An (orthogonal) factorization system in a category consists of two classes of morphisms such that every morphism in factors as an -morphism followed by an -morphism, and we have satisfies the left lifting property –uniquely for the orthogonal case, , see orthogonality – for any and .
To enrich this, we should first consider enriched orthogonality. The statement for maps and can be rephrased by saying that the square
More generally, the statement that there exists at least one lift is to say that the canonical morphism
It is then clear that if is enriched over some monoidal category , to say that in an enriched sense, we should instead require this square to be a pullback of enriched hom-objects in . Note, though, that and are still maps in the underlying ordinary category of . Likewise, the factorization property still only makes sense for maps in .
Therefore, we define an enriched (orthogonal) factorization system on an enriched category to consist of two classes of maps in such that
By the definition of , enriched orthogonality implies ordinary orthogonality. Therefore, an enriched factorization system on induces an ordinary factorization system on . Conversely, if has powers that preserve the maps in , or copowers that preserve the maps in , then unenriched orthogonality in implies enriched orthogonality by a Yoneda lemma argument.
Moreover, the factorization functor can be made into an enriched functor in the following way. There is a -category whose objects are morphisms in and whose hom-objects are defined by, for and , a pullback
(This is the power of by in the 2-category , and also the -functor category , where denotes the free -category on .)
Likewise, we have whose objects are composable pairs of morphisms in , and whose hom-objects are defined by pullbacks
By functoriality we then mean that the factorization is given by a functor which, when composed with the “composition” functor , gives the identity of .
The interesting part of the enrichment of this functor is the following: given and in , with factorizations and , by enriched orthogonality we have a pullback
and we also have a commutative square
inducing a map . It is then straightforward to construct the rest of the functor.
This argument, as it depends crucially on the universality of the pullback and hence the uniqueness part of orthogonality, fails for weak factorization systems. Although they can be made functorial in many cases, rarely can their functoriality be made enriched (as far as is known).
Enriched lifting and enriched factorization are discussed around from page 133 on (section “April 3”) in