# Coronas of Spaces and the Borsuk Shape Category

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?