Path category for free - Open morphisms from coalgebras with non-deterministic branching

There are different categorical approaches to variations of transition systems and their bisimulations. One is coalgebra for an functor G, where a bisimulation is defined as a span of G-coalgebra homomorphism. Another one is in terms of path categories and open morphisms, where a bisimulation is defined as a span of open morphisms. This similarity is no coincidence: given a functor G, fulfilling certain conditions, we derive a path-category for pointed G-coalgebras and lax homomorphisms, such that the open morphisms turn out to be precisely the G-coalgebra homomorphisms. The above construction provides path-categories and trace semantics for free for different flavours of transition systems: (1) non-deterministic tree automata (2) regular nondeterministic nominal automata (RNNA), an expressive automata notion living in nominal sets (3) multisorted transition systems. This last instance relates to Lasota's construction, which is in the converse direction.


Introduction
Coalgebras [24] and open maps [14] are two main categorical approaches to transition systems and bisimulations. The former describes the branching type of systems as an endofunctor, a system becoming a coalgebra and bisimulations being spans of coalgebra homomorphisms. Coalgebra theory makes it easy to consider state space types in different settings, e.g. nominal sets [15,16] or algebraic categories [4,10,19]. The latter, open maps, describes systems as objects  of a category and the execution types as particular objects called paths. In this case, bisimulations are spans of open morphisms, namely morphisms reflecting extensions of executions. Open maps are particularly adapted to extend bisimilarity to history dependent behaviors, e.g., true concurrency [7,6], timed systems [21] and weak (bi)similarity [8].
Coalgebra homomorphisms and open maps are then key concepts to describe bisimilarity categorically. They intuitively correspond to functional bisimulations, that is, those maps between states whose graph is a bisimulation.
We are naturally interested in the relationship between those two categorical approaches to transition systems and bisimulations. A reduction of open maps situations to coalgebra was given by Lasota using multi-sorted transition systems [18]. In this paper, we give the reduction in the other direction: from the category Coalg l (T F ) of pointed T F -coalgebras and lax homomorphisms, we construct the path-category Path and a functor J : Path −→ Coalg l (T F ) such that Path-open morphisms coincide with strict homomorphisms, hence functional bisimulations. This development is carried out with the case where T is a powerset-like functor, and covers transition systems allowing non-deterministic branching.
The key concept in the construction of Path is precise maps. Roughly speaking, an object P ∈ Path represents a generic shape of finite-depth deterministic transitions. The property required for such a shape is the unique path property: every non-initial state has exactly one previous state, and precise maps are designed to capture this property. Then we represent a finite-depth deterministic transition system by a finite sequence f i : X i −→ F X i+1 + 1 of precise maps (we add 1 for termination). J converts such a data into a pointed T F -coalgebra.
Once we set up the situation J : Path−→Coalg l (T F ), we are on the framework of open map bisimulations. Our construction of Path using precise maps is justified by the characterisation theorem: Path-open morphisms and strict coalgebra homomorphisms coincide (Theorem 3.20 and Theorem 3.24). This coincidence relies on the concept of path-reachable coalgebras, namely, coalgebras such that every state can be reached by a path. Under mild conditions, path-reachability is equivalent to an existing notion in coalgebra, defined as the non-existence of a proper sub-coalgebra (Section 3.5). Additionally, this characterization produces a canonical trace semantics for free, given in terms of paths (Section 3.6).
We illustrate our reduction with several concrete situations: different classes of non-deterministic top-down tree automata using analytic functors (Section 4.1), Regular Nondeterministic Nominal Automata (RNNA), an expressive automata notion living in nominal sets (Section 4.2), multisorted transition sys-tems, used in Lasota's work to construct a coalgebra situation from an open map situation (Section 4.3).
Notation We assume basic categorical knowledge and stick to standard notation (see e.g. [1,3]). In particular, the cotupling of morphisms f : A −→ C, g : B −→ C is denoted by [f, g] : A + B −→ C, and the unique morphsim to the terminal object is ! : X −→ 1 for every X.

Two categorical approaches for bisimulations
We introduce the two formalisms involved in the present paper: the open maps (Section 2.1) and the coalgebras (Section 2.2). Those formalisms will be illustrated on the classic example of Labelled Transition Systems (LTSs).
Definition 2.1. Fix a set A called the alphabet. A Labelled Transition System is a triple (S, i, ∆) with S a set of states, i ∈ S the initial state, and ∆ ⊆ S ×A×S the transition relation. When ∆ is obvious from the context, we write s a − → s ′ to mean (s, a, s ′ ) ∈ ∆.
To formalise LTSs as a category, one has to describe morphisms, that is, those functions that preserves the structure of LTSs: is a function f : S −→ S ′ such that for every (s, a, s ′ ) ∈ ∆, (f (s), a, f (s ′ )) ∈ ∆ ′ . LTSs and morphisms of LTSs form a category, which we denote by LTS A .
Some authors choose other notions of morphisms (e.g., [14]), allowing them to operate between LTSs with different alphabets for example. The usual way of comparing LTSs is by using simulations and bisimulations [22]. The former describes what it means for a system to have at least the behaviours of another, the latter describes that two systems have exactly the same behaviours. Concretely: Definition 2.3. A simulation from T = (S, i, ∆) to T ′ = (S ′ , i ′ , ∆ ′ ) is a relation R ⊆ S × S ′ such that (1) initial states are related, that is, (i, i ′ ) ∈ R, and (2) transitions of T can be simulated by transitions of T ′ , that is, for every s a − → t and (s, Morphisms of LTSs are functional simulations, i.e., functions between states whose graph is a simulation. So how to model (1) systems, (2) functional simulations and (3) functional bisimulations categorically? In the next two sections, we will describe known answers to this question, with open maps and coalgebra. In any cases, it is possible to capture similarity and bisimilarity of two LTSs T and T ′ . A simulation is a (jointly monic) span of a functional bisimulation and a functional simulation, and a bisimulation is a simulation whose converse is also a simulation, as depicted in Table 1. Consequently, to understand similarity and bisimilarity, it is enough to understand functional simulations and bisimulations.

