Joseph Abdou
email: abdou@univ-paris1.fr.

Hans Keiding
email: hans.keiding@econ.ku.dk.

A Qualitative Theory of Conflict Resolution and Political Compromise

Keywords: Delegation, compromise, simplicial complex, contiguity, strong homotopy JEL Classification: C70, C79, AMS Classification: 91A70

We view political activity as an interaction between forces seeking to achieve a political agenda. The viability of a situation depends on the compatibility of such agendas. However even in a conflictual situation a compromise may be possible. Mathematically a political structure is modeled as a simplicial complex and a viable configuration as a simplex. A represented compromise is a viable configuration obtained by the withdrawal of some agents in favor of some friendly representatives. A delegated compromise is a sophisticated version of a compromise obtained by the iteration of the withdrawal process. Existence of such solutions depends on the discrete topology of the simplicial complex. In particular we prove that the existence of a delegated compromise is equivalent to the strong contractibility of the simplicial complex.

Introduction

We model political activity as an interaction between forces seeking to achieve a political agenda. A stable government allows the implementation of the political agenda of the party or the coalition holding power. In normal circumstances, the struggle for power is regulated by constitutional rules that guarantee a peaceful and consensual outcome of the process. This includes commonly admitted mechanisms for breaking deadlocks once they occur (for instance elections, referendum or justice ruling). However many political entities, under special circumstances, incur a blocked governance process while the current order fails to provide a clue for a solution. This is a political crisis or a stalemate. For instance, one can witness the formation of a crisis in a place where a military conflict left the entity with a pre-state, pre-constitutional configuration, that is, where a universally accepted rule does not exist; or where the threat of a violent action impedes the regular unfolding of the political process; or else where the risk of disrupting the ongoing process is wielded by a force that accumulates discontent with the current establishment.

The general question that we ask is therefore : What happens in a crisis configuration that is unsolvable by the current institutions, or put more explicitly what can be expected if a configuration composed by many parties with incompatible political agendas lacks the institutional mechanism that enforces a settlement?

Modeling politics has a long history starting from the early greek philosophers. One of the main concepts that marked the modern analysis of conflict is the notion of enemy. According to political thinkers that promote this view the essential moment of politics is the discrimination friend/enemy (cf. e.g. [START_REF] Schmitt | The concept of the political: Expanded edition[END_REF]). This binary choice erupts in a situation of disruption, when the political body is in danger, war or civil strife. In this model we admit that political action is by essence produced in a disrupted situation, that is a preinstitutional moment, but we depart from the idea that politics is by essence bi-polar: political action is not drawing a line of separation from (or the destruction of) an enemy but the search of a viable situation from a disrupted one. Political forces in presence can be either compatible for an inclusive governance or not, but this compatibility concerns each subgroup of forces. The current situation is therefore described by this compatibility relation. Mathematically a configuration is represented by a simplicial complex.

The fundamental notion in our approach to crisis resolution in simplicial complexes is delegation: An agent can delegate power to another agent when the latter is at least as well situated in the power hierarchy as the delegating agent. The result of delegation will be a power structure which is simplified to some extent, since some of the sources of incommensurability have been removed. We investigate in detail what can emerge as a result of delegation, giving rise to the notion of representations and represented compromises, which in particular cases may resolve the instabilities fully. Turning next to cases of iterated delegation, where delegation may result in loss of influence as the power originally handed over to another agent is then further delegated. On the other hand, such repeated delegation increases the possibility of avoiding stalemates in decision making.

From a formal point of view, the model falls within the field of discrete topology; both the representations and the delegations can be studied using the theory of homotopy in finite simplicial complexes, and the relevant concepts are introduced when when needed.

The paper is organized as follows: In the following Section 2, we introduce the basic notion of a political structure which has the the mathematical structure of a simplicial complex, and we define the notion of a delegation, giving rise to representations and represented compromises. As we proceed, examples show how these notions apply to simple games and to network models of communication. In the following Section 3, the formalism is developed somewhat further so we can use results from homotopy theory of finite complexes to investigate delegation in political structures. The main results on the structure of represented compromises are presented in Section 4, which also points to some of the shortcomings. This leads to the consideration of the somewhat stronger form of delegation and its consequences, the delegated compromises, in Section 5. Finally, Section 6 contains some concluding comments.

Basic definitions

We consider a situation described by a nonempty set E of agents. In the interpretation, agents may be individuals, but it may as well be groups of individuals or even political issues or institutions. A nonempty subset of agents is called a configuration. We assume that there is given a set K of viable configurations, with the property that any singleton {x}, where x ∈ E, is viable, and any non empty subset of a viable configuration is viable, so that formally K is a simplicial complex, with set of vertices E. The pair (E, K) is called a political structure.

Corresponding to what was said about agents, the interpretation of viable configuration may vary according to the circumstances. In general, viability represents a form of compatibility between agents, whereas configurations which are not viable will display some sort of conflict preventing the functioning of the system. Participation in viable configurations gives a first notion of the importance of an agent in the overall structure.

In the sequel the power set of any set X will be denoted P(X), and P(X) \ {∅} will be denoted P 0 (X).

Example 1 (TU games). A TU (Transferable Utility) game is a pair (N, v), where N is a nonempty set of players and v is a map from the set P 0 (N) of nonempty subsets of N (coalitions) a number interpreted as the money or utility gain that the coalition can obtain for its members. An payoff vector in (N, v) is a vector x ∈ R N satisfying

i∈N x i ≤ v(N); let V (N) be the set of payoff vectors. For S ∈ P 0 (N), let A(S) = x ∈ V (N) i∈S x i ≥ v(S)
be the set of payoff vectors which cannot be improved by the coalition S, in the sense its members get at least as much as it they could get from the coalition. A family of coalitions is viable if there is a payoff vector which cannot be improved by any coalition in the family. A political structure (P 0 (N), K v ) is defined by positing :

K v = {σ ⊆ P 0 (N) | ∩ S∈σ A(S) = ∅}.
Note that A(S) = ∅ and therefore the set of vertices is indeed P 0 (N). Moreover P 0 (N) belongs to K v if and only if there is a payoff vector which cannot be improved by any coalition, equivalently if and only if the core of (N, v) is nonempty. A closer study of K v may however be helpful in the nontrivial case where P 0 (N) / ∈ K v . It should be noticed that here the agents are coalitions, and configurations are families of coalitions.

Example 2 (Effectivity structures). Let N and A be finite nonempty sets (of agents and alternatives, respectively). An effectivity structure on (N, A) is a subset of P 0 (N)× P 0 (A), so that elements of E are pairs (S, B) with S ⊆ N, B ⊆ A. In the interpretation, elements of E are potential cases of exercising power: the coalition S can make sure that outcome of any decision made in the community must belong to B, or equivalently, S can prevent choices outside B. One can consider E as a (partially defined) effectivity function: that is E : P 0 (N) → P(P 0 (A)) where E(S) stands for {B ∈ P 0 A|(S, B) ∈ E} (cf Abdou and Keiding [START_REF] Abdou | Effectivity functions in social choice[END_REF]).

In order to make of E a political structure with elements (S, B) as agents we have to describe viable configurations. For that purpose we need to specify the environment in which the power expressed by (S, B) may be exercised: Let L(A) denote the set of linear orders on A, and define (preference) profiles as maps R : N → L(A), so that the set of profiles is L(A) N . If for some i ∈ N, the alternative a is (strictly) preferred to the alternative b in the profile R, then we write a R + i b, and for a ∈ A, the set of alternatives preferred to a by all members of S in the profile R is

P (a, S, R) = {b ∈ A | b R + i a, all i ∈ S}.
With this notation, we get that the coalition S is induced to exercise its objection power at the profile R if an alternative c ∈ A is suggested such that P (c, S, R) contains a subset B with (S, B) ∈ E. We shall say that a configuration s ⊆ E is viable if it is always able to select an alternative against which none of its members will exercise their objection power, that is if for each profile R there is some a ∈ A such that B is not contained in P (a, S, R) for some (B, S) ∈ s. Considering the configuration s ⊆ E as an effectivity function, viability of s is equivalent to stability of s as considered by e.g. Moulin and Peleg [START_REF] Moulin | Cores of effectivity functions and implementation theory[END_REF], Abdou and Keiding [START_REF] Abdou | Effectivity functions in social choice[END_REF]. It is easy to see that the family K of all viable configurations in E is indeed a simplicial complex, so that the ordered pair (E, K) can be considered as a political structure.

Since our overall purpose is the investigation of conflictual situations and their possible resolution, we are mainly interested in ways in which to eliminate non-viable configurations. Our approach will be to allow agents to delegate their influence to other, more centrally placed agents. In our present setup we formulate this by the notion of delegation.

Definition 1. Let y ∈ E and d ⊆ E, y / ∈ d. A delegation from d to y is a map δ d : E → E such that δ d (x) = y if x ∈ d and δ d→y (x) = x if x / ∈ d.
A simple delegation is a delegation δ d→y where d is a singleton, d = {x}, and it is written as δ x→y .

A delegation δ d→y is said to be friendly if s ∪ {y} ∈ K for all s ∈ K such that s ∩ d = ∅.

A delegation from d to y can be seen as a political action by which the agents in d withdraw from the political interaction in favor of the agent y. A delegation is friendly whenever any agent in d can ensure that every viable configuration to which she participates remains viable after delegation, and moreover, the agent receiving the delegation is present in every configuration which contained the delegating agent. A withdrawal in favor of a delegate is more likely to happen if the delegation is friendly. In our analysis, the possibility of friendly delegations constitutes the only driving principle behind potential moves that reduce conflicts in a political structure.

Example 3 (Networks). The use of formal networks has a long history in the social sciences, with its beginning in the 1930s; for a survey of its history, see e.g. [START_REF] Scott | Social network analysis: developments, advances, and prospects[END_REF]. A social structure is a graph G = (V, E), where V is a finite set of edges and E is a set of two-element subsets {v 0 , v 1 } of E. In the interpretations, V are individuals, and two individuals v 0 , v 1 are socially connected if {v 0 , v 1 } ∈ E. A graph is a simplicial complex where simplices have dimension at most 1.

