Cofree Objects in The Centralizer and The Center Categories

We study cocompleteness, co-wellpoweredness and generators in the centralizer category of an object or morphism in a monoidal category, and the center or the weak center of a monoidal category. We explicitly give some answers for when colimits, cocompleteness, co-wellpoweredness and generators in these monoidal categories can be inherited from their base monidal categories. Most importantly, we investigate cofree objects of comonoids in these monoidal categories.


Introduction and Preliminaries
Universal properties are crucially considered as one of the most important concepts in mathematics. Indeed, they can be thought of as the skeleton of all mathematics concepts. They show how the objects and the morphisms nicely relate the whole category that they live in. Many influential concepts, such as kernels, cokernels, products, coproducts, limits, colimits, etc., are essentially involved with universal properties. Perhaps the most important notion concerned with them is the concept of adjoint functors. It is simply because "Adjoint functors arise everywhere" [16, p. (vii)]. Significantly, free and cofree objects play a crucial role in recasting the adjunctions of the forgetful functors in terms of comma categories. For fundamental concepts and examples of adjoint functors, we refer the reader to [15], [16], [3], [20], [19], or [18]. For the basic notions of comma categories, we refer to [15] and [16]. 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, p. 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 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 readers to [14, p. 138-155]. It turns out that a concrete category (A, U) over X has (co)free objects if and only if the functor that constructs (co)free objects is a (right) left adjoint to the faithful functor U : A → X.
Although cofree objects are the dual of free objects, the behavior of cofree objects is more complicated than the one of free objects. Furthermore, studying such behavior cannot be obtained by studying free objects, because "the categories considered are not selfdual generally" [14, p. 149]. In this paper, we are interested in investigating cofree objects in the centralizer category of an object or morphism in a monoidal category and the center or the weak center of a monoidal category. For the basic notions of monoidal categories, we refer the readers to [10], [4], and [8,Chapter 6].
More recently, these monoidal categories play a vibrant role in characterizing and identifying many of the interesting categories. For instance, to show that two finite tensor categories are Morita equivalent, it suffices to show that their centers are equivalent as braided tensor categories [10, p. 222]. Another example is to show that a fusion category is grouptheoretical, it is sufficient to show its center contains a Lagrangian subcategory [10, p. 313]. In addition, there is a special importance for the center of a finite tensor category in finding its Frobenius-Perron dimension. This comes from the fact that for any finite tensor category C , we have F P dim(Z(C )) = F P dim(C ) 2 [10, p. 168]. We refer to [13] for basics on centralizer categories while we refer to [23, p. 76] and [10, p. 162] for basics on center categories.
Explicitly, the problem can be formulated as follows. Let C be a monoidal category. Fix an object X and a morphism h : A → B in C. For any A ∈ {Z h (C), Z X (C), Z(C), Z ω (C)}, let U A : CoM on(A ) → A be the forgetful functor corresponding to A . Does U A have a right adjoint? A reasonably expected machinery for the answer of this question is the dual of Special Adjoint Functor Theorem (D-SAFT).
We start our inspection by studying cocompleteness in A , and we give some answers for the question: under what conditions the colimits of diagrams in A can be obtained from the corresponding construction of diagrams in C. The later implicitly implies that the forgetful functor U A is cocontinuous. Next, we investigate conditions under which the category A inherits the co-wellpoweredness of C. We also show how the braiding forces the category A to inherit generators from its base category C. Finally, we apply the mechanism of D-SAFT for each case. Furthermore, we try to visualize some interesting consequences by studying the braid category.
Let (C, ⊗, I) be a monoidal category, and for every X ∈ C, let P X , Q X be the functors defined by If C is a biclosed monoidal category, then both functors P X and Q X have right adjoints for every object X ∈ C, hence they are cocontinuous for every object X ∈ C. However, this is not true in general; that is, if C is a cocomplete monoidal category, then the tensor product functors P X and Q X needs not be cocontinuous in each object X. Thus, one might need to consider them more carefully.
