Explicit construction of cofree precoalgebras and coalgebras
EExplicit construction of cofree precoalgebrasand coalgebras
Yuki Goto
Abstract
In this paper, we will explicitly construct cofree coalgebras, by firstconstructing cofree precoalgebras (namely those not necessarily coas-sociative or counital). Our approach does not impose any condition tothe coefficient ring, which means, for example, it does not need to be afield. The main technique is to generate homomorphisms from certaintype of infinite trees so that those constitute cofree precoalgebras.
Given a module V , a coalgebra V with a couniversal homomorphism π : V → V is called a cofree coalgebra. Although this notion arose in algebraic topol-ogy, it is also known to be worthwhile in categorical logic. In fact, originatedfrom Lafont’s idea, Bierman showed that in a category with cofree coalge-bras, we can interpret the exponential operator of linear logic as a functorwhich sends an object to the cofree coalgebra over it [8, 3].Until now, several explicit constructions of cofree coalgebras have beenintroduced. For example, Sweedler constructed them by using the notionof finite dual, which was generalised by Anquela and Cort´es to those overarbitrary algebraic varieties [1, 10]. Block and Leroux presented them as thevector space of all representative linear maps [4]. Hazewinkel used the notionof tensor power series, and Murfet used local cohomology and residues [7, 9].However, all of these constructions can be performed in the case that thecoefficent ring of modules are restricted to those which enjoy some conditions(mainly being a field).In this paper, we will present a new approach to explicitly constructcofree coalgebras, which does not impose any condition to the coefficientring. This means that our method can be carried out for any commutativering as the coefficient.In our approach, we will first consider precoalgebras (namely coalgebrasnot necessarily coassociative or counital), and construct cofree precoalge-bras. We will then construct cofree coalgebras from them. The main tech-nique is to consider some sort of infinite trees and generate homomorphismsfrom them. This idea was once adopted by Fox, but his argument was1 a r X i v : . [ m a t h . R A ] S e p ointed out to contain an error by Hazewinkel [5, 7]. This paper is alsomeant to fix this error.The outline of this paper is as follows. In Section 2, we will recall thedefinition of precoalgebras and coalgebras and their cofreeness. We will alsorecall Barr’s result which provides the method of creating cofree coalgebrasfrom cofree precoalgebras [2]. Section 3 is the main part of this paper,which concerns an explicit construction of cofree precoalgebras and cofreecoalgebras. In Section 4, we will briefly compare the Block and Leroux’s con-struction with ours by explicitly presenting an isomorphism between them. In this section, we recall the definition of precoalgebras and coalgebras, andalso that of their cofreeness. Throughout this paper, we fix a commutativering (not necessarily a field) as a coefficient ring of modules, and let K denoteit. We suppose all modules, homomorphisms between them and the tensorproducts of them are all over K . Definition 1.
A module C is called a precoalgebra when it is equipped withtwo homomorphisms δ : C → C ⊗ C and ε : C → K , which satisfy no otheradditional conditions. Definition 2.
A module C is called a coalgebra when it is equipped with twohomomorphisms δ : C → C ⊗ C and ε : C → K , and these homomorphismsmake the following diagrams commute: C C ⊗ CC ⊗ C C ⊗ C ⊗ C ← → δ ←→ δ ←→ δ ⊗ id ← → id ⊗ δ C K ⊗ C C ⊗ C C ⊗ K ←→ δ ← → ←→ ←→ ε ⊗ id ← → id ⊗ ε Here, C → K ⊗ C and C → C ⊗ K are the canonical isomorphisms. Clearly every coalgebra is a precoalgebra. We will sometimes say that aprecoalgebra is admissible when it is also a coalgebra (that is when the twodiagrams above both commute).A morphism between them is defined as follows:
Definition 3.
For two precoalgebras ( C, δ, ε ) and ( D, δ (cid:48) , ε (cid:48) ) , a homomor-phism ϕ : D → C is called a precoalgebra morphism when the following dia-grams both commute: D CD ⊗ D C ⊗ C ← → ϕ ←→ δ (cid:48) ←→ δ ← → ϕ ⊗ ϕ D C K ← → ϕ ← → ε (cid:48) ←→ ε A coalgebra morphism is defined in the exactly same way.
Definition 4.
For a module V , a precoalgebra ( V , δ, ε ) is said to be cofree over V when the following two conditions both hold: • V is equipped with a homomorphism π : V → V ; • for any precoalgebra ( D, δ (cid:48) , ε (cid:48) ) with a homomorphism ϕ : D → V , thereexists a unique precoalgebra morphism ˜ ϕ : D → V which makes thefollowing diagram commute: DV V ←→ ˜ ϕ ← → ϕ ← → π The cofreeness of a coalgebra is also similarly defined.
Barr showed that, once cofree precoalgebras are constructed, cofree coal-gebras can be given as the largest admissible subprecoalgebras of them. Heproved this fact in any cocomplete and well-powered categories, but here werestate it in the category of modules to fit our settings.
Theorem 5 (Barr, Theorem 4.3 [2]) . Assume that there exists the cofreeprecoalgebra ( V , δ, ε ) over any module V . Then the largest admissible sub-precoalgebra of ( V , δ, ε ) is the cofree coalgebra over every module V . In the next section, we will first present an explicit construction of cofreeprecoalgebras. We will furthermore construct the largest admissible subpre-coalgebras of them, which are proved to be cofree coalgebras by this theorem.
In this section, we will introduce two types of trees: one is called a gradingtree, and the other is called a generating tree. The former is used as aparametre of a special type of modules considered throughout this section(see Definitions 6 and 7), and the latter is used to generate a homomorphism(see Definitions 10 and 11).A grading tree is defined as follows:
Definition 6.
By a grading tree , we mean a finite full binary tree, that is,a rooted tree such that the number of nodes are finite and every node hasexactly zero or two child nodes. t and u , we write t ∧ u for the grading tree obtainedby adding a new root which has two edges to the roots of t and u . Moreover,we write • for the trivial grading tree, which has no node other than theroot.Using grading trees, we generalise the notion of graded algebra. Usualgraded algebras are parametrised by natural numbers, while our new typeof algebras are parametrised by grading trees. Definition 7.
