I was thinking about some topological ideas, especially those relating to shape theory, and came across some interesting constructions that seem to relate to shape theory, but that I can't quite situate within my own knowledge - so was wondering if there is any related ideas out there in math.

Let $\beta$ be the Stone-Cech compactification functor. For any space $S$, let us define a new space $CS$ as follows: Consider the space $\beta(S\times [0,1))$ and the projection $\beta \pi_2:\beta(S\times [0,1))\rightarrow [0,1]$ given by extending the projection to the second coordinate. Define $CS$ to be the fiber of $1$ under $\beta \pi_2$ - note that if $S$ is itself compact Hausdorff, this is just $CS=\beta(S\times [0,1))\setminus S\times [0,1)$. This space somehow "expands" $S$ to be large enough that any map $S\times [0,1)$, when thought of as a homotopy missing its endpoint, will in some sense have a limit.

More precisely, if $Q=\prod_J[-1,1]$ for some indexing set $J$ with projections $\pi_j:Q\rightarrow[-1,1]$ for each $j\in J$ and $B$ is some compact subset of $Q$, one may explicitly describe the maps $CA\rightarrow B$: they are continuous functions $f:S\times [0,1)\rightarrow Q$ such that for any open set $N$ containing $B$, there is some $\alpha$ such that if $t>\alpha$ then $f(a,t)\in N$ under the equivalence relation that $f\sim g$ if for every $j\in J$ and $\varepsilon>0$ there is some $\alpha$ such that if $t>\alpha$ then $\pi_j(f(a,t))-\pi_j(g(a,t))<\varepsilon$. Note that every compact Hausdorff space $B$ can be embedded in some such $Q$.

This characterization seems very similar to the definition of morphisms in Borsuk's Shape Category except that morphisms in that category are taken up to homotopies, whereas these continuous functions $CA\rightarrow B$ are more strictly continuous maps - I haven't worked out all the details, but I think that under an appropriate notion of equivalence (e.g. some sort of homotopy in the corona of $A\times [0,1)\times [0,1]$ using the two embedding of $CA$ into this with last coordinate $0$ and $1$), one can get "morphism" in the shape category as an equivalence class of maps $CA\rightarrow B$.

This leaves me with a natural question: can one use this construction to handle some of the nasty spaces (e.g. the Warsaw circle) that shape theory handles without working in a category defined only up to homotopy? More explicitly, suppose we have some set of reasonably nice spaces (maybe something like compact Hausdorff) - can we define a category whose morphisms from $A$ to $B$ are continuous maps $CA\rightarrow B$? The stumbling block is that there's no obvious composition law to use - but maybe there is some clever composition law, or perhaps $C$ can be given the structure of a comonad via some map $CA\rightarrow CCA$ (although no such maps come to mind!), or maybe the explicit description of the maps $CA\rightarrow B$ can somehow be used (although it seems to become difficult to show well-definedness when one tries to do this).

Is there a way to make a category of "shapes" not up to homotopy along these lines? Is there any literature related to this idea?