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 𝑇0-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 𝑇0-reflection ((𝑋 ̃,𝜏 ̃), 𝑓 ̃) in the comma category Q-TOP/(𝑌 ̃, 𝜎 ̃ ) (and in the comma category 𝑄-TOP0/(𝑌 ̃, 𝜎 ̃ ), where 𝑄-TOP0 denotes the category of 𝑇0-𝑄-topological spaces).