The social contacts is represented by a paths in the graph, that is sequences {v 0 , v 1 }, {v 1 , v 2 }, . . . , {v n-1 , v n } of edges in G. If there is a path from v 0 to v n , then v 0 is indirectly socially connected to v n , and the distance of this indirect connection is the shortest distance of a path from v 0 to v n . In addition, the social position of an individual v ∈ V can be measured by the degree of v, the number of edges in E containing v. In Fig. 1 the individual represented by the point v 0 with degree 4 is clearly more centrally positioned than the other individuals. Centrality measures play an important role in network analysis, cf. [START_REF] Freeman | Centrality in social networks: conceptual clarification[END_REF].

There is a simple friendly delegation from v 0 to v 1 if any social relation of v 0 must be with v 1 , so that v 1 is the only social contact of v 0 . Cionsequently, an R-compromise is therefore a subset V ′ of V such that all v ∈ V \V ′ has degree 1. A pure R-compromise will exist only in specially simple networks. However, an extended version of the compromise, to be introduced later, will be more useful.

.

The following property of delegations is immediate.

v 0 • v 4 • v 3 • v 1 • v 2 •
∅ = d = {x 1 , . . . , x n } ⊂ E and y / ∈ d, if δ d→y is a delegation, then δ d→y = δ x 1 →y • • • • • δ x k →y .
Moreover, δ d→y is friendly if and only if δ x j →y is friendly for j ∈ {1, . . . , n}.

As a result of the delegations, there may be fewer agents with conflicting interests. We introduce the notion of a compromise is as a counterpart of real world solutions to political conflicts, where agents (or issues, or power groups) will have to step aside since they will not be accepted as partners in a political deal but on the other hand can be represented by other, more acceptable, agents.

Definition 2. A nonempty subset F of E is a representation of E if for any x /
∈ F , there exists some y ∈ F such that δ x→y is a friendly delegation.

The subset

F of E is a represented compromise (shorthand: an R-compromise) for E if F is a representation and F ∈ K. A pure R-compromise is an element y ∈ E such that {y} is an R-compromise.
Example 4 (Empirical case). The recent stalemate in Lebanese politics and its resolution through compromise provides an illustration of the concepts introduced above. Beyond the formal constitution that was agreed upon in Taef in October 1989, the political configuration in Lebanon, in its principal components, that prevailed after the assassination of Prime Minister Rafic Hariri February 2005, can be described by the simplicial complex of Fig. 2.

The polarizing issue that divided political forces can be described schematically as East-West, and compatibility of the agendas was dictated by that issue. Independently of that issue, the National Pact, partially represented by Taef Agrement, requires the effective inclusion in the governance of the three major confessions.

The major forces in presence in the conflict are the Future Movement (FUT), AMAL (AMA), Hizbullah (HIZ), Free Patriotic Movement (FPM), and Lebanese Forces (LF). In Fig. 2, brown color has been assigned to Christians (Maronites), blue to Sunnis, and LF FUT AMA HIZ FPM Figure 2: Friendly delegation and political compromise in Lebanese politics green to Shias. After a long search, a compromise was found that respects the National Consensus, as represented by the middle shaded triangle. Presumably this compromise is the result of a friendly delegation from LF to either FPM or FUT, and from AMA to either FUT or HIZ. This compromise was formally implemented by the election of Michel Aoun as President (October 2016) and the formation of a goverment by Saad Hariri (December 2016).

In the following we shall study delegation and compromises using tools of finite homotopy theory, and for this purpose it is convenient to considered delegation from another angle, emphasizing that agents are removed from direct participation.

Definition 3. Let ∅ = F ⊂ E. A retraction to F is a map r : E → E such that r(E) = F and r(x) =
x for all x ∈ F , and F is said to be a retract if there is a retraction to F .

A retraction r is friendly if x ∈ s ∈ K implies {r(x)} ∪ s ∈ K, and a friendly retract is a retract F such that there exists a friendly retraction to F .

A retraction is a political action by which the conflictual political structure (E, K) is reduced to the (sub) configuration (F, K F ), withdrawing agents not in F in favor of suitable agents in F , their representatives. A retraction is friendly if any viable configuration containing a retracting agent as a member remains viable when he is replaced by his representative. Clearly, a delegation δ d→y is a retraction to E\d, and a retraction r to F such that r(E\F ) = {y} is a delegation from E\F to y. We have the following characterization:

Proposition 2. F ⊂ E is a representation if and only if F is a friendly retract.
Proof: Let F be a representation. Define a map r : E → E as follows: r is the identity on F , and if x / ∈ F then put r(x) = y where y is some element of F such that there is a friendly delegation from x to y. It is clear that r is a retraction on F . Moreover if x / ∈ F for any s ∈ K such that x ∈ s, we have that{r(x)} ∪ s ∈ K; and for any x ∈ F and s ∈ K such that x ∈ s, we have that {r(x)} ∪ s = s ∈ K. The converse is straightforward.

The notion of friendly retraction characterizes a representation as the result of withdrawal of some agents, in what follows we consider whether a representation can result from of a sequential process of delegations. Definition 4. A progressive delegation is a sequence δ x 1 →y 1 , . . . , δ xp→yp of simple delegations where the elements x 1 , • • • , x p are distinct and y k / ∈ {x 1 , . . . , x k } for all k = 1, . . . , p. A progressive delegation is friendly if each of the simple delegations that compose it is friendly. The configuration

c = δ xp→yp • • • • • δ x 1 →y 1 (E) is called the outcome of the progressive delegation.
Thus, when performing a progressive delegation, agents are successively delegating their influence to other agents, and in this case it seems natural that once an agent has renounced on influence through delegation, she cannot herself be an object of delegation.

Proposition 3. Let δ x 1 →y 1 , . . . , δ xp→yp be a progressive delegation with φ = δ xp→yp • • • • • δ x 1 →y 1 .
Then φ is a retraction to E \ {x 1 , . . . , x p }, and

fix φ = {x ∈ E | φ(x) = x} = φ(E) = E \ {x 1 , . . . , x p } = ∅. (1)
Conversely, any retraction φ : E → E to some F ⊆ E can be obtained through a progressive delegation. Moreover all the involved delegations can be chosen to be friendly if and only if the retraction is friendly.

Proof: (i) Put F = E \ {x 1 , . . . , x p }. It is clear that F ⊆ fix φ .
Conversely, for any k ∈ {1, . . . , p} φ(x k ) = y ℓ with ℓ ≥ k and by progressivity, y ℓ = x k , so that fix φ ⊂ F and consequently fix φ = F . Now, F ⊂ φ(E). In order to prove the opposite inclusion, let x ∈ E. Then either x ∈ F and therefore φ(x) = x ∈ F , or x = x k and φ(x) = y ℓ for ℓ ≥ k. It follows that y ℓ = x k ′ for k ′ ≥ ℓ and by progressivity of the sequence y ℓ = x k ′ for k ′ < ℓ and finally y ℓ ∈ F . We conclude that F = φ(E).

For any k ∈ {1, . . . , p} let

φ k = δ x k →y k • • • • • δ x 1 →y 1 and F k = E \ {x 1 , . . . , x k }. By convention F 0 = E and φ 0 = Id E . Then, by the first part of the proof φ k is a retraction onto F k and one has φ k = δ x k →y k • φ k-1 . We prove by induction that φ k is friendly. Clearly φ 0 = Id E is friendly. Assume that φ k-1 is friendly. If x ∈ s ∈ K then {φ k-1 (x)} ∪ s ∈ K by the induction hypothesis and C ≡ {δ x k →y k (φ k-1 (x))} ∪ (φ k-1 (x) ∪ s) ∈ K since δ x k →y k is a friendly delegation. But {φ k (x)} ∪ s is a subset of C, therefore {φ k (x)} ∪ s is a simplex. Thus our claim is proved.
For the converse implication, if φ is a retraction to F and E\F = {x 1 , . . . , x p }, we consider δ x 1 →xp , . . . , δ xp→yp where

y k = φ(x k ) for k = 1, . . . , p. It is easily seen that φ = f xp→yp • • • • • δ x 1 →y 1 . For any x ∈ E such that x = φ(x) one has x = x k for some k, so that δ x ℓ →y ℓ (x) = φ(x) if ℓ = k and δ x ℓ →y ℓ (x) = x if ℓ = k. It is easily seen that if x ∈ s ∈ K, one has s ∪ δ x ℓ →y k (x) ⊂ s ∪ φ(x). It follows that δ x k →y k is friendly if φ is friendly.
Corollary 4. F ⊂ E is a representation if and only if there exists a friendly progressive delegation with outcome F .

Topology of the political structure and representations

Let K be a simplicial complex on a set E and let s = {x 0 , . . . , x r } be a simplex in K. The closed simplex s is the set of formal convex combinations r i=0 λ i x i with λ i ≥ 0 for i = 0, . . . , r and r i=0 λ i = 1. Each closed simplex is a metric space under the metric based on Euclidean distance. The geometric realization of |K| of the simplicial complex K is the union of all closed simplices s for s ∈ K endowed with the topology for which

U ⊂ |K| is open if U ∩ s is open in s for each s ∈ K. Given simplicial complexes K on a set E and L on F , a map φ : E → F is simplicial if it takes simplices in K to simplices in L, that is if φ(s) ∈ L for each s ∈ K,
or, otherwise put, if it extends to a map φ : K → L. We associate to ϕ the map |ϕ| : |K| → |L| obtained from ϕ by linear extension.