Suppose that a module A t is given for each grading tree t .Their direct sum A := (cid:77) t A t is called a tree-graded algebra when it is equipped with a homomorphism ˆ ⊗ : A t ⊗ A u → A t ∧ u for any pair of grading trees ( t, u ) . Definition 8.
For two tree-graded algebras A and B , a homomorphism f : A → B is called tree-graded when we have f ( A t ) ⊆ B t for any gradingtree t . Starting from any module V , we can construct a tree-graded algebra.The construction is similar to a usual tensor algebra. Definition 9.
Take a module V . For each grading tree t , define ˆ T t V := V ⊗ leaf( t ) = V ⊗ · · · ⊗ V (cid:124) (cid:123)(cid:122) (cid:125) leaf( t ) times , where leaf( t ) denotes the number of leaves in t . Moreover, for two gradingtrees t and u , let ˆ ⊗ : ˆ T t V ⊗ ˆ T u V −→ ˆ T t ∧ u V ( v ⊗ · · · ⊗ v n ) ⊗ ( w ⊗ · · · ⊗ w m ) (cid:55)−→ v ⊗ · · · ⊗ v n ⊗ w ⊗ · · · ⊗ w m . With this homomorphism, the direct sum ˆ T V := (cid:77) t ˆ T t V is a tree-graded algebra, which we call a tree-tensor algebra of V . Note that, unlike that of a usual tensor algebra, the operation ˆ ⊗ definedabove is not (and never) associative, since ˆ T ( t ∧ u ) ∧ v V (cid:54) = ˆ T t ∧ ( u ∧ v ) V ⊆ ˆ T V forgrading trees t , u , v .Now consider in particular the case V = K . Since we haveˆ T K = (cid:77) t ˆ T t K = (cid:77) t K ⊗ leaf( t ) ∼ = (cid:77) t K , L(1) L(2) R(1) R(2)L(1)L(1) L(1)R(1) L(2)L(1) L(2)L(2) L(2)R(1)L(2)R(2)R(1)L(1)R(1)R(1)R(2)L(1)R(2)R(1)...
Figure 1: Generating tree and position sequencesthe tree-tensor algebra ˆ T K is the free module generated by all grading trees.This means that, to define a tree-graded homomorphism f : ˆ T K → B where B is some tree-graded algebra, we only need to determine f ( t ) ∈ B t for eachgrading tree t .Given a module V , our cofree precoalgebra over V is presented as asubmodule of Hom( ˆ T K , ˆ T ( V ⊕ K )), whose elements are tree-graded onesgenerated by another type of trees, which are called generating trees: Definition 10.
For a module V , a generating tree for V is an infinite tree σ which satisfies the following two conditions: • each node of σ has an even number of two or more children; if thenode has exactly n ( n ≥ children, they are named L(1) , · · · , L( n ) , R(1) , · · · , R( n ) in order from left to right; • each node of σ is labelled by an element of V ⊕ K . Each node of a generating tree σ can be uniquely specified by a finitesequence of symbols of the form L( i ) or R( i ), by reading the name of thechild successively from the root; the root node is specified by the emptysequence ∅ (see Figure 1). We call such a finite sequence a position sequence ,and often identify it with the node specified by it. For a position sequence r , we write σ (cid:104) r (cid:105) for the label attached to the node specified by r , and σ ↓ r for the subtree under that node.Here we see how a generating tree generates a tree-graded homomor-phism. Definition 11.
Given a generating tree σ for a module V , define a tree-graded homomorphism f : ˆ T K → ˆ T ( V ⊕ K ) , which is said to be generated by σ , as follows. First set f ( • ) := σ (cid:104)∅(cid:105) . For a grading tree t other than • , wedefine f ( t ) in the following way.Just like the naming of children in a generating tree, call the left child L and the right child R . Then every node of t is uniquely specified by a finitesequence of L or R . Let p , · · · , p n be all the leaves of t in order from leftto right. For each inner node q in t , prepare a variable symbol ι (cid:104) q (cid:105) whichranges over nonnegative integers. or ≤ k ≤ n , write p k =: ξ k, · · · ξ k,m k where each ξ k,l is either L or R .Set r k := ξ k, ( ι (cid:104)∅(cid:105) ) ξ k, ( ι (cid:104) ξ k, (cid:105) ) · · · ξ k,m k ( ι (cid:104) ξ k, · · · ξ k,m k − (cid:105) ) , which is a finite sequence of symbols of the form L( i ) or R( i ) . Now define f ( t ) := (cid:88) ι ( σ (cid:104) r (cid:105) ⊗ σ (cid:104) r (cid:105) ⊗ · · · ⊗ σ (cid:104) r n (cid:105) ) , where the sum ranges over ( r , r , · · · , r n ) such that every r k (1 ≤ k ≤ n ) specifies some nodes in σ when ι (cid:104) q (cid:105) ’s run through nonnegative integers. Thissum is obviously finite, and thus determines an element of ˆ T t ( V ⊕ K ) . For example, consider the case t = . Since t has three leaves p := LL, p := LR, p := R, we have r = L( ι (cid:104)∅(cid:105) )L( ι (cid:104) L (cid:105) ) r = L( ι (cid:104)∅(cid:105) )R( ι (cid:104) L (cid:105) ) r = R( ι (cid:104)∅(cid:105) ) . Thus the value f ( t ) is defined as f ( t ) = (cid:88) ι ( σ (cid:104) r (cid:105) ⊗ σ (cid:104) r (cid:105) ⊗ σ (cid:104) r (cid:105) )= (cid:88) ι ( σ (cid:104) L( ι (cid:104)∅(cid:105) )L( ι (cid:104) L (cid:105) ) (cid:105) ⊗ σ (cid:104) L( ι (cid:104)∅(cid:105) )R( ι (cid:104) L (cid:105) ) (cid:105) ⊗ σ (cid:104) R( ι (cid:104)∅(cid:105) ) (cid:105) ) . Rewriting i := ι (cid:104)∅(cid:105) and j := ι (cid:104) L (cid:105) for readability, this can be also equivalentlypresented as f ( t ) = (cid:88) i,j ( σ (cid:104) L( i )L( j ) (cid:105) ⊗ σ (cid:104) L( i )R( j ) (cid:105) ⊗ σ (cid:104) R( i ) (cid:105) ) . If the generating tree is that given by Figure 1, this sum is expanded as f ( t ) = σ (cid:104) L(1)L(1) (cid:105) ⊗ σ (cid:104) L(1)R(1) (cid:105) ⊗ σ (cid:104) R(1) (cid:105) + σ (cid:104) L(2)L(1) (cid:105) ⊗ σ (cid:104) L(2)R(1) (cid:105) ⊗ σ (cid:104) R(2) (cid:105) + σ (cid:104) L(2)L(2) (cid:105) ⊗ σ (cid:104) L(2)R(2) (cid:105) ⊗ σ (cid:104) R(2) (cid:105) . Here note that the range through which ι (cid:104) q (cid:105) runs may depend on ι (cid:104) q (cid:48) (cid:105) for shorter q (cid:48) . In the example above, ι (cid:104) L (cid:105) (which is also written as j ) canonly be 1 when ι (cid:104)∅(cid:105) (also written as i ) is equal to 1, while ι (cid:104) L (cid:105) can be 1 and2 when ι (cid:104)∅(cid:105) is 2. Definition 12.
