nLab local abelian category

An abelian category is local if it has the smallest topologizing subcategory.

An alternative description is the following.

Recall the preorder $\succ$ on the collection of nonzero objects of an abelian category $A$. Namely, $M\succ N$ if $N$ is a subquotient of finitely many copies of $M$. In other words, $M\succ N$ if there exists a positive number $k$ and a subobject $U$ of a direct sum $\oplus_{i = 1}^k M$ of $k$ copies of $M$ and an epimorphism from $U$ to $N$.

A nonzero object $M$ of an abelian category $A$ is said to be quasifinal if $M\succ N$ for any nonzero object $N$ of $A$. The quasifinal object clearly belongs to the spectrum of the abelian category $A$. An abelian category is called local if it has a quasifinal object.

Local abelian categories in the theory of noncommutative spectra (in the sense of noncommutative algebraic geometry) have a role similar to local rings in the theory of usual spectra of rings and in the theory of locally ringed spaces. Moreover there is a “center” functor from the category of topological spaces equipped with stacks of local categories to the locally (commutatively) ringed spaces. It is basically induced by the construction of the center of abelian category? which to an essentially small abelian category attaches the ring of endomorphisms of the identical functor, thanks to the functoriality of that construction with respect to the localizations.

• A. L. Rosenberg, Topics in noncommutative algebraic geometry, homological algebra and K-theory, preprint MPIM Bonn 2008-57 pdf (mainly Lec. 1.6 around page 30)

• A. L. Rosenberg, The spectrum of abelian categories and reconstructions of schemes, in Rings, Hopf Algebras, and Brauer groups, Lectures Notes in Pure and Appl. Math. 197, Marcel Dekker, New York, 257–274, 1998; MR99d:18011; and Max Planck Bonn preprint Reconstruction of Schemes, MPIM1996-108, pdf (1996).

• A. L. Rosenberg, Noncommutative local algebra, Geom. Funct. Anal. 4 (1994), no. 5, 545–585.