Relating coalgebraic notions of bisimulation

The theory of coalgebras, for an endofunctor on a category, has been proposed as a general theory of transition systems. We investigate and relate four generalizations of bisimulation to this setting, providing conditions under which the four different generalizations coincide. We study transfinite sequences whose limits are the greatest bisimulations.


Introduction
Notions of bisimulation play a central role in the theory of transition systems. The theory of coalgebras provides a setting in which different notions of system can be understood at a general level. In this article I investigate notions of bisimulation at this general level.
To explain the generalization from transition systems to coalgebras, we begin with the traditional presentation of a labelled transition system, (X, (→ X ) ⊆ X × L × X) (for some set L of labels). A labelled transition system can be considered 'coalgebraically' as a set X of states equipped with a function X → P(L × X), into the powerset of (L × X), assigning to each state x ∈ X the set {(l, x ′ ) | x l − → X x ′ }. Generalizing, we are led to consider an arbitrary category C and an endofunctor B on it; then a coalgebra is an object X ∈ C of 'states', and a 'next-state' morphism X → B(X).
Coalgebras in different categories. Coalgebras appear as generalized transition systems in various settings. For instance: transition systems for name and value passing process calculi have been studied in terms of coalgebras in categories of presheaves (e.g. [19,22,48]); probabilistic transition systems have been modelled by coalgebras for a probabilitydistribution monad (e.g. [6,53]); descriptive frames and concepts from modal logic have been studied in terms of coalgebras over Stone spaces (e.g. [1,9,32]); basic process calculi with recursion have been modelled using coalgebras over categories of domains [31,44]; and stochastic transition systems have been studied in terms of coalgebras over metric and measurable spaces (see e.g. [12,16,52,53]). Finally, there are questions about the conventional theory of labelled transition systems in a more constructive universe of sets (e.g. [7]).
Notions of bisimulation. Once coalgebras are understood as generalized transition systems, we can consider bisimulation relations for these systems. Recall that, for labelled transition systems (X, → X ) and (Y, → Y ), a relation R ⊆ X × Y is a bisimulation if, whenever x R y, then for all l ∈ L: How should the notion of bisimulation be generalized to the case of coalgebras for endofunctors on arbitrary categories? In this article, we identify four notions of bisimulation that have been proposed in the coalgebraic context.
(1) A relation over which a suitably compatible transition structure can be defined, as proposed by Aczel and Mendler [2]; (2) A relation that is compatible for a suitable 'relation-lifting' of the endofunctor, as proposed by Hermida and Jacobs [25]; (3) A relation satisfying a 'congruence' condition, proposed by Aczel and Mendler [2] and used to obtain their general final coalgebra theorem; (4) A relation which is the kernel of a common compatible refinement of the two systems. The four notions coincide for the particular case of labelled transition systems. Under certain conditions, the notions are related in the more general setting of coalgebras.
Relationship with the terminal sequence. Various authors have constructed terminal coalgebras as a limit of a transfinite sequence; the initial part of the sequence is: 1 ! Acknowledgements. It has been helpful to discuss the material in this article with numerous people over the last eight years, particularly Marcelo Fiore. Benno van den Berg gave some advice on algebraic set theory. Many of the results in this article are well-known in the case where C = Set. In other cases, some results may be folklore; I have tried to ascribe credit where it is due.

