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I see shape theory being brought up at the Cafe. So did people ever think of a coshape theory? Yes, I see from a talk
The notion of coshape of a space was introduced by T. Porter.
Still, it doesnâ€™t seem to be nearly so well studied. Is it less interesting for some reason?
In many ways classical coshape is the same as singular homotopy theory. The original form of shape with Borsuk etc. looked at the properties of a space in terms of mapping it into polyhedra (up to homotopy). It was Cech homotopy. The coshape then would be mapping polyhedra into the space (up to homotopy) so was related (sort of universally) to mapping the singular complex in. There are categorical coshape situations that do give useful results but I do not now what the strong coshape theory would be nor how it might adapt to the topos situation. I did write one paper whch looked at both at the same time. I do not think coshape is less interesting in general but classically it was (sort of ) known.
As no doubt you found there is a paper Yu T Lisitsa 1980 Russ. Math. Surv. 35 250
on Strong Coshape Theory.
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