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I have a certain construction relating to subobjects in an arbitrary category. Now the nlab article on subobjects says:

More generally, in some contexts we may take “subobject” to mean an isomorphism class of morphisms $i: a\to c$ satisfying some suitable condition other than being a monomorphism (usually a stronger one).

And this makes perfect sense, now when trying to generalize the construction to some subclass of monics (in order to find the 'correct' construction for a category when there's a better notion of subobjects), I came to realize that I needed some pseudo-functors to be in this "subclass" of monics. For example, I needed monics of presheaves to be calculated pointwise, and some 2-morphisms of Grothendieck fibrations to be 2-monic in this stronger sense.

This leads me to my question: is there a (universally accepted?) definition of "classes of monic" that behaves well in higher categories as well? I really don't want to reinvent the wheel here.

One such definition could be a small category $D$ such that $n$-monics are $n$-limits of $n$-functors from $D$. This seems like a good definition, only it misses some classes such as effective monics which are not given like this. Is there a better one?

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    $\begingroup$ Perhaps being half of some kind of factorisation system? Sometimes it is easier to define a class of categorified epimorphisms... $\endgroup$
    – David Roberts ♦
    Commented Mar 8, 2018 at 7:04

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I think I found an answer: $n+1$-categories can be viewed as categories enriched in $n$-categories, and if we had a notion of "monomorphism" for $n$-categories, whatever that notion would be for the higher version, the $hom$ bifunctor should preserve this structure (similar to the fact that fully faithful functors are the monics in $Cat$).

So we are left with defining the enriching 2-category of monoidal categories equipped with "monomorphism" structure and this would determine a "good" definition for higher "monomorphisms".

I believe the definiton for this category should be $(\mathcal{V}, m)$ where $\mathcal{V}$ is monoidal and $m$ is a lax monoidal endofunctor $m:\mathcal{V}\to\mathcal{V}$ that is an identity on objects. $m$ is to be thought of as the "inclusion of the subcategory of monomorphisms"

The morphisms (at least for the same $\mathcal{V}$) should be factorizations of the endofunctors which correspond to 'enlarging' the class of monomorphisms.

Now this definition allows degenerate classes, such as given by the $id$ or ones remembering only isomorphisms, so this could use some refinement.

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    $\begingroup$ I agree with your first paragraph, but the third feels like it might be too general (as you point out). Maybe write to an expert in 2-categories, like Steve Lack or Ross Street. Also, are you aware of the theory of enriched weak factorization systems? It seems very relevant to your idea. Experts would be Emily Riehl and Richard Garner. $\endgroup$
    – David White
    Commented Mar 11, 2018 at 14:01
  • $\begingroup$ Thank you for the references! I was not aware of those concepts, which seem very much like 'generalized monomorphisms'. As for the generality, there is a simple approach I'm looking at, which is dual to the factorization approach, only simpler: take $m$ to be a functor on pointed arrow categories, with the yoneda embedding as the marked point - This excludes the trivial cases. Now, using this I think we can prove the existance of a factorization system for the morphisms by lifting through the yoneda embedding Note that a functorial factorization system yields an endofunctor like $m$ by taking $\endgroup$
    – Omer Rosler
    Commented Mar 11, 2018 at 22:17
  • $\begingroup$ a morphism into a $\cal R$ morphism it factors through So this is actually a different approach, where I believe it's advantage is simplicity. When these details are more flashed out, I will edit the answer (hopefully in a few days) $\endgroup$
    – Omer Rosler
    Commented Mar 11, 2018 at 22:19

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