Spectrum of an abelian category via premonoform objects

Let $\text{cA}$ be an abelian category. In this paper, we study $(\mathrm{n})\mathrm{P}\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\text{cA}$, a topological space formed by equivalence classes derived from an equivalence relation on (noetherian) premonoform objects. We classify torsion classes of $\text{cA}$ via closed subclasses of $\text{nPSpec}\text{cA}$. We introduce a new topology on $\text{PSpec}\text{cA}$ and we classify Serre subcategories of $\text{noeth}A$ and localizing subcategories of $\text{cA}$ using this topology. If $A$ is a commutative noetherian ring, we show that $\text{nPSpec}A$ is homeomorphic to $\text{Spec}A$. Moreover, there is a one-to-one correspondence between the closed subsets of $\text{nPSpec}A$ and the open subsets of $\text{ASpec}A$, the atom spectrum of $A$. Finally, we explore the relationships between the new subctegories of $\text{Mod}A$ and subsets of $\text{nPSpec}A$ introduced in this paper, and the known subcategories of $\text{Mod}A$ and subsets of other spectra of $A$.