What if homotopy were expanded to allow any connected space instead of $[0,1]$?

Question

What would happen to homotopy theory if we used a more general definition of homotopy, based on general connected spaces rather than $[0,1]$?

Given continuous $f,g:X\to Y$, define $f$ and $g$ to be C-homotopic if there exists a connected space $Z$ and points $z_0,z_1\in Z$ such that there exists a continuous $h:Z\times X\to Y$ such that $h(z_0,x)=f(x)$ and $h(z_1,x)=g(x)$.

Then obviously the notion of C-homotopy behaves a lot like homotopy; it's an equivalence relation, it respects composition, etc. And so we can talk about C-nulhomotopy, C-homotopy equivalence, C-contractibility, and so forth. All these are coarser notions than homotopy; so one could define C-homotopy groups, which would be quotients of the usual homotopy groups. (I know there have been attempts to broaden the notion of homotopy groups, or at least the fundamental group, to allow them to detect "loops" that ordinary homotopy groups can't; I'm not trying to do that here -- the only part I'm changing is which maps are homotopic. So it will just be a quotient.)

Edit: I forgot to explicitly define this the first time, though obviously I used it implicitly in the notion of "C-homotopy group", but one could also consider relative C-homotopies, where as usual $h$ is a homotopy relative to $A\subseteq X$ if, for any $a\in A$, the value of $h(z,a)$ does not depend on $z$ (obviously this requires $f|_A=g|_A$).

My question then, is, what does this change? I.e.:

1. Can we prove that for sufficiently nice spaces/maps, C-homotopy is the same as homotopy, or the same for any other corresponding concept (homotopy equivalence, contractibility, homotopy groups, etc.)?
2. Can we give examples, ideally not too trivial, of where they're not the same? (One class of examples is below.)

Obviously a trivial example of something different is that if we have two points in different path-components of a connected space, then considered as maps from the one-point space, these are C-homotopic but not homotopic.

A less trivial example is linear continua. This question asked about related things; his "h-contractible" is my "C-contractible". (I'm not using his notion of "h-path-connected" in this question.) Then the argument there shows that, for any bounded linear continuum $X$, $\min: X\times X\to X$ is a C-contraction of $X$; and it's easy to adapt the argument to work when $X$ has only one endpoint. (If $X$ has only a lower endpoint, for instance, then one can use $\min: (X\cup\{\infty\})\times X\to X$.) Then, picking a "midpoint", that argument can be applied twice to show $X$ is C-contractible even if it has no endpoints, even though it may not be contractible. Are there examples not based on this? Could the topologist's sine curve, for instance, be C-contractible?

(One could also consider a notion intermediate between homotopy and C-homotopy -- call it L-homotopy -- where, instead of an arbitrary connected space and any two points in it, one uses a bounded linear continuum and its two endpoints. Obviously the above example doesn't distinguish between C-homotopy and L-homotopy, though the trivial example still does; e.g. picking two points in the topologist's sine curve works here, unless I've made a mistake. That is, it's not "L-connected", which is what goes inbetween connected and path-connected. So if one cares about L-notions rather than C-notions, then L-connectedness, L-components, and local L-connectedness can be added to the list of things to compare.)

Apologies if this question is too broad; I'm not really a topologist and this is basically an idle question, but it seemed worth asking. Thank you all!

Answer 1

In the direction of question 1: if $X$ is compact Hausdorff and $Y$ has the homotopy type of a CW complex, then two maps $X\to Y$ are C-homotopic if and only if they are homotopic. In particular, C-homotopy groups agree with usual homotopy groups for CW complexes. Edit: As Harry pointed out, the definition of homotopy groups uses relative homotopy, so an equivalence of C-homotopy and homotopy does not imply C-homotopy groups agree with homotopy groups.

For $X$, $Y$ arbitrary, let $\mathcal{C}(X,Y)$ be the space of continuous functions from $X$ to $Y$, equipped with the compact-open topology. If $X$ is locally compact Hausdorff, then for every $Z$ there is a bijection $$ \{\text{continuous maps }Z\to\mathcal{C}(X,Y)\}\leftrightarrow\{\text{continuous maps }X\times Z\to Y\}. $$ This implies that for $X$ locally compact Hausdorff, C-homotopy agrees with homotopy for maps $X\to Y$ if and only if every connected component of $\mathcal{C}(X,Y)$ is path-connected.

When $X$ is compact Hausdorff and $Y$ has the homotopy type of a CW complex, it is a theorem of Milnor that $\mathcal{C}(X,Y)$ has the homotopy type of a CW complex. It follows that the components of $\mathcal{C}(X,Y)$ are path-connected (this is true for CW complexes because they are locally path-connected, and a homotopy equivalence induces a bijection both on components and on path-components).

That said, connected components being path-connected is a much weaker condition than having the homotopy type of a CW complex, so probably there are much weaker hypotheses on $X$ and $Y$ under which C-homotopy agrees with homotopy.