nLab
schedule

Schedules

Idea

Let XX be a topological space and 𝒰\mathcal{U} an open cover thereof. A (continuous) path γ:IX\gamma \colon I \to X can pass through many of the elements of 𝒰\mathcal{U} as it winds its way around XX. We can decompose that path into segments such that each segment lies wholly inside one of the open sets in 𝒰\mathcal{U}. The Schedule Theorem says that this can be done continuously over all paths in XX.

This was proved by Dyer and Eilenberg and applied to the question of fibrations over numerable spaces.

Schedules

The idea of a schedule is that it is a way of decomposing a length into pieces and then assigning a label to each piece. This clearly fits with the stated purpose of these things since we wish to decompose a path into pieces and assign an open set to each piece.

To make this precise, we start with a set of labels. Following Dyer and Eilenberg, let us write this as AA. Lengths are positive real numbers and so we also need the set of such, Dyer and Eilenberg denote this by TT; thus T 0T \coloneqq \mathbb{R}_{\ge 0}.

Definition

The schedule monoid of AA is the free monoid on the set A×TA \times T. It is written SAS A. Its elements are schedules in AA.

A schedule in AA is thus a finite ordered list of pairs (a,t)(a,t) where aAa \in A and tTt \in T.

There are two notions of length for a schedule. There is the word length which simply counts the number of pairs. Then there is the function l:SATl \colon S A \to T defined by l((a 1,t 1)(a k,t k))=t 1++t kl((a_1,t_1) \cdots (a_k,t_k)) = t_1 + \cdots + t_k. There is also a right action of TT on SAS A which simply multiplies all of the lengths: ((a 1,t 1)(a k,t k))t=((a 1,t 1t)(a k,t kt))((a_1,t_1) \cdots (a_k,t_k)) \cdot t = ((a_1,t_1 t) \cdots (a_k,t_k t)). Then l(st)=l(s)tl(s t) = l(s) t.

Definition

A schedule is said to be reduced if all of its terms, (a,t)(a,t), has non-zero length, i.e. t>0t \gt 0. The set of reduced schedules forms a submonoid of SAS A which is written RSAR S A.

The empty schedule is reduced.

There is a retraction map ρ:SARSA\rho \colon S A \to R S A defined by removing all terms with zero length part.

The schedule monoid is given a topology so that the labels are discrete and the lengths topologised as usual. More concretely, given a word a 1a 2a ka_1 a_2 \dots a_k of elements in AA, the set of schedules of the form (a 1,t 1)(a 2,t 2)(a k,t k)(a_1,t_1) (a_2,t_2) \cdots (a_k,t_k) is in bijection with T kT^k and we make that bijection a homeomorphism. Then SAS A is topologised by taking the coproduct over the set of words in AA. The reduced schedule monoid is topologised as the quotient of this.

Paths

Let XX be a topological space. Let PXP X denotes its Moore path space. Suppose that we have a family 𝒰\mathcal{U} of subsets of XX indexed by some set AA. Then we consider a schedule in AA as giving an ordered list of these subsets together with the times to be spent in each. For a path in XX, and a schedule of the appropriate length, then we can ask whether or not the path fits (or obeys) the schedule. We make that precise as follows.

Definition

Suppose that we have αPX\alpha \in P X and sSAs \in S A, and suppose that s=(a 1,t 1)(a k,t k)s = (a_1, t_1) \cdots (a_k,t_k). Then we say that α\alpha fits the schedule s, written αs\alpha \Vert s, if the following conditions hold:

  1. l(α)=l(s)l(\alpha) = l(s)
  2. We can split α\alpha into subpaths according to the times {t i}\{t_i\}. Let α i\alpha_i be the iith segment. Then α iPU a i\alpha_i \in P U_{a_i}.

Here, l:PXTl \colon P X \to T is the function that assigns to a Moore path its length. The schedule designates a decomposition of [0,l][0,l] into subintervals with t it_i being the length of the iith subinterval. Then saying that α\alpha fits the schedule ss means that α\alpha spends the iith subinterval in the open set U a iU_{a_i}.

Schedule Theorem

We can now state the main theorem.

