On two new types of Moore subcategories

Abstract

Abstract We introduce and investigate two new classes of Moore subcategories of the concrete category $$\textbf{S}[T]$$ S [ T ] associated with an inclusion-preserving endofunctor $$T:{\textbf {Set}}\longrightarrow {\textbf {Set}}$$ T : Set ⟶ Set . We call these kinds of Moore subcategories the nonsingular and the mono Moore $$(T,{\textbf {Set}})$$ ( T , Set ) -subcategories, respectively. These notions refine the general theory of "Equation missing" -Moore subcategories developed in some recent research and arise naturally when addressing specific problems related to the behavior of monomorphisms. For nonsingular Moore $$(T,{\textbf {Set}})$$ ( T , Set ) -subcategories we prove unique transportability and repleteness. Next, we also exhibit explicit descriptions of subobjects and extremal subobjects in terms of the domains of the monomorphisms arising in suitable $$(\text {Epi},\text {Mono})$$ ( Epi , Mono ) -factorizations in $$\textbf{S}[T]$$ S [ T ] and a version of the Final Coalgebra Theorem for bounded endofunctors preserving extremal monomorphisms. On the other hand, in the case of mono Moore $$(T,{\textbf {Set}})$$ ( T , Set ) -subcategories we show that these subcategories are automatically nonsingular, that the corresponding faithful functor is a Grothendieck "Equation missing" -fibration, that embeddings coincide with extremal monomorphisms, and that an extremal subobject classifier exists.