nLab ternary factorization system

Idea

Just as an (orthogonal/unique) factorization system $(E,M)$ on a category $C$ gives a way to factor every morphism of $C$ as an $E$-map followed by an $M$-map, a ternary (orthogonal) factorization system $(E,F,M)$ gives a way to factor every map of $C$ as an $E$-map followed by an $F$-map followed by an $M$-map.

This is a special case of a notion of k-ary factorization system.

Definition

It turns out that a convenient way to state the definition is in terms of a pair of ordinary (orthogonal) factorization systems. We define a ternary factorization system on $C$ to consist of a pair $(L_1,R_1)$ and $(L_2,R_2)$ of ordinary orthogonal factorization systems such that $L_1 \subseteq L_2$ (or equivalently $R_2 \subseteq R_1$).

The three classes of map $(E,F,M)$ are then defined by $E=L_1$, $F = L_2\cap R_1$, and $M=R_2$. This is justified by:

Proposition

Given a ternary factorization system as above, any morphism $f:A\to B$ factors as

$$A \overset{L_1}{\to} im_2(f) \overset{L_2 \cap R_1}{\to} im_1(f) \overset{R_2}{\to} B$$

in an essentially unique way.

Proof

Consider the two ternary factorizations of $f$ obtained by

1. First factoring $f$ into an $L_1$-map followed by an $R_1$-map, then factoring the $R_1$-part into an $L_2$-map followed by an $R_2$-map; and
2. First factoring $f$ into an $L_2$-map followed by an $R_2$-map, then factoring the $L_2$-part into an $L_1$-map followed by an $R_1$-map.

Note that both start with an $L_1$ map and end with an $R_2$ map. By a straightforward exercise in orthogonality, we can get comparison maps in both directions between these two factorizations which make them isomorphic. Therefore, since the first produces a middle map which is in $L_2$ and the second produces a middle map which is in $R_1$, this middle map must in fact be in $L_2\cap R_1$. Finally, any other such ternary factorization of $f$ induces an $(L_1,R_1)$ and $(L_2,R_2)$ factorization by composing pairwise, and uniqueness of these two implies uniqueness of the ternary factorization.

More explicitly, we factor $f$ as

with $\lambda_i, \ell_i\in L_i, \rho_j,r_j\in R_j$. Then since $R_2\subseteq R_1$, we have $r_2 \rho_1 \in R_1$, so that $(\ell_1,r_1)$ and $(\lambda_1,r_2\rho_1)$ are both $(L_1,R_1)$-factorizations of $f$ and thus we have a unique compatible isomorphism $C_1\cong D_1$. Similarly, $(\lambda_2 \ell_1, \rho_2)$ and $(\ell_2,r_2)$ are both $(L_2,R_2)$-factorizations, so we have a unique compatible isomorphism $C_2\cong D_2$. This gives a diagram

with two commutative triangles, and the middle square also commutes since both sides are lifts in a lifting problem of $\ell_1$ against $r_2$. Finally, since $\lambda_2\in L_2$ is isomorphic to $\rho_1\in R_1$ in the arrow category, both are in fact in $L_2\cap R_1$.

Conversely, just as for a binary factorization system, the extra requirement of orthogonality can be deduced from uniqueness of the factorizations, a unique and functorial ternary factorization implies that it “splits” into a pair of binary factorization systems, i.e. a ternary factorization system as defined here. This is remarked on here.

One can also characterize the notion in terms of a ternary factorization with a “ternary orthogonality” property; see the paper of Pultr and Tholen referenced below.

The sixth class of maps

In addition to $L_1$, $R_1$, $L_2$, $R_2$, and $L_2\cap R_1$, a ternary factorization system also determines a sixth important class of morphisms, namely those whose $(L_2\cap R_1)$-part is an isomorphism, or equivalently those that can be factored as an $L_1$-map followed by an $R_2$-map. We therefore call this class $R_2 L_1$.

Proposition

In a ternary factorization system, $L_1 = L_2 \cap R_2L_1$ and $R_2 = R_1 \cap R_2L_1$.

Proof

In both cases $\subseteq$ is obvious. Conversely, if $f \in L_2 \cap R_2 L_1$, say $f = m e$ for $m\in R_2$ and $e\in L_1$, then orthogonality in the square

$$\array{a & \overset{e}{\to} & c\\ ^f \downarrow && \downarrow ^m\\ b & \underset{id}{\to} & b}$$

exhibits $f$ as a retract of $e$ in $Arr(C)$, whence $f\in L_1$ since $L_1$ is closed under retracts.

In many examples, the class $R_2 L_1$ is closed under composition, but this is not always the case.

Proposition

In a ternary factorization system, the class $R_2L_1$ is closed under composition if and only if $L_1R_2 \subseteq R_2L_1$.

Proof

This is Proposition 3.6 of Pultr and Tholen.

“Only if” is obvious, since $L_1R_2 \subseteq (R_2L_1)(R_2L_1)$ which is contained in $R_2L_1$ if it is closed under composition. Conversely, if $L_1R_2 \subseteq R_2L_1$ then $(R_2L_1)(R_2L_1) = R_2 (L_1 R_2) L_1 \subseteq R_2 (R_2 L_1) L_1 \subseteq R_2 L_1$.

Corollary

In a ternary factorization system, if every morphism factors as an $R_2$-map followed by an $L_1$-map, then $R_2L_1$ is closed under composition if and only if it consists of all morphisms, and if and only if $L_2 \cap R_1$ consists of only the isomorphisms.

Proof

This is remarked after Examples 3.12 in Pultr and Tholen.

