On injective objects and existence of injective hulls in 𝑄-TOP/(𝑌, 𝜎)

Abstract

. In this paper, motivated by Cagliari and Mantovani, we have obtained a characterization of injective objects (with respect to the class of embeddings in the category 𝑄 - TOP of 𝑄 -topological spaces) in the comma category 𝑄 - TOP /( 𝑌, 𝜎 ) , when ( 𝑌, 𝜎 ) is a stratified 𝑄 -topological space, with the help of their 𝑇 0 -reflection. Further, we have proved that for any 𝑄 -topological space ( 𝑌, 𝜎 ) , the existence of an injective hull of (( 𝑋, 𝜏 ) , 𝑓 ) in the comma category 𝑄 - TOP /( 𝑌, 𝜎 ) is equivalent to the existence of an injective hull of its 𝑇 0 -reflection (( ˜ 𝑋, ˜ 𝜏 ) , ˜ 𝑓 ) in the comma category 𝑄 - TOP /( ˜ 𝑌, ˜ 𝜎 ) (and in the comma category 𝑄 - TOP 0 /( ˜ 𝑌, ˜ 𝜎 ) , where 𝑄 - TOP 0 denotes the category of 𝑇 0 - 𝑄 -topological spaces).