Open maps
The categorical framework of open maps [14] assumes functional simulations to be already modeled as a category. For example, for LTS A , objects are systems, and morphisms are functional simulations. The main part is to describe functional bisimulations in form of functional simulations. The main idea is to describe the notion of "run" in a system using morphisms. A run labelled by a = a 1 · · · a n ∈ A ⋆ in a LTS T = (S, i, ∆) consists of states s 1 , · · · , s n ∈ S satisfying: i To represent such a run by a LTS-morphism, we use a particular class of LTSs called linear systems. A linear sytem P a over a represents the "common shape" of runs along the string a, and is defined to be ({0, · · · , n}, 0, We then easily see that runs labelled by a in the LTS T bijectively correspond to LTS-morphisms of type P a −→ T . Moreover, we can represent transfers and extensions of runs: -The transfer of a run r : P a −→ T by a LTS-morphism f : T −→ T ′ is the composite f · r : P a −→ T ′ , which is a run in T ′ .
-Consider the full subcategory Lin A of LTS A consisting of all linear systems. A morphism in Lin A is naturally seen as an embedding of a run shape to a longer one. Thus by composing a run r ′ : P b −→ T and a Lin A -morphism Φ : P a −→ P b , we obtain the truncation r ′ ·Φ : P a −→T of r ′ by Φ. We say that a run r ′ : P b −→T is an extension of a run r : These concepts are readily expressible in a more general situation of a category M together with a specified category P mapped into M [14]. Evidently, Lin A ֒→ LTS A is an open map situation. We note that Lin A is isomorphic to the prefix order over A ⋆ .
We have already seen that morphisms between LTSs are functional simulations. Which ones are functional bisimulations? A sufficient condition is for the graph of a morphism f : T −→ T ′ to satisfy the properties of a bisimulation: It is not a necessary condition because bimisimilarity is a property only on the reachable part of the system, so it suffices to quantify over the s that are reachable by a run. Equivalently, we can quantify over runs: In other words, for any run p and extension q of f · p along Φ : P a −→ P ab , there exists an extension d of p along Φ such that f · d = q.

Coalgebras
The theory of G-coalgebras is another categorical framework to study bimulations. The type of systems is modelled using an endofunctor G : C −→ C and a system is then a coalgebra for this functor, that is, a pair of an object S of C (modeling the state space), and of a morphism of type S −→ GS (modeling the transitions). Let us describe this on LTSs. The main part of a LTS is the transition relation ∆ ⊆ S × A × S. Equivalently, this can be defined as a function ∆ : S −→ P(A × S), where P is powerset. In other words, the transition relation is a coalgebra for the Set-functor P(A × ). Intuitively, this coalgebra gives the one-step behaviour of a LTS: S describes the state space of the system, P describes the "branching type" as being non-deterministic, A × S describe the "computation type" as being linear, and the function itself lists all possible futures after one-step of computation of the system. Now, changing the underlying category or the endofunctor, allows to model different types of systems. This is the usual framework of coalgebra, as described for example in [24]. In this paper, we will treat branching types and initial states as follows. As in [11], we decompose the functor G, as in the case of LTSs, as T F where T is the branching type (non-deterministic, probabilistic, weighted), and F is the computation type (linear, tree-like, ...). Secondly, initial states are modelled as in [2, Sec. 3B], where we describe initial states as a pointing of the system. For example, an initial state of a LTS is the same as a function from the singleton set { * } to the state space S. More generally, a system will be modelled as a coalgebra α : S −→ GS as previously, together with a morphism i : I −→ S from a fixed object I of C describing the "type of initial states". This object I will often be the final object of C, but we will see other examples later. We call them I-pointed G-coalgebras.
In this framework, it is actually simpler to model functional bisimulations, or more precisely, to describe functions that satisfy the equation (2), of the previous section. The intuition is that those functions are those that preserve the initial state, and preserve and reflect the one-step relation.
Definition 2.6. An I-pointed G-coalgebra homomorphism For instance, when G = P(A × −), one can easily see that a function f being a G-coalgebra morphism is stronger than being a functional simulation; the preservation of transitions actually corresponds to the pointwise inequality Gf (α(s)) ⊆ α ′ (f (s)). To express this condition for general G-coalgebras, we introduce a partial order ⊑ X,Y on each homset C(X, GY ) in a functorial manner. Definition 2.7. A partial order on G-homsets is a functor ⊑ : C op × C −→ Pos such that U · ⊑ = C(−, G−); here U : Pos −→ Set is the forgetful functor from the category Pos of posets and monotone functions.
Definition 2.8. A coalgebra situation is given by a category C (of state spaces), an object I of C (type of initial states), an endofunctor G : C −→ C (transition type) and a partial order ⊑: C op × C −→ Pos on G-homsets. Example 2.9. Set, with { * }, P(A × ) and the order f ⊑ g in Set(X, P(A × Y )) iff for every x ∈ X, f (x) ⊆ g(x) is a coalgebra situation, modelling LTS.
is a morphism f : S −→ S ′ making the right-hand diagram commute. In a coalgebra situation, the I-pointed G-coalgebras and lax homomorphisms form a category, denoted by Coalg l (I, G).
The open map situation for categories of coalgebras Fix a category C with pullbacks, functors T, F : C −→ C, an object I ∈ C and a partial order ⊑ T on T -homsets. They determine a coalgebra situation (C, I, T F, ⊑) where ⊑ is the partial order on T F -homsets defined by ⊑ X,Y = ⊑ T X,F Y . Here, F handles the linear behaviour of coalgebras, e.g. F X = A×X, and T models the branching behaviour, e.g. powerset construction P : Set −→ Set.
Under some conditions on T and F , we construct a path-category Path(I, F + 1) and an open map situation Path(I, F +1) ֒→ Coalg l (I, T F ) where (strict) T Fcoalgebra homomorphisms and Path(I, F + 1)-open morphisms coincide.

