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.



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