Just as an (orthogonal/unique) factorization system on a category gives a way to factor every morphism of as an -map followed by an -map, a ternary (orthogonal) factorization system gives a way to factor every map of as an -map followed by an -map followed by an -map.
This is a special case of a notion of k-ary factorization system.
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 to consist of a pair and of ordinary orthogonal factorization systems such that (or equivalently ).
The three classes of map are then defined by , , and . This is justified by:
Given a ternary factorization system as above, any morphism factors as
in an essentially unique way.
Consider the two ternary factorizations of obtained by
Note that both start with an map and end with an 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 and the second produces a middle map which is in , this middle map must in fact be in . Finally, any other such ternary factorization of induces an and factorization by composing pairwise, and uniqueness of these two implies uniqueness of the ternary factorization.
More explicitly, if we factor as
with , then we obtain from the lifting problems
arrows : now it’s easy to see that , and that .
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.
In addition to , , , , and , a ternary factorization system also determines a sixth important class of morphisms, namely those whose -part is an isomorphism, or equivalently those that can be factored as an -map followed by an -map. We therefore call this class .
In a ternary factorization system, and .
In both cases is obvious. Conversely, if , say for and , then orthogonality in the square
exhibits as a retract of in , whence since is closed under retracts.
In Top, let quotient maps, injective continuous maps, surjective continuous functions, and subspace embeddings. Here 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 is the class of monic epics, sometimes called bimorphisms. The maps in are sometimes called strict morphisms.
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 is the class of eso+faithful functors, while is the class of full functors. This factorization system plays an important role in the study of stuff, structure, property.
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 have no name other than “localic surjections,” and those in have no established name (although they are briefly mentioned in A4.6.10 of the Elephant).
Suppose that has a binary factorization system and that is an ambifibration? relative to : i.e. every arrow in has an opcartesian lift and every arrow in has a cartesian lift. (In particular, could be a bifibration.) Then there is a ternary factorization system on for which is the class of opcartesian arrows over , is the class of cartesian arrows over , and is the class of vertical arrows (those lying over identities). See this comment.
A similar example is given by a span of categories where is a fibration whose cartesian morphisms are -vertical and is an opfibration whose opcartesian morphisms are -vertical (that is, the span is both a left and a right fibration in the sense of Street). Then the two factorization systems on given by the -opcartesian and -vertical morphisms on the one hand, and the -vertical and -cartesian morphisms on the other, satisfy the condition above, so that every morphism in factors as a -opcartesian morphism followed by a morphism that is both - and -vertical, followed by a -cartesian morphism.
Such a span is a two-sided fibration if , that is if the three-way factorization of the composite of a -cartesian morphism followed by a -opcartesian one has its middle term an isomorphism.
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 is important, however: it is precisely the class of weak equivalences.
The notion of k-ary factorization system is a generalization to factorizations into morphisms.
Just as strict factorization systems can be identified with distributive laws in the bicategory of spans, so “strict” ternary (and k-ary) factorization systems can be identified with iterated distributive laws in .
A. Pultr and W. Tholen, Free Quillen Factorization Systems. Georgian Math. J.9 (2002), No. 4, 807-820