Precise morphisms
While the path category is intuitively clear for F X = A × X, it is not for inner functors F that model tree languages. For example for F X = A + X × X, a PF -coalgebra accepts binary trees with leaves labelled in A as the input.
If F preserves weak pullbacks, then we can assume g = id wlog.
Example 3.3. Intuitively speaking, for a polynomial Set-functor F , a map s : S−→ F R is F -precise iff every element of R is mentioned precisely once in the definition of the map f . For example, for F X = A × X + {⊥}, the case needed later for LTSs, a map f : Figure 1 is not F -precise, because one element of Y is used three times, and two elements do not occur in f at all. However, f ′ : Also note that f ′ defines a forest where X is the set of roots, which is closely connected to the intuition that, in the F -precise map f ′ , from every element of Y ′ , there is precisely one edge up to a root in X.
So when transforming a non-precise map into a precise map, one duplicates elements that are used multiple times and drops elements that are not used. We will cover functors F for which this pattern provides F -precise maps: Definition 3.4. Fix a class of objects S ⊆ obj C closed under isomorphism. We say that F admits precise factorizations w.r.t. S if for every morphism f : Intuitively S contains objects that have a certain choice principle (see (Ax5) later). Hence for C = Set, S contains all sets. However for the category of nominal sets, S will only contain the strong nominal sets (see details in subsection 4.2).
Remark 3.5. Precise morphisms are essentially unique. If f 1 : X −→ F Y 1 and is F -precise and F -admits precise factorizations, then Y ∈ S.
Functors admitting precise factorizations are closed under basic constructions: Proposition 3.6. The following functors admit precise factorizations w.r.t. S: 1. Constant functors, if C has an initial object 0 and 0 ∈ S.

3.
i∈I F i , if all (F i ) i∈I do so and S is closed under I-coproducts.

4.
i∈I F i , if all (F i ) i∈I do so, C is I-extensive and S is closed under I-coproducts.

Right-adjoint functors, if and only if its left-adjoint preserves S-objects.
Example 3.7. Every polynomial Set-functor admits precise factorizations. The bag functor B, where B(X) is the set of finite multisets, also admits precise factorizations (as we will see later). In contrast, F = P does not admit precise factorizations, and if f :

Path categories in pointed coalgebras
We define a path for I-pointed T F -coalgebras as a tree according to F . Following the observation in Example 3.3, one layer of the tree is modelled by a F -precise morphism and hence a path in a T F -coalgebra is defined to be a finite sequence of (F + 1)-precise maps, where the + 1 comes from the dead states w.r.t. T ; the argument is given later in Remark 3.23 when reachability is discussed. Since the + 1 is not relevant yet, we define Path(I, F ) in the following and will use Path(I, F + 1) later. We simplify the notation when working with families: Notation. We write X n for finite families (X k ) 0≤k<n Definition 3.8. The category Path(I, F ) consists of the following. An object is (P n+1 , p n ) for an n ∈ N with P 0 = I and p n a family of F -precise maps Example 3.9. Paths for F X = A × X + 1 and I singleton are as follows. First, a map f : I −→ F X is precise iff (up-to isomorphism) either X = I and f ( * ) = (a, * ) for some a ∈ A; or X = ∅ and f ( * ) = ⊥. Then a path is isomorphic to an object of the form: and p k ( * ) = ⊥. A path is the same as a word, plus some "junk", concretely, a word in A ⋆ .⊥ ⋆ . For LTSs, an object in Path(I, F ) with F X = A × X is simply a word in A ⋆ . For a more complicated functor, Figure 2 depicts a path of length 4, which is a tree for the signature with one unary, one binary symbol, and a constant. The layers of the tree are the sets P 4 . Also note that since every p i is F -precise, there is precisely one route to go from every element of a P k to the root * .
Remark 3.10. The inductive continuation of Remark 3.5 is as follows. Given a morphism φ n+1 in Path(I, F ), since φ 0 is an isomorphism, then φ k is an isomorphism for all 0 ≤ k ≤ n. If F admits precise factorizations and if I ∈ S, then for every path (P n+1 , p n ), all P k , 0 ≤ k ≤ n, are in S.
Remark 3.11. If in Definition 3.4, the connecting morphism h : Y ′ −→Y uniquely exists, then it follows by induction that the hom-sets of Path(I, F ) are at most singleton. This is the case for all polynomial functors, but not the case for the bag functor on sets (discussed in subsection 4.1).
Definition 3.12. The path poset PathOrd(I, F ) is the set 0≤n C(I, F n 1) equipped with the order: for u : This matches the morphisms in Path(I, F ) that witnesses that one path is prefix of another: is full, and reflects isomorphisms.
2. If F admits precise factorizations w.r.t. S and I ∈ S, then Comp is surjective on objects.

If additionally the connecting morphism h in Definition 3.4 is unique, then
Comp has a right-inverse.
In particular, PathOrd(I, F ) is Path(I, F ) up to isomorphism. In the instances, it is often easier to characterize PathOrd(I, F ). This also shows that Path(I, F ) contains the elements -understood as morphisms from I -of the finite start of the final chain of F : 1 Example 3.14. When F X = A × X + 1, F n 1 is isomorphic to the set of words in A ⋆ .⊥ ⋆ of length n. Consequently, PathOrd(I, F ) is the set of words in A ⋆ .⊥ ⋆ , equipped with the prefix order. In this case, Comp is an equivalence of categories.