Coalgebras: Definitions and examples
Recall the definition of a coalgebra for an endofunctor: Definition 1.1. Consider an endofunctor B on a category C. A B-coalgebra is given by an object X of C together with morphism X → B(X) in C.
A homomorphism of B-coalgebras, from (X, h) to (Y, k), is a morphism f : X → Y that respects the coalgebra structures, i.e. such that Bf • h = k • f .
Coinductive datatypes. Coinductive datatypes can be understood in terms of coalgebras for polynomial endofunctors. A polynomial endofunctor on a category with sums and products is a functor of the following form. (See e.g. Rutten [46,Sec. 10].) X → i∈I A i × X n i (Here, each A i is an object of the category, and each n i is a natural number.) Transition systems. In the introduction we discussed the correspondence between labelled transition systems and coalgebras for the endofunctor P(L × (−)). Here, P is the powerset functor, that acts by direct image. For finite non-determinism, and image-finite transition systems, one can instead consider the endofunctor where P f is the finite powerset functor, the free semilattice.
Transition systems in toposes and name-passing calculi. Recall that a topos is a category with finite limits and a powerobject construction. By definition, the powerobjects classify relations, and so the coalgebraic characterization of labelled transition systems is relevant in any topos.
In process calculi such as the π-calculus [41], transitions occur between terms with free variables, and those free variables play an important role. Conventional labelled transition systems in the category of sets are inadequate for such calculi. Instead, one can work in a category of covariant presheaves (C → Set). Various categories have been proposed for C. We will focus on two examples: the category I of finite sets and injections between them, and the category F + of non-empty finite sets and all functions between them. More sophisticated models of process calculi are found by taking presheaves over more elaborate categories (see e.g. [10,11,22,49]).
In this setting, the object X of states is no longer a set, but a presheaf. For instance, if X : I → Set, we think of X(C) as the set of states involving the free variables in the set C, and the functorial action of X describes injective renaming of states.
The appropriate endofunctor on these presheaf categories typically has the following form (1.1) with the summands of B ′ representing input, output, and silent actions respectively. The presheaf N is a special object of names. The interesting question is: what is P? • A natural choice is to let P be the powerobject functor in the presheaf topos [C, Set].
For any presheaf Y ∈ [C, Set], and any object C ∈ C, (P(Y ))(C) is the set of subpresheaves of (C(C, −) × Y ). A coalgebra X → P(B ′ (X)) is a natural transformation between presheaves, that assigns a behaviour to each state x ∈ X(C), for C ∈ C. This behaviour is not only a set of future states for x, but also the future states of Xf (x) for any morphism f : C → C ′ in C.
To understand this more formally, recall that the powerobject P classifies relations in the following sense. For presheaves X and Y in [C, Set], there is a bijective correspondence between natural transformations r : X → P(Y ) and subpresheaves R ⊆ X × Y . A subpresheaf R ⊆ X × Y determines a natural transformation r : X → P(Y ); for C ∈ C and x ∈ X(C), we have r C (x) ∈ (P(Y ))(C): • The powerobject P(X) accommodates infinite branching transition systems. To focus on finite branching, we can find a 'finite' subfunctor of P. The approach taken by Fiore and Turi [19] is to let P be the free semi-lattice (henceforth P f ). This is sometimes called 'Kuratowski finiteness'. For any presheaf Y ∈ [C, Set], and any object C ∈ C, (P f (Y ))(C) is the set of finite subsets of Y (C). We have an natural monomorphism i Y : P f (Y ) P(Y ) into the full powerobject: for C ∈ C, S ⊆ (Y (C)), we define a subpresheaf i Y,C (S) ∈ (P(Y ))(C): • The free semilattice is too naive on the presheaf category [F + , Set]. For example, the π-calculus process (ā | b) cannot perform a τ -step, but it can perform a τ -step after the substitution {a → b, b → b}. The construction P f (X) is too small to allow this information to be recorded. Indeed, the π-calculus can be described as a coalgebra for the functor P(B ′ (−)) on [F + , Set], but this coalgebra does not factor through the free semilattice, P f (B ′ (−)). In this situation, a more appropriate finite powerset is the sub-join-semilattice of the powerobject P(Y ) that is generated by the partial map classifier. We will write P pf (Y ) for this -Freyd [20] writesK. I gave an algebraic description of this construction in [50], and it has been used by Miculan in his model of the fusion calculus [37].
Frames in modal logic. Let the base category C be the category of Stone spaces and continuous maps. (A Stone space is a compact Hausdorff space in which the clopen sets form a basis.) Let K(X) be the space of compact subsets of X, with the finite (aka Vietoris) topology. The construction K is made into a functor, acting by direct image. Coalgebras for K can be understood as descriptive general frames. Just as the category of Stone spaces is dual to the category of Boolean algebras, the category of K-coalgebras is dual to the category of modal algebras (i.e., Boolean algebras equipped with a meet-preserving operation) -see e.g. [1,32].
Powersets in algebraic set theory. A general treatment of powersets is suggested by the algebraic set theory of Joyal and Moerdijk [29]. A model of algebraic set theory is a category C together with a class of 'small' maps S in C, all subject to certain conditions. An intuition is that a map f : X → Y is small if its fibres f −1 (y) are all small.
In such a situation, an S-relation is a relation R ⊆ X × Y for which the projection R → X is in S. An endofunctor P S on C is said to be the S-powerset if there is an S-relation (∋ Y ) ⊆ P S (Y ) × Y inducing a bijective correspondence between S-relations (R ⊆ X × Y ) and morphisms X → P S (Y ). Further details are given in the appendix.
These ideas cater for the notions of power set discussed so far. For instance: • Let C be the category of sets, in the classical sense, and say that a function f : X → Y is small if for every y ∈ Y the set f −1 (y), i.e. {x ∈ X | f (x) = y}, is finite. This class S of maps satisfies all the axioms for small maps given in the appendix. An S-relation is precisely an image-finite one, and the S-powerset is the finite powerset. • For a presheaf category [C, Set], the free semilattice construction P f is an S-powerset where S is the class of natural transformations between presheaves, φ : This class S of morphisms always satisfies Axioms A1-A6 and A9 for small maps, but not (M): monos are not small unless C is a groupoid.
In the presheaf category [F + , Set], the free semilattice generated by the partial map classifier, P pf , is more liberal: it classifies the natural transformations between presheaves that satisfy the finiteness condition (i) but the requirement on naturality (ii) is weakened to the situation when f is an injection. This class of morphisms satisfies all the axioms for small maps, including (M). (This argument is quite specific to F + .) (Although the Vietoris construction can be considered over more general spaces, the characterization of K as an S-powerset is specific to Stone spaces. This raises the question of how best to treat more expressive positive/topological set theories [36] in an algebraic setting.) Probabilistic transition systems. For any set X, let D f (X) be the set of sub-probability distribution functions on X, viz., functions from d : X → [0, 1] into the unit interval for which {x ∈ X | d(x) = 0} is finite and x∈X d(x) ≤ 1. This construction extends to an endofunctor on Set, with the covariant action given by summation. Coalgebras for D f are discrete probabilistic transition systems [6,53].
Systems where the state space has more structure. For continuous stochastic systems, researchers have investigated coalgebras for probability distribution functors on categories of metric or measurable spaces (see e.g. [12,16,52,53]).
For recursively defined systems, it is reasonable to investigate coalgebras for powerdomain constructions on a category of domains (in the bialgebraic context, see [31,44]).
Levy and Worrell [35,54] have considered endofunctors on categories of preorders, posets, and categories enriched in quantales, in their investigations of similarity.