For a module V , a homomorphism f : ˆ T K → ˆ T ( V ⊕ K ) is said to be representative when f is generated by some generating tree for V . Let Rep( V ) ⊆ Hom( ˆ T K , ˆ T ( V ⊕ K )) denote the set of all representativehomomorphisms. V ) is thecofree precoalgebra over every module V . First we will show that Rep( V )is a module. Lemma 13.
For a module V , the set Rep( V ) is a submodule of Hom( ˆ T K , ˆ T ( V ⊕ K )) .Proof. Take two generating trees σ and τ . We define a new tree σ + τ in thefollowing way. First set ( σ + τ ) (cid:104)∅(cid:105) := σ (cid:104)∅(cid:105) + τ (cid:104)∅(cid:105) . For a nonempty positionsequence r and s of σ and τ respectively, write r =: ξ ( i ) r (cid:48) and s =: η ( j ) s (cid:48) where ξ is either L or R and so is η . Set( σ + τ ) (cid:104) ξ ( i ) r (cid:48) (cid:105) := σ (cid:104) ξ ( i ) r (cid:48) (cid:105) ( σ + τ ) (cid:104) η ( j + n ) s (cid:48) (cid:105) := τ (cid:104) η ( j ) s (cid:48) (cid:105) , where n is half the number of the children of the root of σ . In other words, σ + τ is the tree obtained by the following procedure: • remove the root of σ to obtain σ ↓ L(1) , · · · , σ ↓ L( n ) , σ ↓ R(1) , · · · , σ ↓ R( n ) ; • remove the root of τ to obtain τ ↓ L(1) , · · · , τ ↓ L( m ) , τ ↓ R(1) , · · · , τ ↓ R( m ) ; • create a new root which is labelled σ (cid:104)∅(cid:105) + τ (cid:104)∅(cid:105) ; • create edges from the new root to σ ↓ L(1) , · · · , σ ↓ L( n ) , τ ↓ L(1) , · · · , τ ↓ L( m ) , σ ↓ R(1) , · · · , σ ↓ R( n ) , τ ↓ R(1) , · · · , τ ↓ R( m ) in order from left to right.Next take a generating tree σ and a scalar λ . To define a new tree λσ ,set for each position sequence r of σ ,( λσ ) (cid:104) r (cid:105) := (cid:40) λ σ (cid:104) r (cid:105) ( r is of the form L( i ) · · · L( i a ), a ≥ σ (cid:104) r (cid:105) (otherwise) . If σ and τ generate f and g respectively, then σ + τ generates f + g .Similarly, if σ generates f , then λσ generates λf . This proves the lemma.Now recall the following well-known fact: Proposition 14.
For modules C and D , the homomorphism Ψ : Hom( C, D ) ⊗ n −→ Hom( C ⊗ n , D ⊗ n ) g ⊗ · · · ⊗ g n (cid:55)−→ [ c ⊗ · · · ⊗ c n (cid:55)−→ g ( c ) ⊗ · · · ⊗ g n ( c n )] is injective. To define a precoalgebra structure on Rep( V ), we need the followingthree lemmata: The term “representative” is borrowed by Block–Leroux, but its meaning here is notthe same as the original. emma 15. For a module V , the homomorphism defined by Φ : Hom( ˆ T K , ˆ T ( V ⊕ K )) ⊗ −→ Hom( ˆ T K , ˆ T ( V ⊕ K )) g ⊗ h (cid:55)−→ (cid:20) • (cid:55)−→ t ∧ u (cid:55)−→ g ( t ) ˆ ⊗ h ( u ) (cid:21) is injective.Proof. By Proposition 14, the morphism Ψ : Hom( ˆ T K , ˆ T ( V ⊕ K )) ⊗ −→ Hom( ˆ T K ⊗ ˆ T K , ˆ T ( V ⊕ K ) ⊗ ˆ T ( V ⊕ K )) g ⊗ h (cid:55)−→ [ t ⊗ u (cid:55)−→ g ( t ) ⊗ h ( u )]is injective. In addition, Ψ : Hom( ˆ T K ⊗ ˆ T K , ˆ T ( V ⊕ K ) ⊗ ˆ T ( V ⊕ K )) −→ Hom( ˆ T K , ˆ T ( V ⊕ K ) ⊗ ˆ T ( V ⊕ K )) p (cid:55)−→ (cid:20) • (cid:55)−→ t ∧ u (cid:55)−→ p ( t ⊗ u ) (cid:21) is also clearly injective. Now consider ψ : ˆ T ( V ⊕ K ) ⊗ ˆ T ( V ⊕ K ) −→ ˆ T ( V ⊕ K ) a ⊗ b (cid:55)−→ a ˆ ⊗ b, which induces Ψ : Hom( ˆ T K , ˆ T ( V ⊕ K ) ⊗ ˆ T ( V ⊕ K )) −→ Hom( ˆ T K , ˆ T ( V ⊕ K )) q (cid:55)−→ ψ ◦ q. Since ψ is clearly injective, so is Ψ . Thus the composite Φ = Ψ ◦ Ψ ◦ Ψ isshown to be injective. Lemma 16.
