A categorical version of the Lefschets-Nِbeling-Pontryagin theorem on embedding compacta in Rn

For a category K we use Ob(K) to denote the class of all objects of K; if X, Y ϵ Ob(K), then MorK(X, Y) is the set of all K-morphisms from X into Y. Let A and B be subcategories of the category of all topological spaces and their continuous maps. We say that a covariant functor F:A → B is an embedding functor if there exists a class {ix: X ϵ Ob(A)} satisfying the following conditions: 1. : X → F(X) is a homeomorphic embedding for every X ϵ Ob(A), and f X, Y ϵ Ob(A) and f ϵ MorK(X, Y), then F(f) o ix = iy o f. natural number n let C(n) denote the category of all n-dimensional compact metric spaces and their continuous maps. Let G(< ∞) be the category of all Hausdorff finite-dimensional topological groups and their continuous group homomorphisms. We prove that there is no embedding covariant functor F:C(1) → G(< ∞), but there exists a covariant embedding functor F:C(0) → G(0), where G(0) is the category consisting of the single (zero-dimensional) compact metric group Z2ω and all its continuous group homomorphisms into itself, i.e., Ob(G(0)) = {Z2ω} and MorG(0)(Z2ω, Z2ω) is the set of all continuous group homomorphisms from Z2ω into Z2ω.