Ladders of recollements, categories of monomorphisms and singularity categories

In this paper we show that the (un)bounded derived categories$\colon$(i) of the monomorphism category, (ii) of the morphism category and (iii) of the double morphism category, admit a periodic infinite ladder of recollements. These results are based on a characterization that we provide for a recollement of (compactly generated) triangulated categories to admit a ladder of some height going either upwards or downwards. Moreover, we introduce and study the singularity category of the monomorphism category over an Artin algebra $\Lambda$ and show that there is a periodic infinite ladder that connects this triangulated category with the standard singularity category of $\Lambda$. We also provide sufficient conditions for the monomorphism categories of two algebras to be singularly equivalent. The last aim of this paper is to study monomorphisms where the domain has finite projective dimension. We show that the latter category is a Gorenstein subcategory of the monomorphism category when $\Lambda$ is a Gorenstein Artin algebra. Finally, we consider the category of coherent functors over the stable category of this Gorenstein subcategory, and show that it carries a structure of a Gorenstein abelian category.