enriched factorization system




An (orthogonal) factorization system in a category CC consists of two classes of morphisms (E,M)(E,M) such that every morphism in CC factors as an EE-morphism followed by an MM-morphism, and we have ee satisfies the left lifting property –uniquely for the orthogonal case, eme\perp m, see orthogonality – for any eEe\in E and mMm\in M .

To enrich this, we should first consider enriched orthogonality. The statement eme\perp m for maps e:ABe:A\to B and m:XYm:X\to Y can be rephrased by saying that the square

C(B,X) C(A,X) C(B,Y) C(A,Y) \array{C(B,X) & \to & C(A,X)\\ \downarrow && \downarrow \\ C(B,Y) & \to & C(A,Y)}

is a pullback in Set.

More generally, the statement that there exists at least one lift is to say that the canonical morphism

C(B,X)C(A,X) C(A,Y)C(B,Y) C(B,X) \to C(A,X) \prod_{C(A,Y)} C(B,Y)

into the pullback is a split epimorphism.

It is then clear that if CC is enriched over some monoidal category VV, to say that eme\perp m in an enriched sense, we should instead require this square to be a pullback of enriched hom-objects in VV. Note, though, that ee and mm are still maps in the underlying ordinary category C 0C_0 of CC. Likewise, the factorization property still only makes sense for maps in C 0C_0.

Therefore, we define an enriched (orthogonal) factorization system on an enriched category CC to consist of two classes of maps (E,M)(E,M) in C 0C_0 such that

  1. eme\perp m in the above enriched sense for every eEe\in E and mMm\in M, and
  2. Every map in C 0C_0 factors as an EE-map followed by an MM-map.

By the definition of C 0C_0, enriched orthogonality implies ordinary orthogonality. Therefore, an enriched factorization system on CC induces an ordinary factorization system on C 0C_0. Conversely, if CC has powers that preserve the maps in MM, or copowers that preserve the maps in EE, then unenriched orthogonality in C 0C_0 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 VV-category C 2C^{\mathbf{2}} whose objects are morphisms in C 0C_0 and whose hom-objects are defined by, for f 1:X 1Y 1f_1:X_1\to Y_1 and f 2:X 2Y 2f_2:X_2\to Y_2, a pullback

C 2(f 1,f 2) C(X 1,X 2) C(Y 1,Y 2) C(X 1,Y 2). \array{C^{\mathbf{2}}(f_1,f_2) & \to & C(X_1,X_2)\\ \downarrow && \downarrow \\ C(Y_1,Y_2) & \to & C(X_1,Y_2).}

(This is the power of CC by 2{\mathbf{2}} in the 2-category VCatV-Cat, and also the VV-functor category [2 V,C][{\mathbf{2}}_V,C], where 2 V{\mathbf{2}}_V denotes the free VV-category on 2{\mathbf{2}}.)

Likewise, we have C 3C^{\mathbf{3}} whose objects are composable pairs XfYgZX\overset{f}{\to} Y \overset{g}{\to} Z of morphisms in C 0C_0, and whose hom-objects are defined by pullbacks

C 3((f 1,g 1),(f 2,g 2)) C 2(f 1,f 2) C 2(g 1,g 2) C(Y 1,Y 2). \array{C^{\mathbf{3}}((f_1,g_1),(f_2,g_2)) & \to & C^{\mathbf{2}}(f_1,f_2)\\ \downarrow && \downarrow \\ C^{\mathbf{2}}(g_1,g_2) & \to & C(Y_1,Y_2).}

By functoriality we then mean that the factorization is given by a functor C 2C 3C^{\mathbf{2}} \to C^{\mathbf{3}} which, when composed with the “composition” functor C 3C 2C^{\mathbf{3}} \to C^{\mathbf{2}}, gives the identity of C 2C^{\mathbf{2}}.

The interesting part of the enrichment of this functor is the following: given f 1:X 1Z 1f_1:X_1\to Z_1 and f 2:X 2Z 2f_2:X_2\to Z_2 in C 2C^{\mathbf{2}}, with factorizations X 1m 1Y 1e 1Z 1X_1 \overset{m_1}{\to} Y_1 \overset{e_1}{\to} Z_1 and X 2m 2Y 2e 2Z 2X_2 \overset{m_2}{\to} Y_2 \overset{e_2}{\to} Z_2, by enriched orthogonality we have a pullback

C(Y 1,Y 2) C(X 1,Y 2) C(Y 1,Z 2) C(X 1,Z 2) \array{C(Y_1,Y_2) & \to & C(X_1,Y_2)\\ \downarrow & & \downarrow\\ C(Y_1,Z_2) & \to & C(X_1,Z_2)}

and we also have a commutative square

C 2(f 1,f 2) C(X 1,X 2) C(X 1,Y 2) C(Z 1,Z 2) C(Y 1,Z 2) C(X 1,Z 2) \array{C^{\mathbf{2}}(f_1,f_2) & \to & C(X_1,X_2) & \to & C(X_1,Y_2)\\ \downarrow &&&& \\ C(Z_1,Z_2) &&&& \downarrow\\ \downarrow &&&& \\ C(Y_1,Z_2) & & \longrightarrow & & C(X_1,Z_2) }

inducing a map C 2(f 1,f 2)C(Y 1,Y 2)C^{\mathbf{2}}(f_1,f_2) \to C(Y_1,Y_2). 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

Last revised on April 6, 2012 at 05:18:55. See the history of this page for a list of all contributions to it.