For a module V and a representative homomorphism f : ˆ T K → ˆ T ( V ⊕ K ) , there exist two finite families ( g i ) i ∈ I and ( h i ) i ∈ I of representativehomomorphisms such that, for any grading trees t and u , we have f ( t ∧ u ) = (cid:88) i ∈ I ( g i ( t ) ˆ ⊗ h i ( u )) . Proof.
Suppose that f is generated by σ . Let L(1) , · · · , L( n ) , R(1) , · · · , R( n )be all the children of the root of σ . For each 1 ≤ i ≤ n , let g i be thehomomorphism generated by σ ↓ L( i ) , and h i that generated by σ ↓ R( i ) . Then( g i ) ≤ i ≤ n and ( h i ) ≤ i ≤ n satisfy the desired property. Lemma 17.
For a module V and a representative homomorphism f : ˆ T K → ˆ T ( V ⊕ K ) , suppose that we have finite families ( g i ) i ∈ I , ( h i ) i ∈ I , ( g (cid:48) j ) j ∈ J , ( h (cid:48) j ) j ∈ J of representative homomorphisms such that, for any grading trees t and u ,we have f ( t ∧ u ) = (cid:88) i ∈ I ( g i ( t ) ˆ ⊗ h i ( u )) = (cid:88) j ∈ J ( g (cid:48) j ( t ) ˆ ⊗ h (cid:48) j ( u )) . hen we have (cid:88) i ∈ I ( g i ⊗ h i ) = (cid:88) j ∈ J ( g (cid:48) j ⊗ h (cid:48) j ) as elements of Rep( V ) ⊗ Rep( V ) .Proof. Set p := (cid:88) i ∈ I ( g i ⊗ h i ) p (cid:48) := (cid:88) j ∈ J ( g (cid:48) j ⊗ h (cid:48) j ) , and let Φ denote the homomorphism defined in Lemma 15. For any gradingtrees t and u , we have Φ ( p )( • ) = 0 Φ ( p )( t ∧ u ) = (cid:88) i ∈ I ( g i ( t ) ˆ ⊗ h i ( u )) = f ( t ˆ ⊗ u ) , and p (cid:48) also satisfies the exactly same formulae. Thus we have Φ ( p ) = Φ ( p (cid:48) ).Since Φ is injective, we have p = p (cid:48) , which proves the lemma.For a representative homomorphism f : ˆ T K → ˆ T ( V ⊕ K ), take ( g i ) i ∈ I and ( h i ) i ∈ I as in Lemma 16 and define δ ( f ) := (cid:88) i ∈ I ( g i ⊗ h i ) . By Lemma 17, this is independent of the choice of ( g i ) i and ( h i ) i and dependsonly on f . This concludes that the definition above yields a homomorphism δ : Rep( V ) → Rep( V ) ⊗ Rep( V ). In addition, letting ε ( f ) := (the K -component of f ( • )) π ( f ) := (the V -component of f ( • ))yields ε : Rep( V ) → K and π : Rep( V ) → V . We will show that these datagive the cofree precoalgebra: Theorem 18.
For a module V , the precoalgebra (Rep( V ) , δ, ε, π ) is cofreeover V . Before the proof, we introduce a new notation. For a precoalgebra(
D, δ (cid:48) , ε (cid:48) ) and an element d ∈ D , since δ (cid:48) ( d ) is an element of D ⊗ D , itcan be written as a finite sum of the tensor products of two elements of D .Write δ (cid:48) ( d ) =: (cid:88) i ( d L( i ) ⊗ d R( i ) )9or that sum, where i runs through some range of nonnegative integers.Note that such representation of δ (cid:48) ( d ) is not unique, so we fix one suchrepresentation for each d . Moreover, for each i , similarly write δ (cid:48) ( d L( i ) ) =: (cid:88) j ( d L( i )L( j ) ⊗ d L( i )R( j ) ) δ (cid:48) ( d R( i ) ) =: (cid:88) j ( d R( i )L( j ) ⊗ d R( i )R( j ) ) , and proceed recursively to define d r where r is a position sequence.Using this notation, for another precoalgebra ( C, δ, ε ) and a homomor-phism ϕ : D → C , the commutativity of the diagram D CD ⊗ D C ⊗ C ← → ϕ ←→ δ (cid:48) ←→ δ ← → ϕ ⊗ ϕ is rephrased as (cid:88) i ( ϕ ( d ) L( i ) ⊗ ϕ ( d ) R( i ) ) = (cid:88) i ( ϕ ( d L( i ) ) ⊗ ϕ ( d R( i ) )) . Hence in particular, if ϕ is a precoalgebra morphim, then ϕ satisfies thisformula. Proof.