Theorem 1.1. [11, 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. [21, p. 284] The (left) weak center of C denoted Z ω (C) is a category whose objects are pairs (A, σ A,− ), where A ∈ C and σ A,− : A⊗− → − ⊗ A is a natural transformation such that the following conditions hold and The category Z ω (C) is monoidal with [23, p. 76] The center of C, denoted by Z(C), is a category whose objects are pairs (A, σ A,− ), where A ∈ C and σ A,− : A ⊗ − ∼ − → − ⊗ A is a natural isomorphism such that the following conditions hold: for all X, Y ∈ C.
is an arrow f : A → B in C such that, for all X ∈ C, the following diagram is commutative The category Z(C) is braided via [13, p. 46-47] The centralizer Z X (C) of an object X ∈ C is the category whose objects are pairs (A, α), where A ∈ C and α : is an arrow f : A → B in C such that the following diagram is commutative: This becomes a monoidal category with (1.14) Definition 1.4. [13, p. 49] The centralizer Z h (C) of an arrow h : A → B in C is the category whose objects are triples (X, α, β), where X ∈ C and α : Remark 1.5. The category Z h (C) was introduced in [13, p. 49] as an essential part of the proof of Lemma 7, and the authors implicitly indicated that it is a monoidal category. For convenience, we explicitly show that the category Z h (C) is monoidal. Proposition 1.6. Let h : A → B be an arrow in C. Then the category whereᾱ,β are given respectively by the compositions . Therefore, X ⊗ Y ∈ Z h (C) and, hence, the category Z h (C) is monoidal. Remark 1.7. (i) For all X ∈ C, we have the following evaluation functor , for every X ∈ C. However, to show that an object (A, σ A,− ) ∈ Z(C), it suffices to show that (A, σ A,X ) ∈ Z X (C) for every X ∈ C, σ is a natural transformation and the condition 1.7 holds as well (since the condition 1.6 on objects of Z(C) is redundant [23, p. 76]).
(ii) For all A ∈ C, if (X, α, β) ∈ Z id A (C), then Definition 1.4 implies that α = β. This gives rise to an isomorphism given by . It turns out that the centralizer category of an object A in C can be identified as the centralizer category of the identity morphism of A in C. However, we will explicitly study the centralizer category of an object due to the discussion of part (i).

Cocompleteness
Recall that a category C is cocomplete when every functor F : D → C, with D a small category has a colimit [7]. For the basic notions of cocomplete categories and examples, we refer to [2], [7], or [22]. A functor is cocontinuous if it preserves all small colimits [11, p. 142]. Proof. Let D be a small category, and let F : D → Z h (C) be a functor. Since C is a cocomplete category, the functor U F has a colimit (C, (φ D ) D∈D ).
Since P A is cocontinuous, P A U F has a colimit (P A (C), (P A (φ D )) D∈D ). Equivalently, First, we note that the functor F : D → Z h (C) assigns to each object D ∈ D an object (F D, α F D , β F D ) ∈ Z h (C). We also have F f : is an arrow in Z h (C) for every arrow f : D → D in D. Thus, we have the following commutative diagrams: The last equality comes from the fact that (C, (φ D ) D∈D )) is a cocone on U F . Therefore, (C ⊗A, Therefore, we get the commutative diagram Next, we show thatᾱ is an invertible arrow. From the commutativity of the diagrams 2.2 and 2.3, we havē From the commutativity of the diagrams 2.2 and 2.3, we have Therefore, the arrowᾱ is invertible andᾱ −1 =ᾱ . Replacing the object A by B and following the same strategy we did to getᾱ, we can similarly get an invertible arrowβ : B ⊗ C ∼ − → C ⊗ B and the commutative diagrams To show that (C,ᾱ,β) ∈ Z h (C), we need to show that the following diagram is commutative: To show this, consider the following diagram for any D in D. Furthermore, for any D in D, we have The proof is complete whence we show that g is a morphism in Z h (C). Explicitly, we need to show that the diagrams We also have Similarly, replacing A by B and considering the following diagram From (2.10) and (2.12), we have g is a morphism in Z h (C), and thus ((C,ᾱ,β), (φ D ) D∈D ) is a colimit of F , and the proof is complete.
