 2 )... V n (X n )). Proof. The proof is straightforward. For any m {, 2,..., n}, let U m : CoAlg(Rep (Q m )) Rep (Q m ) be the obvious forgetful functor with a right adjoint V m. Proposition 5.4 and Remar 3.2 immediately imply the following consequence. Corollary 5.5. The product functor V V 2... V n : Rep (Q ) Rep (Q 2 )... Rep (Q n ) CoAlg(Rep (Q )) CoAlg(Rep (Q 2 ))... CoAlg(Rep (Q n )) is a right adjoint of the obvious product forgetful functor U U 2... U n : CoAlg(Rep (Q )) CoAlg(Rep (Q 2 ))... Ḡ.

36 36 A. H. Abdulwahid CoAlg(Rep (Q n )) Rep (Q ) Rep (Q 2 )... Rep (Q n ). Furthermore, for any (M, M 2,..., M n ) Rep (Q ) Rep (Q 2 )... Rep (Q n ), the cofree object over (M, M 2,..., M n ) is exactly (V (M ), V 2 (M 2 ),...,V n (M n )). For any m {, 2,..., n}, let U m : CoAlg(Rep (Q m )) Rep (Q m ) be the obvious forgetful functor with a right adjoint V m. By Proposition 5.9, for any n 2, the category Rep (Q,Q 2 has a generating set,...,qn) G. For any m {, 2,..., n}, let G m be a generating set of Rep (Q m ). The definition of n-representations of quivers implies that each element of G taes the form (G, G 2,..., G n, χ, χ 2,..., χ n ) where G m G m for every m {, 2, 3,..., n}. Furthermore, Remar 5.3 and Proposition 4.4 imply the following proposition which gives an explicit description for cofree coalgebras in the concrete category (Rep (Q,Q 2, U ), where U :,...,Qn) CoAlg(Rep ) Rep (Q,Q 2,...,Qn) (Q,Q 2 is the obvious forgetful functor,...,qn) with a right adjoint V. Proposition 5.6. Let M = (M (), M (2),..., M (n), ψ, ψ 2,..., ψ n ) Rep (Q,Q 2,...,Qn). Then the cofree object over M is exactly (V (M () ), V 2 (M (2) ),..., V n (M (n) ), ψ, ψ 2,..., ψ n ), for some unique -linear maps ψ, ψ 2,..., ψ n. Proof. This can be proved by applying Remar 5.3 and Proposition 4.4. Explicitly, if Ḡ = (G (), G (2),..., G (n), χ, χ 2,..., χ n ), f = (f (), f (2),..., f (n) ),

37 A Universal Investigation of n-representations of n-quivers 37 we have the following. V ( M) = lim Ḡ [ f:ḡ M] Rep (Q,Q 2,...,Qn), Ḡ CoAlg(Rep (Q,Q 2,...,Qn) ), Ḡ rep (Q,Q 2,...,Qn) = (lim G (), [f () :G () M () ] Rep (Q ), G () CoAlg(Rep (Q )), G () rep (Q )..., lim G (n), [f (n) :G (n) M (n) ] Rep (Q n), G (n) CoAlg(Rep (Q n)), G (n) rep (Q n) ψ, ψ 2,..., ψ n ) = (V (M () ), V 2 (M (2) ),..., V n (M (n) ), ψ, ψ 2,..., ψ n ), for some unique -linear maps ψ, ψ 2,..., ψ n. We end this paper by pointing out that the universal investigation above can be adjusted to study cofree objects in the centralizer and the center categories of Rep (Q,Q 2. Indeed, one can describe them in terms of cofree,...,qn) objects in centralizer and center categories of Rep (Q j ), j {, 2,..., n}. References [] Abdulwahid, A.H. and Iovanov, M.C., Generators for comonoids and universal constructions, Arch. Math. 6 (26), [2] Adame, J., Herrlich, H., and Strecer, G.E., Abstract and Concrete Categories: The Joy of Cats, Dover Publication, 29. [3] Assem, I., Sowronsi, A., and Simson, D., Elements of the Representation Theory of Associative Algebras : Techniques of Representation Theory, London Math. Soc. Student Texts 65, Cambridge University Press, 26. [4] Auslander, M., Reiten, I., S.O. Smalø, S.O., Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, 995. [5] Awodey, S., Category Theory, Oxford University Press, 2.