Bisimulation: four definitions
We now recall four notions of bisimulation on the state spaces of coalgebras. The four notions generalize the standard notion of bisimulation for labelled transition systems (i.e. coalgebras for P(L × (−)) on Set), due to Milner [40] and Park [43]. For all four notions, the maximal bisimulation is the usual notion of strong bisimilarity for labelled transition systems.
To some extent, the different notions of bisimulation have arisen from the examples in Section 1.1, as authors sought coalgebraic notions of bisimulation that were appropriate to the base category and endofunctor under consideration, as well as to the intended applications. For name-passing calculi, coalgebraic bisimulation can be used to capture the bisimulations of Milner, Parrow and Walker [41] and also the open bisimulation of Sangiorgi [47] (see e.g. [19,50,Sec. 6]); for discrete-space probabilistic systems, coalgebraic bisimulation can describe the probabilistic bisimulation of Larsen and Skou [34] (see e.g. [53]).

Relations in categories.
For objects X, Y of a category C, we let Rel C (X, Y ) be the preorder of relations, viz. jointly-monic spans When C has products, the preorder Rel C (X, Y ) coincides with the preorder of monos into (X × Y ). Relations are most well-behaved in regular categories (see Appendix).
Context. In this section we fix a category C and consider an endofunctor B on C. We fix two B-coalgebras, h : X → BX and k : Y → BY .
RELATING COALGEBRAIC NOTIONS OF BISIMULATION 7 2.1. The lifting-span bisimulation of Aczel and Mendler [2]. This notion is categorically the simplest. It directly dualizes the concept of congruence from universal algebra.
is an AM-bisimulation between (X, h) and (Y, k) if there exists a B-coalgebra structure on R that lifts it to a span of coalgebra homomorphisms, as in the following diagram.
The relation-lifting bisimulation of Hermida and Jacobs [25]. The following is a simplification of the bisimulation of Hermida and Jacobs, who work in a more general fibrational setting.
When C has pullbacks, let Φ HJ (R) be the following pullback: For illustration, we briefly return to the situation of transition systems, where C = Set Thus the operator Φ HJ is the construction F considered by Milner [39,Sec. 4].  [2].

