# Orthogonality relations and accessibility?

Suppose I have a pair of locally presentable categories connected by an accessible functor. Then the preimage of an accessible subcategory of the codomain is an accessible subcategory of the domain. In this way, we can see, for instance, that the class of maps in a locally presentable category orthogonal to a fixed (set of) "projective" object(s) forms an accessible subcategory of the arrow category: the functors in question are covariant representables, and isomorphisms form an accessible subcategory of Set^2.

But suppose I'm interested in a class of maps orthogonal to a fixed (set of) "injective" object(s)? Now the relevant representable functors are contravariant and Set^op is of course not locally presentable. Is there anything I can say about the class of maps that are left orthogonal to a set of objects? I'd be particularly interested in "solution set" type conditions for the functor including these maps into the arrow category.

• By "orthogonal", do you mean having the left/right lifting property? "Orthogonal" usually means a unique lifting property, but then epimorphisms wouldn't matter. – Mike Shulman Aug 20 '13 at 20:56
• Oops. You're right. Secretly I'm interested in classes that are characterized by a left lifting property. But because I'm thinking "algebraically" I'm optimistic that ideas arising in the orthogonal case will be relevant. I'll edit the question to fix my mistake. Thanks! – Emily Riehl Aug 21 '13 at 2:55