Journal of Pure and Applied Algebra 215 (2011) 697–704
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Journal of Pure and Applied Algebra
journal homepage: www.elsevier.com/locate/jpaa
Generalized lax epimorphisms in the additive case
George Ciprian Modoi
‘‘Babeş-Bolyai’’ University, Faculty of Mathematics and Computer Science, Chair of Algebra, 1, M. Kogălniceanu, RO-400084, Cluj-Napoca, Romania
a r t i c l e i n f o
Received 11 January 2010
Received in revised form 7 June 2010
Available online 15 July 2010
Communicated by J. Adámek
MSC: 18E15; 18A20; 18E35
a b s t r a c t
In this paper we call generalized lax epimorphism a functor defined on a ring with several
objects, with values in an abelian AB5 category, for which the associated restriction functor
is fully faithful. We characterize such a functor with the help of a conditioned right
cancellation of another functor, constructed in a canonical way from the initial one. As
consequences we deduce a characterization of functors inducing an abelian localization
and also a necessary and sufficient condition for a morphism of rings with several objects to
induce an equivalence at the level of two localizations of the respective module categories.
© 2010 Elsevier B.V. All rights reserved.
All categories which we deal with are preadditive, i.e. there exists an abelian group structure on the hom sets, such
that the composition of the morphisms is bilinear. For a category C we denote by C(−, −) : C
× C → Ab the bifunctor
assigning to every pair of objects the abelian group of all maps between them. All functors between preadditive categories are
additive i.e. preserve the addition of maps. Consider a small preadditive category U. Recall that a preadditive category with
exactly one object is nothing but an ordinary ring with identity, therefore small preadditive categories are also called rings
with several objects. As in the case of ordinary rings, a (right) module over U (or simply, a U-module) is a functor U
All U-modules together with natural transformations between them form an abelian, AB5 category denoted Mod(U), where
limits and colimits are computed pointwise. Moreover the Yoneda functor
U → Mod(U), given by U → U(−, U)
is an embedding and its image form a set of (small, projective) generators for Mod(U), therefore Mod(U) is a Grothendieck
category. This embedding allows us to identify an object U ∈ U with its image in Mod(U), that is with the functor
U(−, U). In what follows we freely use this identification. We denote by Hom
(X, Y ) the set of all U-linear maps (i.e.
natural transformations) between the U-modules X and Y ; that is Hom
(X, Y ) = Mod(U)(X, Y ).
Following , a functor between small non-additive categories T : U → V is called a lax epimorphism, provided that the
, Set] → [U
, Set], T
X = X ◦ T
is fully faithful (Here [U
, Set] denotes the category of all contravariant functors from U to the category of sets.). We shall
use the same terminology in the additive case (consequently, replacing [U
, Set] with Mod(U)). We now consider a functor
T : U → C, where U is a ring with several objects and C is any cocomplete, abelian category. Then there is a unique, up
to a natural isomorphism, colimit preserving functor T
: Mod(U) → C such that T
U(−, U) = TU, for all U ∈ U. The
has a right adjoint, namely the functor
: C → Mod(U), T
C = C(T −, C) for all C ∈ C.
During the work of the first version of this paper the author was supported by the grant CEEX 47/2005. The final version of this work was supported
by CNCSIS UEFISCSU, project no. PN2CD-ID-489/2007.
E-mail address: email@example.com.
0022-4049/$ – see front matter © 2010 Elsevier B.V. All rights reserved.