Take a precoalgebra (
D, δ (cid:48) , ε (cid:48) ) and a homomorphism ϕ : D → V .Suppose that there exists a precoalgebra morphism ˜ ϕ : D → Rep( V ) suchthat D Rep( V ) V ←→ ˜ ϕ ← → ϕ ← → π commutes. We will first prove the uniqueness of such ˜ ϕ and then prove itsexistence.For d ∈ D , by the definition of ε and π , we have˜ ϕ ( d )( • ) = π ( ˜ ϕ ( d )) + ε ( ˜ ϕ ( d ))= ϕ ( d ) + ε (cid:48) ( d ) . Moreover, for any grading trees t and u , by the definition of δ , we have˜ ϕ ( d )( t ∧ u ) = (cid:88) i ( ˜ ϕ ( d ) L( i ) ( t ) ˆ ⊗ ˜ ϕ ( d ) R( i ) ( u ))= (cid:88) i ( ˜ ϕ ( d L( i ) )( t ) ˆ ⊗ ˜ ϕ ( d R( i ) )( u )) . d ∈ D and grading trees t , these two formulae inductivelydetermine ˜ ϕ ( d )( t ). This implies that such ˜ ϕ is unique if it exists.For any d ∈ D , these formulae indeed define a homomorphism ˜ ϕ ( d ) : ˆ T K → ˆ T ( V ⊕ K ), because such a homomorpism can be defined by setting the imageof each grading tree. Thus it remains to show that ˜ ϕ ( d ) is representative.Define a generating tree σ ( d ) as follows. First let σ ( d ) (cid:104)∅(cid:105) := ε (cid:48) ( d ) + ϕ ( d ).For each position sequence r , let σ ( d ) (cid:104) r (cid:105) := ε (cid:48) ( d r ) + ϕ ( d r ) if d r is defined.Then σ ( d ) generates ˜ ϕ ( d ), which implies that ˜ ϕ ( d ) is representative. In this subsection, we will present cofree coalgebras by constructing thelargest admissible subprecoalgebras of cofree coalgebras.Fix a module V and a generating tree σ . Take a grading tree t with n :=leaf( t ), and let p , · · · , p n be the leaves of t . As in the previous subsection,for each leaf p k (1 ≤ k ≤ n ), let p k =: ξ k, · · · ξ k,m k denote the representationas a finite sequence of L or R, and then set r k := ξ k, ( ι (cid:104)∅(cid:105) ) ξ k, ( ι (cid:104) ξ k, (cid:105) ) · · · ξ k,m k ( ι (cid:104) ξ k, · · · ξ k,m k − (cid:105) ) , to obtain a position sequence, where the variable of the form ι (cid:104) q (cid:105) rangesover nonnegative integers. In addition to it, take α k which is either theprojection α V : V ⊕ K → V or α K : V ⊕ K → K . Then we have choosen the n -tuple (cid:126)α := ( α , · · · , α n ), which we call a projection tuple . Let a := deg( (cid:126)α )denote the number of α V ’s among them. Now define σ (cid:104) t | (cid:126)α (cid:105) := (cid:88) ι ( α ( σ (cid:104) r (cid:105) ) ⊗ · · · ⊗ α n ( σ (cid:104) r n (cid:105) )) . This is an element of the tensor product of a copies of V and n − a copiesof K . Since such an element can be canonically identified with an elementof V ⊗ a , we regard σ (cid:104) t | (cid:126)α (cid:105) as an element of V ⊗ a . Definition 19.
Fix a module V . A generating tree σ is said to be weakly nor-mal when, for any grading trees t and u , and for any projection tuples (cid:126)α :=( α , · · · , α leaf( t ) ) and (cid:126)β := ( β , · · · , β leaf( u ) ) which satisfy deg( (cid:126)α ) = deg( (cid:126)β ) ,we have σ (cid:104) t | (cid:126)α (cid:105) = σ (cid:104) u | (cid:126)β (cid:105) .Moreover, a generating tree σ is said to be normal when, for any positionsequence s , the generating tree σ ↓ s is weakly normal. Definition 20.
For a module V , a homomorphism f : ˆ T K → ˆ T ( V ⊕ K ) is said to be normal when f is generated by a normal generating tree. Let Nor( V ) denote the set of all normal homomorphisms. Here recall that, for generating trees σ, τ and a scalar λ , we defined σ + τ and λσ in the proof of Lemma 13. 11 emma 21. Fix a module V , and take generating trees σ and τ . For agrading tree t with n := leaf( t ) , a projection tuple (cid:126)α := ( α , · · · , α n ) and aposition sequence s , we have ( σ + τ ) ↓ s (cid:104) t | (cid:126)α (cid:105) = σ (cid:104) t | (cid:126)α (cid:105) + τ (cid:104) t | (cid:126)α (cid:105) ( s = ∅ ) σ ↓ ξ ( i ) s (cid:48) (cid:104) t | (cid:126)α (cid:105) ( s = ξ ( i ) s (cid:48) , i ≤ m ) τ ↓ ξ ( i − m ) s (cid:48) (cid:104) t | (cid:126)α (cid:105) ( s = ξ ( i ) s (cid:48) , i > m ) , where m is half the number of the children of the root of σ .Proof. First consider the case s = ∅ . Write ρ := ( σ + τ ) ↓ s = σ + τ forsimplicity. If t = • , we have ρ (cid:104) t | (cid:126)α (cid:105) = α ( ρ (cid:104)∅(cid:105) )= α ( σ (cid:104)∅(cid:105) + τ (cid:104)∅(cid:105) )= σ (cid:104) t | (cid:126)α (cid:105) + τ (cid:104) t | (cid:126)α (cid:105) . Otherwise, for each 1 ≤ k ≤ n , let r k := ξ k, ( ι (cid:104)∅(cid:105) ) ξ k, ( ι (cid:104) ξ k, (cid:105) ) · · · ξ k,m k ( ι (cid:104) ξ k, · · · ξ k,m k − (cid:105) ) r (cid:48) k := ξ k, ( ι (cid:104) ξ k, (cid:105) ) · · · ξ k,m k ( ι (cid:104) ξ k, · · · ξ k,m k − (cid:105) ) , where ξ k,l is as above. Now we have ρ (cid:104) t | (cid:126)α (cid:105) := (cid:88) ι ( α ( ρ (cid:104) ξ , ( ι (cid:104)∅(cid:105) ) r (cid:48) (cid:105) ) ⊗ · · · ⊗ α n ( ρ (cid:104) ξ n, ( ι (cid:104)∅(cid:105) ) r (cid:48) n (cid:105) )) . By the definition of σ + τ , we have ρ (cid:104) ξ k, ( ι (cid:104)∅(cid:105) ) r (cid:48) k (cid:105) = (cid:40) σ (cid:104) ξ k, ( ι (cid:104)∅(cid:105) ) r (cid:48) k (cid:105) ( ι (cid:104)∅(cid:105) ≤ m ) τ (cid:104) ξ k, ( ι (cid:104)∅(cid:105) ) r (cid:48) k (cid:105) ( ι (cid:104)∅(cid:105) > m ) . Thus the sum above is decomposed as ρ (cid:104) t | (cid:126)α (cid:105) = (cid:88) ι ( α ( σ (cid:104) ξ , ( ι (cid:104)∅(cid:105) ) r (cid:48) (cid:105) ) ⊗ · · · ⊗ α n ( σ (cid:104) ξ n, ( ι (cid:104)∅(cid:105) ) r (cid:48) n (cid:105) ))+ (cid:88) ι ( α ( τ (cid:104) ξ , ( ι (cid:104)∅(cid:105) ) r (cid:48) (cid:105) ) ⊗ · · · ⊗ α n ( τ (cid:104) ξ n, ( ι (cid:104)∅(cid:105) ) r (cid:48) n (cid:105) )) , which implies ρ (cid:104) t | (cid:126)α (cid:105) = σ (cid:104) t | (cid:126)α (cid:105) + τ (cid:104) t | (cid:126)α (cid:105) .The statement in the case s (cid:54) = ∅ is obvious by the definition of σ + τ . Lemma 22.