3. Let C be a cocomplete category and X an object in C. If P X , Q X are cocontinuous, then Z X (C) is cocomplete and the forgetful functor U : Z X (C) → C is cocontinuous. Moreover, the colimits of diagrams in Z X (C) can be obtained by the corresponding construction of diagrams in C.
Proof. The proof follows from Remark 1.7 (ii) and Proposition 2.2 by setting h = id X .
Proposition 2.4. Let (C, ⊗, I) be monoidal category with C cocomplete. If P X and Q X are cocontinuous ∀X ∈ C, then Z(C) is cocomplete and the forgetful functor U : Z(C) → C is cocontinuous. Further, the colimits of diagrams in Z(C) can be obtained by the corresponding construction of diagrams in C.
Proof. Let D be a small category, and let F : D → Z(C) be a functor.
For any X ∈ C, let U X : Z X (C) → C be the corresponding forgetful functor.
Since U is cocontinuous, U X is cocontinuous for every X ∈ C. Moreover, we have H X F :: D → Z X (C) is a (small) functor and U X H X = U for any X ∈ C, where H X is the functor defined in Remark (1.7). Since C is a cocomplete category, the functor U F has a colimit (C, (φ D ) D∈D ). For every X ∈ C, (P X (C), (P X (φ D )) D∈D ) is a colimit of the functor P X U F because P X is cocontinuous. Thus, for any X ∈ C, Further, the functor F : D → Z ( C) assigns to each object D ∈ D an ob- for every arrow f : D → D in D. By 1.12, we have the commutative diagram The last equality comes from the fact that (C, for any D in D, where the last equality is coming from the fact that Next, we show that µ C,X is an invertible arrow. From the commutativity of the diagrams 2.14 and 2.15, we have In a similar way, from the commutativity of the diagrams 2.14 and 2.15, we have Clearly, (C ⊗ X, (ν C,X µ C,X (φ D ⊗ id X )) D∈D ) is a cocone on Q X U X H X F . Since (C ⊗X, (φ D ⊗id X ) D∈D ) is a colimit of Q X U X H X F , we have ν C,X µ C,X = id C⊗X . Therefore, the arrow µ C,X is invertible and µ −1 C,X = ν C,X . It follows that (C, µ C,X ) ∈ Z X (C), and φ D is an arrow in Z X (C), ∀D ∈ D. Hence, ((C, µ C,X ), (φ D ) D∈D ) is a cocone on H X F . It remains to show that ((C, µ C,X ), (φ D ) D∈D ) is a colimit of H X F . Let ((C , η), (ψ D ) D∈D ) be a cocone on H X F . Since (C, (φ D ) D∈D ) is a colimit of U X H X F , there exists a unique morphism g : C → C in C with gφ D = ψ D for every D ∈ D. Clearly, all we need is to show that g is a morphism in Z X (C). Indeed, we need to show that the diagram (2.18) Thus, we obtain ((C, µ C,X ), (φ D ) D∈D ) is a colimit of H X F , ∀X ∈ C, and thus, we obtain a family of invertible arrows {µ C,X } X∈C in C, where µ C,X is the map in diagram 2.14 for all X ∈ C. Therefore, the proof is complete whence we show that {µ C,X } X∈C are natural in X, for any X ∈ C, and the conditions (1.6) and (1.7) hold.
To show that µ C,− : C ⊗ − → − ⊗ C is a natural transformation, let ζ : A → B be an arrow in C. We need to show that the following diagram is commutative: Thus, for any D ∈ D, the diagram commutes. Now, consider the following diagram Furthermore, for any D ∈ D, we have in place of X and α F D , respectively.) Hence, µ C,− : C ⊗ − −→ − ⊗ C is a natural transformation. By Remark 1.7, it remains to show that the condition (1.7) holds. Consider the following diagram is a colimit of Q X⊗Y U F , it follows that the condition (1.7) is satisfied. Therefore, (C, µ C,− ) is a colimit of F and the proof is complete.
The following is an immediate consequence of the proof of Proposition 2.4 and Remark 1.7.
