A criterion for the existence of subobject classifiers
The author observes that if \(\mathcal E\) is a cocomplete category having a small dense subcategory \(\mathcal C\), `the subobject classifier of \(\mathcal E\)' may always be considered to exist as a presheaf on \(\mathcal C\). He shows that \(\mathcal E\) has a subobject classifier if and only if this presheaf lies in \(\mathcal E\) considered as a reflective subcategory of the presheaf category. A couple of variants of this condition are given. These results are closely related to those of \textit{F. Borceux} and \textit{B. Veit} [``Subobject classifier for algebraic structures'', J. Algebra 112, No. 2, 306-314 (1988; Zbl 0636.18003)].
scientific article; zbMATH DE number 1292207