Embedding paths into pointed coalgebras
The paths (P n+1 , p n ) embed into Coalg l (I, T F ) as one expects it for examples like Figure 2: one takes the disjoint union of the P k , one has the pointing I = P 0 and the linear structure of F is embedded into the branching type T .
During the presentation of the results, we require T , F , and I to have certain properties, which will be introduced one after the other. The full list of assumptions is summarized in Table 2: (Ax1) -The main theorem will show that coalgebra homomorphisms in Coalg l (I, T F ) are the open maps for the path category Path(I, F + 1). So from now on, we assume that C has finite coproducts and to use the results from the previous sections, we fix a class S ⊆ obj C such that F + 1 admits precise factorizations w.r.t. S and that I ∈ S.
(Ax2) -Recall, that a family of morphisms (e i : X i −→ Y ) i∈I with common codomain is called jointly epic if for f, g : Y −→Z we have that f ·e i = g ·e i ∀i ∈ I implies f = g. For Set, this means, that every element y ∈ Y is in the image of some e i . Since we work with partial order on T -homsets, we also need the generalization of this property if f ⊑ g are of the form Y −→ T Z ′ .
(Ax3) -In this section, we encode paths as a pointed coalgebra by constructing a functor J : Path(I, F + 1) ֒→ Coalg l (I, T F ). For that we need to embed the linear behaviour F X + 1 into T F X. This is done by a natural transformation [η, ⊥] : Id +1 −→ T , and we require that ⊥ : 1 −→ T is a bottom element for ⊑.
Example 3.15. For the case where T is the powerset functor P, η is given by the unit η X (x) = {x}, and ⊥ is given by empty sets ⊥ X ( * ) = ∅.
Definition 3.16. We have an inclusion functor J : Path(I, F +1) ֒→ Coalg l (I, T F ) that maps a path (P n+1 , p n ) to an I-pointed T F -coalgebra on P n+1 := 0≤k≤n P k . The pointing is given by in 0 : I = P 0 −→ P n+1 and the structure by: Example 3.17. In the case of LTSs, a path, or equivalently a word a 1 ...a k .⊥...⊥ ∈ A ⋆ .⊥ ⋆ , is mapped to the finite linear system corresponding to a 1 ...a k (see Section 2.1), seen as a coalgebra (see Section 2.2).
Proposition 3.18. Given a morphism [x k ] k≤n : P n+1 −→X for some system (X, ξ, x 0 ) and a path (P n+1 , p n ), we have Also note that the pointing x 0 of the coalgebra is necessarily the first component of any run in it. In a run [x k ] k≤n , p k corresponds to an edge from x k to x k+1 .
Example 3.19. For LTSs, since the P k are singletons, x k just picks the kth state of the run. The right-hand side of this lemma describes that, this is a run iff there is a transition from the kth state and the k + 1th state.

Open morphisms are exactly coalgebra homomorphisms
In this section, we prove our main contribution, namely that Path(I, F + 1)open maps in Coalg l (I, T F ) are exactly coalgebra homomorphisms. For the first direction of the main theorem, that is, that coalgebra homomorphisms are open, we need two extra axioms: (Ax4) -describing that the order on C(X, T Y ) is essentially point-wise. This holds for the powerset because every set is the union of its singleton subsets.
(Ax5) -describing that C(X, T Y ) admits a choice-principle. This holds for the powerset because whenever y ∈ h[x] for a map h : X −→ Y and x ⊆ X, then there is some The converse is not true in general, because intuitively, open maps reflect runs, and thus only reflect edges of reachable states, as we have seen in Section 2.1. The notion of a state being reached by a path is the following: Example 3.22. For LTSs, this means that every state in X is reached by a run, that is, there is a path from the initial state to every state of X.
Remark 3.23. In Definition 3.21, it is crucial that we consider Path(I, F +1) and not Path(I, F ) for functors incorporating 'arities ≥ 2'. This does not affect the example of LTSs, but for I = 1, F X = X × X and T = P in Set, the coalgebra (X, ξ, x 0 ) on X = {x 0 , y 1 , y 2 , z 1 , z 2 } given by is path-reachable for Path(I, F + 1). There is no run of a length 2 path from Path(I, F ), because y 2 has no successors, and so there is no path to z 1 or to z 2 .  Table 2

Connection to other notions of reachability
There is another concise notion for reachability in the coalgebraic literature [2].
Under further assumptions, this notion coincides with the path-based definition of reachability (Definition 3.21). We assume for the present subsection 3.5.
The first direction follows directly from Theorem 3.20: For the other direction it is needed that T F preserves arbitrary intersections, that is, wide pullbacks of monomorphisms. In Set, this means that for a family (X i ⊆ Y ) i∈I of subsets we have that i∈I T F X i and T F i∈I X i are isomorphic subsets of T F Y . All those technical assumptions are satisfies in the case of LTSs, and will also be satisfied in all our instances in the section 4.

Trace semantics for pointed coalgebras
The characterization from Theorem 3.20 and Theorem 3.24 points out a natural way of defining a trace semantics for pointed coalgebras. Indeed, the paths category Path(I, F + 1) provides a natural way of defining the runs of a system. A possible way to go from runs to trace semantics is to describe accepting runs as the subcategory J ′ : Path(I, F ) ֒→ Path(I, F + 1). We can define the trace semantics of a system (X, ξ, x o ) as the set: Let us look at two LTS-related examples (we will describe some others in the next section). First, for F X = A × X. The usual trace semantics is given by all the words in A ⋆ that are labelled of a run of a system. This trace semantics is obtained because PathOrd(I, F ) = n≥0 A n and because Comp maps every path to its underlying word. Another example is given for F X = A × X + { }, where stands for a final state. In this case, a path in Path(I, F ) of length n is either a path that can still be extended or encodes less than n steps to an accepting state . This obtains the trace semantics containing the set of accepted words, as in automata theory, plus the set of possibly infinite runs.