It is easily seen that friendly retractions are simplicial. Two simplicial maps ϕ, ψ : (E, K) → (F, L) are said to be contiguous if for any s ∈ K, φ(s) ∪ ψ(s) ∈ L. Denote by ≈ the contiguity relation in E. ≈ is symmetric but generally not transitive. Let ∼ be the transitive closure of ≈. By definition, f ∼ g if there exists a sequence f = f 0 , . . . , f p = g such that f k ≈ f k+1 for k = 0, . . . , p -1. Then ∼ is transitive and symmetric, hence an equivalence relation, the classes of which are called contiguity classes. Moreover ≈ and therefore also ∼ are compatible with composition. A simplicial map ϕ : (E, K) → (F, L) is a strong equivalence if there exists ψ : (F, L) → (E, K) such that ψ • ϕ ∼ Id E and ϕ • ψ ∼ Id F . It is worthwhile noting that two simplicial maps that have the same contiguity class are homotopic, but that the converse is not true (cf. e.g. Spanier [START_REF] Spanier | Algebraic topology[END_REF] Chap. 3).

Let (E, K) be a political structure. For any configuration F we define K F as the set of all elements of K that are included in F . It is clear that (F, K F ) is a political structure, the agents of which are the elements of F . Moreover K F is a full subcomplex of K. Clearly one has K = K.

Lemma 5. A map ϕ : E → E is contiguous to Id E if and only if for all x ∈ s ∈ K one has {ϕ(x)} ∪ s ∈ K.
Proof. Assume that ϕ has the announced property, and {x 1 , . . . x p } = s ∈ K. Then by induction on k, s ∪ {ϕ(x 1 ), . . . , ϕ(

x k )} = s ∪ {ϕ(x 1 ), • • • , ϕ(x k-1 )} ∪ {ϕ(x k )} is in K, therefore s ∪ ϕ(s) ∈ K and ϕ is contiguous to Id E . Conversely If ϕ is contiguous to Id E and if x ∈ s ∈ K then {ϕ(x)} ∪ s ⊂ ϕ(s) ∪ s ∈ K
In particular we have the following: Proposition 6. A retraction is friendly if and only if it is contiguous to Id E .

Remark 1. To a retraction r to F we associate the map r F : E → F which is just r with restricted range F . If i : F → E is the inclusion map, then by definition r F • i = Id F . Conversely, any r F : E → F such that r F • i = Id F gives rise to the retraction r by putting r = i • r F . We use same terminology of a retraction to r F when no confusion is possible.

x 0 • x 4 • x 3 • x 1 • x 2 • x 1 x 2 x 3 x 0 x 1 x 2
x 3

x 0 In both of the above political structures, the retraction onto x 0 is contiguous to the identity. Consequently x 0 is a pure R-compromise.

In view of Prop 2 and lemma 5, we can have the following characterization of representations: Proposition 7. F is a representation of (E, K) if and only if r F : (E, K) → (F, K F ) is a strong equivalence.

Proof: In view of Lemma 5, r is friendly if and only if r is contiguous to Id E , and in view of the Remark 1 this happens if and only if

i • r F = r ∼ Id E . Since r F • i = Id F , this implies that r F is a strong equivalence.
The set of simple friendly delegations in a political structure can be used to define a binary relation ⊲ on E as follows,

x, y ∈ E, x ⊲ y if either x = y or the delegation δ x→y is friendly. Lemma 8. For any x ∈ E, let σ x be the set of all maximal simplices containing x. Then x ⊲ y if and only if σ x ⊆ σ y .

Proof: If x ⊲ y and x = y, then there is a friendly delegation δ x→y , so that every maximal simplex containing x contains y as well. Conversely, if σ x ⊆ σ y , then the map δ taking x to y and leaving all other elements of E unchanged is a friendly delegation.

It follows from Lemma 8 that that ⊲ is transitive and reflexive so that (E, ⊲) is a preordered set. Definition 5. A representation F of (E, K) is said to be minimal if there is no representation that is a strictly included in F Proposition 9. A representation F of (E, K) is minimal if and only if for any x, y ∈ F , x ⊲ y implies x = y.

Proof: If F is not minimal for inclusion then there exists some F ′ ⊂ F , x ∈ F \ F ′ , and y ∈ F ′ such that x ⊲ y. Conversely If there exists x, y ∈ F x = y such that x ⊲ y then F ′ = F \ {x} is easily seen to be a representation contradicting minimality of F . Lemma 10. Let ∅ = F ⊂ E be arbitrary.

(i) If x ⊲ y for all x, y ∈ F , then F ∈ K.

(ii) E ∈ K if and only if for any x, y ∈ E, x ⊲ y.

Proof. Assume that F / ∈ K and let s be a maximal simplex included in F . Since s = F there exists some y ∈ F, y / ∈ s. Let x ∈ s. Since, by the maximality of s, s ∪ {y} / ∈ K one has that δ x→y is not a friendly delegation. This proves (i). (ii) follows immediately from (i) One may exploit constructions developed for simplicial complexes to obtain a measure of the possibility of delegation. For this purpose, we introduce (cf. Barmak [START_REF] Barmak | Algebraic topology of finite topological spaces and applications[END_REF]) the nerve N (K) of K as the simplicial complex in which the set of vertices is the set of maximal simplices in K, say K M , and the set of simplices is the set of subsets {s 0 , . . . , s r } ⊆ K M such that ∩ r i=0 s i = ∅. Repeating the construction, one gets N 2 (K) = N (N (K)) and in generel,

N k (K E ) = N (N k-1 (K)).
We consider the second nerve N 2 (K) in more detail since it plays a role in the characterization of minimal representations. A vertex in N 2 (K) is a maximal set σ = {s 0 , . . . , s r } of maximal simplices in K such that ∩ r i=0 s i = ∅. Let Φ be the set of all maps ϕ from N 2 (K) to E assigning to σ = {s 0 , . . . , s r }, a vertex of N 2 (K), some element of ∩ r i=0 s i . Notice that the set Φ is nonempty since ∩ s∈σ s = ∅ for every σ ∈ N 2 (K).

Proposition 11. The set Φ has the following properties: (i) Any ϕ ∈ Φ is a simplicial map from N 2 (K) to K and an isomorphism onto its image. (ii) For any ϕ ∈ Φ the image ϕ(N 2 (K)) is a minimal representation. (iii) For each minimal representation F ⊂ E, there is ϕ ∈ Φ such that F = ϕ(N 2 (K)).

Proof: (i) Let ϕ ∈ Φ. If {σ 0 , . . . , σ k } is a simplex in N 2 (K), then ∩ k i=0 σ i = ∅, so that there exists a maximal simplex s in K which belong to all the sets σ 0 , . . . , σ k . By its definition, ϕ(σ i ) belongs to s for i = 0, . . . , k, therefore {ϕ(σ 1 ), . . . , ϕ(σ k )} is a simplex in K. This shows that ϕ is a simplicial map.

Next, we show that ϕ is an injective map. Suppose that ϕ(σ 1 ) = ϕ(σ 2 ) = x. If σ 1 = (s 1 0 , . . . , s 1 r ), σ 2 = (s 2 0 , . . . , s 2 p ), then by our construction x ∈ s 1 i for i = 0, 1, . . . , r and x ∈ s 2 i for i = 0, 1, . . . , p, so that σ 1 ∪ σ 2 is a family of maximal simplices from K with nonempty intersection. Since σ 1 and σ 2 are already maximal collections of simplices from K with nonempty intersection, we have that σ 1 = σ 2 , so that f is indeed injective.

Finally we prove that the inverse map ϕ -1 defined on the image F of ϕ is simplicial. Let s = {x 1 , . . . , x k } ⊂ F be a simplex (i.e. an element of K F ) and let σ i = ϕ -1 (x i ) i = 1, . . . k. Denote by s ′ a maximal element of K containing s. For any i ∈ {1, . . . , n], any element of σ i has x i as a member, therefore any element of σ i ∪ {s ′ } has x i as a member. In particular σ i ∪ {s ′ } is a simplex in N (K). Since σ i is maximal, we have that s ′ ∈ σ i . Since every σ i i = 1, . . . , k, contains s ′ it follows that {σ 1 , . . . , σ k } is a simplex.

(ii) Let ϕ ∈ Φ and let F be the image by ϕ of the vertices of N 2 (K). For any x ∈ E, let σ x be the set of all maximal simplices in K containing x. σ x is thus a simplex of N (K). If x / ∈ F , let σ be a maximal element of N (K) containing σ x , and let y = ϕ(σ). Then σ x ⊂ σ = σ y , so that by Lemma 8, x ⊲ y. This proves that F is a representation. If x ∈ F , there exists σ maximal such that x = ϕ(σ) so that σ ⊂ σ x and therefore σ = σ x . Similarly, if y ∈ F then σ y is maximal and ϕ(σ y ) = y, but if x ⊲ y, then σ x ⊂ σ y , and by maximality of σ x one has σ x = σ y and therefore y = x. It follows from Proposition 9 that F is a minimal representation.

(iii) Conversely, let F be a minimal representation. We claim that for any y ∈ F the set σ y is a maximal element of N 2 (K). Suppose not; then there must be x = y with σ y ⊂⊂ σ x . By Lemma 8, y ⊲ x, and by Proposition 9, x / ∈ F . But then there must be z ∈ F , such that x ⊲ z. By transitivity of ⊲ we get that y ⊲ z, and by Proposition 9 y = z. It follows that σ x ⊂ σ y , a contradiction. We conclude that σ y is indeed maximal for each y ∈ F .

Remark that for any maximal σ one has σ x = σ for all x ∈ ∩ s∈σ s. We claim that there exists some x ∈ F such that σ = σ x . If not then σ = σ y only for some y / ∈ F . But y ⊲ z for some z ∈ F and therefore σ ⊂ σ z and since σ is maximal σ z = σ, a contradiction. In fact such an x is unique since σ x = σ y where x, y ∈ F implies x = y (Proposition 9). One therefore may define ϕ by putting ϕ(σ) = x where x is the unique F such that σ = σ x . By our construction, ϕ belongs to Φ and by the preceding paragraph every element of F is an image of some maximal σ, that is ϕ(N 2 (K)) = F . It follows from proposition 11 that there is a bijection between Φ and the set of minimal representations of (E, K).