Theorem

Let XX be a topological space. Let 𝒰\mathcal{U} be a locally finite open covering of XX by numerable open sets with indexing set AA. Then there is a covering \mathcal{F} of PXP X by closed sets and a family of continuous functions f:FSAf \colon F \to S A, indexed by FF \in \mathcal{F} such that:

  1. for each αPX\alpha \in P X, there some finite subfamily {F 1,,F k}\{F_1, \dots, F_k\} \subseteq \mathcal{F} such that α\alpha is in the interior of F j\bigcup F_j,
  2. for each αF\alpha \in F, αf F(α)\alpha \Vert f_F(\alpha), and
  3. for each αFF\alpha \in F \cap F', ρ(f F(α))=ρ(f F(α))\rho(f_F(\alpha)) = \rho(f_{F'}(\alpha))

The first condition is purely about the covering. Dyer and Eilenberg use the term local covering for a covering by closed sets with this property.

Corollary

There exists a continuous function h:PXRSAh \colon P X \to R S A such that αh(α)\alpha \Vert h(\alpha) if l(α)>0l(\alpha) \gt 0 and h(α)=Λh(\alpha) = \Lambda if l(α)=0l(\alpha) = 0.

Here, ΛRSA\Lambda \in R S A is the empty word.

Globalisation Theorem

The original motivation for the notion of schedules was to prove the globalisation theorem for (Hurewicz) fibrations.

Theorem

Let p:YBp \colon Y \to B be a continuous function. Suppose that 𝒰\mathcal{U} is a locally finite covering of BB by numerable open sets with the property that for each U𝒰U \in \mathcal{U} then the restriction p U:Y UUp_U \colon Y_U \to U is a fibration. Then pp is a fibration.

The link between the globalisation theorem and the schedule theorem is the characterisation of Hurewicz fibrations in terms of Hurewicz connections.

Proof of the Schedule Theorem

Let XX be a topological space. Let 𝒰\mathcal{U} be a locally finite open covering of XX by numerable open sets and indexing set AA.

Let us write A *A^* for the free monoid on AA. Then there is a function A *×TSAA^* \times T \to S A which takes (a 1a 2a k,t)(a_1 a_2 \cdots a_k, t) to the schedule (a 1,t/k)(a 2,t/k)(a k,t/k)(a_1,t/k)(a_2,t/k)\cdots (a_k,t/k). We say that a path αPX\alpha \in P X evenly fits sA *s \in A^*, and write this as α es\alpha \Vert_e s, if it fits the schedule corresponding to (s,l(α))(s,l(\alpha)).

We need an initial technical result.

Lemma

There is a locally finite covering 𝒲={W ssA *}\mathcal{W} = \{W_s \mid s \in A^*\} of PXP X by numerable open sets such that for αW s\alpha \in W_s then α\alpha evenly fits the word ss.

Remark

Let us explain why this is a reasonable result. Consider a path, α\alpha, of length ll. We pull back the cover 𝒰\mathcal{U} to a cover of [0,l][0,l]. Using compactness of [0,l][0,l] we can replace the pull-back cover by a finite family of open subintervals of [0,l][0,l] which cover [0,l][0,l]. Each subinterval is labelled by an element of 𝒰\mathcal{U} (though a label might be reused). As the family is finite, the intersections are finite and therefore have a minimum length. Choose nn big enough so that l/nl/n is less than this minimum length. Then consider the subdivision of [0,l][0,l] given by {0,1/n,,l/n}\{0,1/n,\dots,l/n\}. Our conditions on nn guarantee that every intersection of subintervals contains at least one of these division points. We can therefore assign to each subinterval of the form [k/n,(k+1)/n][k/n, (k+1)/n] one of the original family of subintervals that contains it. Then we can assign the corresponding element of 𝒰\mathcal{U}. Thus α\alpha fits evenly the corresponding word.

Thus the sets Y s{α:α es}Y_s \coloneqq \{\alpha : \alpha \|_e s\} cover PXP X. That they are open follows from the fact that the condition for membership depends on certain compact sets lying in certain open sets and we use the compact-open topology on PXP X.

What is more complicated is reducing the family to a locally finite one.

As 𝒲\mathcal{W} is locally finite and its elements are numerable, we can choose a numeration that is also a partition of unity. That is, we can choose continuous functions q s:X[0,1]q_s \colon X \to [0,1] with the property that q s 1((0,1])=W sq_s^{-1}((0,1]) = W_s and sq s=1\sum_s q_s = 1.

