Nonconstant continuous maps of spaces and of their \(\beta\)- compactifications (Q1823500)

The main theorem claims that all faithful functors \(\psi\) : \({\mathcal K}_ 1\to {\mathcal K}_ 2\) between small categories \({\mathcal K}_ 1\) and \({\mathcal K}_ 2\) (and, clearly, only such) can be ``almost fully'' represented by the \(\beta\)-compactification \(\beta\) : Tych\(\to Comp\) of Tychonoff (i.e. completely regular and \(T_ 1)\) spaces, in the sense that there is a pair of almost full embeddings \(\Phi_ 1: {\mathcal K}_ 1\to Tych\) and \(\Phi_ 2: {\mathcal K}_ 2\to Comp\) with \(\Phi_ 2\circ \psi =\beta \circ \Phi_ 1\) (a functor \(\Phi\) : \(K\to Tych\) is an almost full embedding if it is faithful and for every pair A, B of \({\mathcal K}\)-objects it takes the class Hom(A,B) of \({\mathcal K}\)-morphisms onto the class of all nonconstant continuous mappings from \(\Phi\) A to \(\Phi\) B). Some consequences are stated explicitly and a possibility of generalization mentioned.