On Well-Founded and Recursive Coalgebras

This paper studies fundamental questions concerning category-theoretic models of induction and recursion. We are concerned with the relationship between well-founded and recursive coalgebras for an endofunctor. For monomorphism preserving endofunctors on complete and well-powered categories every coalgebra has a well-founded part, and we provide a new, shorter proof that this is the coreflection in the category of all well-founded coalgebras. We present a new more general proof of Taylor’s General Recursion Theorem that every well-founded coalgebra is recursive, and we study conditions which imply the converse. In addition, we present a new equivalent characterization of well-foundedness: a coalgebra is well-founded iff it admits a coalgebra-to-algebra morphism to the initial algebra.


Introduction
What is induction? What is recursion? In areas of theoretical computer science, the most common answers are related to initial algebras. Indeed, the dominant trend in abstract data types is initial algebra semantics (see e.g. [ ]), and this approach has spread to other semantically-inclined areas of the subject. The approach in broad slogans is that, for an endofunctor F describing the type of algebraic operations of interest, the initial algebra µF has the property that for every F -algebra A, there is a unique homomorphism µF → A, and this is recursion. Perhaps the primary example is recursion on N, the natural numbers. Recall that N is the initial algebra for the set functor F X = X + 1. If A is any set, and a ∈ A and α : A → A + 1 are given, then initiality tells us that there is a unique f : N → A such that for all n ∈ N, f (0) = a f (n + 1) = α(f (n)).

( . )
A full version of this paper including full proof details is available on arXiv [ ]. Then the first additional problem coming with this approach is that of how to "recognize" initial algebras: Given an algebra, how do we really know if it is initial? The answer -again in slogans -is that initial algebras are the ones with "no junk and no confusion." Although initiality captures some important aspects of recursion, it cannot be a fully satisfactory approach. One big missing piece concerns recursive definitions based on well-founded relations. For example, the whole study of termination of rewriting systems depends on well-orders, the primary example of recursion on a well-founded order. Let (X, R) be a well-founded relation, i.e. one with no infinite sequences · · · x 2 R x 1 R x 0 . Let A be any set, and let α : PA → A. (Here and below, P is the power set functor, taking a set to the set of its subsets.) Then there is a unique f : X → A such that for all x ∈ X, f (x) = α({f (y) : y R x}).
( . ) The main goal of this paper is the study of concepts that allow one to extend the algebraic spirit behind initiality in ( . ) to the setting of recursion arising from well-foundedness as we find it in ( . ). The corresponding concepts are those of well-founded and recursive coalgebras for an endofunctor, which first appear in work by Osius [ ] and Taylor [ , ], respectively. In his work on categorical set theory, Osius [ ] first studied the notions of well-founded and recursive coalgebras (for the power-set functor on sets and, more generally, the power-object functor on an elementary topos). He defined recursive coalgebras as those coalgebras α : A → PA which have a unique coalgebra-to-algebra homomorphism into every algebra (see Definition . ). Taylor [ , ] took Osius' ideas much further. He introduced well-founded coalgebras for a general endofunctor, capturing the notion of a well-founded relation categorically, and considered recursive coalgebras under the name 'coalgebras obeying the recursion scheme'. He then proved the General Recursion Theorem that all well-founded coalgebras are recursive, for every endofunctor on sets (and on more general categories) preserving inverse images. Recursive coalgebras were also investigated by Eppendahl [ ], who called them algebra-initial coalgebras. Capretta, Uustalu, and Vene [ ] further studied recursive coalgebras, and they showed how to construct new ones from given ones by using comonads. They also explained nicely how recursive coalgebras allow for the semantic treatment of (functional) divide-and-conquer programs. More recently, Jeannin et al. [ ] proved the General Recursion Theorem for polynomial functors on the category of many-sorted sets; they also provide many interesting examples of recursive coalgebras arising in programming.
Our contributions in this paper are as follows. We start by recalling some preliminaries in Section and the definition of (parametrically) recursive coalgebras in Section and of well-founded coalgebras in Section (using a formulation based on Jacobs' next time operator [ ], which we extend from Kripke polynomial set functors to arbitrary functors). We show that every coalgebra for a monomorphism preserving functor on a complete and well-powered category has a well-founded part, and provide a new proof that this is the coreflection in the category of well-founded coalgebras (Proposition . ), shortening our previous proof [ ]. Next we provide a new proof of Taylor's General Recursion Theorem (Theorem . ), generalizing this to endofunctors preserving monomorphisms on a complete and well-powered category having smooth monomorphisms (see Definition . ). For the category of sets, this implies that "well-founded ⇒ recursive" holds for all endofunctors, strengthening Taylor's result. We then discuss the converse: is every recursive coalgebra well-founded? Here the assumption that F preserves inverse images cannot be lifted, and one needs additional assumptions. In fact, we present two results: one assumes universally smooth monomorphisms and that the functor has a pre-fixed point (see Theorem . ). Under these assumptions we also give a new equivalent characterization of recursiveness and well-foundedness: a coalgebra is recursive if it has a coalgebra-to-algebra morphism into the initial algebra (which exists under our assumptions), see Corollary . . This characterization was previously established for finitary functors on sets [ ]. The other converse of the above implication is due to Taylor using the concept of a subobject classifier (Theorem . ). It implies that 'recursive' and 'well-founded' are equivalent concepts for all set functors preserving inverse images. We also prove that a similar result holds for the category of vector spaces over a fixed field (Theorem . ).
Finally, we show in Section that well-founded coalgebras are closed under coproducts, quotients and, assuming mild assumptions, under subcoalgebras.

