That makes me remember Exercice 3.3.1 of Tom Leinster's book that I mention in another answer: "Choose a mathematician at random. Ask them whether they can accurately state any axiomatization of sets (without looking it up). If not, ask them what operating principles they actually use when handling sets in their day-to-day work."

Could you give the page number please ? In case you need a model category structure, I suggest that you start from the nLab page http://ncatlab.org/nlab/show/model+structure+on+chain+comple‌​xes.

Cubical homotopy is only a beginning for the author above :-). It is preferrable to start from the bibliography of this book : pages.bangor.ac.uk/~mas010/nonab-a-t.html. In particular the book Kamps, K. H., and Porter, T., Abstract homotopy and simple homotopy theory. World Scientific Publishing Co. Inc., River Edge, NJ (1997) which explores the subject very well. Unfortunately, it is not available on the Web.

I am not sure if it is a possible answer so I leave it in comment. The notion of weak $\omega$-categories has many acceptations. Everyone agrees that it is something with higher dimensional morphisms but there are a lot of possible axioms.

@AlexanderKörschgen The answer above answers your question. It says that $Top(T,-)$ is right adjoint to $T\times -$ as endofunctor of the category CGWH. In other terms, it is because CGWH is cartesian closed.

@AndrejBauer To be sure (intuitionistic meaning without LEM for me). In the implication "constructive => continuous", constructive = intuitionistic or it is two different adjectives ?

@AndrejBauer OK. I had already read the implication "constructive=>continuous" without paying attention to this assertion very much :-). I guess that constructive analysis can make the difference in some way between the step function $f$ above and e.g. the parabola $x\mapsto x^2$ (the behaviour at $0$). What I mean is that that means that continuity is not an appropriate notion for constructive analysis. Am I wrong ? Or does it mean that the map $f$ above does not exist in constructive analysis ? (sorry I wanted to edit my comment to add something, and I had to cancel it)

@AndrejBauer I am not familiar with constructive mathematics: why is it impossible to exhibit a discontinuous map in constructive mathematics ? If I define $f:\mathbb{R}\to \mathbb{R}$ by $f(x)=0$ if $x<0$ and $f(x)=1$ if $x\geq 0$, this map is not continuous on $0$. And for me, $f$ looks like something constructive. I can easily write a program calculating $f$. Where do I use excluded middle ? I guess that I am missing something.

Could you expand the proof please and/or give some references about e.g. duality of modules over Boolean semiring ? A basic course for a beginner I mean.