The congruences of Aczel and Mendler
The definition might appear clumsy and unmotivated, but AM-precongruences are of primary interest because of their connection with terminal coalgebras in a general setting, as will become clear in Theorem 4.1.
If C has pushouts, then it is sufficient to check the case where Z is the pushout of R. If C also has pullbacks, let Φ AM (R) be the pullback of the cospan By definition, a relation R is an AM-precongruence if and only if R ≤ Φ AM (R).
(Note that Φ AM is different from Φ HJ , even when B is the identity functor on Set.) Our definition differs from that of [2] in that we consider relations between different coalgebras. The connection is as follows: if (X, h) = (Y, k), then an equivalence relation is an AM-precongruence exactly when it is a congruence in the sense of Aczel and Mendler [2] (see Section 4.2).

2.4.
Terminal coalgebras and kernel-bisimulations. Many authors have argued that when the category of coalgebras has a terminal object, equality in the terminal coalgebra is the right notion of bisimilarity. Suppose that the category C has pullbacks, and suppose for a moment that there is a terminal B-coalgebra, (Z, z). This induces a relation in Rel C (X, Y ) as the pullback of the unique terminal morphisms X → Z ← Y . The relation is sometimes called 'behavioural equivalence'.
We can formulate a related notion of bisimulation without assuming that there is a terminal coalgebra. Definition 2.6. Let C have pullbacks. A relation R is a kernel-bisimulation if there is a B-coalgebra (Z, z) and a cospan of homomorphisms, (X, h) → (Z, z) ← (Y, k), and R is the pullback of (X → Z ← Y ).
Aside from the great many works involving terminal coalgebras, various authors (e.g. [24,31,33]) have used kernel-bisimulations (though not by this name). (The term 'cocongruence' is sometimes used to refer directly to the cospan involved.)

Properties of endofunctors
All the definitions of the previous section are relevant when C is a regular category. From a categorical perspective, one might restrict attention to endofunctors that are regular, i.e. that preserve limits and covers. But none of the endofunctors in Section 1.1 are regular. We now recall five weaker conditions that might be assumed of our endofunctor B.
(1) The image of a relation under a functor need not again be a relation, and one can restrict attention to endofunctors that preserve relations, i.e., for which a jointly-monic span is mapped to a jointly-monic span. The remaining restrictions have to do with weak forms of pullback-preservation. To introduce them, we consider a cospan (A 1 → Z ← A 2 ) in C, and in particular the mediating morphism m : B(A 1 × Z A 2 ) → (BA 1 ) × BZ (BA 2 ) from the image of the pullback to the pullback of the image: Here are some conditions on B, listed in order of decreasing strength.
(2) B preserves pullbacks if m is always an isomorphism.
(3) B preserves weak pullbacks, if m is always split epi. Gumm [23] describes several equivalent definitions of this term. (4) B covers pullbacks if m is always a cover. The terminology is due to [51]. Note that a split epi is always a cover. also covers pullbacks.