Preliminaries
We start by recalling some background material. Except for the definitions of algebra and coalgebra in Subsection . , the subsections below may be read as needed. We assume that readers are familiar with notions of basic category theory; see e.g. [ ] for everything which we do not detail. We indicate monomorphisms by writing and strong epimorphisms by . .

Algebras and Coalgebras.
We are concerned throughout this paper with algebras and coalgebras for an endofunctor. This means that we have an underlying category, usually written A ; frequently it is the category of sets or of vector spaces over a fixed field, and that a functor F : We usually drop the functor F . Given two algebras (A, α) and (B, β), an algebra homomorphism from the first to the second is h : We denote by Coalg F the category of all coalgebras for F .
Example . . ( ) The power set functor P : Set → Set takes a set X to the set PX of all subsets of it; for a morphism f : X → Y , Pf : PX → PY takes a subset S ⊆ X to its direct image f [S]. Coalgebras α : X → PX may be identified with directed graphs on the set X of vertices, and the coalgebra structure α describes the edges: b ∈ α(a) means that there is an edge a → b in the graph.
( ) Let Σ be a signature, i.e. a set of operation symbols, each with a finite arity. The polynomial functor H Σ associated to Σ assigns to a set X the set where Σ n is the set of operation symbols of arity n. This may be identified with the set of all terms σ(x 1 , . . . , x n ), for σ ∈ Σ n , and x 1 , . . . , x n ∈ X. Algebras for H Σ are the usual Σ-algebras. ( ) Deterministic automata over an input alphabet Σ are coalgebras for the functor F X = {0, 1} × X Σ . Indeed, given a set S of states, a next-state map S × Σ → S may be curried to δ : S → S Σ . The set of final states yields the acceptance predicate a : S → {0, 1}. So an automaton may be regarded as a coalgebra a, δ : S → {0, 1} × S Σ . ( ) Labelled transitions systems are coalgebras for F X = P(Σ × X). ( ) To describe linear weighted automata, i.e. weighted automata over the input alphabet Σ with weights in a field K, as coalgebras, one works with the category Vec K of vector spaces over K. A linear weighted automaton is then a coalgebra for F X = K × X Σ .

. Preservation Properties.
Recall that an intersection of two subobjects s i : S i A (i = 1, 2) of a given object A is given by their pullback. Analogously, (general) intersections are given by wide pullbacks. Furthermore, the inverse image of a subobject s : S B under a morphism f : A → B is the subobject t : T A obtained by a pullback of s along f . All of the 'usual' set functors preserve intersections and inverse images: Example . . ( ) Every polynomial functor preserves intersections and inverse images.
( ) The power-set functor P preserves intersections and inverse images. ( ) Intersection-preserving set functors are closed under taking coproducts, products and composition. Similarly, for inverse images. ( ) Consider next the set functor R defined by RX = {(x, y) ∈ X × X : x = y} + {d} for sets X. For a function f : , and d otherwise. R preserves intersections but not inverse images.

