### Injectivity in a category: an overview on smallness conditions

#### Abstract

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.

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