Proposition 3.2. Every polynomial endofunctor on an extensive category preserves pullbacks and covers.
Regarding powerset functors from algebraic set theory (see Appendix), we have the following general results. (1) The functor P S preserves pullbacks along monomorphisms.
(3) Let C also be extensive, and let S satisfy the axioms for extensive categories (axioms (A1-6)) and also collection (A9), but not necessarily (M). The functor P S covers pullbacks.
Proof of Propn. 3.3: I will sketch proofs in the set-theoretic notation. Translation into categorical language is straightforward. For item (1), consider a pullback square, and its image under P S .
To see that the right-hand square is a pullback, consider S in P S (A), for which the direct image P S f (S) is in P S (Z ′ ); then, by definition, S is in P S (f −1 (Z ′ )). For items (2) and (3), consider a cospan (A 1 → Z ← A 2 ). For item (2), we must exhibit a section s : Here we have used the separation axiom, which is valid when all monomorphisms are small.
For item (3), we show that the canonical morphism is a cover. Consider (S 1 , S 2 ) in P S (A 1 ) × P S (Z) P S (A 2 ); we must show that there is S in P S (A 1 × Z A 2 ) whose direct image is (S 1 , S 2 ). By the (strong) collection axiom, we have T 1 in P S (A 1 × Z A 2 ) such that π 1 (T 1 ) = S 1 and π 2 (T 1 ) ⊆ S 2 . Similarly, we also have T 2 in P S (A 1 × Z A 2 ) such that π 2 (T 2 ) = S 2 and π 1 (T 2 ) ⊆ S 1 . Thus (T 1 ∪ T 2 ) is in P S (A 1 × Z A 2 ), and its direct image is (S 1 , S 2 ), as required. Thus Propn. 3.3 is proved.
The free semi-lattice functor P f on any topos covers pullbacks, but will only preserve weak pullbacks if the topos is Boolean. In general, the corresponding class of small maps does not contain all monomorphisms, as Johnstone et al. [27,Ex. 1.4] have observed. The counterexample of [27] is easily adapted to the settings of the presheaf categories for name passing, correcting oversights in [19,22,48].
On the other hand, the endofunctor P pf on the presheaf category [F + , Set] does preserve weak pullbacks.
The compact-subspace endofunctor K on Stone spaces covers pullbacks (a consequence of Propn. 3.3) although it does not preserve weak pullbacks, as observed by Bezhanishvili et al [9].
The probability distribution functor D f on Set preserves weak pullbacks [42,53]. More sophisticated continuous settings are problematic. Counterexamples to the weakpullback-preservation of probability distributions on measurable spaces are discussed in [52]. Plotkin [44] discusses problems with coalgebraic bisimulation in categories of domains: the convex powerdomain does not even preserve monomorphisms. The endofunctors on posets that Levy considers [35] typically do not preserve monomorphisms either.

Relating the notions of bisimulation
The purpose of this section is to relate the four notions of bisimulation introduced in Section 2. (1) Every AM-bisimulation is an HJ-bisimulation.
(3) Every AM-precongruence is contained in a kernel bisimulation that is an AM-precongruence, provided C has pushouts. (4) Every kernel bisimulation is an AM-bisimulation, provided B preserves weak pullbacks. In summary: AM-bisim. (1) / / HJ-bisim. For item (2), let R be an HJ-bisimulation. Let (X → Z ← Y ) be a cone over the span (X ← R → Y ). Consider the following commuting diagrams. The left-hand squares of diagram (a) say that R is an HJ-bisimulation. The right-hand square of diagram (a) commutes since diagram (b) commutes, and BR B R is epi. Thus the whole of (a) commutes, and R is an AM-precongruence.
For item (3), let R be an AM-precongruence. Let (X → Z ← Y ) be the pushout of the span (X ← R → Y ). The following diagram commutes; the dotted morphism follows from universality of the pushout.
For items (4), (5) and (6), let R be a kernel bisimulation, the pullback of a cospan (X → Z ← Y ), for some coalgebra (Z, z). Note that the following diagram commutes. For item (4), we must show that R is an AM-bisimulation. We construct a coalgebra structure on R by considering the morphism R → BR induced since BR is a weak pullback, as in the following diagram. For item (5), we must show that R is an HJ-bisimulation. This follows from the following fact, which is immediate from the definition ofBR, and which is worth recording: Z is a pullback, so isB R  For item (6), we must show that R is an AM-precongruence. This is more involved. We make use of the following lemma. Lemma 4.3. Consider objects S, S ′ , V , V ′ , and morphisms p, q, p ′ , q ′ , f, g, f ′ , g ′ , making the following three diagrams commute. If the left-hand diagram is a pullback, and B preserves pullbacks along monos, and C is regular, then the following diagram also commutes.

Proof of Lemma 4.3:
Write im(f ) for the image of f : X → V , etc.. Subdivide the pullback (a) as follows.
Since C is regular, the composite S → W is regular epi. In this situation, the leftmost pullback is also a pushout, as indicated (see e.g. [13,Thm 5.2].) Since diagram (b) commutes, and since B preserves pullbacks along monos, we have unique morphisms S ′ → B(im(p)) and S ′ → B(im(q)) making the following diagram commute.
Now consider diagram (c). Since W is a pushout, we have a unique morphism W → V ′ making the following diagram commute.

