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, we factor as
with . Then since , we have , so that and are both -factorizations of and thus we have a unique compatible isomorphism . Similarly, and are both -factorizations, so we have a unique compatible isomorphism . This gives a diagram
with two commutative triangles, and the middle square also commutes since both sides are lifts in a lifting problem of against . Finally, since is isomorphic to in the arrow category, both are in fact in .
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 many examples, the class is closed under composition, but this is not always the case.
In a ternary factorization system, the class is closed under composition if and only if .
This is Proposition 3.6 of Pultr and Tholen.
“Only if” is obvious, since which is contained in if it is closed under composition. Conversely, if then .
In a ternary factorization system, if every morphism factors as an -map followed by an -map, then is closed under composition if and only if it consists of all morphisms, and if and only if consists of only the isomorphisms.
This is remarked after Examples 3.12 in Pultr and Tholen.
By the Proposition , if is all morphisms and is closed under composition, then it is also all morphisms, and by uniqueness of ternary factorizations this is equivalent to being the isomorphisms.
Concrete examples are below.
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 under composition.
In a triple distributive law there are three monads and distributive laws , , and satisfying the Yang-Baxter equation. Thus, in particular there are three composite monads , , and , and the Yang-Baxter equation ensures we get distributive laws and and a unique induced monad structure on .
In a ternary factorization system, we would naturally take , , and . We then get distributive laws and , since and are closed under composition. However, if is not closed under composition, we do not get a distributive law .
It is now natural to conjecture that this is the only obstacle, though: strict ternary factorization systems for which is closed under composition should be equivalent to triple distributive laws in . 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.
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. The maps in 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 as where the second factor is even split epi. This is the case in Top, so that is a concrete example where 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 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.
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 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.
For instance, the above factorization system on is induced in this way via the forgetful functor from the (epi,mono) factorization system on Set.
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.
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
Last revised on December 30, 2023 at 14:15:10. See the history of this page for a list of all contributions to it.