Categorical Realizations of Quivers

Abstract

Abstract. We introduce and study categorical realizations of quivers. This construction generalizes comma categories and includes representations of quivers on categories, twisted representations of quivers (in the sense of [9]) and bilinear pairings as special cases. We prove a Krull-Schmidt Theorem in this general context, which results in a Krull-Schmidt Theorem for the special cases just mentioned. We also show that cancelation holds under milder assumptions. Using similar ideas we prove a version of Fitting's Lemma for natural transformations between functors. OverviewThroughout, all rings are assumed to have a unity and ring homomorphisms are required to preserve it. Subrings are assumed to have the same unity as the ring containing them. The Jacobson radical of a ring R is denoted by Jac(R).In this paper, we introduce categorical realizations of quivers and representations of quivers on these realizations. This construction generalizes comma categories (which are a categorical realizations of the quiver • → • ; see [1, 3K]). Among its special cases are representations of quivers on additive categories (vector spaces in particular), twisted representations of quivers in the sense of [9], and bilinear pairings.Using semi-centralizer subrings (also called semi-invariant subrings), introduced in [8], we establish a Krull-Schmidt Theorem for representations of quivers on their realizations, thus yielding Krull-Schmidt Theorems for the special cases mentioned above. We note that the categories in question are usually not abelian, so no "finite length" considerations (see Example 2.1) can be applied. Using similar ideas, we also prove a version of Fitting's Lemma for natural transformations between functors. Finally, different techniques are used to prove cancelation (from direct sums) of representations of quivers on their realizations under milder assumptions. Again, this yields cancelation theorems for the previous special cases. This paper can be viewed as a continuation of [8] presenting further applications of semi-invariant subrings.Section 2 recalls the Krull-Schmidt Theorem for pseudo-abelian categories. Section 3 introduces categorical realizations of quivers and section 4 defines linearly topologized categories. Section 5 recalls Fitting's Property (called quasi-π ∞ -regular