38 38 A. H. Abdulwahid [6] Baalov, B. and Kirillov, A., Jr., Lectures on Tensor Categories and Modular Functor, University Lecture Series 2, American Math. Soc., 2. [7] Barot, M., Introduction to the Representation Theory of Algebras, Springer, 25. [8] Benson, D.J., Representations and Cohomology I: Basic Representation Theory of Finite Groups and Associative Algebras, Cambridge Stud. Adv. Math. 3, Cambridge University Press, 99. [9] Borceux, F., Handboo of Categorical Algebra : Basic Category Theory, Cambridge University Press, 994. [] Borceux, F., Handboo of Categorical Algebra 2: Categories and Structures, Cambridge University Press, 994. [] Buan, A.B., Reiten, I., and Solberg, O., Algebras, Quivers and Representations, Springer, 23. [2] Dăscălescu, S., Iovanov, M., Năstăsescu, C., Quiver algebras, path coalgebras and co-reflexivity, Pacific J. Math. 262 (23), [3] Etingof, P., Gelai, S., Nishych, D., and Ostri, V., Tensor Categories. Mathematical Surveys and Monographs 25, American Math. Soc., 25. [4] Etingof, P., Golberg, O., Hensel, S., Liu, T., Schwendner, A., Vaintrob, D., Yudovina, E., Introduction to Representation Theory, Student Mathematical Library 59, American Math. Soc., 2. [5] Freyd, P.J. and Scedrov, A., Categories, Allegories, Elsevier Science Publishing Company, 99. [6] Gabriel, P., Unzerlegbare Darstellungen I, Manuscripta Math. 6 (972), 7-3. [7] Kilp, M., Knauer, U., Mihalev, A.V., Monoids, Acts, and Categories: With Applications to Wreath Products and Graphs, De Gruyter exposition in Math. 29, 2. [8] Leinster, T., Basic Category Theory, London Math. Soc. Lecture Note Series 298, Cambridge University Press, 24. [9] Leinster, T., Higher Operads, Higher Categories, Lecture Note Series 298, London Math. Soc., Cambridge University Press, 24. [2] Mac Lane, S., Categories for the Woring Mathematician, Graduate Texts in Math. 5, Springer-Verlag, 998. [2] McLarty, C., Elementary Categories, Elementary Toposes, Oxford University Press, 25.

39 A Universal Investigation of n-representations of n-quivers 39 [22] Mitchell, B., Theory of Categories, Academic Press, 965. [23] Pareigis, B., Categories and Functors, Academic Press, 97. [24] Rotman, J.J., An Introduction to Homological Algebra, Springer, 29. [25] Schiffler, R., Quiver Representations, CMS Boos in Math. Series, Springer International Publishing, 24. [26] Schubert, H., Categories, Springer-Verlag, 972. [27] Sergeichu, V.V., Linearization method in classification problems of linear algebra, São Paulo J. Math. Sci. (2) (27), [28] Street, R., Quantum Groups: a Path to Current Algebra, Lecture Series 9, Australian Math. Soc., Cambridge University Press, 27. [29] Wisbauer, R., Foundations of Module and Ring Theory: A Handboo for Study and Research, Springer-Verlag, 99. [3] Zimmermann, A., Representation Theory: A Homological Algebra Point of View, Springer-Verlag, 24. Adnan H. Abdulwahid Department of Mathematics, University of Thi-Qar, Nasiriyah, Thi- Qar, Iraq. adnan-al-hafaji@uiowa.edu

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