The Jacobson radical that I have been mentioning in the last several communications, and which is defined in terms of the circle composition product x * y = x + y - xy, is related to category theory by M. W. Gray (Pacific Jour. Math 1967, vol. 23, 79-89 and her book A radical approach to algebra, Addison-Wesley: Reading, 1970. In particular, the semiabelian category was introduced by her to include various of the radicals including the Jacobson radical. A very accessible detailed presentation of her work and those of others up to 1994 is T. W. Palmer Banach algebras and the general theory of *-algebras volume I: algebras and Banach algebras, Cambridge University Press: Cambridge, which is part of the exceptionally valuable Encyclopedia of Mathematics and Its Applications of the Cambridge University Press (not of course a usual encyclopedia - the name is used for numerous separate books of the highest quality on crucial mathematical subjects important in the latest rsearch). See especially sections 4.3 and 4.7.