Analytic functors and variety of tree automata
In Section 3.1, Proposition 3.6, we have seen that when the category C has enough well-behaved coproducts, then (Ax1) is automatically satisfied for all polynomial functors. That is the case when C = S = Set, which already allows us to consider many models like LTSs, or top-down non-deterministic tree automata (as we will see soon). However, in Set, we can do much better by proving that (Ax1) is satisfied for a larger class of functors, namely the analytic functors: Definition 4.1 [12,13]. An analytic Set-functor is a functor of the form F X = σ/n∈Σ X n /G σ where Σ is a signature, and for every σ/n ∈ Σ we have a subgroup G σ of the permutation group S n on ar(σ) = n elements.
Example 4.2. Every polynomial functor is analytic. The bag-functor is analytic, where Σ = N has one operation symbol per arity and G σ = S ar(σ) is the full permutation group on ar(σ) elements. It is the archetype of a analytic functor, in the sense that for every analytic functor F : Set −→ Set, there is a natural transformation into the bag functor α : F −→ B. If F is given by Σ and G σ as above, then α X is given by  Much as the case of LTS (with possibly final states) from Section 3.6, we can model tree automata with pointed coalgebras. More precisely, we can model topdown non-deterministic bounded tree automata as in [5]. Given a signature Σ, such a tree automata is the same as a { * }-pointed P( σ/n∈Σ X n )-coalgebra. Since F X = σ/n∈Σ X n is a polynomial functor, we can use our machinery to study tree automata. In particular, we can recover the language of this automata as a trace semantics: elements of Path({ * }, F +1) are essentially the same as multicontexts, that is trees with possibly several holes, plus some junk. Considering Path(I, F ) its full subcategory of such paths with holes only on the last level, and PathOrd(I, F ) the category of multicontexts ordered by prefix, we obtain that the trace semantics of a system is exactly finite truncations of its tree language. We can also model many kinds of symbols: (1) bounded symbols: the ones we have seen so far, given with an arity n, and their component in the functor F is given by X n , (2) unbounded symbols: those can have any number of children, so their component is given by n∈N X n , (3) commutative bounded symbols: they have a fixed number of children, but those are unordered. Their component is given by X n /S n , (4) commutative unbounded symbols: combination of the last two. Their component is given by the bag functor.
We can imagine many other types (cyclic, alternate, ...), using the proper subgroup of the the permutation group.

Nominal Sets: Regular Nondeterministic Nominal Automata
We derive an open map situation from the coalgebraic situation for regular nondeterministic nominal automata (RNNAs) [25]. They are an extension of automata to accept words with binders, consisting of literals a ∈ A and binders | a for a ∈ A; the latter is counted as length 1. An example of such a word of length 4 is a| c bc, where the last c is bound by | c . The order of binders makes difference: | a | b ab = | a | b ba. RNNAs are coalgebraically represented in the category of nominal sets [9], a formalism about atoms (e.g. variables) that sit in more complex structures (e.g. lambda terms), and gives a notion of binding. Because the choice principle (Ax4) and (Ax5) are not satisfied by every nominal sets, we instead use the class of strong nominal sets for the precise factorization (Definition 3.4).
Definition 4.4 [9,23]. Fix a countably infinite set A, called the set of atoms.
For the group S f (A) of finite permutations on the set A, a group action (X, ·) is a set X together with a group homomorphism · : S f (A) −→ S f (X), written in infix notation. An element x ∈ X is supported by S ⊆ A, if for all π ∈ S f (A) with π(a) = a ∀a ∈ S we have π · x = x. A nominal sets is a group action for S f (A) such that every x ∈ X is finitely supported, i.e. supported by a finite The category of nominal sets and equivariant maps is denoted by Nom. A nominal set (X, ·) is called strong if for all x ∈ X and π ∈ S f (A) with π · x = x we have π(a) = a for all a ∈ supp(x).
Intuitively, the support of an element is the set of free literals. An equivariant map can forget some of the support of an element, but can never introduce new atoms, i.e. supp(f (x)) ⊆ supp(x). The intuition behind strong nominal sets is that all atoms appear in a fixed order, that is, A n is strong, but P f (A) (the finite powerset) is not. We set S to be the class of strong nominal sets. In the application, we fix the set I = A #n of distinct n-tuples of atoms (n ≥ 0) as the pointing. The hom-sets Nom(X, P f Y ) are ordered point-wise. As for F , we study a LTS-like functor, extended with the binding functor [9]:  An element in PathOrd(A #n , F ) may be regarded as a word with binders under a context a ⊢ w, where a ∈ A #n , all literals in w are bound or in a, and w may end with . Moreover, two word-in-contexts a ⊢ w and a ′ ⊢ w ′ are identified if their closures are α-equivalent, that is, | a1 · · · | an w = | a ′ 1 · · · | a ′ n w ′ . The trace semantics of a RNNA T contains all the word-in-contexts corresponding to runs in T . This trace semantics distinguishes whether words are concluded by .

Subsuming arbitrary open morphism situations
Lasota [18] provides a translation of a small path-category P ֒→ M into a functor F : Set obj P −→ Set obj P defined by F X P P = ( Q∈P P(X Q )) P(P,Q) P ∈P . So the hom-sets Set obj P (X, FY ) have a canonical order, namely the point-wise inclusion. This admits a functor Beh from M to F-coalgebras and lax coalgebra homomorphisms, and Lasota shows that f ∈ M(X, Y ) is P-open iff Beh(f ) is a coalgebra homomorphism. In the following, we show that we can apply our framework to F by a suitable decomposition F = T F and a suitable object I for the initial state pointing. As usual in open map papers, we require that P and M have a common initial object 0 P . Observe that we have F = T · F where T (X P ) P ∈P = P(X P ) P ∈P and F (X P ) P ∈P = Q∈P P(P, Q) × X Q P ∈P .
Lasota considers coalgebras without pointing, but one indeed has a canonical pointing as follows. For P ∈ P, define the characteristic family χ P ∈ Set obj P by χ P Q = 1 if P = Q and χ P Q = ∅ if P = Q. With this, we fix the pointing I = χ 0P . Proposition 4.8. T , F and I satisfy the axioms from Table 2, with S = Set obj P .
The path category in Coalg l (I, T F ) from our theory can be described as follows.

