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Stream: theory: category theory

Topic: A property of monadic functors

Ari Rosenfield (Jan 22 2025 at 18:41):

Say that a functor $U : \mathcal{C} \to \mathcal{D}$ is order-reflecting on monomorphisms if it reflects monomorphisms and satisfies the following condition:

If $f \colon X \to Z$ and $g \colon Y \to Z$ are morphisms in $\mathcal{C}$ such $U(f)$ and $U(g)$ are monomorphisms with a factorization $U(f) = U(g) \circ \sigma$ in $\mathcal{D}$, then there exists $\sigma_0 \colon X \to Y$ in $\mathcal{C}$ such that $U(\sigma_0) = \sigma$ and $f = g \circ \sigma_0.$

The forgetful functor from abelian groups to sets is order-reflecting on monomorphisms, and it's not hard to show that any monadic functor has this property.

I have a feeling that being order-reflecting on monomorphisms is a consequence of some more general property a functor might have (weaker than monadicity), but I have had some trouble finding it. I thought someone here might be able to spot it!

Mike Shulman (Jan 22 2025 at 19:04):

I'm not sure if this is the sort of thing you're looking for, but one property that lies between monadicity and OROM is that every monomorphism is cartesian, i.e. that its domain has the "$C$-structure" induced universally from its codomain.

Mike Shulman (Jan 22 2025 at 19:06):

For instance, I think categories that are intuitively "algebraic" but fail to be monadic for size reasons, like the category of complete lattices, will have this property.

Ari Rosenfield (Jan 22 2025 at 22:48):

Yes, this is helpful; I hadn't seen cartesian morphisms before. Thanks!