A Finiteness Condition for Locally Compact Abelian Groups

A map f: A→B in category is called monic if fg = fh implies that g = h for all maps g, h: C → A; it is called epic if gf = hf implies that g = h for all maps g, h: B → C. An object A ∈ is called an S-object if every monic map f: A → A is also epic; it is called a Q-object if every epic map f: A → A is also monic. If A is both an S-object and a Q-object then A is called an SQ-object. In the category of sets the SQ-sets are the finite sets. In the category of vector spaces over a field F the SQ-spaces are precisely the finite dimensional spaces. In the light of these simple examples, it seems reasonable to view the SQ-objects of a category as being of ‘finite type’. We shall be chiefly concerned with investigating the SQ-objects in certain subcategories of the category of locally compact abelian groups.