Example 5 (Simple games). A simple game is a TU game (N, v) (cf. Example 1) with v(S) ∈ {0, 1} for all S ∈ P 0 (N), so that it is characterized by the set W = {S ∈ P 0 (N) | v(S) = 1} of winning coalitions. Simple games arise in many contexts and has been studied very intensely since their first appearance in [START_REF] Von Neumann | Theory of games and economic behavior[END_REF], see e.g. [START_REF]Simple games: An outline of the descriptive theory[END_REF][START_REF] Peleg | Game theoretic analysis of voting in committees[END_REF]. There are several ways of associating a political structure with a simple game. The most immediate one is (P 0 (N), K v ) introduced in Example 1. It is easy to see that K v is the set of all σ ⊂ P 0 (N) such that ∩ S∈σ∩W S = ∅. We are going to show that this political structure is redundant by showing that delegation may lead to trivial representations. Assume that W = P 0 (N). If S 0 / ∈ W then any delegation to S 0 is friendly: actually if s ∈ K v then s ∪ {S 0 }. It follows that S 0 is a pure R-compromise ! One can discard this difficulty by associating to W the more natural structure (W, K) where K the set of all s ⊂ W such that ∩ S∈σ S = ∅. Configurations in K may be interpreted as situations where a common agreement is possible, whereas conflicts arise when a set {S 0 , . . . , S r } is not a simplex. The smallest number r for which this occurs is known as the Nakamura number of W, cf. [START_REF] Nakamura | The vetoers in a simple game with ordinal preferences[END_REF]. Assume N ∈ W, then any delegation to N is friendly and as a consequence N is an R-compromise, and this solution is again trivial ! There are however other political structures that can be associated to W.

(1) Let W min ⊂ W be the set of winning coalitions which are minimal for inclusion. We assume for simplicity in the following that each i ∈ N belongs to some W ∈ W min . Let (N, K 1 W ) be the political structure with N as set of agents and where a simplex is any nonempty subset of some element of W min . By our definition, K 1 W is a simplicial complex. A delegation δ i→j in (N, K 1 W ) is friendly when j is in all the minimal winning coalitions containing i (so that j is a partner of i in the terminology of [START_REF] Reny | The partnered core of a game without side payments[END_REF]).

(2) In the political structure (W min , K 2 W ) the agents are the minimal winning coalitions, and the configurations are sets of coalitions {S 0 , . . . , S r } from W min such that ∩ r i=0 S i = ∅.. Note that K 2 W is a full subcomplex of K. A delegation from S to T is friendly if for all configurations {S 0 , . . . , S r } in K 2 W containing S, the configuration {T, S 0 , . . . , S r } belongs to K 2 W . Delegating power to T means that S will not exercise its own power and therefore may be considered as removed from W.

Having dealt with nerves of simplicial complexes in the previous sections, we can see that the poltiical structures (a) and (b) are closely interrelated. Indeed, if

∪W min = N, then K 2 W = N (K 1 W ).
To see this, we notice that in the nerve of K 1 W , vertices are the maximal simplices in K 1 W , which are the minimal winning coalitions in W, that is exactly the vertices in K 2 W , and simplices are collections of minimal winning coalitions with nonempty intersection, which are the simplices of K 2 W .

Irreducible political structures. We end this section by the study of the particular case where no friendly delegations exist. Definition 6. A political structure (E, K) is irreducible if it has no friendly delegation.

It is clear that (E, K) is irreducible if and only if it has no nontrivial friendly retract or equivalently that E is a minimal representation in (E, K). The following result can be found in Barmak [START_REF] Barmak | Algebraic topology of finite topological spaces and applications[END_REF], Chap.5.

Lemma 12. Let (E, K) be irreducible and let f : E → E be contiguous to Id E . Then f = Id.

Proof: Let x ∈ E and let s be a maximal simplex containing x. Then f (s) ∪ s is a simplex, and f (x) ∈ f (s) ∪ s = s, where the last equality follows from the maximality of s. Thus, every maximal simplex containing x contains f (x) as well. If x = f (x), there would be a friendly delegation δ x→f (x) and E\{x} contradicting the irreducibility. We conclude that f = Id E .

The lemma has a useful consequence, which is stated below as a corollary.

Corollary 13. A strong equivalence between two irreducible political structures

(F 1 , K F 1 ) and (F 2 , K F 2 ) is an isomorphism.
As a corollary of Proposition 11 we have :

Corollary 14. A political structure (E, K) is irreducible if and only if it is isomorphic to N 2 (K).
In partiacular if (E, K) is irreducible then Φ is a singleton.

Existence and Structure of R-compromises

In this section we are interested in the structure of R-compromises and the existence of such solutions.

Lemma 15. Z is a representation and if Z ⊂ Z ′ then Z ′ is representation.

Proof: (i) Let r be a friendly retraction to Z, and let r ′ be the map defined by :

r ′ (x) = r(x) x ∈ E\Z ′ , x x ∈ Z ′ .
Clearly r ′ is a retraction to Z ′ . Moreover for any x ∈ s ∈ K, either x ∈ E\Z ′ and s ∪ {r( ′ x)} = s ∪ {r(x)} ∈ K, or x ∈ F and s ∪ {r( ′ x)} = s ∪ {x} = s ∈ K. We conclude that r is friendly.

Remark 2. It is not true that a subset of a R-compromise is a R-compromise. Figure 4 presents an example where there is a represented compromise but where there is no pure represented compromise.

Corollary 16. Any viable configuration that includes a R-compromise is a R-compromise.

Let R E be the set of all R-compromises and let R E be the set of all pure Rcompromises.

Proposition 17. If R E = ∅, then R E ∈ K: that is the set of all pure R-compromises is a R-compromise Proof: Assume that R E is nonempty, and let s be a maximal (for inclusion) subset of R E that also belongs to K. Let x ∈ R E be arbitrary, and let r be the retraction to {x}. Since r is contiguous to Id E , one has that r(s) ∪ s is a simplex in K. Since r(s) = {x}, it follows that {x} ∪ s is a simplex contained in R E . By the maximality of s, we have that x ∈ s. Since x is arbitrary we conclude that s = R E .

Corollary 18. R E = E if and only if E ∈ K.
The following result gives some more information about the structure of the Rcompromises even in the absence of pure R-compromises.

Proposition 19. Let I be an arbitrary subset of R E . Then there exists a subset of ∪ F ∈I F which belongs to R E and intersects all members of I.

In particular, if R E = ∅, then there exists an element of R E that intersects all elements of R E .

Proof: Let F ∈ I. The collection of simplices included in ∪ F ′ ∈I F ′ and containing F is nonempty and has a maximal (for inclusion) element which we denote c. Let F ′ ∈ I and let r be a retraction to F ′ that is contiguous to Id E , so that r(c) ∪ c is a simplex. Since r(c) ∪ c ⊂ ∪ F ′ ∈I F ′ , it follows from the maximality of c that r(c) ⊆ c. Since ∅ = r(c) ⊂ F ′ , we conclude that c ∩ F ′ = ∅. Now we prove that c is a friendly retract. Let r c be defined by:

r c (x) = x x ∈ c r F (x) x / ∈ c. If s ∈ K, then r c (s) ∪ s = r c (s\c) ∪ s ⊂ r F (s) ∪ s ∈ K. We conclude that r c is contiguous to Id E , so that c ∈ R E .
We now introduce an important concept of the theory of simplicial complexes: The political structure (E, K) is strongly contractible if (E, K) ∼ {x 0 }, for some x 0 ∈ E; this is equivalent to saying that the identity map on E belongs to the same contiguity class as some constant map. It is interesting to compare this notion with the standard concept of contractibility. (E, K) is contractible if there exists a homotopy from the identity map of |K| to some constant map. It is known that strong contractibility implies contractibility (cf. [START_REF] Spanier | Algebraic topology[END_REF] Chap. 3, Section 5) but not the converse (cf. [START_REF] Barmak | Strong homotopy types, nerves and collapses[END_REF], Example 2.13).

Proposition 20. If E has a R-compromise then E is strongly contractible.

Proof. Let x ∈ F , where F is a R-compromise. Let r be a contraction to F that is contiguous to Id E , and let r x be the contraction to {x}, that is r x is the constant map taking all elements in E to x. For any s ∈ K, r(s) ∪ r x (s) ⊂ F , so r ≈ r x . Since r ≈ Id E , one has r x ∼ Id E . We conclude that E is strongly contractible.

The converse of Theorem 20 is not true. A configuration may be strongly contractible but fail to have a R-compromise. In the example to the right in Fig. 4, no represented compromise exists in E, however it is clear that E is strongly contractible.

x 1 • x 2 • x 3 • x 4 • x 5 • x 6 • x 1 • x 0 • x 2 • x 3 • x 4 • x 5 • x 6 • Figure 4:
In the political structure to the left there is no pure represented compromise, but the configuration {x 1 , x 2 } is a R-compromise. In the political structure to the right no R-compromise exists. Both structures are strongly contractible.

A configuration may have an R-compromise but fail to have a pure R-compromise: In the example to left in Fig. 4 neither δ x 1 →x 2 nor δ x 2 →x 1 are contiguous to identity, so one can verify that no retraction to a singleton exists. Therefore no pure R-compromise exists in

E. Let φ = δ x 4 →x 1 • δ x 3 →x 1 and η = δ x 6 →x 2 • δ x 5 →x 2 . Then η • φ is a retraction to {x 1 , x 2 }.
It can be checked that φ and Id E are contiguous, and that also η and Id E are contiguous, so that η • φ is contiguous to Id E . Therefore η • φ is a friendly retraction to {x 1 , x 2 }, and {x 1 , x 2 } is a R-compromise.

x 1 • x 2 • x 3 • x 1 • x 2 • x 3 • x 4 • x 1 • x 2 • x 3 • x 4 • x 5 • Figure 5:
To the left the central point x 2 is a represented pure compromise; in the middle the central 1-simplex {x 2 , x 3 } is an R-compromise and no pure R-compromise exists; to the right there is no R-compromise.