By the Proposition , if $L_1R_2$ is all morphisms and $R_2L_1$ is closed under composition, then it is also all morphisms, and by uniqueness of ternary factorizations this is equivalent to $L_2 \cap R_1$ being the isomorphisms.

Concrete examples are below.

Relation to iterated distributive laws

Since strict factorization systems can be identified with distributive laws in the bicategory of spans, it is natural to think that “strict” ternary factorization systems can be identified with iterated distributive laws in Span. However, this is not true in general, due to the lack of closure of $R_2L_1$ under composition.

In a triple distributive law there are three monads $A,B,C$ and distributive laws $B A \to A B$, $C A \to A C$, and $C B\to B C$ satisfying the Yang-Baxter equation. Thus, in particular there are three composite monads $A B$, $A C$, and $B C$, and the Yang-Baxter equation ensures we get distributive laws $(B C)A\to A(B C)$ and $C(A B) \to (A B)C$ and a unique induced monad structure on $A B C$.

In a ternary factorization system, we would naturally take $A = R_2$, $B = L_2 \cap R_1$, and $C = L_1$. We then get distributive laws $B A \to A B$ and $C B \to B C$, since $A B = R_1$ and $B C = L_2$ are closed under composition. However, if $A C = R_2L_1$ is not closed under composition, we do not get a distributive law $C A \to A C$.

It is now natural to conjecture that this is the only obstacle, though: strict ternary factorization systems for which $R_2L_1$ is closed under composition should be equivalent to triple distributive laws in $Span$. Similarly, the non-strict case should correspond to triple distributive laws over the groupoid of isomorphisms in Prof, as for ordinary orthogonal factorization systems and ordinary distributive laws.

Examples

• In Top, let $L_1=$ quotient maps, $R_1=$ injective continuous maps, $L_2=$ surjective continuous functions, and $R_2=$ subspace embeddings. Here $L_2\cap R_1=$ bijective continuous maps, and the two intermediate objects in the ternary factorization of a continuous map are obtained by imposing the coarsest and the finest compatible topologies on its set-theoretic image.

• More generally, if a category has both (epi, strong mono) and (strong epi, mono) factorizations, then since strong epis are epi, we have a ternary factorization. Here $L_2\cap R_1$ is the class of monic epics. The maps in $R_2 L_1$ are sometimes called strict morphisms.

  By Corollary , if every morphism factors as a strong mono followed by a strong epi, the strict morphisms are only closed under composition if they are the isomorphisms. For instance, this is true if the category has coproducts whose injections are strong mono, since we can then factor $A\to B$ as $A \to A+B \to B$ where the second factor is even split epi. This is the case in Top, so that is a concrete example where $R_2L_1$ is not closed under composition.

• On Cat there is a 2-categorical version of a ternary factorization system, determined by the 2-categorical factorization systems (eso+full, faithful) and (eso, full and faithful). Here $L_2\cap R_1$ is the class of eso+faithful functors, while $R_2 L_1$ is the class of full functors. This factorization system plays an important role in the study of stuff, structure, property.

  Restricted to groupoids this is the 1-image-2-image factorization, the 3-stage Postnikov system of groupoids.

• On Topos there is also a 2-categorical ternary factorization system composed of the binary 2-categorical factorization systems (hyperconnected, localic) and (surjection, inclusion). Here the maps in $L_2\cap R_1$ have no name other than “localic surjections,” and those in $R_2 L_1$ have no established name (although they are briefly mentioned in A4.6.10 of the Elephant).

• Suppose that $C$ has a binary factorization system $(E,M)$ and that $p\colon A\to C$ is an ambifibration? relative to $(E,M)$: i.e. every arrow in $E$ has an opcartesian lift and every arrow in $M$ has a cartesian lift. (In particular, $p$ could be a bifibration.) Then there is a ternary factorization system on $A$ for which $L_1$ is the class of opcartesian arrows over $E$, $R_2$ is the class of cartesian arrows over $M$, and $L_2\cap R_1$ is the class of vertical arrows (those lying over identities). See this comment.

  For instance, the above factorization system on $Top$ is induced in this way via the forgetful functor $Top\to Set$ from the (epi,mono) factorization system on Set.

• A similar example is given by a span $A \overset{p}{\leftarrow} E \overset{q}{\to} B$ of categories where $p$ is a fibration whose cartesian morphisms are $q$-vertical and $q$ is an opfibration whose opcartesian morphisms are $p$-vertical (that is, the span $(p,q)$ is both a left and a right fibration in the sense of Street). Then the two factorization systems on $E$ given by the $q$-opcartesian and $q$-vertical morphisms on the one hand, and the $p$-vertical and $p$-cartesian morphisms on the other, satisfy the $L_1 \subseteq L_2$ condition above, so that every morphism in $E$ factors as a $q$-opcartesian morphism followed by a morphism that is both $p$- and $q$-vertical, followed by a $p$-cartesian morphism.

  Such a span is a two-sided fibration if $L_1R_2 \subseteq R_2L_1$, that is if the three-way factorization of the composite of a $p$-cartesian morphism followed by a $q$-opcartesian one has its middle term an isomorphism.

Related concepts

• The notion of model category involves a pair of weak factorization systems called (acyclic cofibration, fibration) and (cofibration, acyclic fibration) which are compatible in the same sense as above. However, non-uniqueness of these factorizations means that the resulting “ternary factorization” of a morphism is not unique. The class corresponding to $R_2 L_1$ is important, however: it is precisely the class of weak equivalences.

• The notion of k-ary factorization system is a generalization to factorizations into $k$ morphisms.

References

Ternary factorisation systems are introduced as double factorisation systems in:

• A. Pultr and W. Tholen, Free Quillen Factorization Systems. Georgian Math. J.9 (2002), No. 4, 807-820

• Cafe discussion