Corollary 2.5. Let C be a cocomplete category. If P X , Q X are cocontinuous ∀X ∈ C, then Z ω (C) is cocomplete and the forgetful functor U : Z ω (C) → C is cocontinuous. Moreover, the colimits of diagrams in Z ω (C) can be obtained by the corresponding construction of diagrams in C.
The following theorem is well-known. Following [8, p. 293, 294], a monoidal category C is biclosed when, for each object X ∈ C, both functors P X = X ⊗ − and Q X = − ⊗ X have a right adjoint. A biclosed symmetric monoidal category is called a symmetric monoidal closed category. Since in a symmetric monoidal category, both functors P X = X ⊗ − and Q X = − ⊗ X are naturally isomorphic, it follows that a symmetric monoidal category C is closed if and only if, for each object X ∈ C, the functor Q X = − ⊗ X : C → C has a right adjoint [8, p. 294]. Therefore, by using Proposition 2.1 together with Proposition 2.2, Corollary 2.3, Proposition 2.4, Corollary 2.5 and Theorem 2.6, we have the following immediate consequence. Proposition 2.7. Let C be a monoidal category. Fix an object X and a morphism h : A → B in C. For any A ∈ {Z h (C), Z X (C), Z(C), Z ω (C)}, let U A : CoM on(A ) → A be the forgetful functor corresponding to A . If C is a cocomplete biclosed monoidal category, then A is cocomplete and the forgetful functor U A is cocontinuous for any A ∈ {Z h (C), Z X (C), Z(C), Z ω (C)}. Furthermore, the colimits of diagrams in A can be obtained by the corresponding construction of diagrams in C.

Example 2.8.
(1) The category Set of sets and mappings is cocomplete [22, p. 66], and it can also be seen to be cartesian closed [8, p. 296], hence symmetric monoidal closed. By Proposition 2.7, Z h (Set), Z X (Set), Z(Set), and Z ω (Set) are cocomplete for any X ∈ Set and a morphism h in Set.
(2) The category Cat of small categories and functors is cocomplete [22, p. 66], and it can also be seen as cartesian closed [8, p. 296], hence symmetric monoidal closed. By Proposition 2.7, Z h (Cat), Z X (Cat), Z(Cat), and Z ω (Cat) are cocomplete for any X ∈ Cat and a morphism h in Cat. (7) The category C inf SL of complete inf semi-lattices is evidently complete and therefore cocomplete, and it can also be seen as a symmetric monoidal closed category [5]. By Proposition 2.7, Z h (C inf SL ), Z X (C inf SL ), Z(C inf SL ), and Z ω (C inf SL ) are cocomplete for any X ∈ C inf SL and a morphism h in C inf SL .
(8) Grothendieck topos GT Y P is evidently complete and therefore cocomplete, and it is also a symmetric monoidal closed category [5]. By Proposition 2.7, Z h (GT Y P ), Z X (GT Y P ), Z(GT Y P ) and Z ω (GT Y P ) are cocomplete for any X ∈ GT Y P and a morphism h in GT Y P .
(9) The category LCA of locally compact abelian groups for which the duality reduces to the standard duality for those groups [6]. By Proposition 2.7, Z h (LCA), Z X (LCA), Z(LCA), and Z ω (LCA) are cocomplete for any X ∈ LCA and a morphism h in LCA.
(10) If D is a small category, the category F un(D, Set) of functors and natural transformations is cartesian closed, hence a symmetric monoidal closed [8, p. 297]. The category F un(D, Set) is also cocomplete [22, p. 66]. By Proposition 2.7, the categories Z h (F un(D, Set)), Z X (F un(D, Set)), Z(F un(D, Set)) and Z ω (F un(D, Set)) are cocomplete for any X ∈ F un(D, Set) and a morphism h in F un(D, Set). (2) Let G be a monoid (which we will usually take to be a group), and let A be an abelian group (with operation written multiplicatively). Let C G = C G (A) be the category whose objects δ g are labeled by elements of G (so there is only one object in each isomorphism class), Hom C G (δ g 1 , δ g 2 ) = ø if g 1 = g 2 , and Hom C G (δ g , δ g ) = A, with the functor ⊗ defined by δ g ⊗ δ h = δ gh , and the tensor product of morphisms defined by a ⊗ b = ab. Then C G is a monoidal category with the associativity isomorphism being the identity, and the identity object I = being the unit element of G.
