A Formal Space of Paths Introduction 1 Covering System - CiteSeerX

A Formal Space of Paths Thierry Coquand Chalmers University Abstract, April 1996

Introduction In general, a connected locally connected space may fail to be path-connected. This means that if X is a connected locally connected space, the map 7?! ( (0); (1)) that to a path 2 C([0; 1]; X); seen as a continuous map from [0; 1] to X; associates the pair of its endpoints may fail to be be onto. (This \well-known" topological fact is recalled in [4].) It is remarkable, and was conjectured by A. Joyal, that this map is actually (open) surjective, when de ned as a map between formal spaces. This conjecture is proved in [4] but we present here a proof that seems to us more direct. Such a proof is actually sketched in the reference [4], but rejected as being less intuitive and at least as complicated as the proof presented there. Contrary to what is said in [4], we don't think that the resulting proof is less intuitive, and it is de nitively more elementary. Furthermore, to carry out this proof leads naturally to the consideration of a dierent \space of path" P(X) whose points may be thought of as paths de ned independently of any non canonical parametrisation. We think that the consideration of this space of paths is interesting in itself. We divide thus the proof that C([0; 1]; X)!X 2 is (open) surjective in two parts. First, we build a formal space P(X) of paths in X with a surjective open map P(X)!X 2 ; and then a surjective open map C([0; 1]; X)!P(X): Intuitively P(X) is the space of paths where we \forget" about the parametrisation. This abstract is organised as follows. In a rst section, we x our terminology/notation about formal spaces, and recall some facts about covering system [3]. We present then the space P(X); and next the space C([0; 1]; X) using the notation of covering system. The last section presents the proof that P(X)!X 2 is open surjective.

1 Covering System The notion of covering systems appears in the work of Dragalin [1]. It seems to be well-known in the framework of locales and appears also essentially in Johnstone's book [2] and in the exercice 5, chapter IX of [3]. A poset is a set X with a relation  that is re exive and transitive. If A is a subset of X we let A0 be the subset fw 2 X j 9u 2 A w  ug: If A; B are subsets of X; we write A  B i for all a 2 A; there exists b 2 B such that a  b; that is A  B 0 : If A is a subset of X, A is monotone i u 2 A and v  u implies v 2 A; this is equivalent to A0  A: We let M(X) be the collection of all monotone subsets of X. A formal space is a structure (X; ; Cov) such that

 (X; ) is a poset,  for each x 2 X; Covx is a family of subsets of X, such that 1. y  x if y 2 S and S 2 Covx ; 2. if y  x and S 2 Covx ; then there exists T 2 Covy such that T  S: The elements of X are the basic open of the space X. Given such a structure, and U a subset of X; we de ne inductively u
U by the clauses:

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 (C1) u
U if u 2 U;  (C2) u
U if u  v and v
U;  (C3) u
U if S
U for one S 2 Covx ; where S
U means s
U for all s 2 S:

In term of spaces as set of points, the relation u
U means that basic open u is covered by the collection U of basic open. We write u
v for u
fvg if u; v 2 X: If U  X; we de ne rU = fu 2 X j u
U g: We have then that U
V i U  rV: We say that U is a formal open of the space X i U = rU: We let O(X) be the collection of all formal open of the space X. Lemma: The formal open of a space X form a complete W Heyting algebra, with the operations of set-theoretic intersection U \ V; and of arbitrary union Ui = r([Ui). The product X  X is de ned as follow. As a poset, it is the product poset. If U 2 Covu in X then U  fvg 2 Cov(u;v) in X  X and if V 2 Covv in X then fug  V 2 Cov(u;v) in X  X: A space (X; ; Cov) is positive i each element of Covu are inhabited, for any u 2 X: Given a positive space X, we say that two basic open u and v of X are overlapping i there exists a basic open w which is included in both u and v. A chain is a non empty sequence u1 : : :un of basic open of X such that ui ; ui+1 are overlapping for i < n: We say that X is connected i X is positive and whenever X
U and u; v 2 U there exists a chain u = u1; : : :; un = v of elements of U joining u and v: We say that X is locally connected i X is positive and whenever U 2 Covu and v; w 2 U; there exists a chain u1 = v; : : :; un = w of elements of U joining v and w. A continuous map f : Y !X is a function f  : X !O(Y ) such that the following conditions hold:  Y
f  X;  if u  v then f  u
f  v;  if U 2 Covu then f  u
f  U;  if f  u1 \ f  u2
f  fu j u  u1 ^ u  u2 g: W If U  X; we write f  U for u2U f  u. It can then be shown that U
V implies f  U
f  V: This map f is surjective i the converse implication holds: f  U
f  V implies U
V: The main tool here for proving that a map is (open) surjective is provided by the following proposition, extracted from proposition 1, IX 8 of [3]. Proposition 1: If  : Y !X is a monotone map of posets such that   is onto, and if u   v; then there exists w 2 Y such that w  v and  w = u;  if U 2 Cov v then v
?1 U;  if V 2 Covv then  v
 V; then the map f  U = ?1 (rU) de nes a continuous surjective map Y !X:

2 A Space of Paths Let X be a formal space connected and locally connected. The basic open of the space P(X) are the chains u1 : : :uk of X: We write m; n; : : : for nite sequences of basic open of X (may be empty) and u; v; : : : for basic open of X. Any sequence m = u1 : : :uk 2 P(X) has a tail tl(m) = uk and a head hd(m) = u1: We write m + n the concatenation of the two sequences m and n: We say that u appears in m i m can be written m1 + u + m2 : Finally, if M  P(X) and m; n are two nite sequences of basic open of X, we denote by m + M + n the set of all chains m + m0 + n such that m0 2 M:

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If U  X then n 2 P(X) is said to respects U i for all v appearing n, we have v 2 U: If U  X; we write P(U) the set of n 2 P(X) that respects U. The order relation m  n is de ned as follows. If m = u then m  n means that u  v for all v that appears in n: Otherwise, m = u + m1 ; n = v + n1 ; u  v and m1  n or m1  n1 or m  n1: This de nes a re exive transitive relation on P(X): The basic covering are de ned by the rules:

 u + v is covered by the the collection of elements u + w + v such that w  u and w  v,  if U 2 Covu in X then P(U) 2 Covu in P(X);  nally, if M 2 Covm then n1 + M + n2 2 Covn1 +m+n2 :

3 A Presentation of

C ([0;

1]; X )

We give now a possible presentation of C([0; 1]; X) that emphasizes the connection with the space P(X): Actually, we present simultaneously all spaces Yp;q = C([p; q]; X) with 0  p < q  1: By convention, we take Yp;q to be the set consisting of the empty sequence if p = q: The basic open of Yp;q are nite sequences m = ([p; p1]; u1) : : :([pk?1; q]; uk) with p < p1 < : : : < pk?1 < q a nite increasing sequence of rationals, and u1 : : :uk a chain of X: We de ne then hd(m) = u1 and tl(n) = uk : We write m; n; : : : for the elements of Yp;q : It is clear how to de ne m + n 2 Yp;r if m 2 Yp;q and n 2 Yq;r are such that tl(m) and hd(n) are overlapping. As before, we say that m = ([p; p1]; u1) : : :([pk?1; q]; uk) respects U  X i ui 2 U for all i  k: We de ne m  n in Yp;q as follows

 ([p; q]; u)  n if u  v for all open v in n;  ([p; p1]; u) + m  ([p; q1]; v) + n i u  v and { either p1 < q1 and m  ([p1; q1]; v) + n in Yp1;q ; { or p1 = q1 and m  n in Yp1 ;q ; { or p1 > q1 and ([q1; p1]; u) + m  n in Yq1;q : The basic covering are de ned by the rules:

 ([p; q]; u)+([q; r]; v) is covered by the the collection of elements ([p; s]; u)+([s; t]; w)+([t;r]; v) such that w  u; w  v, and p < s < q < t < r,  if U 2 Covu in X; then ([p; q]; u) is covered by the collection of all elements of Yp;q that respects U,  nally, if M 2 Covm in Yp;q then n1 + M + n2 2 Covn1 +m+n2 in Yr;s for r < p < q < s and n1 2 Yr;p ; n2 2 Yq;s :

4 Main Results We can now state the two main propositions. Proposition 2: The map  : m 7?! (hd(m); tl(m)) satis es all conditions of proposition 1, and hence de ne an open surjection P(X)!X 2 : Proof: The fact that  is onto follows from the fact that X is connected. If (m)  (v; w) then we have v + m + w  m and (v + m + w) = (v; w): It is clear that if W 2 Cov(m) then m
?1(W). Finally, if (m) = (u; v) and M 2 Covm , then we prove that (u; v)
(M): There are only two cases that are not direct, and both can be treated in the same way. One such case is when m = u + n and M is of the form P(U) + n, with U 2 Covu : Since X is locally connected, for each w 2 U; there exists

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n1 2 P(U) such that hd(n1) = w and tl(n1 ) overlaps n. We have then (n1 + n) = (w; v): This shows that (M) contains U  fvg and hence that (m)
(M). Proposition 3: The map ([0; p1]; u1) : : :([pk?1; 1]; uk) 7?! u1 : : :uk satis es all conditions of proposition 1, and hence de ne an open surjection C([0; 1]; X)!P(X). Proof: This follows from the density of the poset of rationals.

Conclusion After reading a rst draft, Icke Moerdijk suggested that a test to our notion of paths would be to prove that the space P(X) is the same as the quotient C([0; 1]; X)=G by the group G of reparametrisation (monotone bijection of [0; 1] into itself). If this conjecture holds, this would justify to consider the space P(X) as the space of paths de ned independently of any non canonical parametrisations. In any case, we think that the intuition of a chain as an \approximation" of a path is suggestive.

References [1] A.G. Dragalin. Mathematical Intuitionism. Translations of Mathematical Monographs, AMS, Volume 67 [2] P.Johnstone. Stone Sapces. Cambridge University Press [3] S. Mac Lane and I. Moerdijk. (1992) Sheaves in Geometry and Logic. Universitext, Springer-Verlag [4] I. Moerdijk and G. Wraith. (1986) Connected Locally Connected Toposes are Path-Connected. Transactions of the American Mathematical Society, Vol.295, p. 849-859

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