Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58532120140701Injectivity in a category: an overview on smallness conditions831126800ENM. MehdiEbrahimiDepartment of Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran.MahdiehHaddadiDepartment of Mathematics, Statistics and Computer Science, Semnan University, Semnan, Iran.MojganMahmoudiDepartment of Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran.Journal Article20140601Some 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 first 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 $mod(Sigma, {mathcal E})$ of models of a set $Sigma$ of equations in a suitable category, say a Grothendieck topos ${mathcal E}$, given by M.Mehdi Ebrahimi. We close the paper by some examples to support the results.http://cgasa.sbu.ac.ir/article_6800_3a21602701c668271925317f72f7ea0a.pdf