Proposition . [ ].
For every set functor F there exists an essentially unique set functorF which coincides with F on nonempty sets and functions and preserves finite intersections (whence monomorphisms).
Remark . . ( ) In fact, Trnková gave a construction ofF : she definedF ∅ as the set of all natural transformations C 01 → F , where C 01 is the set functor with C 01 ∅ = ∅ and C 01 X = 1 for all nonempty sets X. For the empty map e : ∅ → X with X = ∅,F e maps a natural transformation τ : C 01 → F to the element given by τ X : 1 → F X.
( ) The above functorF is called the Trnková hull of F . It allows us to achieve preservation of intersections for all finitary set functors. Intuitively, a functor on sets is finitary if its behavior is completely determined by its action on finite sets and functions. For a general functor, this intuition is captured by requiring that the functor preserves filtered colimits [ ]. For a set functor F this is equivalent to being finitely bounded, which is the following condition: for each element x ∈ F X there exists a finite subset M ⊆ X such that x ∈ . Chains. By a transfinite chain in a category A we understand a functor from the ordered class Ord of all ordinals into A . Moreover, for an ordinal λ, a λ-chain in A is a functor from λ to A . A category has colimits of chains if for every ordinal λ it has a colimit of every λ-chain. This includes the initial object 0 (the case λ = 0).

Definition . . ( ) A category
A has smooth monomorphisms if for every λ-chain C of monomorphisms a colimit exists, its colimit cocone is formed by monomorphisms, and for every cone of C formed by monomorphisms, the factorizing morphism from colim C is monic. In particuar, every morphism from 0 is monic. ( ) A has universally smooth monomorphisms if A also has pullbacks, and for every morphism f : X → colim C, the functor A / colim C → A /X forming pullbacks along f preserves the colimit of C. This implies that initial object 0 is strict, i.e. every morphism f : X → 0 is an isomorphism. Indeed, consider the empty chain (λ = 0).
( ) Vec K has smooth monomorphisms, but not universally so because the initial object is not strict. ( ) Categories in which colimits of chains and pullbacks are formed "set-like" have universally smooth monomorphisms. These include the categories of posets, graphs, topological spaces, presheaf categories, and many varieties, such as monoids, groups, and unary algebras. ( ) Every locally finitely presentable category A with a strict initial object (see Remark . ( )) has smooth monomorphisms. This follows from [ , Prop. . ]. Moreover, since pullbacks commute with colimits of chains, it is easy to prove that colimits of chains are universal using the strictness of 0.
( ) The category CPO of complete partial orders does not have smooth monomorphisms. Indeed, consider the ω-chain of linearly ordered sets A n = {0, . . . , n}+ { } ( a top element) with inclusion maps A n → A n+1 . Its colimit is the linearly ordered set N + { , } of natural numbers with two added top elements < . For the sub-cpo N + { }, the inclusions of A n are monic and form a cocone. But the unique factorizing morphism from the colimit is not monic.

Notation . . For every object
Remark . . If A is a complete and well-powered category, then Sub(A) is a complete lattice. Now suppose that A has smooth monomorphisms. Remark . . Recall [ ] that every endofunctor F yields the initial-algebra chain, viz. a transfinite chain formed by the objects F i 0 of A , as follows: F 0 0 = 0, the initial object; F i+1 0 = F (F i 0), and for a limit ordinal i we take the colimit of the chain (F j 0) j<i . The connecting morphisms w i,j : F i 0 → F j 0 are defined by a similar transfinite recursion.