This example has a "linear" version. Namely, let K be a field and K − V ec G denote the category of G-graded vector spaces over K, that is, vector spaces V with a decomposition V = g∈G V g . Morphisms in this category are linear maps which preserve the grading. Define the tensor product on this category by the formula (V ⊗W ) g = x,y∈G:xy=g V x ⊗W y , and the unit object I by I 1 = K and I g = 0 for g = 1. Then, defining the associativity constraint and left and right unit constraints in an obvious way, we equip K − V ec G with the structure of a monoidal category. Similarly one defines the monoidal category f.d.K−V ec G of finite dimensional G-graded K-vector spaces. When no confusion is possible, we will denote the categories K − V ec G , f.d.K−V ec G simply by V ec G , f.d.V ec G [10, p. 27]. Then the category V ec G is rigid if and only if the monoid G is a group [10, p. 43]. Furthermore, if G is a finitely generated infinite simple group (it is known that such groups exist), then Z(V ec G ) is equivalent to the category f.d.V ec of finite dimensional spaces [10, p. 207]. Thus, Z(V ec G ) is not cocomplete.

Co-wellpoweredness
Let E be the 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 [22, p. 92, 95]. We refer the reader to [2] basics on quotients and co-wellpowered categories.
Proposition 3.1. [1, p. 5] Let CoM on(C) be the category of comonoids of C and U : CoM on(C) → C be the forgetful functor. If C is co-wellpowered, then so is CoM on(C).
Proposition 3.2. Let C be a co-wellpowered category, and let h : A → B be an arrow in C. If P J is cocontinuous ∀J ∈ {A, B}, then Z h (C) is cowellpowered.
Proof. It is enough to show that if p : (X, α, β) → (Y, α , β ) and q : (X, α, β) → (Z, α , β ) are in Z h (C) and equivalent as epimorphisms in C, then they are equivalent (as epimorphisms) in Z h (C). Let θ : Y → Z be an isomorphism in C for which θp = q. We show that θ is in fact an isomorphism in Z h (C).
Since p and q are arrows in Z h (C), the following diagrams are commutative.
Consider the following diagrams: Since P A is cocontinuous, it preserves epimorphisms [16, p. 72]. Hence, Similarly, from diagram (3.4), we get α (id B ⊗ θ) = (θ ⊗ id B )α . Therefore, θ is an isomorphism in Z h (C). Proposition 3.3. Let C be a co-wellpowered category and X an object in C. If Q X is cocontinuous, then Z X (C) is co-wellpowered.
Proof. As in Proposition 3.2, it suffices to show that if p : (A, α) → (B, β) and q : (A, α) → (B , β ) are in Z X (C) and equivalent as epimorphisms in C, then they are equivalent (as epimorphisms) in Z X (C). Let θ : B → B be an isomorphism in C with θp = q. We show that θ is in fact an isomorphism in Z X (C).
Since p and q are arrows in Z X (C), the following diagrams are commutative: Since Q X is cocontinuous, it preserves epimorphisms. Hence, Q X (p) = (p ⊗ id X ) is an epimorphism. Thus, β (θ ⊗ id X ) = (id X ⊗ θ)β. Therefore, θ is an isomorphism in Z X (C), and the proof is complete.
Corollary 3.4. Let C be a co-wellpowered category. If Q X is cocontinuous, ∀X ∈ C, then Z(C) and Z ω (C) are co-wellpowered.
Proof. This immediately follows from the proof of Proposition 3.3 and Remark 1.7.
Using Theorem 2.6 implies the following immediate consequence.
Proposition 3.5. Let C be a monoidal category. Fix an object X and a morphism h : Example 3.6.
(1) The category Set of sets and mappings is co-wellpowered [22, p. 66], and it can also be seen as cartesian closed [8, p. 296], hence symmetric monoidal closed. By Proposition 3.5, the categories Z h (Set), Z X (Set), Z(Set), and Z ω (Set) are co-wellpowered for any X ∈ Set and a morphism h in Set.
