In the classical theory of $(1, 1)$ categories, the chain of classes of mono/epi morphisms is well known: plain $\leftarrow$ strong $\leftarrow$ effective $\leftarrow$split ((I assume that the category is at least finitely bicomplete and that is why I do not distinguish here from the classes of extremal, strict and regular morphisms). Strong ones are defined as right orthogonal epimorphisms, and effective ones as those that arise as equalizers (or equivalently those that are the equalizer of their cokernel pair). In addition to (indicated by arrows) embeddings of these classes, there are also known easy sufficient conditions (well-poweredness and completeness) guaranteeing that the pair (strong epi, mono) forms a factorization system in the category (and dually). We also know that in regular categories strong epi = effective epi (and thus, in particular, the last class is closed under compositions).
When passing to general $1$-categories (that is, $(\infty, 1)$ / not truncated), in place of plain mono we have a nested sequence of concepts of $n$-truncated morphisms (in which the ordinary monomorphism is numbered $(-1)$). Dualizing it and considering orthogonal classes, we have the concepts
- $n$-monomorphism (I use this here as a synonym for $n$-truncated morphism in order to apply the usual epithets because I do not yet see a more understandable and consistent terminology within the framework of the question)
- $n$-strong monomorphism (right orthogonal n-epimorphism, which in turn, by definition, are n-monomorphisms in the dual category)
In addition there is
- regular monomorphism (those that are the limit of some cosimplicial diagram)
- effective monomorphism (those that are the limit of their coCech nerve)
- split monomorphism (section)
And also all their dual concepts.
What is known about inclusions between these classes and soft sufficient conditions for orthogonal pairs to be factorization systems?
The main case in which I am interested in these concepts is $(\infty, 1)$-topoi. In addition to what has already been stated, in this context I am concerned about:
Which of the described classes stick together in them? For example, we know that effective epimorphisms = $(-1)$-strong epimorphisms (and more generally, Lurie introduces the concept of a regular category, where this is done).
Is it true in topoi that all epimorphisms are effective epimorphisms? As far as I understand, this is true in $\infty\text{-}\mathrm{Groupoid}$, where effective epimorphisms are morphisms in which all fibers are not empty, and epimorphisms are morphisms in which all fibers are acyclic in the sense that of their suspensions are contractible (and therefore the fibers are not empty). See, G. Raptis - Some characterizations of acyclic maps
Are there any useful texts on these issues, other than those we know at HTT/Keredon?