Recursive Coalgebras
Assumption . . We work with a standard set theory (e.g. Zermelo-Fraenkel), assuming the Axiom of Choice. In particular, we use transfinite induction on several occasions. (We are not concerned with constructive foundations in this paper.) Throughout this paper we assume that A is a complete and well-powered category A and that F : A → A preserves monomorphisms.
For A = Set the condition that F preserves monomorphisms may be dropped. In fact, preservation of non-empty monomorphism is sufficient in general (for a suitable notion of non-empty monomorphism) [ , Lemma . ], and this holds for every set functor.
The following definition of recursive coalgebras was first given by Definition . . A coalgebra α : A → F A is called recursive if for every algebra e : F X → X there exists a unique coalgebra-to-algebra morphism e † : A → X, i.e. a unique morphism such that the square on the left below commutes: there is a unique morphism e † : A → X such that the square on the right above commutes.
Example . . ( ) A graph regarded as a coalgebra for P is recursive iff it has no infinite path. This is an immediate consequence of the General Recursion Theorem (see Corollary . and Example . ( )). ( ) Let ι : F (µF ) → µF be an initial algebra. By Lambek's Lemma, ι is an isomorphism. So we have a coalgebra ι −1 : µF → F (µF ). This algebra is (parametrically) recursive. By [ , Thm. . ], in dual form, this is precisely the same as the terminal parametrically recursive coalgebra (see also [ , Prop. ]).
( ) Colimits of recursive coalgebras in Coalg F are recursive. This is easy to prove, using that colimits of coalgebras are formed on the level of the underlying category.
( ) It follows from items ( )-( ) that in the initial-algebra chain from Remark . all coalgebras w i,i+1 : ( ) Every parametrically recursive coalgebra is recursive. (To see this, form for a given e : F X → X the morphism e = e · π, where π : F X × A → F X is the projection.) In Corollaries . and . we will see that the converse often holds.
Here is an example where the converse fails [ ]. Let R : Set → Set be the functor defined in Example . ( ). Also, let C = {0, 1}, and define γ : C → RC by γ(0) = γ(1) = (0, 1). Then (C, γ) is a recursive coalgebra. Indeed, for every algebra α : However, (C, γ) is not parametrically recursive. To see this, consider any morphism e : RX × {0, 1} → X such that RX contains more than one pair [ ] showed that recursivity semantically models divide-andconquer programs, as demonstrated by the example of Quicksort. For every linearly ordered set A (of data elements), Quicksort is usually defined as the recursive function q : A * → A * given by where A * is the set of all lists on A, ε is the empty list, is the concatenation of lists and w ≤a denotes the list of those elements of w which are less than or equal than a; analogously for w >a . Now consider the functor We shall see that this coalgebra is recursive in Example . . Thus, for the F -algebra m : there exists a unique function q on A * such that q = m · F q · s. Notice that the last equation reflects the idea that Quicksort is a divide-and-conquer algorithm. The coalgebra structure s divides a list into two parts w ≤a and w >a . Then F q sorts these two smaller lists, and finally in the combine-(or conquer-) step, the algebra structure m merges the two sorted parts to obtain the desired whole sorted list.
Jeannin et al. [ , Sec. ] provide a number of recursive functions arising in programming that are determined by recursivity of a coalgebra, e.g. the gcd of integers, the Ackermann function, and the Towers of Hanoi.

The Next Time Operator and Well-Founded Coalgebras
As we have mentioned in the Introduction, the main issue of this paper is the relationship between two concepts pertaining to coalgebras: recursiveness and well-foundedness. The concept of well-foundedness is well-known for directed graphs (G, →): it means that there are no infinite directed paths g 0 → g 1 → · · · . For a set X with a relation R, well-foundedness means that there are no backwards sequences · · · R x 2 R x 1 R x 0 , i.e. the converse of the relation is well-founded as a graph. Taylor [ , Def. . . ] gave a more general category theoretic formulation of well-foundedness. We observe here that his definition can be presented in a compact way, by using an operator that generalizes the way one thinks of the semantics of the 'next time' operator of temporal logics for non-deterministic (or even probabilistic) automata and transitions systems. It is also strongly related to the algebraic semantics of modal logic, where one passes from a graph G to a function on PG. Jacobs [ ] defined and studied the 'next time' operator on coalgebras for Kripke polynomial set functors. This can be generalized to arbitrary functors as follows.
Recall that Sub ( In more detail: we define s and α(s) by the pullback in ( . ). (Being a pullback is indicated by the "corner" symbol.) In words, assigns to each subobject s : S A the inverse image of F s under α. Since F s is a monomorphism, s is a monomorphism and α(s) is (for every representation s of that subobject of A) uniquely determined.
Example . . ( ) Let A be a graph, considered as a coalgebra for P : Set → Set. If S ⊆ A is a set of vertices, then S is the set of vertices all of whose successors belong to S. ( ) For the set functor F X = P(Σ × X) expressing labelled transition systems the operator for a coalgebra α : A → P(Σ × A) is the semantic counterpart of the next time operator of classical linear temporal logic, see e.g. Manna and Pnüeli [ ]. In fact, for a subset S → A we have that S consists of those states all of whose next states lie in S, in symbols: ( ) If µF exists, then as a coalgebra it is well-founded. Indeed, in every pullback ( . ), since ι −1 (as α) is invertible, so is β. The unique algebra homomorphism from µF to the algebra β −1 : F B → B is clearly inverse to m. ( ) If a set functor F fulfils F ∅ = ∅, then the only well-founded coalgebra is the empty one. Indeed, this follows from the fact that the empty coalgebra is a fixed point of . For example, a deterministic automaton over the input alphabet Σ, as a coalgebra for F X = {0, 1} × X Σ , is well-founded iff it is empty. ( ) A non-deterministic automaton may be considered as a coalgebra for the set functor F X = {0, 1} × (PX) Σ . It is well-founded iff the state transition graph is well-founded (i.e. has no infinite path). This follows from Corollary . below. ( ) A linear weighted automaton, i.e. a coalgebra for F X = K × X Σ on Vec K , is well-founded iff every path in its state transition graph eventually leads to 0. This means that every path starting in a given state leads to the state 0 after finitely many steps (where it stays).
Notation . . Given a set functor F , we define for every set X the map τ X : F X → PX assigning to every element x ∈ F X the intersection of all subsets m : M → X such that x lies in the image of F m: Recall that a functor preserves intersections if it preserves (wide) pullbacks of families of monomorphisms.
Gumm [ , Thm. . ] observed that for a set functor preserving intersections, the maps τ X : F X → PX in ( . ) form a "subnatural" transformation from F to the power-set functor P. Subnaturality means that (although these maps do not form a natural transformation in general) for every monomorphism i : X → Y we have a commutative square: Definition . . Let F be a set functor. For every coalgebra α : A → F A its canonical graph is the following coalgebra for P: Thanks to the subnaturality of τ one obtains the following results.
Proposition . . For every set functor F preserving intersections, the next time operator of a coalgebra (A, α) coincides with that of its canonical graph.