Some other simple cases of political structures are shown in Fig. 5. Again it may be noticed that all are strongly contractible but that R-compromises exist only in two of them, and only one has a pure R-compromise. Lemma 11 provides the necessary and sufficient conditions for the existence of an R-compromise. Precisely:

Theorem 21. Let (E, K) be a political structure. The following assertions are equivalent:

(i) There exists a R-compromise, (ii) One (and therefore any) minimal representation is viable, (iii) The second nerve N 2 (K) is a simplex.

Corollary 22. In a political structure (E, K) a pure R-compromise exists if and only if N 2 (K) is a singleton.

Example 6 (Networks continued). It is easily seen that an agent with degree 1 can delegate to the only other agent to which she is connected by a configuration. If an agent x with degree > 1 can delegate to another agent y, then by definition any agent connected to x is also connected to y; if z = y is another agent connected to x, then {x, z} is viable but {y} ∪ {x, z} is not a viable configuration.

An acyclic connected graph is a tree. For (E, K) be a connected graph, if (E, K) has a pure R-compromise, then (E, K) is a tree where all points except one have degree 1. This follows immediately from the preceding remarks. If (E, K) has an R-compromise, then there are at most two agents x 1 , x 2 with degree ≥ 1, and a {x 1 , x 2 } is an Rcompromise. Indeed, it is seen immediately that if there are exactly two vertices x 1 , x 2 with degree greater than one, then {x 1 , x 2 } is a R-compromise. If there are more than two vertices with degree greater than one, then neither of these can delegate to any other. Consequently the case of two vertices with degree ≥ 1 is the only case which is compatible with the existence of an R-compromise. The two cases are illustrated in Fig. 6, where the graph to the left has a pure Rcompromise and the graph to the right represents the case where the R-compromise is an edge. It follows from our discussion that except for the number of edges to points with degree 1, these are the only possible cases of nonempty R-compromises. in the sense that the representing agent enters into all the viable configurantions where the agent who withdrew might have been present. Unfortunately, R-compromises may not always exist; indeed we saw that the existence of R-compromises imply that the political structure, viewed as a simplicial complex, is strongly contractible, in itself a rather restrictive property, but this was not enough, since even a strongly contractible political structure may fail to exhibit R-compromises.

x 0 • x 4 • x 3 • x 1 • x 2 • x 0 • y 0 • x 1 • x 2 • y 1 • y 2 •
This shortcoming has to do with the restrictions that we have put on a delegation. In an R-compromise the delegation δ x→y amounts to removing x from the outcome, but it does not imply the withdrawal of x from the search for a compromise. This happens since any other other delegation δ x ′ →y ′ , in order to be friendly, must secure that y ′ ∪ s be viable for all s that contain x ′ , and the simplex s may contain x. In this sense x is not totally absent from the seach for a compromise even after the delegation.

In the following we present a slightly weaker notion called delegated compromise, in which a delegation from x to y implies that x withdraws from the process, so that further delegations need not take x into account. This clearly makes the search for a solution easier.

Definition 7. A configuration F of (E, K) is a narrowing of E if there exists a decreas- ing sequence E = F 0 ⊃ F 1 ⊃ • • • ⊃ F n = F and friendly retractions r i of (F i-1 , K F i-1 ) onto F i for i = 1, . . . , n
Notice that the retraction r i are relative to the simplicial complex (F i-1 , K F i-1 ). It follows from Proposition 7 that this is equivalent to require that the associated ri : (F i-1 , K F i-1 ) → (F i , K F i ) is a strong equivalence for i = 1, . . . , n. Therefore we have the following:

Proposition 23. If F is a narrowing of (E, K) then (F, K F ) is strongly equivalent to (E, K).
In view of Proposition 3 any narrowing F can be obtained by a progressive delegation δ x 1 →y 1 , . . . , δ xp→yp such that δ xp→yp • • • • • δ x 1 →y 1 (E) = F and such that the restrictions of δ x k →y k to E \ {x 1 , . . . , x k-1 }, k = 1, . . . , p, are friendly. We are ready to introduce the main concept of solution of this section:

Definition 8. A configuration F of (E, K) is a delegated compromise (D-compromise for short) if F a narrowing of (E, K) and F ∈ K.
The obvious way of looking at delegated compromises is to see them as repeated compromises; in the first step, F 1 is an R-compromise for E, in the next step an R-compromise F 2 in F 1 is obtained, etc. It should be noted however, that the sets F i themselves need not be R-compromises in the original structure since we do not demand that they belong to K. It is clear that an R-compromise is a delegated compromise, but the converse need not be true. Note that a D-compromise must be a viable configuration (a simplex).

The D-core. Given a political structure, one can try to search for a D-compromise following a process of successive friendly delegations, but is there a narrowing that leads to a positive outcome, namely a viable configuration? In order to capture the best narrowing that can be obtained from such a search, we need to know what are the minimal configurations that can be achieved as outcome.

The idea of best narrowing is captured by the following: Definition 9. A configuration F of (E, K) is a delegated core (shorthand: a D-core1 ) of (E, K) if F is a narrowing of (E, K) and (F, K F ) is irreducible. Now we can state and prove the main result about D-cores of political structures:

Proposition 24. Let (E, K) be a political structure. Then the following hold:

(i) (E, K) has a D-core, (ii) (E, K) is strongly equivalent to its D-core, (iii) the D-core is unique up to isomorphism.

Proof: (i) Either (E, K) is irreducible, in which case it is itself a D-core, or one can find a non trivial friendly retract F 1 . If F 1 , K F 1 is irreducible, otherwise we repeat the operation on (F 1 , K F 1 ) and so on we thus construct a narrowing

E = F 0 ⊃ F 1 ⊃ • • • ⊃ F k ⊃ • • • .
Since the sequence is strictly decreasing and E is finite we reach eventually an index n with irreducible simplicial complex (F n , K Fn ). Clearly this is a D-core. (ii) follows from Proposition 23. It remains only to prove (iii): Suppose that (F 1 , K F 1 ) and (F 2 , K F 2 ) are two D-cores of (E, K). Then by (ii) we have that

(F 1 , K F 1 ) ∼ (E, K) ∼ (F 2 , K F 2 ),
and by the Corollary to Lemma 12, (F 1 , K F 1 ) and (F 2 , K F 2 ) are isomorphic.

Using the D-core as a formal description of conflict resolution may seem attractive in view of these results. On the other hand, it should be remembered that the D-core does not point to a single configuration, rather it is an indication of the limits beyond which conflicts cannot be solved by delegation alone.

Existence of D-compromises. We now investigate the existence of a D-compromise in a political structure. By definition, if the D-core is viable, then it is a D-compromise. The D-core always exists (Proposition 24); on the other hand, being irreducible, if it is viable it must be a singleton. These considerations are made precise by the following propostiion.

(ii) each x ∈ E is a pure delegated compromise in (E, K), (iii) the simplicial complex K is strongly contractible, (iv) the topological space |K| is contractible.

Proof: (i)⇒(ii). Let x ∈ E be arbitrary, and for each y ∈ E, define the distance from y to x, d(y, x), as the number of edges in a path from x to y; since (E, K) is a tree, d(x, y) is well-defined. Let D = {x 1 , . . . , x k } ⊂ E be the set of all vertices in E with maximal distance from x; then each x ∈ D has degree 1. Let E 1 = E\D. Each x ∈ D is connected to some y(x) ∈ E which does not belong to D since (E, K) is a tree. There is a friendly delegation δ x→y(x) from x to the vertex y(x), and the composition δ x k →y(x k ) • • • • • δ x 1 →y(x 1 ) is progressive and defines a narrowing from (E, K) to (E 1 , K E 1 ). The political structure (E 1 , K E 1 ) is a graph which is connected and has no cycles, hence it is a tree. Moreover E 1 has strictly less elements than E. Repeating the above procedure, we successively obtain substructures (E i , K E i ) and narrowings (E i , K E i ) → (E i+1 , K E i+1 ), for i = 1, . . . , p, untill we obtain a structure (E p+1 , K E p+1 ) in which no agent has degree 1. But a tree without vertices of degree 1 consists of the point x, which consequently is a delegated compromise of (E, K).

(ii)⇔(iii) is a consequence of Theorem 25, (iii)⇒(iv) is a consequence of Lemma 2 in Chapter 3, Section 5 in Spanier [START_REF] Spanier | Algebraic topology[END_REF], and finally (iv)⇒(i) is Lemma 1 in Chapter 3, Section 7 in Spanier [START_REF] Spanier | Algebraic topology[END_REF].

It is seen that delegated compromises exist if and only if the political structure is strongly contractible. The topology of the political structure, and precisely its strong homotopy, is the basic determinant for the possibility of a compromise. When delegated compromises exist, they may take the form of single agents (pure D-compromise) or a viable configuration. The question whether some D-compromise may seem unacceptable (dictatorial) is irrelevant in our setting. This is due to the fact that in our model we are concerned only by the viability of the solution. In this sense the model may look underdetermined: It may be the case that the political environment requires some further conditions for viable configurations to be acceptable, as seen in the empirical case in Example 4, where the confessional composition of any compromise is an indispensable pre condition which must be satisfied by a viable configuration. Future research must therefore consider viability with some formal kind of acceptability.

Concluding remarks

In the present paper, we have presented a formal model of compromise in contexts of political decision making. This model is that of a political structure, mathematically a simplicial complex in which vertices are interpreted as political agents and simplices as viable configurations. In this context, the fundamental motive of action is founded on the notion of delegation. Delegation can be formulated as a mapping taking the delegating agent to another agent, to whom power is to be transferred, and the idea of handling over influence to another agent is formalized through the notion of friendliness. The analysis of delegation and its consequences can now exploit the theory of maps between simplicial complexes, and due the discrete character of the model the relevant mathematics appear to be that of strong homotopy.

Two fundamental types of delegation are of interest. In the first one, influence may be transferred but the receiver must be able to respond back and cannot delegate further. This gives rise to the notions of representations and R-compromises. Allowing for repeated delegation opens up for further concentration of power, expressed by notions of D-cores and D-compromise.