Thus Lemma 4.3 is proved.
Returning to the proof of Theorem 4.1, notice that item (6)   Case (7i), where all epis split, is thus trivial, and in case (7ii), where B preserves relations, the cover is an isomorphism. In cases (7iii) and (7iv), we define a section BR → BR = P S (B ′ R) by defining the following composite relation (BR ← • → B ′ R): v v l l l l l l l l l l In case (7iii), we must check that the composite relation is an S-relation. By the axioms of open maps, it is sufficient to check that all the leftwards morphisms in the composite are in S. The right-most leftwards morphism is in S because B ′ preserves relations, hence it is monic. This concludes our proof of Theorem 4.1.

4.2.
A note about equivalence relations. When (X, h) = (Y, k) then the maximal bisimulation, when it exists, is often an equivalence relation. Some authors focus attention on those bisimulation relations that are equivalence relations. In this setting, it is reasonable to adjust the definition of kernel bisimulation, so that the two coalgebra homomorphisms X → Z are required to be equal. An appropriate adjustment of Theorem 4.1 still holds even when the pullback-preservation requirements are weakened as follows: • In item (3), it is not necessary for C to have pushouts, it is sufficient for C to have effective equivalence relations (i.e., that every equivalence relation arises as a kernel pair); • In item (4), it is not necessary for B to preserve weak pullbacks, it is sufficient for B to weakly preserve kernel pairs; • In item (5), it is not necessary for B to cover pullbacks, it is sufficient that B covers kernel pairs; • In item (6), it is not necessary for B to preserve pullbacks along monos, it is sufficient for B to preserve monos. (In diagram (4.2), if f = g then W = im(f ) = im(g).) The conditions are connected: in an extensive regular category, a functor covers pullbacks if and only if it covers kernel pairs and preserves pullbacks along monos. (Gumm and Schröder [24] showed this for Set, and their argument is readily adapted to this more general setting.).
In summary, we have the following situation, when focusing on equivalence relations:

Constructing bisimilarity
In this section we consider a procedure for constructing the maximal bisimulation. We relate it with the terminal sequence, which is used for finding final coalgebras.
Context. In this section we assume that the ambient category C is complete and regular.
We fix an endofunctor B on C, and fix two B-coalgebras, h : X → BX and k : Y → BY .

5.1.
The relation refinement sequence. The greatest HJ-bisimulations can be understood as greatest fixed points of the operator Φ HJ on Rel C (X, Y ). We define an ordinal indexed cochain (r β,α : R HJ β R HJ α ) α≤β in Rel C (X, Y ), in the usual way: • Limiting case: If λ is limiting, then let R HJ λ be α<λ R HJ α , i.e. the limit of the cochain (r β,α : R HJ β R HJ α ) α≤β<λ . In particular, let R HJ 0 = X × Y . • Inductive case: let R HJ α+1 = Φ HJ (R HJ α ). We call this cochain the relation refinement sequence. If this sequence is eventually stationary then it achieves the maximal post-fixed point of Φ HJ , the greatest HJ-bisimulation. (NB. The sequence always converges when Rel C (X, Y ) is small, e.g. when C is well-powered.) For the case of the endofunctor P(L × (−)) on Set, the relation refinement sequence is a transfinite extension of Milner's sequence ∼ 0 ≥ ∼ 1 ≥ . . . ≥ ∼ (e.g. [38,Sec. 5.7]), studied from an algorithmic perspective by Kanellakis and Smolka [30].
If C has pushouts, we can also consider a cochain (R AM β R AM α ) α≤β corresponding to the operator Φ AM . As will be seen, the two sequences of relations often coincide.
(The other notions of bisimulation, AM-bisimulation and kernel bisimulation, cannot be characterized as post-fixed points, in general.) 5.2. The terminal sequence. There is another sequence that is often studied in the coalgebraic setting. The terminal sequence is an ordinal-indexed cochain (z β,α : Z β → Z α ) α≤β that can be used to construct a final coalgebra for an endofunctor (e.g. Worrell [55]). The idea is to begin with the terminal object, and then successively find B-algebra structures, so that if the sequence converges, i.e. the B-algebra structure is an isomorphism, then we have a B-coalgebra structure, and indeed the final such. The cochain commutes and satisfies the following conditions: • Limiting case: Z λ = lim {z β,α : Z β → Z α | α ≤ β < λ }, if λ is limiting, in which case the cone {z λ,α : Z λ → Z α | α < λ } is the limiting one; • Inductive case: Z α+1 = B(Z α ); and z β+1,α+1 = B(z β,α ) : Z β+1 → Z α+1 .
These cones determine another ordinal indexed cochain (R β R α ) α≤β in Rel C (X, Y ). For every ordinal α, let R α be the following pullback. (1) R α contains all the kernel bisimulations and all the AM-precongruences.
(2) If B covers pullbacks, then R α = R HJ α . (3) If B preserves pullbacks along monos, and C has pushouts, then R α = R AM α . Proof. To see that R α contains all the kernel bisimulations, notice that for a cospan of coalgebras, (X → Z ← Y ), the coalgebra Z determines a cone over the terminal sequence, just as X and Y do. The relation R α contains all the AM-precongruences by transfinite induction on α: the limit step is vacuous and the inductive step uses the definition of AMprecongruence. Statements (2) and (3) are also proved by transfinite induction on α: the limit steps use the fact that limits commute with limits; the inductive steps follow from Fact 4.2, and from Lemma 4.3, respectively.
Note. A consequence of Item (3) of Theorem 5.1 is that, if the sequence (R β R α ) α≤β converges, then the result is the greatest AM-precongruence, provided the endofunctor preserves pullbacks along monos and C has pushouts. In fact, this corollary still holds even if C does not have all pushouts. This can be proved directly by transfinite induction; the inductive step uses Lemma 4.3.

