l know that K functor is continous with respect to direct limit, how about inverse limit ？does inverse limit exist in general in the caegory of topological space(or C star algebra). is there a textbook giving a proof or a counterexample？and how about homology and homotopy functor, are they continous？
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A K functor is the image of the diffeomorphism which is the 'inverse limit' of a multi-linear function to some chosen manifold. Homology is then constructed around the boundaries of two such functions so as to consider them similar, the same, or equal. This equality is technically a hologram of a chosen differential form, thus an endofunctoid is the complete complement of two functions and is never rigorous in symbolism. – McTaffy Jul 10 '18 at 18:05