Fix a module V , and take a generating tree σ and a scalar λ .For a grading tree t with n := leaf( t ) , a projection tuple (cid:126)α := ( α , · · · , α n ) and a position sequence s , we have ( λσ ) ↓ s (cid:104) t | (cid:126)α (cid:105) = (cid:40) λ σ ↓ s (cid:104) t | (cid:126)α (cid:105) ( s is of the form L( i ) · · · L( i a ) , a ≥ σ ↓ s (cid:104) t | (cid:126)α (cid:105) ( otherwise ) . roof. Write ρ := λσ . For each 1 ≤ k ≤ n , take r k as above. Now we have ρ ↓ s (cid:104) t | (cid:126)α (cid:105) = (cid:88) ι ( α ( ρ ↓ s (cid:104) r (cid:105) ) ⊗ · · · ⊗ α n ( ρ ↓ s (cid:104) r n (cid:105) ))= (cid:88) ι ( α ( ρ (cid:104) sr (cid:105) ) ⊗ · · · ⊗ α n ( ρ (cid:104) sr n (cid:105) )) . If s is of the form L( i ) · · · L( i a ), only sr among sr , · · · , sr n is of the formL( i ) · · · L( i b ). Thus by the definition of λσ , we have ρ ↓ s (cid:104) t | (cid:126)α (cid:105) = (cid:88) ι ( α ( λσ (cid:104) sr (cid:105) ) ⊗ α ( σ (cid:104) sr (cid:105) ) · · · ⊗ α n ( σ (cid:104) sr n (cid:105) ))= λ (cid:88) ι ( α ( σ ↓ s (cid:104) r (cid:105) ) ⊗ α ( σ ↓ s (cid:104) r (cid:105) ) ⊗ · · · ⊗ α n ( σ ↓ s (cid:104) r n (cid:105) ))= λ σ ↓ s (cid:104) t | (cid:126)α (cid:105) . If s is not of the form L( i ) · · · L( i a ), none of sr , · · · , sr n is of the formL( i ) · · · L( i b ). Thus by the similar calculation, we have ρ ↓ s (cid:104) t | (cid:126)α (cid:105) = σ ↓ s (cid:104) t | (cid:126)α (cid:105) . Now to prove the cofreeness of Nor( V ), we generalise the notion of pro-jection tuple and show one property of weakly normal generating trees.Consider an n -tuple (cid:126)α := ( α , · · · , α n ) such that now each α k is either α V or α K or id V ⊕ K . Let can( (cid:126)α ) denote the tuple obtained by removing α K ’sfrom ( α , · · · , α n ). For a generating tree σ and a grading tree t , we similarlydefine σ (cid:104) t | (cid:126)α (cid:105) := (cid:88) ι ( α ( σ (cid:104) r (cid:105) ) ⊗ · · · ⊗ α n ( σ (cid:104) r n (cid:105) )) , where r , · · · , r n is the same as before. Writing a and a (cid:48) for the numberof α V ’s and id V ⊕ K ’s in ( α , · · · , α n ) respectively, we regard σ (cid:104) t | (cid:126)α (cid:105) as anelement of the tensor product of a copies of V and a (cid:48) copies of V ⊕ K .For example, in the case n = 4 and (cid:126)α = ( α V , α K , id V ⊕ K , α V ), we havecan( (cid:126)α ) = ( α V , id V ⊕ K , α V ) and σ (cid:104) t | (cid:126)α (cid:105) ∈ V ⊗ ( V ⊕ K ) ⊗ V .Note that, in particular when (cid:126)α = (id , · · · , id), we have σ (cid:104) t | (cid:126)α (cid:105) := (cid:88) ι ( σ (cid:104) r (cid:105) ⊗ · · · ⊗ σ (cid:104) r n (cid:105) ) . Thus letting f be the homomorphism generated by σ , we have f ( t ) = σ (cid:104) t | (id , · · · , id) (cid:105) as elements of ( V ⊕ K ) ⊗ n . Lemma 23.
Fix a module V , and take a weakly normal generating tree σ .For any grading trees t and u , and for any generalised projection tuples (cid:126)α :=( α , · · · , α leaf( t ) ) and (cid:126)β := ( β , · · · , β leaf( u ) ) which satisfy can( (cid:126)α ) = can( (cid:126)β ) , wehave σ (cid:104) t | (cid:126)α (cid:105) = σ (cid:104) u | (cid:126)β (cid:105) . roof. Let a and a (cid:48) be the number of α V ’s and id V ⊕ K ’s in ( α , · · · , α n )respectively. By binomial expansion, σ (cid:104) t | (cid:126)α (cid:105) can be regarded as an elementof (cid:77) ≤ k ≤ a (cid:48) C ( a (cid:48) , k ) V ⊗ ( a + k ) , where C ( a (cid:48) , k ) is the binomial coefficient, and C ( a (cid:48) , k ) V ⊗ ( a + k ) is the directsum of C ( a (cid:48) , k ) copies of V ⊗ ( a + k ) . Then its V ⊗ ( a + k ) -component is of the form σ (cid:104) t | (cid:126)α ◦ (cid:105) for some (not generalised) projection tuple (cid:126)α ◦ with a + k = deg( (cid:126)α ◦ ).Since σ is weakly normal, such σ (cid:104) t | (cid:126)α ◦ (cid:105) depends only on a + k . This impliesthat σ (cid:104) t (cid:105) depends only on a and a (cid:48) , which proves the lemma.Now we will prove that the set Nor( V ) gives the cofree coalgebra over amodule V . This will be done by three steps. Lemma 24.