(2) The category Ab of abelian groups with its tensor product of abelian groups is a biclosed monoidal category. By Proposition 3.5, Z h (Ab), Z X (Ab), Z(Ab), and Z ω (Ab) are co-wellpowered for any X ∈ Ab and a morphism h in Ab.
(3) The category Top of topological spaces and continuous mappings is wellpowered [22, p. 66], and it can be provided with the structure of a symmetric monoidal closed category (See [8, p. 299]). By Proposition 3.5, Z h (Top), Z X (Top), Z(Top), and Z ω (Top) are co-wellpowered for any X ∈ Top and a morphism h in Top.

Generators
Following [16, 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 [16], [2], or [11].
Let C be a monoidal category with a generating set G, and let A ∈ {Z h (C), Z X (C), Z(C), Z ω (C)}. Fix an object X and a morphism h : A → B in C. Our inspection in the previous sections gives rise to the following question. When can the category A inherit a generating set involved with G from C?
Under the assumption above, let f, g : Z → W be any parallel pair of morphisms in A with f = g. Since G is a generating set for C, there is an G ∈ G and an arrow α : G → X in C with f α = gα. Now, if we want to show that A has a generating set G whose underlying is G, we need to show that G ∈ A and the morphism α : G → X is in A . Although, this is not true in general, it perfectly works when C is a braided. For the basic notions of braided monoidal categories, we refer to [23], [12] and [17]. It turns out that we have the following theorem.
Theorem 4.1. Let C be a braided monoidal category with a braiding Ψ, and let G be a generating set for C. Fix an object X and a morphism h in C. Then the category A has a generating set for any Proof. Consider the following diagram It is well-known that there is an embedding Φ 1 : C → Z(C) via W → (W, Ψ W,− ) [9, p. 264]. Define the functors Clearly, Φ i is embedding for all i = 2, 3, 4. Therefore, the category C can be viewed as a subcategory of the category A , Now, let f, g : Z → W be any parallel pair of morphisms in A with f = g. Since G is a generating set for C, there is an G ∈ G and an arrow α : G → X in C with f α = gα. From the diagram 4.1, we have G ∈ A , and the morphism α : G → X is in A . Thus, for every A ∈ {Z h (C), Z X (C), Z(C), Z ω (C)}, A has a generating set G A whose underlying is G.
The following assertion is important in characterizing the cofree objects in CoM on(A ), for all A ∈ {Z h (C), Z X (C), Z(C), Z ω (C)}.

Investigating cofree objects
In this section, we use Theorem 1.1 and Propositions 2.1, 3.1, and the consequences we have to show that the concrete category (CoM on(A ), U A ) has cofree objects, ∀A ∈ {Z h (C), Z X (C), Z(C), Z ω (C)}.
Theorem 5.1. Let U : CoM on(Z h (C)) → Z h (C) be the forgetful functor and h : A → B be an arrow in C, and let P J and Q J be cocontinuous ∀J ∈ {A, B}. If C is cocomplete, co-wellpowered and if CoM on(Z h (C)) has a generating set, then U has a right adjoint or, equivalently, the concrete category (CoM on(Z h (C)), U ) has cofree objects.
Corollary 5.2. Let (C, ⊗, I) be a braided monoidal category and h : A → B an arrow in C. Let U : CoM on(Z h (C)) → Z h (C) be the forgetful functor and P J , Q J be cocontinuous ∀J ∈ {A, B}. If C is cocomplete, co-wellpowered and if CoM on(C) has a generating set, then U has a right adjoint, and hence the concrete category (CoM on(Z h (C)), U ) has cofree objects.
Proof. It is immediate from Theorem 5.1 and Corollary 4.2.
Similarly, the following are immediate consequences of Proposition 2.1, Corollary 2.3, 3.3 and Theorem 1.1.
Theorem 5.3. Let X be an object in C. Let U : CoM on(Z X (C)) → Z X (C) be the forgetful functor and P X , Q X cocontinuous. If C is cocomplete, cowellpowered and if CoM on(Z X (C)) has a generating set, then the functor U has a right adjoint, hence, the concrete category (CoM on(Z X (C)), U ) has cofree objects.
