# About the Moore composition of paths

1) QUESTION

I work with weak Hausdorff $$k$$-spaces (so all spaces are $$T_1$$). The internal hom is denoted by $$\mathbf{TOP}(-,-)$$. Let $$\mathcal{G}$$ be the topological group of nondecreasing homeomorphisms from $$[0,1]$$ to itself (the composition law being the composition of maps). I denote by $$\mu_\ell:[0,\ell]\to [0,1]$$ the homothetie $$x\mapsto x/\ell$$ with $$\ell>0$$.

I consider a triple $$(X,P_{a,b},P_{b,c})$$ where $$X$$ is a topological space, $$a,b,c$$ are three distinct points of $$X$$, $$P_{a,b}$$ is an arbitrary subspace of $$\mathbf{TOP}([0,1],X)$$ such that for all $$\gamma\in P_{a,b}$$, $$\gamma(0)=a$$ and $$\gamma(1)=b$$, $$P_{b,c}$$ is an arbitrary subspace of $$\mathbf{TOP}([0,1],X)$$ such that for all $$\gamma\in P_{b,c}$$, $$\gamma(0)=b$$ and $$\gamma(1)=c$$. I suppose $$P_{a,b}$$ and $$P_{b,c}$$ closed under the reparametrization by $$\mathcal{G}$$.

I consider the space of paths $$P_{a,b}*P_{b,c} \subset \mathbf{TOP}([0,1],X)$$ defined as follows: every path $$\gamma$$ of $$P_{a,b}*P_{b,c}$$ is of the form :

1. We choose $$(\ell_1,\ell_2)$$ such that $$0<\ell_1,\ell_2<1$$, $$\ell_1+\ell_2=1$$
2. $$\gamma = (\gamma_1.\mu_{\ell_1}) * (\gamma_2.\mu_{\ell_2})$$ with $$\gamma_1\in P_{a,b}$$ and $$\gamma_2\in P_{b,c}$$ where $$*$$ is the Moore composition.

I consider another triple $$(X',P_{a',b'},P_{b',c'})$$ as above such that there exists a continuous map $$f:X\to X'$$ such that $$f(a)=a'$$, $$f(b)=b'$$, $$f(c)=c'$$ inducing weak homotopy equivalences $$P_{a,b} \to P_{a',b'}$$ and $$P_{b,c} \to P_{b',c'}$$.

Is it true that the continuous map $$g:P_{a,b}*P_{b,c} \to > P_{a',b'}*P_{b',c'}$$ induced by $$f$$ is a weak homotopy equivalence ?

The difficulty of this question is to understand $$P_{a,b}*P_{b,c}$$. There is a continuous map $$\mathrm{Int}(\Delta^1) \times P_{a,b}\times P_{b,c} \to P_{a,b}*P_{b,c} \ \ (1)$$ where $$\mathrm{Int}(\Delta^1)$$ is the interior of the $$1$$-simplex which takes $$((\ell_1,\ell_2),\gamma_1,\gamma_2)$$ to $$(\gamma_1.\mu_{\ell_1}) * (\gamma_2.\mu_{\ell_2})$$ which is surjective. The map (1) is actually a quotient map. It is not necessarily one-to-one because the paths of $$P_{a,b}*P_{b,c}$$ may stop at $$b$$. If none of the paths of $$P_{a,b}*P_{b,c}$$ stops at $$b$$ (i.e. the inverse image is one point of $$]0,1[$$, then the map (1) above is a homeomorphism. And if none of the paths of $$P_{a',b'}*P_{b',c'}$$ stops at $$b'$$ as well, then the map $$g$$ is a weak homotopy equivalence.

2) MOTIVATION (EDIT 02/13/2020)

I add the motivation in the hope that someone could have a suggestion by reading it. The proof of the left properness of the q-model category of multipointed $$d$$-spaces given in Left properness of multipointed d-spaces is incomplete. I forgot to treat the case of the generating cofibration $$R:\{0,1\}\to \{0\}$$ identifying two states, i.e. that the pushout of a weak equivalence along $$R:\{0,1\}\to \{0\}$$ is a weak equivalence. The problem above is a particular case. The similar fact for the category of flows is trivial. The generating cofibration $$R:\{0,1\}\to \{0\}$$ is not necessary to build the cellular objects so it would not be a real problem if the category of multipointed $$d$$-spaces was only "almost" left proper.

3) ABOUT THE WEAK HAUSDORFF CONDITION (EDIT 02/13/2020)

If the spaces are not $$T_1$$, then I run into several complicated point-set topology obstacles because there are three non-homeomorphic topologies on two points: the discrete one, the indiscrete one and the Sierpinski topology. To avoid pointless complicated mathematical arguments, it is sufficient to work with the correct separability condition for $$\Delta$$-generated spaces as explained in the model category structure and its left determinedness (Section 3). The question above is written down in the framework of weak Hausdorff $$k$$-spaces because I do not think that the local presentability condition has any role in this problem.

• I don't understand how $\mathcal{G}$ is a group? Shouldn't the inverse of a nondecreasing function be nonincreasing? – Jeff Strom Jan 22 at 22:18
• @JeffStrom $\mathcal{G}$ is a group for the composition of maps. It contains only homeomorphisms. – Philippe Gaucher Jan 22 at 22:34
• Of course. Too silly. – Jeff Strom Jan 23 at 3:22