The Raĭkov conjecture fails for simple analytical reasons (Q714120)

Raikov's conjecture states that preabelian categories for which the pushout (pullback) of a (co-)kernel along any morphism is monic (epic) satisfy the stronger property that the latter monomorphism (epimorphism) is even a (co-)kernel. The conjecture was disproved implicitly by Bonet and Dierolf. A counterexample by a Krull-Schmidt category with six objects was given by the reviewer. The paper gives a new analysis of the Bonet-Dierolf counterexample and relates the main point in it to Köthe and Grothendieck's early examples of (LB) spaces. Furthermore, a wide class of (LF) spaces which are not \(\alpha\)-regular are derived from partial differential operators on a space of Schwartz distributions. In a 1975 paper of Dierolf, locally convex spaces are represented as quotients of complete spaces for which the bounded sets are finite-dimensional. This result is combined with a criterion of the reviewer to obtain other counterexamples.