In Spanier's Algebraic Topology we read the following:
"The problem of classifying topological spaces and continuous maps up to topological equivalence does not seem amenable to attack directly by means of computable algebraic functors... many of the computable functors, because they are computable, are invariant under continuous deformation."
I understand the informal idea he is trying to convey; I would be curious to see if it has been formalized. Have computable topologists set down notions of "computable functor" from a good class of topological spaces (equipped with some notion of presentation) into something like Ab or an Abelian category and proven that such a functor is necessarily invariant under homotopy?
This seems very strong so I would be happy to read about any arguments of a weaker level of rigor than invoking Type II turing machines or whatnot that convincingly argue why a functor is generally not effectively computable if it is not invariant up to homotopy type. I find Spanier's terse explanation somewhat cryptic.