Let \mathcal{B} be the set of finite subsets of A *ΛA^* \setminus \Lambda (where Λ\Lambda is the empty word). For bb \in \mathcal{B} we define

D b {αPX sbq s(α)=1} ={αPXq s(α)=0for allsb} \begin{aligned} D_b &\coloneqq \{\alpha \in P X \mid \sum_{s \in b} q_s(\alpha) = 1 \} \\ &=\{ \alpha \in P X \mid q_s(\alpha) = 0 \; \text{for all}\; s \notin b\} \end{aligned}

This is a covering of PXP X by closed sets. As 𝒲\mathcal{W} is locally finite, for αPX\alpha \in P X there is some neighbourhood VV which meets only a finite number of the 𝒲\mathcal{W}. These are indexed by elements of A *A^*, indeed of A *ΛA^* \setminus \Lambda, and so the set of indices is an element, say bb, of \mathcal{B}. Then for sbs \notin b, q sV=0q_s \mid V = 0 and so for βV\beta \in V, sbq s(β)=1\sum_{s \in b} q_s(\beta) = 1, whence VD bV \subseteq D_b. Thus each α\alpha is contained in the interior of some D bD_b.

Now let us put a total ordering on A *A^*. This induces a total ordering on each bb \in \mathcal{B} and thus allows us to define the partial sums of the summation sbq s\sum_{s \in b} q_s. Write these as Q iQ_i, with Q 0Q_0 as the zero function.

Fix bb \in \mathcal{B} and write it as b={s 1,s 2,,s k}b = \{s_1,s_2,\dots,s_k\} in the inherited ordering. Let e=(l 1,r 1,,l k,r k)e = (l_1,r_1,\dots,l_k,r_k) be a list of integers with the property that 1l ir i#s i1 \le l_i \le r_i \le \#s_i where #s i\#s_i is the word length of s is_i. Define:

D (b,e)={αD bl i1#s iQ i1(α)l i#s iandr i1#s iQ i(α)r i#s i}. D_{(b,e)} = \left\{ \alpha \in D_b \mid \frac{l_i -1}{\# s_i} \le Q_{i - 1}(\alpha) \le \frac{l_i}{\# s_i} \; \text{and} \; \frac{r_i - 1}{\# s_i} \le Q_i(\alpha) \le \frac{r_i}{\# s_i} \right\}.

This is closed in D bD_b and the collection {D (b,e)}\{D_{(b,e)}\} is a finite cover of D bD_b. The family {D (b,e)}\{D_{(b,e)}\} ranging over all bb \in \mathcal{B} and suitable ee is the family \mathcal{F} that we are looking for. It has the required covering property since the interiors of the D bD_b cover PXP X.

Define f (b,e):D (b,e)SAf_{(b,e)} \colon D_{(b,e)} \to S A as follows:

f (b,e)(α)=σ 1σ kl(α) f_{(b,e)}(\alpha) = \sigma_1 \cdots \sigma_k l(\alpha)

where σ i\sigma_i is the schedule with #σ i=r il i+1\# \sigma_i = r_i - l_i + 1 and l(σ i)=q s i(α)l(\sigma_i) = q_{s_i}(\alpha), and if s i=a 1a ns_i = a_1 \cdots a_n then if l i<r il_i \lt r_i we have

σ i=(a l 1,l inQ i1(α))(a l i+1,1n)(a r i1,1n)(a r i,Q i(α)r i1n) \sigma_i = \left( a_{l_1}, \frac{l_i}{n} - Q_{i - 1}(\alpha)\right) \left(a_{l_i+1}, \frac{1}{n} \right) \cdots \left(a_{r_i - 1}, \frac{1}{n} \right) \left( a_{r_i}, Q_i(\alpha) - \frac{r_i - 1}{n} \right)

otherwise, σ i=(a l i,Q i(α)Q i1(α))\sigma_i = (a_{l_i}, Q_i(\alpha) - Q_{i-1}(\alpha)).

This is continuous and for αD (b,e)\alpha \in D_{(b,e)} then α\alpha fits f (b,e)(α)f_{(b,e)}(\alpha). Moreover, for αD (b,e)D (b,e)\alpha \in D_{(b,e)} \cap D_{(b',e')} then ρf (b,e)(α)=ρf (b,e)(α)\rho f_{(b,e)}(\alpha) = \rho f_{(b',e')}(\alpha).

References

  • Dyer, E. and E., Samuel. (1988). Globalizing fibrations by schedules. Fund. Math., 130, 125–136. MR0963792

  • Dyer, Eilenberg, MR0963792

Revised on March 9, 2012 13:28:16 by Andrew Stacey (80.203.115.55)