Further work
We proved that coalgebra homomorphisms for systems with non-deterministic branching can be seen as open maps for a canonical path-category, constructed from the computation type F . This limitation to non-deterministic systems is unsurprising: as we have proved in Section 4.3 on Lasota's work [18], every open map situation can been encoded as a coalgebra situation with a powerset-like functor, so with non-deterministic branching. As a future work, we would like to extend this theory of path-category for categories of coalgebras for further kinds of branchings, especially probabilistic and weighted. This will require (1) to adapt open maps to allow those kinds of branchings (2) adapt the axioms from Table 2, by replacing the "+1" part of (Ax1), which is the linear subfunctor of a powerset-like functor, to something depending on the branching type.
Proof. Sufficiency is clear, because one can fix D = R and g = id R in the definition of F -precise. For necessity, consider s, f, g, h as in the definition of F -precise. The pullback of g along h is weakly preserved by F , and so we have the following commuting diagram: Hence, s induces some d : R −→ P with F d · s = f ′ and π 2 · d = id R . The witness that s is F -precise is π 1 ·d : R−→C, because F (π 1 ·d)·s = f and h·π 1 ·d = g. ⊓ ⊔ Proof of Remark 3.5 Apply that f 2 is F -precise and one obtains a lifting d : Since d has both a left and right inverse, it is an isomorphism, and so is its left-inverse h.
For the second part, we a have a precise factorisation of f , that is, a F -precise morphism f ′ : S −→ Y ′ with Y ′ ∈ S, and a morphism h : By the previous point, h is an isomorphism, and since S is closed under isomorphisms, Y ∈ S.
Proof of Proposition 3.6 1. Fix Z an object of C, and assume given a morphism f : S −→ Z = C Z Y , with S ∈ S. Since a morphism of the form f ′ : S −→ F 0 is always F -precise, then Taking h as the unique morphism from 0 to Y , F h · f ′ = id Z · f = f . 2. Let us start with the following lemma: Proof. Assume given the following situation: Let us prove this point now. We start with a morphism f : Proof. Consider a square

⊓ ⊔
Let us prove this point now. Consider f i i∈I : X −→ i∈I F i Y and consider the F i -precise factorizations: and it is the i∈I F i -precise factorization of f i i∈I : 4. By I-extensive we mean 'extensive' if I is finite and 'infinitary extensive' if I is infinite. In any case, C is I-extensive if for for all families ( the following equivalence holds: (x i : X i −→ P ) i∈I is a coproduct ⇐⇒ for all i ∈ I the above square is a pullback Let us start with the following lemma: Lemma A.4. Given functors F i : C−→C, i ∈ I, on an I-extensive category, and morphisms f i : Proof. Consider a commutative square Note that by the extensivity of C we can assume that the top morphism Let us prove this point now. Given a morphism h : X −→ ( i∈I F i )(Y ), construct the pullbacks in the I-extensive category C: So we have X = i∈I X i with the coproduct injections as in the top row of the above pullback diagram, and in particular h = [in i · f i ] i∈I = i∈I f i . Let f ′ i be the F i -precise morphisms through which f i factors for every i ∈ I: 5. We first show that η X : X −→ R(LX) is R-precise for every X ∈ C. Since right adjoints preserve limits, R preserves (weak) pullbacks and it suffices by the universal mapping property of L ⊣ R and by the naturality of the isomorphism C(LX, Z) ∼ = C(X, RZ) respectively. Now, we can prove this point: -For necessity and X ∈ S, we have that η X : X −→ RLX is R-precise by the previous argument, and hence LX ∈ S by Remark 3.5. -For sufficiency, consider a morphism f : is R-precise as shown above. Since L preserves objects in S, we have that LX ∈ S.

Proof of Example 3.7
Given an P-precise f : The first equation implies that for every y ∈ Y , d(y) is inl(y) or inr(y). So for x ∈ X and for every y ∈ Y , either inl(y) or inr(y) is in d[f (x)] = (Pd · f )(x) = f ′ (x), which is a contradiction unless X or Y is empty. Hence, f (x) = ∅ for all x ∈ X.

Proof of Remark 3.10
The proof is by induction on the length. Since P 0 = I = Q 0 , φ 0 : P 0 −→ Q 0 is an isomorphism. Assume that φ k : P k −→ Q k is isomorphic for k < n. Since q k · φ k and p k are F + 1-precise, we have that φ k+1 is isomorphic by Remark 3.5. The second part of the remark is a consequence of the second part of Remark 3.5.
There is nothing to do in the base case φ 0 := id I . For the step, assume some φ k : P k −→ Q k making the following diagram commute: So, φ k+1 is indeed a morphism in Path(I, F ). 2. For every u : I −→ F n 1 we can define a path (P n+1 , p n ) inductively, starting with P 0 := I ∈ S. Given some C-morphism u ′ : P k −→ F n−k 1, k ≤ n, P k ∈ S, we are done with r = id 1 if k = n. If k < n consider the F -precise factorization P k F P k+1 providing P k+1 ∈ S and p k : P k −→ F P k+1 and some u ′′ : P k+1 −→ F n−k−1 1. This defines the families P n+1 and p n with Comp(P n+1 , p n ) = u. 3. By Remark 3.11.

Proof of Proposition 3.18
For all k < n, the following diagram commutes: For sufficiency, assume that [x k ] k≤n is a morphism in Coalg l (I, T F ). Hence for all k < n: For necessity, we have for all k < n: ⊑ and P n j≤n P j X F j≤n P j + 1 1 Since the family of coproduct injections in k is jointly epic we have by (Ax2) that [x k ] k≤n is indeed a weak homomorphism.