For a module V , the set (Nor( V ) , δ, ε ) is a subprecoalgebra of (Rep( V ) , δ, ε ) .Proof. Take normal generating trees σ, τ and a scalar λ . To prove thatNor( V ) is closed under summation and scalar multiplication, we will showthat σ + τ and λσ are also normal. It immediately follows by Lemmata 21and 22.It remains to show that δ ( f ) ∈ Nor( V ) ⊗ Nor( V ) holds for any element f ∈ Nor( V ). Suppose f is generated by σ . For each i , let g i be the homor-phism generated by σ ↓ L( i ) and h i by σ ↓ R( i ) . By the definition of normality,it follows that σ ↓ L( i ) and σ ↓ R( i ) are normal, and thus g i and h i are alsonormal. Now by the definition of δ , we have δ ( f ) = (cid:88) i ( g i ⊗ h i ) , and thus δ ( f ) is in Nor( V ) ⊗ Nor( V ). Lemma 25.
For a module V , the precoalgebra (Nor( V ) , δ, ε ) is admissible.Proof. To prove the coassociativity of δ , we need to show that, for anynormal homomorphism f , we have (cid:88) i,j ( f L( i )L( j ) ⊗ f L( i )R( j ) ⊗ f R( i ) ) = (cid:88) i,j ( f L( i ) ⊗ f R( i )L( j ) ⊗ f R( i )R( j ) ) . By Proposition 14, it suffices to show that, for any grading trees t, u, v , wehave (cid:88) i,j ( f L( i )L( j ) ( t ) ⊗ f L( i )R( j ) ( u ) ⊗ f R( i ) ( v )) = (cid:88) i,j ( f L( i ) ( t ) ⊗ f R( i )L( j ) ( u ) ⊗ f R( i )R( j ) ( v )) ,
14s elements of ( V ⊕ K ) ⊗ (leaf( t )+leaf( u )+leaf( v )) . Supposing that f is generatedby a normal generating tree σ , this equation is rewritten as σ (cid:104) ( t ∧ u ) ∧ v | (id , · · · , id) (cid:105) = σ (cid:104) t ∧ ( u ∧ v ) | (id , · · · , id) (cid:105) , which holds by Lemma 23.To prove the counitality of ε , we need to show that, for any normalhomomorphism f , we have f = (cid:88) i α K ( f L( i ) ( • )) f R( i ) = (cid:88) i α K ( f R( i ) ( • )) f L( i ) To this end, it suffices to show that, for any grading tree t , f ( t ) = (cid:88) i α K ( f L( i ) ( • )) f R( i ) ( t ) = (cid:88) i α K ( f R( i ) ( • )) f L( i ) ( t )as elements of ( V ⊕ K ) ⊗ leaf( t ) . Supposing that f is generated by a normalgenerating tree σ , this equation is rewritten as σ (cid:104) t | (id , · · · , id) (cid:105) = σ (cid:104)• ∧ t | ( α K , id , · · · , id) (cid:105) = σ (cid:104) t ∧ • | (id , · · · , id , α K ) (cid:105) , which again holds by Lemma 23. Theorem 26.
For a module V , the coalgebra (Nor( V ) , δ, ε ) is cofree over V . Before the proof, we recall the notation defined before. For a subpre-coalgebra (
F, δ, ε ) of (Nor( V ) , δ, ε ) and an element f ∈ F , we have written δ ( f ) =: (cid:88) i ( f L( i ) ⊗ f R( i ) ) . Suppose that f is generated by σ . For each i , let g i be the homorphismgenerated by σ ↓ L( i ) and h i by σ ↓ R( i ) . Now since we have δ ( f ) =: (cid:88) i ( g i ⊗ h i ) , we can choose f L( i ) := g i and f R( i ) := h i . In the next proof, we adopt thischoice for f L( i ) ’s and f R( i ) ’s. Then we have f L( i ) ( • ) = σ ↓ L( i ) (cid:104)∅(cid:105) = σ (cid:104) L( i ) (cid:105) and similarly f R( i ) ( • ) = σ (cid:104) R( i ) (cid:105) . Recall also that we have defined f r for aposition sequence r . Adopting the choice above, we have f r ( • ) = σ ↓ r (cid:104)∅(cid:105) = σ (cid:104) r (cid:105) .We also introduce a new notation here. Take an admissible subprecoalge-bra ( F, δ, ε ) of (Nor( V ) , δ, ε ). Consider a morphism F → F ⊗ n ( n ≥
1) whichis given as a composite of morphisms of the form id ⊗ a ⊗ δ ⊗ id ⊗ b : F ⊗ ( a + b +1) → F ⊗ ( a + b +2) ( a, b ≥ δ : F → F ⊗ F is coassociative, such a morphismis independent of the way it is constructed, and depends only on n . Thuswe write { δ } n : F → F ⊗ n for that morphism.15 roof. By Theorem 5, it suffices to show that (Nor( V ) , δ, ε ) is the largestadmissible subprecoalgebra of (Rep( V ) , δ, ε ). To this end, for a admissiblesubprecoalgebra ( F, δ, ε ) and an element f ∈ F , we will prove that f isnormal. Let σ be a generating tree which generates f .First we will prove that f is weakly normal. Take a grading tree t with n := leaf( t ) and a projection tuple (cid:126)α := ( α , · · · , α n ) with a := deg( (cid:126)α ). Wehave f ( t ) = (cid:88) ι ( σ (cid:104) r (cid:105) ⊗ σ (cid:104) r (cid:105) ⊗ · · · ⊗ σ (cid:104) r n (cid:105) )= (cid:88) ι ( f r ( • ) ⊗ f r ( • ) ⊗ · · · ⊗ f r n ( • ))= θ ⊗ n ( { δ } n ( f )) , where θ : F → V ⊕ K is the morphism defined by θ ( f ) := f ( • ). Thus itfollows that σ (cid:104) t | (cid:126)α (cid:105) is an image of f by the morphism obtained