Corollary 5.4. Let (C, ⊗, I) be a braided monoidal category and X an object in C. Let U : CoM on(Z X (C)) → Z X (C) be the forgetful functor and P X , Q X be cocontinuous. If C is cocomplete, co-wellpowered and if CoM on(C) has a generating set, then U has a right adjoint, hence, the concrete category (CoM on(Z X (C)), U ) has cofree objects.
Proof. It follows immediately from Theorem 5.3 and Corollary 4.2.
By Propositions 2.1, 2.4, Corollary 3.4 and Theorem 1.1, we have the following version for the existence of cofree objects in the monoidal center. (respectively U : CoM on(Z ω (C)) → Z(C)) be the forgetful functor, and let P X , Q X be cocontinuous ∀X ∈ C. If C is cocomplete, co-wellpowered and if CoM on(Z(C)) (resp. CoM on(Z ω (C))) has a generating set, then U (resp. U ) has a right adjoint. It turns out that, equivalently, the concrete category (CoM on(Z(C)), U ) (resp. (CoM on(Z ω (C)), U )) has cofree objects.
Corollary 5.6. Let (C, ⊗, I) be a braided monoidal category and U : CoM on(Z(C)) → Z(C) (resp. U : CoM on(Z ω (C)) → Z(C)) the forgetful functor, and let P X , Q X be cocontinuous ∀X ∈ C. If C is cocomplete, co-wellpowered and if CoM on(C) has a generating set, then U (resp. U ) has a right adjoint. It turns out that, equivalently, the concrete category (CoM on(Z(C)), U ) (resp. (CoM on(Z ω (C)), U )) has cofree objects.
Proof. The required statement follows from Theorem 5.5 and Corollary 4.2.
(2) Following [23, p. 69-70], the braid category B has as objects the natural numbers 0, 1, 2, ... and as arrows α : n → n the braids on n strings; there are no arrows n → n for m = n. A braid α on n strings can be regarded as an element of the Artin braid group B n with generators s 1 , ..., s n−1 subject to the relations Composition of braids is just multiplication in this group, represented diagrammatically by vertical stacking of braids with the same number of strings. Tensor product of braids adds the number of strings by placing one braid next to the other longitudinally. This makes B a strict monoidal category. A braiding c m,n : m + n → n + m is given by crossing the first m strings over the remaining n. Then B is braided monoidal category. Indeed, it is a balanced monoidal category. To see how the braid s i , the composition of braids, tensor product of braids and the braiding c m,n can be depicted, we refer the reader to [23, p. 69-70].
Proposition 5.8. The category B is not cocomplete.
Proof. Let D be a small category, and let F : D → B be a functor. By the way of contradiction, let B be a cocomplete category. It follows that F has a colimit (t, (φ D ) D∈D ). The definition of B implies that F D = t, for all D ∈ D. In particular, we have F is a constant functor, for every functor F : D → B with D a small category. It is clear that this is a contradiction because we can always define a nonconstant functor from a small category to B. Therefore, the category B is not cocomplete. Proof. It follows immediately from Theorem 5.9 that CoM on(A ) is cocomplete, co-wellpowered, and with a generating set. Thus, using Theorem 1.1 completes the proof. Composition inB is vertical stacking of diagrams, and tensor product for B is horizontal placement of diagrams, much as for B. The braiding c m,n : m + n → n + m forB is obtained by placing the first m ribbons over the remaining n without introducing any twists. ThenB is a braided monoidal category. Indeed, it is a balanced monoidal category. To see how s n and the braiding c m,n can be visualized, we refer the reader to [23, p. 74-75].
The identification ofB is similar to that of B. Thus, for any (fixed) object X and an arrow h inB, Proposition 5.8 and Theorems 5.9 and 5.10 imply the following consequences. (iii) For any A ∈ {Z h (B), Z X (B), Z(B), Z ω (B)}, the corresponding forgetful functor U A has a right adjoint, and hence the corresponding concrete category (CoM on(A ), U A ) has cofree objects). acknowledgement I would like to thank Prof. Miodrag Iovanov for his support and the referees for their valuable suggestions.