Since a general theory must be assessed in terms of the insights, which it offers in particular applications, we have briefly considered some such applications of the theory, namely to TU and in particular simple games, to effectivity functions, and to network models of political influence. The notion of delegation indicates how to analyse situations where decisions are not immediately obtainable so that compromises are called for. We have only touched upon this area of research where there are several directions of future developments of the theory. One of such directions is to consider situations where acceptability conditions in addition to viability are required for a compromise.

The approach to political structures through simplicial complexes may be applied to other aspects of decision making than delegation, as it has been done in the present paper. Instead of power being handed over from some agents to other, one may consider the exclusion of some agents by the others, a process which can also be treated using suitable notions of homotopy. This again is a topic of future research.

Figure 1 :Proposition 1 .

 11 Figure 1: Point centrality in a network: The vertex v 0 with degree 4 is connected to all the other vertices with an edge, whereas the other vertices are connected only to v 0 .

Figure 3 :

 3 Figure 3: Political structures represented as points connected by lines (1-simplices) or triangles (2-simplices).In both of the above political structures, the retraction onto x 0 is contiguous to the identity. Consequently x 0 is a pure R-compromise.

Figure 6 :

 6 Figure 6: The graph in the left panel represents the class of graphs with a pure R-compromise, and the one in the right panel represents those with an R-compromise containing two agents. Any graphs with a nonempty R-compromise belongs to one of the two classes.

Delegated compromiseIn the previous section we have seen that R-compromises, where the agents that are withdrawn from the political contentions remain represented by suitable representatives,

The term "core" is used byBarmak [2], Chap.5, for the same notion in the context of abstract finite spaces.

Proposition 25. Let (E, K) be a political structure. Then the following hold:

(i) (E, K) has a D-compromise, (ii) (E, K) has a pure D-compromise, (iii) (E, K) is strongly contractible, (iv) The D-core is a singleton.

Proof: (i)⇔(ii). If F is a D-compromise, then, since F is a simplex, there exists a friendly retraction to any of its points {x}. It follows that {x} is a narrowing of (E, K) an therefore that {x} is a pure D-compromise.

(ii)⇒(iii). Since {x} is a narrowing, we have that (E, K) ∼ {x}, so that (E, K) is strongly contractible.

(iii)⇒(iv). Let F be a D-core. From (E, K) ∼ (F, K F ) and (E, K) ∼ {x}, we get that (F, K F ) ∼ {x}. Since both (F, K F ) and {x} are irreducible, they are isomorphic by the Corollary to Lemma 12.

(iv)⇒(ii) is straightforward, since the D-core being a singleton is also a pure Dcompromise.

We may exploit the results on nerves of simplicial complexes obtained earlier to obtain additional information about narrowings of the political structure.

If (E, K) is not irreducible, N 2 (K) is not isomorphic to (E, K), but the number of vertices in N 2 (K) is no greater than that of (E, K), and one may proceed to the derived nerves N k (K)) for k > 3. Clearly, there will be some number d ≥ 1 such that N 2d+2 (K) and N 2d (K) have the same number of vertices. By convention N 0 (K) = K. Definition 10. Let (E, K) be a political structure. The depth of delegation of (E, K), written d (E,K) , is the smallest number d such that the N 2d+2 (K) and N 2d (K) have the same number of vertices.

Using Lemma 11, we get the following proposition.

Clearly the depth provides the smallest number of iterated representations in a narrowing to achieve the D-core.

In the remaining part of this section, we consider a political structure which is a graph, as considered in Example 2. We saw the existence of a pure R-compromise had very far-reaching consequences. With delegated compromises, the situation is quite different. Here the existence of a delegated compromise has rather far-reaching consequences.

Proposition 27. Let (E, K) be a political structure which is a graph. The following are equivalent:

(i) (E, K) is a tree,


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INTRODUCTION

The Large Ultraviolet/Optical/Infrared Surveyor (LUVOIR) [START_REF] Bolcar | The Large UV/Optical/Infrared Surveyor (LUVOIR): Decadal Mission concept design update[END_REF] is one of four large mission concepts, which were selected for consideration by the 2020 Astronomy and Astrophysics Decadal Survey. The current baseline architecture implies building of a large space telescope with an unprecedented 15-m primary mirror providing a huge collecting area. The LUVOIR mission will cover a number of groundbreaking scientific targets. It will perform a search and the characterization of habitable conditions and signs of life on dozens of potentially habitable worlds beyond our Solar System. LUVOIR will have the ability to characterize hundreds of transiting and directly imaged planets and will revolutionize our understanding of all classes of extrasolar planets and the common threads that connect them. When used for observations of the bodies within the Solar System, LUVOIR can provide up to about 25 km imaging resolution in visible light for Jupiter, monitor atmospheric dynamics in the outer planets, provide high-resolution imaging and spectroscopy of Solar System comets, asteroids, moons, and Kuiper Belt objects and perform many other tasks. In the field of cosmology LUVOIR can address such questions as the nature of dark matter and galaxy formation processes on small size scales by imaging fainter and smaller structures in the universe than ever before. From the point of galaxy evolution studies, LUVOIR's unprecedented resolution will resolve stellar populations in star-forming regions of galaxies at distances up to 10-25 Mpc, accessing more diverse galaxy morphologies, sizes, and cluster environments. In addition, the UV capability provided by LUVOIR will be vital for discriminating among our theories of star and planet formation. Finally, the UV spectropolarimetry capability of LUVOIR will allow us to study, e.g., linear polarization and depolarization effects from circumstellar disks and stellar magnetic fields. This list of science goals is not exhaustive, but it clearly shows the huge scientific impact, which the mission may have.

According to the current baseline the telescope payload consists of 4 instruments. In the current paper we consider the optical design of high-resolution UV spectropolarimeter called POLLUX, which represents the European contribution to the mission.

BASELINE OPTICAL ARCHITECTURE

1.1.

Top-level requirements Based on the key science goals of the mission the science groups of the POLLUX consortium have defined the following essential top-level requirements for the instrument (see Table1). The only type of spectrograph optical design that allows to reach the target high spectral resolving power is an echelle spectrograph [START_REF] Eversberg | Spectroscopic Instrumentation: Fundamentals and Guidelines for Astronomers[END_REF] . Such a spectrograph consists of collimating and focusing optics and two dispersing elements. The main disperser is an echelle gratinga grating with relatively low spatial frequency, working with a high incidence angle in a large number of higher diffraction orders. It forms the corresponding number of overlapped spectral images stretched in the X direction. The second disperser or cross-disperser is an element which separates the orders of the echelle grating. Its dispersion operates along the perpendicular Y axis.

1.2.

Basic design solutions The instrument optical design architecture is based on the following assumptions about the components and technologies. We must note that most of the points below are provided by the LUVOIR project.

1) We assume that the telescope is a three-mirror anastigmate (TMA) with a collecting area of 139-173.4 m 2 and it operates with an f-ratio of 20.48. We also assume the instrument picks off the beam close to the field of view centerthe field X and Y coordinates are (0.015°, -0.020°). The telescope residual aberrations are neglected.

2) It is assumed that a pinhole is placed at the instrument entrance. When a pinhole is used instead of a slit, a simpler aberration correction conditions may be used and all the channels can be rotated around the chief ray. The latter feature makes the design more flexible.

3) The working spectral range should be split into sub-ranges, because it exceeds one full octave. Such an approach also allows to achieve high spectral resolving power with feasible values of the detector length, the camera optics field of view and the overall size of the instrument. In addition, it allows to use dedicated optical elements, coatings and detectors for each band and obtain a gain in efficiency. The entire spectral range is thus divided into far ultraviolet (FUV), medium ultraviolet (MUV) and near ultraviolet (NUV). Each of the bands is recorded by a dedicated channel.

4) The MUV and NUV channel are separated by means of a dichroic splitter. It was demonstrated that such a splitter can have a high efficiency [START_REF]GALEX: Chapter 1 -Instrument overview[END_REF] . Also, in comparison with other separation techniques such as a spatial splitter or a rotating mirror, a dichroic splitter allows to work in two bands simultaneously and use the full aperture thus achieving the high resolving power with relatively a small collimator focal length.

5) Due to the channels separation method, chosen above, it is impossible and unnecessary to provide a considerable overlap between the MUV and NUV bands. It is assumed to be a sharp boundary. The longwave limit for the NUV is the same as that for the entire instrument, i.e. 390 nm. Thus, to have one full octave in the NUV we set the MUV/NUV boundary at 195 nm.

6) Currently there are no dichroic splitters operating in the FUV below the Lyα line and there is no data that such an element will become possible in the future. So we propose to use a flip mirror to feed the FUV channel. It should be located right before the dichroic splitter, because the dichroic substrate cannot transmit the FUV irradiation.

7) Accounting for the chosen FUV splitting method, the FUV and MUV boundaries are set as follows. One of the most important spectral lines in the UV is the Lyα line centered at ~121.5 nm. Depending on the conditions its width can reach several nm. Therefore, the shortest limit for the MUV band is set at 118.5 nm, while the longest one for the FUV is 124.5 nm. Thus, the Lyα line appears in both channels.

8) The shortest wavelength for the FUV is the same as that for the entire instrument and equals to 90 nm. This value strongly depends on the main telescope throughput and may be reconsidered in the future.

9)

In each channel the beam is collimated by an ordinary off-axis parabolic (OAP) mirror. The off-axis shift and the corresponding ray deviation angle are chosen in such a way that the distance between the entrance pinhole and the echelle is large enough to place the polarimeter and corresponding mechanical parts etc.

10) Echelle grating works in a quasi-Littrow mounting. The exact values of the grooves frequency and the blazing angle are computed to obtain the required dispersion and subsequently the spectral resolving power. No boundary conditions were directly applied to these computations.

