Injectivity in a category: an overview on smallness conditions

, , , , ,


Abstract. Some of the so called smallness conditions in algebra as well
as in category theory, are important and interesting for their own and also
tightly related to injectivity, are essential boundedness, cogenerating set, and
residual smallness.
In this overview paper, we rst try to refresh these smallness condition
by giving the detailed proofs of the results mainly by Bernhard Banaschewski
and Walter Tholen, who studied these notions in a much more categorical
setting. Then, we study these notions as well as the well behavior of injectivity,
in the class mod(; E) of models of a set of equations in a suitable
category, say a Grothendieck topos E, given by M.Mehdi Ebrahimi. We close
the paper by some examples to support the results.


Cogenerating set, essential extension, residual smallness, injective. Mathematics Subject Classication [2010]: 08-02, 08B30, 18-02, 18A20, 18E15, 18G05, 20M30, 20M50.

Full Text:




J. Adamek, H. Herrlich, and G.E. Strecker, Abstract and Concrete Categories",

John Wiley and Sons, Inc., 1990.

F.W.. Anderson and K.R. Fuller, Rings and Categories of Modules", Springer, New

York, 1974.

B. Banaschewski, Injectivity and essential extensions in equational classes of algebras,

Queen's Paper in Pure Appl. Math. 25 (1970), 131-147.

H. Barzegar and M.M. Ebrahimi, Sequentially pure monomorphisms of acts over

semigroups, Eur. J. Pure Appl. Math. 1(4) (2008), 41-55.

H. Barzegar, M.M. Ebrahimi, and M. Mahmoudi, Essentiality and injectivity relative

to sequential purity of acts, Semigroup Forum 79(1) (2009), 128-144.

P. Berthiaume, The injective envelope of S-Sets, Canad. Math. Bull. 10(2) (1967),


K.R. Bhutani, Injectivity and injective hulls of abelian groups in a localic topos, Bull.

Austral. Math. Soc. 37 (1988), 43-59.

S. Burris and H.P. Sankapanavar, A Course in Universal Algebra", Graduate Texts

in Math. No. 78, Springer-Verlag, 1981.

D. Dikranjan and W. Tholen, Categorical Structure of Closure Operators, with

Applications to Topology, Algebra, and Discrete Mathematics", Mathematics and

its Applications, Kluwer Academic Publisher, 1995.

Injectivity in a category: smallness conditions 111

M. M. Ebrahimi, M. Haddadi, and M. Mahmoudi, Injectivity in a category: An

overview of well behaviour theorems, Algebra, Groups, and Geometries 26 (2009),


M.M. Ebrahimi, Algebra in a Grothendieck Topos: Injectivity in quasi-equational

classes, J. Pure Appl. Algebra 26 (1982), 269-280.

M.M. Ebrahimi, Equational compactness of sheaves of algebras on a Noetherian

locale, Algebra Universalis 16 (1983), 318-330.

M.M. Ebrahimi, Internal completeness and injectivity of Boolean algebras in the

topos of M-sets, Bull. Austral. Math. Soc. 41(2) (1990), 323-332.

M.M. Ebrahimi, On ideal closure operators of M-sets, Southeast Asian Bull. Math.

(2006), 439-444.

M.M. Ebrahimi and M. Mahmoudi, Purity and equational compactness of projection

algebras, Appl. Categ. Structures 9(4) (2001), 381-394.

M.M. Ebrahimi and M. Mahmoudi, The category of M-sets, Italian J. Pure Appl.

Math. 9 (2001), 123-132.

M.M. Ebrahimi and M. Mahmoudi, Baer criterion and injectivity of projection algebras,

Semigroup Forum 71(2) (2005), 332-335.

M.M. Ebrahimi, M. Mahmoudi, and Gh. Moghaddasi Angizan, Injective hulls of

acts over left zero semigroups, Semigroup Forum 75(1) (2007), 212-220.

M.M. Ebrahimi, M. Mahmoudi, and Gh. Moghaddasi Angizan, On the Baer criterion

for acts over semigroups, Comm. Algebra 35(12) (2007), 3912-3918.

M.M. Ebrahimi, M. Mahmoudi, and L. Shahbaz, Proper behaviour of sequential

injectivity of acts over semigroups, Comm. Algebra 37(7) (2009), 2511-2521.

R. Goldblatt, Topoi: The Categorial Analysis of Logic", North Holland, 1986.

M. Kilp, U. Knauer, and A. Mikhalev, Monoids, Acts and Categories", Walter de

Gruyter, Berlin, New York, 2000.

S. Maclane, Categories for theWorking Mathematicians", Graduate Texts in Mathematics,

No. 5, Springer-Verlag, 1971.

M. Mahmoudi, Internal injectivity of Boolean algebras in MSet, Algebra Universalis

(1999), 155-175.

M. Mahmoudi and Gh. Moghaddasi Angizan, Injective hulls of acts over idempotent

semigroups, Semigroup Forum 74(2) (2007), 240-246.

M.M. Ebrahimi, M. Haddadi, M. Mahmoudi

M. Mahmoudi and L. Shahbaz, Sequentially dense essential monomorphisms of acts

over semigroups, Appl. Categ. Structures 18(5) (2010), 461-471.

M. Mahmoudi and L. Shahbaz, Characterizing semigroups by sequentially dense

injective acts, Semigroup Forum 75(1)(2007), 116-128.

W. Taylor, Residually small varieties, Algebra Universalis 2 (1972), 33-53.

B.R. Tennison, Sheaf Theory", Cambridge University Press, 1975.

W. Tholen, Injective Objects and Cogenerating sets, J. Algebra 73(1) (1981), 139-

W. Wechiler, Universal Algebra for Computer Scientists", EATCS Monographs on

Theoretical Computer Science, Springer-Verlag, 1992.


  • There are currently no refbacks.