5.4.
Convergence for the relation refinement sequences. By Theorem 5.1, if the terminal sequence converges, then the relation refinement sequence does too. Of course, this is not a sufficient condition. Indeed, even when there is a final coalgebra, the relation refinement sequence may converge before the terminal sequence: Proposition 5.2. If, for some ordinal α, the morphism z α+1,α : Z α+1 → Z α is monic, then R α = R α+1 .
Proof. We have the following situation.
The left-hand square is a pullback -this is a rearrangement of the definition of R α+1 . The right-hand square is a pullback if and only if z α+1,α is monic. Thus the outer square is a pullback. Now x α = (x α+1 · z α+1,α ) and y α = (y α+1 · z α+1,α ), which means that R α+1 = R α .
If C is Set and B preserves filtered colimits, then the terminal sequence does not converge until (ω + ω), but it becomes monic at ω [55]. As is well-known, the relation refinement sequence for image-finite transition systems converges at ω.
For the case when C = Set and sets X and Y are both finite, the relation refinement sequence will converge before ω because the skeleton of Rel C (X, Y ) is finite. This is relevant in a slightly more general setting: In a Boolean Grothendieck topos, every descending ω-cochain of subobjects from a finitely presentable object is eventually constant.
The presheaf topos [I, Set], used to model name-passing, is not Boolean. It does, however, have Sh ¬¬ (I) as a Boolean subcategory, and this is perhaps a more appropriate universe for name-passing calculi (see e.g. [18]). There, the finitely presentable objects are exactly those objects that can be described by finite 'named-sets with symmetry' [18,21]. Thus the techniques of this article provide a general foundation for the coalgebra-inspired verification procedures of Ferrari, Montanari and Pistore [17].
Powersets. Given a regular category C and a class of open maps S, an S-relation is a jointly monic span (I ← R → A) whose left projection is in S.
An S-powerset for an object A of C is an object P S (A) together with an S-relation (P S (A) ← ∋ A → A) such that for every S-relation (I ← R → A) there is a unique morphism I → P S (A) making (R I × A) a pullback of (∋ A P S (A) × A): It follows from axiom (A2) that morphisms I → P S (A) are in bijective correspondence with S-relations (I ← R → A).
If every object of C has an S-powerset, then the construction P S extends straightforwardly to a covariant endofunctor on C, as follows. For any morphism f : A → B, the action P S (f ) : P S (A) → P S (B) corresponds to the S-relation in the image of the span (P S (A) ← ∋ A → A → B), using axiom (A6).

Separation and collection.
There are various axioms and axiom schema that can be assumed as principles for defining sets in a constructive setting. The notes by Aczel and Rathjen [3] provide an overview.
The separation axiom (also known as bounded comprehension) amounts to the following axiom on S. (M) All monomorphisms in C are in S. The (strong) collection axiom has the following categorical counterpart, when there is an S-powerset: (A9) The endofunctor P S preserves covers.
Further axioms. The axioms above are all that we need in this article. As a foundation of mathematics, these axioms are too weak: one would typically also require that there is a small natural numbers object; that each powerset P S (X) is small; and that there is a class of all sets, a universal small map.