11) The cross-disperser in each channel operates also as a camera mirror, so it is a concave reflection grating. This approach allows to decrease the number of optical components and increase the throughput. In order to correct the aberrations we suppose that the cross-disperser's surface is a freeform and it has a complex pattern of grooves formed by holographic recording (i.e. interferometric technique using two coherent point sources).

12) The detector is represented only by dimensions of its sensitive area and the pixel size. The baseline configuration uses δ-doped CCD's in all channels. The achievable parameters were recently reported in a number of publications [START_REF] Nikzad | High-efficiency UV/optical/NIR detectors for large aperture telescopes and UV explorer missions: development of and field observations with delta-doped arrays[END_REF][START_REF] Nikzad | Single Photon Counting UV Solar-Blind Detectors Using Silicon and III-Nitride Materials[END_REF] .

13) Polarimeters are placed right after the splitters in each channel. The polarimeters should be retractable at least in the MUV and NUV channels. Such a solution will provide a considerable gain of throughput in the pure spectroscopic mode. For the FUV polarimeter only the modulator part is removable.

14) The change of optical path caused by the polarimeter removal is compensated by means of a change of the OAP mirror.

15) The polarimeter principle design was chosen separately for each channel accounting for the technological feasibility and the expected performance: a birefringent modulator and a Wollaston prism for the NUV (similar to the design for the Arago instrument 6 ; see also Le Gal et al., these proceedings); a three-mirror modulator and a Wollaston prism for the MUV; and a three-mirror modulator and a Brewster angle analyzer for the FUV.

Accounting for all the assumptions and basic design solutions listed in the previous paragraph we propose the baseline optical architecture shown in Fig. 1 (note that the linear and angular values on this schematic drawing are not to scale).

OPTICAL DESIGN

For the development of the optical design with real optical components we made two additional assumptions. First, in order to account for possible misalignments and manufacturing errors as well as for the resolution changes due to switching from the pure spectroscopic mode to the spectropolarimetric one, the target spectral resolving power was set to 130 000 (i.e. 110% of the requirement). Second, the MUV and NUV optical schemes were unified as much as possible: they have identical collimators, the same camera focal length, the same detector format and similar echelle blazing angles. It will simplify the manufacturing, optical testing and assembling.

Optical design overview

The optical design of each channel was optimized according to the conditions described above. The polarimeter units were integrated into the same models. The optimization was performed with the standard tools provided by the Zemax software. In order to account for different wavelengths and polarization states 24 configurations representing 4 orders were used. The spot RMS size was used as the main image quality criterion with the weight coefficient of 5 assigned to its X size. The boundary conditions were used to control the image centering, the image lines (i.e. orders and polarization components) separation, the air gap between the echelle grating and detector and the arms lengths in the holographic grating recording geometry. In addition, for the MUV and NUV designs the Wollaston prisms wedges edge thicknesses are constrained.

The MUV channel optical design general view is shown as an example in Fig. 2 (note that the rays propagation direction differs from the actual one because of the software conventions). The NUV and FUV channels repeat the same principal geometry, though the polarization units are different. The key design solutions made for each of the components are described in the next sections.

1.4.

Optical system components The qualitative description for the optical components design is given below in the order of radiation propagation:

1) The splitters are placed as close to the focal point as possible in order to decrease their size: the flip mirror is located 20 mm from the focus and the distance from the focus to the dichroic is 35 mm.

2) The polarimeters should have the minimal size in order to decrease the defocusing and chromatic aberrations and simplify the observation mode switching. The NUV polarimeter is inspired by the Arago design 6 (see Le Gal et al., these proceedings).. The final design of the MUV and FUV polarimeters will depend on the exact materials properties. In the pure spectroscopic mode the MUV and NUV polarimeters are removed from the optical path; the FUV polarimeter is static.

3) All the collimators are usual OAP mirrors. The collimators may introduce additional aberrations because of the polarization components separation. For the defocus compensation when switching from the spectropolarimatric to the spectroscopic mode, the OAP is replaced by another one with a translation platform.

4) The MUV and NUV echelle gratings have similar sizes and blazing angles as well as the frequencies that differ by a factor of ~2. All of these parameters values are achievable with the current technology level [START_REF] Mcentaffer | First results from a next-generation off-plane X-ray diffraction grating[END_REF] . The FUV echelle grating is unique because of its frequency and blazing angle.

5) Each of the cross-dispersers is a concave freeform holographic grating. It is a novel type of optical elements having high aberration correction capabilities. The surface shape is described by a vertex sphere and 5 Zernike terms corresponding to the main aberrations. Such a surface has no axial symmetry, but it is still symmetrical with respect to the YZ plane. The freeform complexity is assessed by deviation from the best fit sphere (BFS). For all the cross-dispersers this deviation is relatively small. The grooves are formed by recording of an interference pattern from two coherent point sources. A sufficient aberration correction is possible only if the spectral components at the cross-disperser's surface are separated. However, it leads to the grating's aperture growth.

6) In the current study the detector is represented as an array of ideal sensitive elements. As it was mentioned before, the MUV and NUV detectors have the same geometry.

The quantitative description of the optical schemes components is given below in Table 2.

L y        ' , (1)
where Δy' is the instrument function full width at the half maximum (FWHM) and ∂λ/∂L is the reciprocal linear dispersion. A typical IF is shown in Fig. 4 (left part).

The spectral resolving power is defined as

,     R ( 2
)
where λ is the wavelength.

At the same time the theoretical limit is given by 8

     tg D tg D R TE col 2 , ( 3
)
Here Dcol is the collimated beam diameter, φ is the echelle blazing angle, DTE is the effective diameter of the telescope primary mirror, δ is the angular size of the entrance slit or pinhole. The actual resolution is defined by the smallest value between the ones obtained from ( 2) and ( 3).

The diagram showing the results of spectral resolving power computations for all the channels is given in Fig. 5. The separate datapoints correspond to the values computed with Equations ( 1)-(2) using the aberrated IFs. The horizontal lines shows the values obtained with Equation (3) as well as the target value (see Table 1) and the corrected target value accounting for possible misalignments and the mode change (see Section 3). As one can see, the spectral resolution requirements are fulfilled for all wavelengths. There is only one exclusion at the edge of the NUV band. This resolution loss appears because of the polarization ray splitting and clearly demonstrates the optical design sensitivity. 1.6. Throughput Below we consider the efficiency of each element type point-by-point along the optical train. The data used for this computations was taken from the literature or, when possible, found with simplified simulations. We must note that in order to avoid issues with sampling difference, noises and high frequency irregularities in the initial data was interpolated. So all the spectral dependences are presented by smoothened envelope curves.

1) The pick-off mirror has to be efficient for all channels (MUV, NUV, and FUV). Therefore, a coating with a SiC capping layer on a metal base layer will be required, namely -Al+MgF2+SiC. It will provide 30% to 40% in the FUV channel and higher values in the NUV and MUV channels [START_REF] Larruquert | Multilayer coatings with high reflectance in the extreme-ultraviolet spectral range of 50 to 121.6 nm[END_REF] .

2) For the dichroic splitter characteristics, we used the data from the GALEX splitter [START_REF]GALEX: Chapter 1 -Instrument overview[END_REF] . For simplicity we assumed that the throughput curves shapes remain the same, but they are shifted to fit with the splitting border of 195 nm (instead of 178 nm for GALEX).

3) The current assumptions concerning the reflective coatings are as follows. For the reflective elements in the FUV a single layer of SiC will be used. Its reflection can exceed 40% in a nearly-normal incidence [START_REF] Fernández-Perea | Transmittance and reflective coatings for the 50-200 nm spectral range[END_REF] . The baseline conception also assumes that the rest of the mirrors use the same coating as the main LUVOIR telescope. It will be, most likely, a multilayer coating consisting of Al+LiF+AlF3. It has high reflectance in the NUV and MUV; also it is notable for its stability [START_REF] Bolcar | UV Coatings and Short-wavelength Cutoff[END_REF][START_REF] Balasubramaniana | Aluminum Mirror Coatings for UVOIR Telescope Optics including the Far UV[END_REF] .

4) The birefringent material used in the MUV and NUV polarimeters is MgF2. Its properties are described in a number of sources [START_REF] Wakaki | Physical Properties and Data of Optical Materials[END_REF] . 5) The three-mirror modulators in the FUV and MUV operate with high incidence angles. Therefore, their efficiencies were computed separately. It was supposed here that the polarimeter's mirrors for the MUV are coated by Al+LiF and those for the FUV are with a SiC layer, which behaves as a dielectric. Also, it is supposed that the Brewster angle analyzer in the FUV is completely lossless. 6) The customized echelle gratings can be produced on a Si substrate by means of photo-lithography and e-beametching [START_REF] Mcentaffer | First results from a next-generation off-plane X-ray diffraction grating[END_REF][START_REF] Marlowe | Modeling and empirical characterization of the polarization response of off-plane reflection gratings[END_REF] . Due to the material properties the peak of the triangular groove profile has an angle of 70.6º instead of 90º. The echelle gratings coating is 17 nm of LiF on Al. In addition, for the efficiency estimation the deviation from the Littrow mounting was neglected. We must emphasize that at the current stage possible signal summation for the overlapping parts of adjacent orders is not accounted for. Thus the minimum efficiency may be increased, though the maxima will remain the same.

7) The cross-dispersers grooves efficiency was computed with use of the RCWA (rigorous coupled-wave analysis) method, implemented in the GD-calc software [START_REF] Johnson | Grating Diffraction Calculator (GD-Calc® ) -Demo and Tutorial Guide[END_REF] . The computation was performed for all the control wavelengths and 2 polarization states over a 3x3 probe grid covering the pupil. It accounted for the actual ray direction and the freefrom surface normal, but the local changes of the grooves frequency and curvature were neglected. It was supposed that the groove profile is triangular blazed for the central wavelength for each subrange and the profile depth is ½ of this wavelength. Below average values computed for the 9 rays and for 2 polarizations are used.