Proof of Theorem 3.20
As usual in open map proofs, this direction of the characterization theorem is shown by considering only Path-morphisms of length difference 1. Indeed every Path-morphism φ from a path of length n to a path of length m ≥ n + 2 can be expressed as a composition of Path-morphisms of smaller length difference φ = ψ · θ. When applying the openness of a morphism h to θ, we obtain a diagonal d : dom(ψ) −→ dom(h) which allows applying the openness of h to ψ, yielding the desired lifting for φ.  Consider a Path(I, F + 1)-morphism φ n+1 : (P n+1 , p n ) −→ (Q n+2 , q n+1 ) and a commuting square in Coalg l (I, T F ) where h is a homomorphism: By precomposition with in n we obtain the following commuting diagram: Applying (Ax5) to the outer part of the diagram yields some f : P n −→ F X + 1 and the commuting diagram: Since q n is F + 1-precise, we obtain the following: is indeed a morphism in Coalg l (I, T F ) and that it makes the desired diagram in Coalg l (I, T F ) commute.
1. That [d k ] k≤n+1 is a weak homomorphism follows from Proposition 3.18 because we have: and for all k < n: is proven because the coproduct injections are jointly epic. For k ≤ n we clearly have

Proof of Theorem 3.24
Having ζ · h ⊒ T F h · ξ already, we only need to show: By path-reachability, it suffices to show that for all runs [x k ] k≤n : J(P n+1 , p n )−→ (X, ξ, x 0 ) we have: By (Ax2), coproduct injections are jointly epic in an ordered-enriched sense, so by induction, it suffices to prove that: By (Ax4), we can prove this by showing that for all p ′ n : P n −→ T F Y we have. We will prove the implication In the following, this implication is proved for all p ′ n . For such a p ′ n : P n −→F Y +1, since P n ∈ S, (Ax1) yields an object P n+1 ∈ S and morphisms p n : P n −→ F P n+1 + 1 and y n+1 : P n+1 −→ Y with (F y n+1 + 1) · p n = p ′ n . So we have a new path (P n+2 , p n+1 ) and a morphism defined by with y k = x k for k ≤ n, and y n+1 as above.
This is indeed a lax homomorphism J(P n+2 , p n+1 ) −→ (Y, ζ, y 0 ) because of Proposition 3.18 and because for all k < n: Nat. and P n X Y Since y k = x k , for k ≤ n, the constructed lax homomorphism makes the following square commute in Coalg l (I, T F ).
Since h is open, we obtain a diagonal lifting, that is, a lax homomorphism [d k ] k≤n+1 : JP n+1 −→ (X, ξ, x 0 ) with h · d k = y k for k ≤ n + 1 and d k = x k for k ≤ n. This finally proves the desired implication: Nat.

Remarks on Assumption 3.26
The category of sets and the category of nominal sets have (epi,mono)-factorizations. However, the assumption is rather unusual, because in most categories, a morphism that is both epi and mono is not necessarily an isomorphism. By the assumption of (epi,mono)-factorizations together with the cocompleteness of C, the following are equivalent for any family (e i : and since h is open by Theorem 3.20, we have a run [d k ] k≤n of (P n+1 , p n ) in (Y, ζ, y 0 ). Since [x k ] k≤n is epic and h · d k = x k for all 0 ≤ k ≤ n, we have that the mono h is an epimorphism and hence an isomorphism.

Proof of Proposition 3.28
For functors H : C −→ C preserving intersections, we have breadth-first-search: 1. m 0 is the mono-part of the (E,mono)-factorization of i. 2. m k+1 is the least subobject of X such that ξ · m k factors through Hm k+1 . 3. The union of the m k , i.e. the image of [m k ] k≥0 : k≥0 X k −→ X is the carrier of a reachable subcoalgebra of (X, ξ, x 0 ). In particular, if (X, ξ, x 0 ) is reachable, then [m k ] k≥0 ∈ E.
3. Consider the (E,mono)-factorization k≥0 X k e this property. In the following, we choose sets F k , k ≥ 0, containing morphisms (E, e) ∈ F k , e : E −→ X k , E ∈ S, inductively: 1. F 0 contains only x ′ 0 : I ։ X 0 , where x 0 = m 0 · x ′ 0 is the pointing I −→ X, as provided by item 1 of Lemma A.5. 2. For every (E, e) ∈ F k , let G (E,e) ⊆ C(E, F X k+1 + 1) be the set of morphisms g : E −→ F X k+1 + 1 with: For each g ∈ G (E,e) , choose some object Y g ∈ S, and morphisms y g : Y g −→ X k+1 and a F +1-precise p g : E −→F Y g +1 with g = (F y g +1)·p g , according We need to prove some properties about the F k -By construction, for every (E ′ , e ′ ) ∈ F k+1 , there is some (E, e) ∈ F k and an F + 1-precise map p g : T F e ′ Consequently by Proposition 3.18, for every (E, e) ∈ F k , k ≥ 0, there is a run [x j ] j≤k : J(P k+1 , p k ) −→ (X, ξ, x 0 ) with P k = E, x k = in k · e. -For every family F k , k ≥ 0, the morphism [e] (E,e)∈F k is an isomorphism; this means that for the factorization: we have that s k is an isomorphism for every k ≥ 0.
Hence, T F s·v (E,e) = ξ k ·e for all (E, e) ∈ F k . Since the e ∈ F k are jointly epic by the induction hypothesis, we have a unique diagonal ξ ′ k in the following square: Since ξ k was constructed to be the least morphism, s k+1 is necessarily an isomorphism.
is an epimorphism. Hence, the familiy (m k · f k ) k≥0 is jointly epic; this family is contained in the family of runs in (X, ξ, x 0 ), and so the family of runs is jointly epic.

