Bimorphisms in pro-homotopy and proper homotopy
A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. The category is balanced if every bimorphism is an isomorphism. In the paper properties of bimorphisms of several categories are discussed (pro-homotopy, shape, proper homotopy) and the question of those categories being balanced is raised. Our most interesting result is that a bimorphism f:X → Y of $tow(H_0)$ is an isomorphism if Y is movable. Recall that $\tow(H_0)$ is the full subcategory of $pro-H_0$ consisting of inverse sequences in $H_0$, the homotopy category of pointed connected CW complexes.