8) All the detectors in the baseline design are δ-doped CCD. They have relatively high quantum efficiency down to the extreme UV [START_REF] Nikzad | High-efficiency UV/optical/NIR detectors for large aperture telescopes and UV explorer missions: development of and field observations with delta-doped arrays[END_REF][START_REF] Nikzad | Single Photon Counting UV Solar-Blind Detectors Using Silicon and III-Nitride Materials[END_REF] . Here we apply the QE data for an uncoated CCD (otherwise, the efficiency in the FUV will be significantly decreased) and use an envelope curve. 9) Finally, we account for the reflection losses inside the telescope. It was assumed that the telescope mirrors are coated by the Al+LiF+AlF3 coating and the working beam experiences 4 bounces with a small angle of incidence [START_REF] Bolcar | The Large UV/Optical/Infrared Surveyor (LUVOIR): Decadal Mission concept design update[END_REF] .

The overall throughput was found as the product of all the elements efficiencies in the corresponding channels. Then it was converted to the equivalent effective area by multiplication by a geometrical factor [START_REF] France | The LUVOIR Ultraviolet Multi-Object Spectrograph (LUMOS): instrument definition and design[END_REF] equal to Ageom=135 000 cm 2 . It allows to account for the LUVOIR's large collecting area. Two cases were consideredcoupling with the current telescope and coupling with an ideal 100%-transmitting telescope. The latter computation is necessary to assess the performance of POLLUX itself. The results are shown in Fig. 6.

Even with the relatively low throughput the instrument can show a high performance due to the huge collecting area of the telescope. The maximum values of the effective area are 546, 3917 and 38569 cm 2 for the FUV, MUV and NUV channels respectively in the cases when all the losses inside the telescope and polarimetric unit are accounted for. These values should be sufficient for the most of our scientific goals. One can note that the FUV throughput drops to zero below 103 nm. This is explained by the telescope's cut-off and represents a point of ongoing discussion with the LUVOIR team and future improvement.

COUPLING WITH THE TELESCOPE

POLLUX will be mounted behind the LUVOIR's deployable primary mirror in one of the 4 dedicated volumes (see Fig. 7). The pick-off mirror directs a portion of the telescope beam to the instrument entrance. The angular coordinates of the entrance pinhole projection to the sky are (-0.015º, -0.020º).

CONCLUSIONS AND FUTURE WORK

In the present paper we presented the preliminary design concept of the POLLUX instrument, a high-resolution UV spectropolarimeter for the LUVOIR space mission project. Thanks to the recent developments in optical components and detectors technologies it will be able to reach the target performance and cover a number of groundbreaking science cases.

During this preliminary study a number of technological risks and critical points, which define the future work, were found. Below we provide a short overview of them:

1) There is a risk related to the reflective coatings properties ambiguity. All the coatings, especially the ones used in the FUV channel should be examined under different conditions, including incidence angles and polarization states and be space-qualified. Also, the deposition technique [START_REF] Fludra | SPICE EUV spectrometer for the Solar Orbiter mission[END_REF] and alternative materials should be considered in details [START_REF] Del Hoyo | Enhanced aluminum reflecting and solar-blind filter coatings for the farultraviolet[END_REF] .

2) It was assumed that the dichroic splitter has a performance similar to that of the GALEX dichroic. In the future, the reflectivity and transmission as well as the working angles of the POLLUX dichroic should be studied separately. Moreover, the FUV channel will be placed in the direct propagation after the flip mirror, while the MUV/NUV will be placed in reflection in order to increase the FUV throughput.

3) The size of the FUV echelle grating exceeds the maximum size of a grating ever produced with the chosen technology. So this is a technological risk and a subject for future studies.

4) The possibility to fabricate a holographic freeform grating with triangular grooves must be demonstrated in practice. Scaling of such an element represents a separate technological challenge. We should note that the backup solution for this element is a variable line spacing ruled freeform grating.

5) The detector parameters are also subject to a further investigation. Use of CMOS instead of CCD is an option. Because the detectors have large dimensions, we should study options of tiling. Furthermore, in the future one may consider the detector's anti-reflective coating properties.

6) Finally, a number of necessary analyses have not been performed yet including tolerance analysis and ghost analysis.

Figure 1 .

 1 Figure 1. POLLUX instrument baseline architecture schematic diagram: 1-pinhole, 2-flip mirror, 3-dichroic splitter, 4birefringent modulator, 5-Wollaston analyzer, 6-OAP collimator (2 mirrors for different operational modes, replaceable by translational mechanism), 7-echelle grating, 8-concave freeform holographic grating, 9-detector array, 10-3-mirror modulator, 11-Brewster angle analyzer.

Figure 2 .

 2 Figure 2. MUV channel optical design: leftgeneral view, rightzoom-in of the polarimetric unit.

  spectral resolutionDuring the optical design development and optimization the image quality is controlled through spot diagrams. They are used for the image centering on the detector and maintain the orders and polarization components separation as well as for the aberrations minimizations. An example of full-frame spot diagram for the MUV channel is given in Fig.3. It clearly shows the detector filling and the lines separation.A typical spot diagram for an individual configuration is shown in Fig.4(right part). Since the resolution is defined only by the line spread function along the X axis, the spot Y size had ~5-time lower weight coefficients during the numerical optimization. Therefore, the spot diagrams are elongated along the Y axis.

Figure 3 .

 3 Figure 3. The MUV channel image: leftthe detector filling diagram, rightthe zoom-in showing the orders separation and the polarization components separation in each order. The final control on the image quality and the spectral resolution computation were performed with use of instrument functions (IF). The IF in this case is a relative distribution of illumination in the pinhole monochromatic image measured along in the XZ section. It is computed as a convolution of the optical system line spread function and the pinhole rectangular function. It was assumed that the spectral resolution in wavelength units equals to

Figure 4 .

 4 Figure 4. Typical optical quality indicators at 377.8 nm: leftinstrument function (FWHM is 31.2 µm), rightspot diagram (RMS radius is 25.1 µm).

Figure 5 .

 5 Figure 5. Spectral resolving power diagram for the POLLUX channels. The theoretical limits are: FUV -129979, MUV-130008, NUV-130855.

Figure 6 .

 6 Figure 6. Overall efficiencies and the corresponding effective areas for the POLLUX channels, top-to-bottom -FUV, MUV, NUV.

Figure 7 .

 7 Figure 7. Location of the POLLUX dedicated volume with respect to the telescope and other payload instruments. The maximum dimensions of the volume are 3600x2147x3688 mm 3 (LxWxH) (courtesy of the Goddard Space Flight Center).

Fig. 8 .

 8 Fig.8. shows the arrangement of channels and optical components inside the volume. This scheme will be used as a starting point for the opto-mechanical design.

Figure 8 .

 8 Figure 8. Optical system of POLLUX arranged inside the dedicated volume.

Table 1 .

 1 High-level requirements to the spectropolarimeter.

	Parameter	Requirement	Goal
	Shortest wavelength	90 nm	90 nm
	Longest wavelength	390 nm	visible
	Spectral resolving power	120 000	200 0000
	Spectral length of the order	6 nm	≥6 nm
	Polarisation mode	Circular+linear (= QUV)
	Aperture size	0.03"	0.01"
	Observing modes	with and without polarimetry,
		i.e. spectropolarimetric and pure spectroscopic modes

Table 2 .

 2 Design parameters of the optical system components.

	Channel	NUV	MUV	FUV
	Wavelength range	195-390 nm	118.5-195 nm	90-124.5 nm
	Polarimeter
	Modulator	3 pairs of MgF2 plates	3 mirrors, the first	3 mirrors, the first
			incidence angle=47º	incidence angle=80º
	Thicknesses, mm	0.8 mm thickness,	50 (x2)	20 (x2)
		0.2 mm separation
	Clear aperture max, mm	4.4	8x12	4.2x23
	Analyzer	MgF2 Wollaston prism	MgF2 Wollaston prism	SiC plate at the
				Brewster angle
	Prism angle, º	5.027	3.175	33.365
	Clear aperture max, mm	6.6	9.6	5.5x6.6
	Wedge thickness	1/0.716	1/0.74	N/A
	center/edge, mm
	Rays separation at	200	200	N/A
	detector, µm
	Collimator
	Focal length, mm	1702.692	1702.692	3109.396
	Clear aperture, mm	88	87	154.5
	Decenter, mm	120.2	120.2	210
	Deviation angle, º	4.041	4.041	3.873
	Echelle
	Frequency, 1/mm	144.95	289.4	589.7
	Angle, º	59.551	59.388	42.779
	Clear aperture XxY, mm	89.2x168.6	87.4x167.4	154.7x210.6
	Orders	31-61	31-50	19-26
	Order spectral length, nm 3.2-12.2	2.4-6.1	3.4-6.3
	Typical orders separation,	0.7	0.9	2.6
	mm
	Cross-disperser
	Focal length, mm	1200	1200	2100
	Clear aperture XxY, mm	216x99.5	215.4x98.3	416.1x165.4
	Asphericity RMS/PTV,	2.15/3.21	2.27/3.36	2.4/3.7
	µm
	Frequency, 1/mm	105.4	212.3	272.6
	Rec. sources coordinates,	(30.399,1614.151)and	(100.194,1913.946)and	(163.034,2534.301) and
	(y,z) mm	(-53.760,1648.412)	(-99.558,1936.898)	(-179.888,2606.124)
	Detector
	Type	δ-doped CCD	δ-doped CCD	δ-doped CCD
	Size XxY, mm	131x24	131x19	202.7x19.4
	Pixel size, µm	13	13	13
	Sampling	2.4	2.4	2.3

ACKNOWLEDGEMENTS

The authors acknowledge the entire LUVOIR and POLLUX consortia for their contributions. We would like to especially thank Kevin France (Univ. of Colorado -Boulder), Matthew Bolcar (NASA GSFC), Aki Roberge (NASA GSFC), Randall McEntaffer (Penn State Univ.), Chris Evans and David Montgomery (UK ATC). Eduard Muslimov acknowledges the support from the European Research council through the H2020 -ERCSTG-2015 -678777 ICARUS program.