Proof of Proposition 4.3
The first step is to describe when a map is B-precise: Proof. Sufficiency is proven by the two inequalities: (≤ 1) Let t X,Y : BY ×X −→B(Y ×X) be the functorial strength defined as for all Set-functors by t X,Y (b, x) = B(y → (y, x))(b). We thus have the commuting diagram Since s is B-precise, we obtain a map d : Y −→ Y × X with π 1 · d = id Y and t X,Y · s, id X = Bd · s, that means B(y → (y, x))(s(x)) = B(y → (y, d ′ (y)))(s(x)), with d ′ = π 2 · d. Hence, every y ∈ Y appears in at most one s(x), namely only in s(d ′ (y)) or not at all; this means for every y ∈ Y , only one summand of x∈S s(x)(y) is non-zero.
Lemma A.7. Let F preserve weak pullbacks and let α : F −→ G be a natural transformation whose naturality squares are weak pullbacks. Then a morphism If G admits precise factorizations w.r.t. S ⊆ obj C, then so does F .
Proof. For sufficiency, let f : X −→F Y be F -precise and consider a commuting diagram Since the naturality square for w is a weak pullback, a morphism g ′ : X −→ F Z exists, making the following diagram commute: Since f is F -precise, we obtain a morphism d : Y −→ W with w · d = y and F d · f = g ′ , and thus also g = α W · g ′ = α W · F d · f = Gd · α Y · f as desired.
For necessity, we use the simplified version of F -precise because F preserves weak pullbacks. So let α Y · f be G-precise and consider a commuting diagram The naturality square for d is a weak pullback, and so we have some f ′ : X −→F Y making the following diagrams commute: For precise factorizations, consider f : X −→ F Y with X ∈ S. For the G-precise factorization Gy ′ · g ′ of α Y · f we have: Since the naturality square is a weak pullback, we obtain some f ′ : To conclude, since analytic functors preserve weak pullbacks, it is enough to prove the following: Lemma A.8. The naturality squares of α : F −→ B are weak pullback squares.
Proof. Consider a map f : X −→ Y and elements t ∈ F Y , s ∈ BX with α Y (t) = Bf (s). We can write t as an equivalence class t = [σ(y 1 , . . . , y n )]. Every y i is in the image of f , and so any t ′ = [σ(x 1 , . . . , x n )] ∈ F X with f (x i ) = y i for all 1 ≤ i ≤ n has the property that α X (t ′ ) = s and F f (t ′ ) = t.
⊓ ⊔ So we obtain Proposition 4.3 as a corollary of Lemma A.7.
Proof of Proposition 4.5 In this section, we will use the following standard lemma on strong nominal sets (see e.g. [20, Prop. 5.10] or [17, p. 3]): Lemma A.9. Given a strong nominal set X, a subset O ⊆ X such that for every x ′ ∈ X there is precisely one x ∈ X with a π ∈ S f (A) fulfilling x ′ = π · x, and a map f For every b ∈ A, there is some π ∈ S f (A) and a ∈ A with b = π · a, and hence x ′ (b) = x ′ (π · a) = π · x ′ (a) ∈ π · x(a) = x(b), i.e. η X · x ′ ⊑ x. By construction of c, we have that h · x ′ = y.
Proof of Proposition 4.7 To prove this, by Proposition 3.6.4, it is enough to prove that the binding functor [A] admits precise factorizations w.r.t. strong nominal sets.
Recall from [23] that [A] has the left adjoint A# , sending X to the nominal set A#X := {(a, x) ∈ A × X | a ∈ supp(x)}. If X is a strong nominal set, then so is A#X, hence [A] admits precise factorizations w.r.t. strong nominal sets by Proposition 3.6.5.
⊓ ⊔ Proof of Proposition 4.8 All the axioms are proved as in Set, component-wise. The only interesting axiom is to prove that F {X P } P ∈P = { Q∈P P(P, Q) × X Q } P ∈P admits precise factorisations. Since F = Q∈P H Q × π Q with: -H Q being the constant functor on (P(P, Q)) P ∈P , -π Q being the functor mapping {X P } P ∈P to {X Q } P ∈P , by Proposition 3.6. (1,3,4), it is enough to prove that π Q admits precise factorisations. For that purpose, given a morphism, i.e. family of maps, (f P ) P ∈P : {X P } P ∈P → π Q ({Y P } P ∈P ) = {Y Q } P ∈P , define (Z P ) P ∈P with Z Q = P ∈P X P and Z P = ∅ for P = Q.
and (in P ) P ∈P : {X P } P ∈P −→ π Q (Z P ) P ∈P in P : X P −→ (π Q (Z P ) P ∈P ) P = Z Q = P ∈P X P and (h P ) P ∈P : (Z P ) P ∈P −→ (Y P ) P ∈P with Obviously, (π Q h) P · in P = f P for all P ∈ P. It remains to show that (in P ) P ∈P is π Q -precise. Consider a commutative square: π Q ((Z) P ∈P ) π Q ((W ) P ∈P ) Obviously, we have the first triangle v P = d Q · in P for all P ∈ P. The other triangle follows by case distinction: for P = Q, z P = w P · d P holds by initiallity of Z P = ∅; we have z Q · in P ( * ) = w Q · v P = w Q · d Q · in P ∀P ∈ P and since the in P : X P −→ Z Q are jointly epic, z Q = w Q · d Q . So (d P ) P ∈P is a diagonal lifting and so (in P ) P ∈P is π Q -precise.
Proof of Proposition 4.9 To prove this proposition, it is enough to prove the following: Lemma A. 10. A morphism f : χ P −→ F Y is F -precise iff Y = χ Q for some Q ∈ P.
Proof. Necessity. Consider f P : 1−→ Q∈P P(P, Q)×Y Q and let f P = in Q (m, x). Define the injective h : χ Q −→ (Y R ) R∈P by h R = ! : ∅ −→ Y R for R = Q and h Q = x : 1 −→ X Q . Furthermore, define by g : χ P −→ F χ Q g P = in Q (m, * ), and so F h · g = f . Since f is F -precise, there is a d : Y −→ χ Q with h · d = id Y , so in total h is an isomorphism.
Sufficiency. For f : χ P −→ F χ Q let m : P −→ Q with f P = in Q (m, * ) and consider a commuting diagram: There are different possibilities how to model a path in such a system: 1. Simply • −→ • −→ • as in LTS for a singleton input alphabet. Of course, this makes the probabilities in the system entirely meaningless for runs. 2. A path and a probability (p, • −→ • −→ •) meaning that there is a path whose weights multiply to at least p. So the above system has a run for such a path if p ≤ 0.08 = 0.2 · 0.4. 3. A path is a probabilistic system in which each state has at most one successor, and so the notion of run of path coincides with a functional simulation from that path to the system. E.g. the above system has a run for • Clearly, there is a functional bisimulation f : A −→ X from the above system to the system X given by While A has no run for the path • 0.5 − − → •, the system X has. So f : A−→X is not an open map, even though it is a functional bisimulation, i.e. coalgebra homomorphism for the subdistribution functor.