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Stream: theory: category theory

Topic: mono in a 2-category

Chetan Vuppulury (Apr 13 2021 at 08:46):

What's the definition of a monomorphism in a (weak) 2-category?

Morgan Rogers (he/him) (Apr 13 2021 at 09:29):

Here's the relevant nLab page, https://ncatlab.org/nlab/show/n-monomorphism
To summarise, if you have an n-category, there is a notion of m-mono for each $0 \leq m \leq n$. In the 2-category of categories, the 0,1 and 2-monomorphisms are the equivalences, fully faithful and faithful functors respectively.

Morgan Rogers (he/him) (Apr 13 2021 at 09:31):

See also https://ncatlab.org/michaelshulman/show/full+morphism
where you have some orthogonal notions defined (eso, eso and full respectively), allowing one to characterize fullness.

Chetan Vuppulury (Apr 13 2021 at 09:48):

Aren't these for $\left(\infty,1\right)$-categories? In a $2$-category we have noninvertible $2$-morphisms, does that not change the definition?

Morgan Rogers (he/him) (Apr 13 2021 at 09:52):

I had to link to that page because I couldn't find a page about generic n-monos.

Morgan Rogers (he/him) (Apr 13 2021 at 09:55):

I think @Mike Shulman explained them on ncatcafe somewhere, maybe he can point you to the right place. In any case, iirc you should be able to extract the definition you're looking for from the $(\infty,1)\text{-category}$ setting.

Amar Hadzihasanovic (Apr 13 2021 at 10:19):

I think the answer is a bit more subtle, in the sense that the notion of monomorphism from 1-categories can also have different generalisations/weakenings that are not captured by the n-monomorphism hierarchy. In particular “1-monomorphisms” seem to generalise from

• A morphism $f: x \to y$ is a monomorphism if and only the $f_*: \mathrm{hom}(z,x) \to \mathrm{hom}(z,y)$ are isomorphisms injections of sets.

to

• A morphism $f: x \to y$ is a monomorphism if and only the $f_*: \mathrm{hom}(z,x) \to \mathrm{hom}(z,y)$ are (whatever the generalisation of injection to categories/higher categories/higher groupoids is, depending on the setting)

Amar Hadzihasanovic (Apr 13 2021 at 10:22):

But one can also try to weaken the “elementary” definition, that is, $f$ is a monomorphism if and only if $g;f = h;f$ implies that $g = h$, more in the direction of 'formal higher category theory'. For example, one weakening that makes sense in 2-categories would be:

“Every 2-isomorphism $\alpha: g;f \simeq h;f$ factors through a 2-isomorphism $\beta: g \simeq h$.”

Amar Hadzihasanovic (Apr 13 2021 at 10:23):

And in the 2-category of categories, I think that corresponds to a pseudomonic functor, which is a functor that is faithful and full on isomorphisms.

Amar Hadzihasanovic (Apr 13 2021 at 10:24):

And that's not the same as a 1-monomorphism in the sense above, which corresponds to a full and faithful functor.

Amar Hadzihasanovic (Apr 13 2021 at 10:27):

So as for many similar questions, the answer is: “There is no single way to generalise this concept from categories to higher categories; you really have to look at the specific aspect that you are trying to capture, and find what works best.”

Mike Shulman (Apr 13 2021 at 14:31):

Amar Hadzihasanovic said:

• A morphism $f: x \to y$ is a monomorphism if and only the $f_*: \mathrm{hom}(z,x) \to \mathrm{hom}(z,y)$ are isomorphisms of sets.

I think you mean "injections of sets".

Amar Hadzihasanovic (Apr 13 2021 at 15:28):

Oops, of course.

Amar Hadzihasanovic (Apr 13 2021 at 15:37):

On second thought, the issue is probably not about what definition we are taking (the “representable” one or the “elementary” one), but rather about the fact that 'monomorphisms of sets' have different generalisations to (higher) categories/groupoids that induce different generalisations on arbitrary higher categories/groupoids via the “representable” definition.

John Baez (Apr 13 2021 at 15:53):

@Chetan Vuppulury

• Pseudomonic morphism

may be what you want; it relies on the definition of "pseudomonic functor".

Mike Shulman (Apr 13 2021 at 17:22):

It's true that there are many factorization systems on $n$-categories for large $n$, but I think only fully-faithful morphisms and pseudomonic morphisms really have a claim to be "the" notion of monomorphism for $n$-categories, since they're the only ones that specialize to monomorphisms of sets considered as discrete $n$-categories.

Mike Shulman (Apr 13 2021 at 17:23):

In general I think when there are noninvertible 2-cells around, fully-faithful morphisms are better-behaved than pseudomonic ones. But the pseudomonic ones have their uses too.