Injective Convergence Spaces and Equilogical Spaces via Pretopological Spaces

Sierpinski space Ω is injective in the category Top of topological spaces, but not in any of the larger cartesian closed categories Conv of convergence spaces and Equ of equilogical spaces. We show that this negative result extends to all sub-cccs of Equ and Conv that are closed under subspaces and contain Top. On the other hand, we study the category PrTop of pretopological spaces that lies in-between Top and Conv/Equ, identify its injective spaces, and show that they are also injective in Conv and Equ.