Corollary . [ , Rem. . . ]. A coalgebra for a set functor preserving intersections is well-founded iff its canonical graph is well-founded.
Example . . ( ) For a (deterministic or non-deterministic) automaton, the canonical graph has an edge from s to t iff there is a transition from s to t for some input letter. Thus, we obtain the characterization of well-foundedness as stated in Example . ( ) and ( ). ( ) Every polynomial functor H Σ : Set → Set preserves intersections. Thus, a coalgebra (A, α) is well-founded if there are no infinite paths in its canonical graph. The canonical graph of A has an edge from a to b if α(a) is of the form σ(c 1 , . . . , c n ) for some σ ∈ Σ n and if b is one of the c i 's. ( ) Thus, for the functor F X = 1 + A × X × X, the coalgebra (A * , s) of Example . ( ) is easily seen to be well-founded via its canonical graph. Indeed, this graph has for every list w one outgoing edge to the list w ≤a and one to w >a for every a ∈ A. Hence, this is a well-founded graph.
Lemma . . The next time operator is monotone: if m ≤ n, then m ≤ n.

Lemma . . Let α : A → F A be a coalgebra and m : B
A a subobject. ( ) There is a coalgebra structure β : B → F B for which m gives a subcoalgebra of (A, α) iff m ≤ m. ( ) There is a coalgebra structure β : B → F B for which m gives a cartesian subcoalgebra of (A, α) iff m = m.

Lemma . . For every coalgebra homomorphism
where α and β denote the next time operators of the coalgebras (A, α) and (B, β), respectively, and ≤ is the pointwise order.

provided that either
( ) f is a monomorphism in A and F preserves finite intersections, or ( ) F preserves inverse images.

Definition . [ ].
The well-founded part of a coalgebra is its largest wellfounded subcoalgebra.
The well-founded part of a coalgebra always exists and is the coreflection in the category of well-founded coalgebras [ , Prop. . ]. We provide a new, shorter proof of this fact. The well-founded part is obtained by the following:

Construction .
[ , Not. . ]. Let α : A → F A be a coalgebra. We know that Sub(A) is a complete lattice and that the next time operator is monotone (see Lemma . ). Hence, by the Knaster-Tarski fixed point theorem, has a least fixed point, which we denote by a * : A * A. By Lemma . ( ), we know that there is a coalgebra structure α * : A * → F A * so that a * : (A * , α * ) (A, α) is the smallest cartesian subcoalgebra of (A, α).

Proposition . . For every coalgebra
Proof. Let m : (B, β) (A * , α * ) be a cartesian subcoalgebra. By Lemma . , a * · m : B → A is a fixed point of . Since a * is the least fixed point, we have a * ≤ a * · m, i.e. a * = a * · m · x for some x : A * B. Since a * is monic, we thus have m · x = id A * . So m is a monomorphism and a split epimorphism, whence an isomorphism.

Proposition . . The full subcategory of Coalg F given by well-founded coalgebras is coreflective. In fact, the well-founded coreflection of a coalgebra (A, α)
is its well-founded part a * : (A * , α * ) (A, α).
For the existence of f , we first observe that ← − f (a * ) is a pre-fixed point of β : indeed, using Lemma . we have β ( It follows that f is a coalgebra homomorphism from (B, β) to (A * , α * ) since f and a * are and F preserves monomorphisms.

