Let $Z$ be a $\Delta$-generated space (a colimit of simplices -not sure that this hypothesis is important but it is the framework I am working in). The set of continuous maps $Z^{[0,1]}$ from $[0,1]$ to $Z$ is equipped with the $\Delta$-kelleyfication of the compact-open topology (the internal hom of the category, see below).

QUESTION : Suppose that the evaluation map at $0$ from $Z^{[0,1]}$ to $Z$ satisfies the right lifting property with respect to any monomorphism of $\Delta$-generated spaces (i.e. injections). Then I suspect that $Z$ must be discrete. Any counter-example ?

Concerning $\Delta$-generated spaces, a short bibliography, just in case that it is important : 1. Notes on Delta-generated spaces by Dugger : http://math.uoregon.edu/~ddugger/delta.html 2. The proof that they assemble to a locally presentable category : A convenient category for directed homotopy by Fajstrup-Rosicky : http://www.tac.mta.ca/tac/volumes/21/1/21-01abs.html 3. A survey of their properties : Section 2 of Homotopical interpretation of globular complex by multipointed d-space http://www.tac.mta.ca/tac/volumes/22/22/22-22abs.html


$\newcommand{\into}{\hookrightarrow}$It seems that if $Z$ has the indiscrete topology, then the evaluation map $ev_0 : Z^I \to Z$ has the right lifting property with respect to any map. That provides a simple counter-example to the question.

So I will assume for the remainder of my answer that $Z$ is $T_1$, i.e. singleton subsets are closed in $Z$. I will show that $Z$ is necessarily discrete if $ev_0 : Z^I \to Z$ has the right lifting property with respect to the inclusion $J \into I$, where $J={}]0,1]$ is a non-closed interval.

By the way, I do not know how to deal with more general classes of spaces, as the argument below uses very strongly the fact that the space $Z$ is generated by simplices, which are path connected spaces.

Claim 0: If $f,g:J\to Z$ are homotopic maps, and $g$ extends to $I$, then $f$ also extends to $I$.

This follows immediately by applying the assumed right lifting property of $ev_0 : Z^I \to Z$ to any homotopy from $g$ to $f$, seen as a map $J\to Z^I$.

Claim 1: Any map $J\to Z$ extends to a map $I\to Z$.

Any map $J\to Z$ is homotopic to a constant map, since $J$ is contractible. The statement now follows from claim 0.

Claim 2: Any continuous map from a simplex to $Z$ is constant.

Since simplices are path connected, it suffices to show that any map $f:I\to Z$ is constant. Let $x,y\in I$.

Consider the continuous map $h:J\to I$ given by $$ h(t) = \frac 1 2 + \frac 1 2 \sin\Bigl(\frac 1 t\Bigr) $$ For any neighbourhood $U$ of zero in $I$, there exist points $a,b\in J\cap U$ such that $h(a)=x$ and $h(b)=y$.

By claim 1, the map $f\circ h$ extends to a continuous map $g:I\to Z$. By our choice of $h$, within any neighbourhood of zero in $I$ there exist points $a$, $b$ such that $g(a)=f(x)$, and $g(b)=f(y)$. Therefore, $g(0)\in \overline{\{f(x)\}}$ and $g(0)\in \overline{\{f(y)\}}$. Since $Z$ is $T_1$, it follows that $f(x)=g(0)=f(y)$.

Conclusion: $Z$ is a discrete space.

Indeed, since $Z$ is $\Delta$-generated, a subset $S$ of $Z$ is open if the inverse image $f^{-1}(S)$ is open for any map $f$ from a simplex to $Z$. Claim 3 then implies that singleton subsets of $Z$ are open, i.e. $Z$ is discrete.

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