Several of the many notions that don't work the same way when passing to $\infty$-categories are the ones mentioned in the title. I'm trying to understand the conceptual picture around these notions in the world of higher category theory (conceptual answers will be more useful to me than model specific answers).


My question are conceptually very simple:



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> 1. What are ($n$-?)monomorphisms and ($n$-?)epimorphisms in an $\infty$-catgory? What are the fundamental differences with $1$-categories?
> 2. What are ($n$-?)coimages and ($n$-?)-images in an $\infty$-category? (it seems that on n-lab the definitions for both of them [[1]](https://ncatlab.org/nlab/show/n-image#ViaColimitOfCechNerve) [[2]](https://ncatlab.org/nlab/show/coimage#InInfCat) are exactly the same which is very confusing) What are the fundamental differences with $1$-categories? Which of these coincides with the Postnikov decomposition in $\infty$-groupoids?
> 3. What is an $n$-ary factorization system for an $\infty$-category? Is this a direct analog of epi-mono factorization? If so In what sense?
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I could elaborate but I tried and concluded that it will probably only serves to confuse me and the readers. I'm still confused by many of these notions. 



Monomorphisms, epimorphisms, (co-)images and factorizations in $\infty$-categories
An **$n$-monomorphism** is a map $A\to B$ for which $Map(X,A)\to Map(X,B)$ has all homotopy fibers **$n$-truncated**, for all $X$; a space (=$\infty$-groupoid) is $n$-truncated if its homotopy groups all vanish in dimensions $>n$.


An **$n$-epimorphism** is an $n$-monomorphism in the opposite $\infty$-category.


When $n=-1$ you recover Denis's definition of **monomorphism** and **epimorphism**: note that a space is **$(-1)$-truncated** if and only if it is empty or contractible.


The phenomenon of "$n$-epi/mono factorization" in an $\infty$-category does not actually involve factorization as ($n$-mono)($n$-epi), but rather as ($n$-mono)($n$-connected map). The class of **$n$-connected** maps is usually defined as a kind of left complement to $n$-monomorphisms, so the existence of such factorizations is basically the assertion that your $\infty$-category admits a factorization system where the right-class is the class of $n$-monos. (These exist in all "reasonable" examples, e.g., presentable $\infty$-categories.)


This same issue shows up in 1-category theory: epi/mono factorization tends to be rare, and more commonly one considers factorizations of the form: (regular epi)(mono).


The factorization ($n$-mono)($n$-connected) can be thought of as factorization through an "$n$-image". It is clearly not a self-dual notion, hence there is a dual but distinct notion of factorization through an "$n$-coimage".