Construction .
[ , Not. . ]. Let (A, α) be a coalgebra. We obtain a * , the least fixed point of , as the join of the following transfinite chain of subobjects a i : A i A, i ∈ Ord. First, put a 0 = ⊥ A , the least subobject of A.
For every limit ordinal j, put a j = i<j a i . Since Sub(A) is a set, there exists an ordinal i such that a i = a * : A * A.
Remark . . Note that, whenever monomorphisms are smooth, we have A 0 = 0 and the above join a j is obtained as the colimit of the chain of the subobject a i : A i A, i < j (see Remark . ).
If F is a finitary functor on a locally finitely presentable category, then the least ordinal i with a * = a i is at most ω, but in general one needs transfinite iteration to reach a fixed point.
Example . . Let (A, α) be a graph regarded as a coalgebra for P (see Example . ). Then A 0 = ∅, A 1 is formed by all leaves; i.e. those nodes with no neighbors, A 2 by all leaves and all nodes such that every neighbor is a leaf, etc. We see that a node x lies in A i+1 iff every path starting in x has length at most i. Hence A * = A ω is the set of all nodes from which no infinite paths start.
We close with a general fact on well-founded parts of fixed points (i.e. (co)algebras whose structure is invertible). The following result generalizes [ , Cor. . ], and it also appeared before for functors preserving finite intersections [ , Theorem . and Remark . ]. Here we lift the latter assumption (see [ , Theorem . ] for the new proof):

Theorem . . Let A be a complete and well-powered category with smooth monomorphisms. For F preserving monomorphisms, the well-founded part of every fixed point is an initial algebra. In particular, the only well-founded fixed point is the initial algebra.
Example . . We illustrate that for a set functor F preserving monomorphisms, the well-founded part of the terminal coalgebra is the initial algebra. Consider F X = A × X + 1. The terminal coalgebra is the set A ∞ ∪ A * of finite and infinite sequences from the set A. The initial algebra is A * . It is easy to check that A * is the well-founded part of A ∞ ∪ A * .

The General Recursion Theorem and its Converse
The main consequence of well-foundedness is parametric recursivity. This is Taylor's General Recursion Theorem [ , Theorem . . ]. Taylor assumed that F preserves inverse images. We present a new proof for which it is sufficient that F preserves monomorphisms, assuming those are smooth.