by thecomposite F F ⊗ n ( V ⊕ K ) ⊗ n V ⊗ a ⊗ K ⊗ ( n − a ) V ⊗ a , ← → { δ } n ← → θ ⊗ n ← → α ⊗···⊗ α n ← → ( (cid:93) )where the last morphism is the canonical isomorphism. We will show thatthis morphism is equal to the composite F F ⊗ a V ⊗ a . ← → { δ } a ← → π ⊗ a ( (cid:93) )If this is done, it follows that σ (cid:104) t | (cid:126)α (cid:105) depends only on a , and thus σ isshown to be weakly normal.For simplicity, we shall only illustrate the case in which α , · · · , α a are α V and α a +1 , · · · , α n are α K , but the argument in the other cases is similar.Conside the diagram F F ⊗ n F ⊗ a F ⊗ a ⊗ K ⊗ ( n − a ) ( V ⊕ K ) ⊗ n V ⊗ a V ⊗ a ⊗ K ⊗ ( n − a ) , ← → { δ } n ←→ { δ } a ← → θ ⊗ n ←→ id ⊗ a ⊗ ε ⊗ ( n − a ) ← → ← → π ⊗ a ← → π ⊗ a ⊗ id ⊗ ( n − a ) ←→ α ⊗···⊗ α n ← → where the unlabelled morphisms are the canonical isomorphisms. The upper-left square commutes by the coassociativity of δ and the counitality of ε ,and the other two parallelograms also commute clearly. Hence the whole16iagram commutes, which means that the two morphisms (cid:93) and (cid:93) areequal.Next we will show that f is normal. For any position sequence s , thehomomorphism f s is that generated by σ ↓ s . Since f s is also in F , the sameargument as above proves that σ ↓ s is weakly normal. This concludes that σ is normal, and thus f is normal. In this section, we will compare our construction of cofree coalgebras withthe other known construction. Since cofree coalgebras are unique up to iso-morphism, our cofree coalgebras are isomorphic to those given by Sweedler,Block–Leroux, Hazewinkel and Murfet [10, 4, 7, 9]. Among them, we willexplicitly present the isomorphism between our cofree coalgebras and thoseby Block–Leroux. First let us recall the construction given by them. Herewe suppose that K is a field. Definition 27.
For a vector space V , a linear map f : K [ X ] → T V is said tobe representative when f preserves degrees and there exist two finite families ( g i ) i ∈ I and ( h i ) i ∈ I of linear maps such that, for any elements t, u ∈ K [ X ] ,we have f ( t · u ) = (cid:88) i ∈ I ( g i ( t ) · h i ( u )) . Here K [ X ] denotes the polynomial algebra over K . Let Rep BL ( V ) denote theset of all representative linear maps . For a linear map f : K [ X ] → T V which is representative in the senceof Block–Leroux, take ( g i ) i ∈ I and ( h i ) i ∈ I as in the definition above. Theyshowed that δ BL ( f ) := (cid:88) i ∈ I ( g i ⊗ h i ) ε BL ( f ) := f (1) π BL ( f ) := f ( X )yield well-defined linear maps δ BL : Rep BL ( V ) → Rep BL ( V ) ⊗ Rep BL ( V ), ε BL : Rep BL ( V ) → K and π BL : Rep BL ( V ) → V , and these data form thecofree coalgebra over V : Theorem 28 (Block–Leroux, Theorem 2 [4]) . For a vector space V , thedata (Rep BL ( V ) , δ BL , ε BL , π BL ) gives the cofree coalgebra over V . In the original paper, Rep BL ( V ) is denoted by K [ X ] V . V , the coalgebras Rep BL ( V ) and Nor( V ) are bothcofree over V , which implies that Rep BL ( V ) and Nor( V ) are mutually iso-morphic. This isomorphism can be explicitly illustrated as follows.If a generating tree σ is normal, for a grading tree t and a projectiontuple (cid:126)α , the value σ (cid:104) t | (cid:126)α (cid:105) depends only on a := deg( (cid:126)α ). Thus we simplywrite σ (cid:104) a (cid:105) for that value.Now let f denote the linear map generated by σ . For each integer n ≥ t n := ... (cid:124) (cid:123)(cid:122) (cid:125) n leaves . Then we have σ (cid:104) n (cid:105) = σ (cid:104) t n | ( α V , · · · , α V ) (cid:105) = α ⊗ nV ( f ( t n )) , where α V : V ⊕ K → V is the projection, and f ( t n ) is regarded as an elementof ( V ⊕ K ) ⊗ n . This ensures that the value σ (cid:104) n (cid:105) is independent of the choiceof σ and determined only by f . Moreover, we have σ (cid:104) (cid:105) = σ (cid:104)• | ( α K ) (cid:105) = α K ( f ( • )) , where α K : V ⊕ K → K is the projection. This means that σ (cid:104) (cid:105) is alsodetermined only by f . Hence the mapping˜ ϕ : Nor( V ) −→ Rep BL ( V ) f (cid:55)−→ (cid:20) K [ X ] −→ T VX n (cid:55)−→ σ (cid:104) n (cid:105) (cid:21) is well-defined and linear by Lemmata 21 and 22. Since σ (cid:104) (cid:105) = α V ( f ( • )),the diagram Nor( V )Rep BL ( V ) V ←→ ˜ ϕ ← → π ← → π BL commutes, and thus ˜ ϕ is the desired isomorphism. References [1] J. A. Anquela, T. Cort´es (1996)
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