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Stream: learning: questions

Topic: making categories from monomorphisms

David Egolf (Sep 09 2022 at 18:30):

Let $M$ be a category. I want to make a new category $C$ from $M$ that includes $M$ as a subcategory, such that the monomorphisms in $C$ are exactly the morphisms in $M$. Further, I would also like to require that each morphism $f$ in $C$ can be factored as the composition of an epimorphism and a monomorphism.

It would be exciting if there were some kind of automated procedure that would construct a $C$ satisfying these conditions from an $M$. My idea is that in some contexts one might have an intuitive idea of what the monomorphisms should be, but a less clear picture of what all the morphisms should be - and maybe a procedure like this could help create a richer context to explore.

As a specific example, to try this out, let $M$ be the category of sets together with injective functions between them. I want to systematically create a new category $C$ containing $M$ as a subcategory where the morphisms of $M$ are exactly the monomorphisms of $C$. One strategy might be to start with the morphisms from $M$ and to try to construct the epimorphisms using the lifting property. This is sounding tricky to me, though. I'd have to think about it more.

How can one form a category $C$ from the category $M$, satisfying the conditions above?

Mike Shulman (Sep 09 2022 at 18:36):

To start with, since a monomorphism in a category is also a monomorphism in every subcategory of it, in order to have any hope of success you'll need to assume that all morphisms in $M$ are monomorphisms already. And once you have this, then $C=M$ satisfies the conditions you've stated.

David Egolf (Sep 09 2022 at 18:40):

Oh, interesting! I did not know that about monomorphisms in subcategories!

Yes, I was just realizing that $C=M$ works. Maybe if we require that there be at least one epimorphism in $C$ that is not also a monomorphism we can get something more interesting?

Mike Shulman (Sep 09 2022 at 18:43):

That sort of requirement doesn't give any information about what that epimorphism might be, so it's unlikely to lead to any canonical construction. My guess is you'll get something more interesting if you can think of a universal property that your $C$ ought to have.

David Egolf (Sep 09 2022 at 19:30):

Huh, that is interesting!
That makes me wonder what universal property the categories of sets with all functions between them has.

Mike Shulman (Sep 09 2022 at 23:40):

I don't know of a universal property it has with respect to the category of sets and injective functions. But as a category, it is the free cocompletion of the terminal category.

Simonas Tutlys (Sep 11 2022 at 02:43):

There's also a paper where the category Set is uniquely defined by a 'chain' of adjunctions between six functors iirc.Regarding the question - maybe look into something stricter than kan extensions,since you (as i see it) want to extend the smallest category with all monomorphisms to some bigger category?

Simonas Tutlys (Sep 11 2022 at 02:56):

or look for the terminal/initial object in a suitably defined category of all kan extensions of the monomorphism category?

Tobias Schmude (Sep 11 2022 at 06:58):

Simonas Tutlys said:

There's also a paper where the category Set is uniquely defined by a 'chain' of adjunctions between six functors iirc.

this

Simonas Tutlys (Sep 11 2022 at 08:30):

yeah,that one :)

David Egolf (Sep 11 2022 at 16:23):

Cool! That paper is a bit advanced for me for now, but it's neat to know it exists. For now, I'm just going to keep an eye out for ways that people use to make bigger categories from smaller ones. And learning that it's possible to make categories by requiring "chains of adjunctions" to exist is an interesting approach that is new to me!

John Baez (Sep 11 2022 at 16:59):

It's a bit circular to "make" the category of sets by requiring the Yoneda embedding has a chain of 6 adjoints, since you need the category of sets to define the Yoneda embedding. I tend to think of this as a wacky and amusing characterization of the category of sets.