Theorem . (General Recursion Theorem). Let A be a complete and wellpowered category with smooth monomorphisms. For F : A → A preserving monomorphisms, every well-founded coalgebra is parametrically recursive.
Proof sketch. ( ) Let (A, α) be well-founded. We first prove that it is recursive. We use the subobjects a i : A i A of Construction . , the corresponding One might object to this use of transfinite recursion, since Theorem . itself could be used as a justification for transfinite recursion. Let us emphasize that we are not presenting Theorem . as a foundational contribution. We are building on the classical theory of transfinite recursion.
morphisms α(a i ) : A i+1 = A i → F A i (cf. Definition . ), and the recursive coalgebras (F i 0, w i,i+1 ) of Example . ( ). We obtain a natural transformation h from the chain (A i ) in Construction . to the initial-algebra chain (F i 0) (see Remark . ) by transfinite recursion. Now for every algebra e : F X → X, we obtain a unique coalgebra-to-algebra morphism f i : F i 0 → X, i.e. we have that f i = e · F f i · w i,i+1 . Since (A, α) is well-founded, we know that α = α * = α(a i ) for some i. From this it is not difficult to prove that f i · h i is a coalgebra-to-algebra morphism from (A, α) to (X, e).
In order to prove uniqueness, we prove by transfinite induction that for any given coalgebra-to-algebra homomorphism e † , one has e † · a j = f j · h j · a j for every ordinal number j. Then for the above ordinal number i with a i = id A , we have e † = f i · h i , as desired. This shows that (A, α) is recursive. ( ) We prove that (A, α) is parametrically recursive. Consider the coalgebra α, id A : A → F A × A for F (−) × A. This functor preserves monomorphisms since F does and monomorphisms are closed under products. The next time operator on Sub(A) is the same for both coalgebras since the square ( . ) is a pullback if and only if the square on the right below is one. Since id A is the unique fixed point of w.r.t. F (see Definition . ), it is also the unique fixed point of w.r.t. F (−) × A. Thus, (A, α, id A ) is a well-founded coalgebra for F (−) × A. By the previous argument, this coalgebra is thus recursive for Theorem . . For every endofunctor on Set or Vec K (vector spaces and linear maps), every well-founded coalgebra is parametrically recursive.
Proof sketch. For Set, we apply Theorem . to the Trnková hullF (see Proposition . ), noting that F andF have the same (non-empty) coalgebras. Moreover, one can show that every well-founded (or recursive) F -coalgebra is a well-founded (recursive, resp.)F -coalgebra. For Vec K , observe that monomorphisms split and are therefore preserved by every endofunctor F .
Example . . We saw in Example . ( ) that for F X = 1 + A × X × X the coalgebra (A, s) from Example . ( ) is well-founded, and therefore it is (parametrically) recursive.
Example . . Well-founded coalgebras need not be recursive when F does not preserve monomorphisms. We take A to be the category of sets with a predicate, i.e. pairs (X, by 1 the terminal object (1, 1). We define an endofunctor F by F (X, ∅) = (X + 1, ∅), and for A = ∅, F (X, A) = 1. For a morphism f : (X, A) → (Y, B), The terminal coalgebra is id : 1 → 1, and it is easy to see that it is wellfounded. But it is not recursive: there are no coalgebra-to-algebra morphisms into an algebra of the form F (X, ∅) → (X, ∅).
We next prove a converse to Theorem . : "recursive =⇒ well-founded". Related results appear in Taylor Since B has only a set of subobjects, there is some λ such that for every i > λ, all of the morphisms β i represent the same subobject of B. Consequently, w λ,λ+1 of Remark . is an isomorphism, due to β λ = β λ+1 · w λ,λ+1 . Then µF = F λ 0 with the structure ι = w −1 λ,λ+1 : F (µF ) → µF is an initial algebra. ( ) Now suppose that (A, α) is a recursive coalgebra. Then there exists a unique coalgebra homomorphism h : (A, α) → (µF, ι −1 ). Let us abbreviate w iλ by c i : F i 0 µF , and recall the subobjects a i : A i A from Construction . . We will prove by transfinite induction that a i is the inverse image of c i under h; in symbols: a i = ← − h (c i ) for all ordinals i. Then it follows that a λ is an isomorphism, since so is c λ , whence (A, α) is well-founded.
In the base case i = 0 this is clear since A 0 = W 0 = 0 is a strict initial object.
For the isolated step we compute the pullback of c i+1 : W i+1 → µF along h using the following diagram: By the induction hypothesis and since F preserves inverse images, the middle square above is a pullback. Since the structure map ι of the initial algebra is an isomorphism, it follows that the middle square pasted with the right-hand triangle is also a pullback. Finally, the left-hand square is a pullback by the definition of a i+1 . Thus, the outside of the above diagram is a pullback, as required. For a limit ordinal j, we know that a j = i<j a i and similarly, c j = i<j c i since W j = colim i<j W j and monomorphisms are smooth (see Remark . ( )). Using Remark . ( ) and the induction hypothesis we thus obtain Corollary . . Let A and F satisfy the assumptions of Theorem . . Then the following properties of a coalgebra are equivalent: ( ) well-foundedness, ( ) parametric recursiveness, ( ) recursiveness, ( ) existence of a homomorphism into (µF, ι −1 ), ( ) existence of a homomorphism into a well-founded coalgebra.
In Theorem . we also proved ( ) ⇒ ( In fact, for the identity functor on Vec K we have µId = (0, id). Hence, every coalgebra has a homomorphism into µId. However, not every coalgebra is recursive, e.g. the coalgebra (K, id) admits many coalgebra-to-algebra morphisms to the algebra (K, id). Similarly, the implication ( ) ⇒ ( ) does not hold.
We also wish to mention a result due to Taylor [ , Rem. . ]. It uses the concept of a subobject classifier originating in [ ] and prominent in topos theory. This is an object Ω with a subobject t : 1 Ω such that for every subobject b : B A there is a uniqueb : A → Ω such that b is the inverse image of t underb. By definition, every elementary topos has a subobject classifier, in particular every category Set C with C small.
Our standing assumption that A is a complete and well-powered category is not needed for the next result: finite limits are sufficient.

Theorem . (Taylor [ ]). Let F be an endofunctor preserving inverse images on a finitely complete category with a subobject classifier. Then every recursive coalgebra is well-founded.
Corollary . . For every set functor preserving inverse images, the following properties of a coalgebra are equivalent: well-foundedness ⇐⇒ parametric recursiveness ⇐⇒ recursiveness.
Example . . The hypothesis in Theorems . and . that the functor preserves inverse images cannot be lifted. In order to see this, we consider the functor R : Set → Set of Example . ( ). It preserves monomorphisms but not inverse images. The coalgebra A = {0, 1} with the structure α constant to (0, 1) is recursive: given an algebra β : RB → B, the unique coalgebra-to-algebra homomorphism h : {0, 1} → B is given by h(0) = h(1) = β(d). But A is not well-founded: ∅ is a cartesian subcoalgebra.
Recall that an initial algebra (µF, ι) is also considered as a coalgebra (µF, ι −1 ). Taylor [ , Cor. . ] showed that, for functors preserving inverse images, the terminal well-founded coalgebra is the initial algebra. Surprisingly, this result is true for all set functors.
Theorem . [ , Thm. . ]. For every set functor, a terminal well-founded coalgebra is precisely an initial algebra.
Theorem . . For every functor on Vec K preserving inverse images, the following properties of a coalgebra are equivalent: well-foundedness ⇐⇒ parametric recursiveness ⇐⇒ recursiveness.

Closure Properties of Well-founded Coalgebras
In this section we will see that strong quotients and subcoalgebras (see Remark . ) of well-founded coalgebras are well-founded again. We mention the following corollary to Proposition . . For endofunctors on sets preserving inverse images this was stated by Taylor [ , Exercise VI. ]: Proposition . . The subcategory of Coalg F formed by all well-founded coalgebras is closed under strong quotients and coproducts in Coalg F . This follows from a general result on coreflective subcategories [ , Thm. . ]: the category Coalg F has the factorization system of Proposition . , and its full subcategory of well-founded coalgebras is coreflective with monomorphic coreflections (see Proposition . ). Consequently, it is closed under strong quotients and colimits.
We prove next that, for an endofunctor preserving finite intersections, wellfounded coalgebras are closed under subcoalgebras provided that the complete lattice Sub(A) is a frame. This means that for every subobject m : B A and every family m i (i ∈ I) of subobjects of A we have m ∧ i∈I m i = i∈I (m ∧ m i ). This property holds for Set as well as for the categories of posets, graphs, topological spaces, and presheaf categories Set C , C small. Moreover, it holds for every Grothendieck topos. The categories of complete partial orders and Vec K do not satisfy this requirement.
Proposition . . Suppose that F preserves finite intersections, and let (A, α) be a well-founded coalgebra such that Sub(A) a frame. Then every subcoalgebra of (A, α) is well-founded.

Corollary . . If a set functor preserves finite intersections, then subcoalgebras of well-founded coalgebras are well-founded.
Trnková [ ] proved that every set functor preserves all nonempty finite intersections. However, this does not suffice for Corollary . : Example . . A well-founded coalgebra for a set functor can have non-wellfounded subcoalgebras. Let F ∅ = 1 and F X = 1 + 1 for all nonempty sets X, and let F f = inl : 1 → 1 + 1 be the left-hand injection for all maps f : ∅ → X with X nonempty. The coalgebra inr : 1 → F 1 is not well-founded because its empty subcoalgebra is cartesian. However, this is a subcoalgebra of id : 1 + 1 → 1 + 1 (via the embedding inr), and the latter is well-founded.
The fact that subcoalgebras of a well-founded coalgebra are well-founded does not necessarily need the assumption that Sub(A) is a frame. Instead, one may assume that the class of morphisms is universally smooth: Theorem . . If A has universally smooth monomorphisms and F preserves finite intersections, every subcoalgebra of a well-founded coalgebra is well-founded.

Conclusions
Well-founded coalgebras introduced by Taylor [ ] have a compact definition based on an extension of Jacobs' 'next time' operator. Our main contribution is a new proof of Taylor's General Recursion Theorem that every well-founded coalgebra is recursive, generalizing this result to all endofunctors preserving monomorphisms on a complete and well-powered category with smooth monomorphisms. For functors preserving inverse images, we also have seen two variants of the converse implication "recursive ⇒ well-founded", under additional hypothesis: one due to Taylor for categories with a subobject classifier, and the second one provided that the category has universally smooth monomorphisms and the functor has a pre-fixed point. Various counterexamples demonstrate that all our hypotheses are necessary. Open Access This chapter is licensed under the terms of the Creative Commons Attribution . International License (http://creativecommons.org/licenses/by/ . /), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.