### Ternary Factorization Systems

#### Posted by Mike Shulman

We’ve been having a bit of discussion on the nForum about ternary and higher-ary factorization systems in categories and higher categories. A $k$-ary factorization system is supposed to give a way of factoring any morphism into a composite of $k$ morphisms of specified types. Ordinary “factorization systems” are *binary* factorization systems, which factor every morphism into a binary composite; for instance, the factorization of a set function into a surjection followed by an inclusion.

The first place we noticed these higher-ary factorization systems is in the “Postnikov decomposition” of higher categories. The basic example is that any functor $f\colon A\to B$ factors as $A \to im_2(f) \to im_1(f) \to B$ where $im_1(f)\to B$ is full and faithful, $im_2(f) \to im_1(f)$ is essentially surjective and faithful, and $A\to im_2(f)$ is essentially surjective and full. Similarly, we expect a functor between $n$-categories to have a natural $(n+2)$-ary factorization. This is called the “layer-cake” view of cohomology as John described here (while I took notes, and somehow ended up with my name on the paper too). However, higher-ary factorization systems (and in particular, ternary ones) also turn up in some other surprising places. But surprisingly, they don’t ever seem to have been precisely defined by anyone, and there are lots of unanswered questions!

So how do we define a ternary or k-ary factorization system? An ordinary “binary” factorization system consists of two classes of maps $(L,R)$, such that $L$ is orthogonal to $R$, and every morphism factors as an $L$-map followed by an $R$-map. The last condition is easy to generalize to a $k$-ary factorization with $k$ classes of maps, but what is the right counterpart of orthogonality?

I think we get a very nice theory by looking at $k$-ary factorizations in a different way. Consider the factorization of a functor described above. In fact, there are two ordinary (binary) factorization systems visible here as well, namely (ess.surj., full + faithful), and (ess.surj + full, faithful). Moreover, the ternary factorization can be obtained in two equivalent ways: we can either

- first factor $f$ as an essentially surjective functor followed by a full + faithful one, and then factor the essentially surjective functor into an ess.surj+full one followed by a faithful one; or
- first factor $f$ as an ess.surj+full functor followed by a faithful one, and then factor the faithful functor into an essentially surjective one followed by a full+faithful one.

This suggests the following definition: a **ternary factorization system** is a pair of ordinary (binary) factorization systems $(L_1,R_1)$ and $(L_2,R_2)$ such that $L_1 \subseteq L_2$ (and hence $R_2 \subseteq R_1$). The above situation generalizes as follows.

**Theorem 1:** Given a ternary factorization system as above, any morphism $f$ 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 as above, by

- 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
- 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.

Toby has proposed a natural generalization of this to $k$-ary factorization systems.

(Note that another way to define ordinary binary factorization systems is by *uniqueness* rather than orthogonality. This is the approach taken at Joyal’s catlab. Does this generalize to the k-ary case?)

The motivating example of factoring a functor is actually a factorization system on a *2-category*, which means that orthogonality and uniqueness only hold up to isomorphism. However, once we write the definition in the above form, we can see that there are also lots of ternary factorization systems on 1-categories: we get one from any pair of ordinary factorization systems, one included in the other.

For instance, in the category of topological spaces, 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 resulting ternary factorization factors a continuous map by imposing the coarsest and the finest compatible topologies on its set-theoretic image.

As another example, some people use the word image to refer to an (epi, regular mono) factorization and *coimage* to refer to a (regular epi, mono) factorization. It is well-known in this context there is a map from the coimage to the image, which is in fact another instance of ternary factorization. In this case $L_2\cap R_1$ is the class of monic epics, sometimes called bimorphisms. (The example of Top is a special case of this one.)

Returning to the theory, so far we can see that a ternary factorization system determines five classes of maps: $L_1$, $R_1$, $L_2$, $R_2$, and $L_2\cap R_1$. In the original motivating example, these are respectively the ess.surj.+full, faithful, ess.surj., full+faithful, and ess.surj+faithful functors. However, there’s a sixth class of functors that’s conspicuously missing from this list: full functors. In fact, any ternary factorization system canonically determines a sixth class: the class of morphisms 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. Thus, let’s call this class $R_2 L_1$.

It’s not hard to see that this gives the right answer in $Cat$: a functor is full iff it can be factored as an ess.surj.+full functor followed by a full+faithful one. In the topological example, these are the continuous maps for which the quotient and subspace topologies on their set-theoretic images coincide. In the image/coimage situation, this class of maps are sometimes called strict morphisms (I have no idea why). We can also generalize the fact that a full+faithful functor is the same as one which is both full and faithful:

**Theorem 2:** 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.

Can this also be generalized to $k$-ary factorization systems? I don’t know.

Let me end with an observation that I find quite suggestive, although I admit I have no idea what it is trying to suggest. The ordinary notion of (binary) factorization system can be generalized in a different direction to a weak factorization system, so it’s natural to think about pairs of WFS $(L_1,R_1)$ and $(L_2,R_2)$ with $L_1\subseteq L_2$.

However, this is precisely the sort of WFS data which underlies a model category, where $L_1=$ acyclic cofibrations, $L_2=$ cofibrations, $R_1=$ fibrations, and $R_2=$ acyclic fibrations. What are the two other classes of maps in this case? Well, $L_2 \cap R_1 =$ maps that are both cofibrations and fibrations; that doesn’t seem very interesting. But $R_2 L_1 =$ maps that can be factored as an acyclic cofibration followed by an acyclic fibration, and those are precisely the *weak equivalences* – the fifth class of maps in a model category.

Of course, Theorem 1 doesn’t hold in the WFS case, where factorizations are never unique, so it doesn’t really make sense to call this a “ternary weak factorization system”. Neither does the characterization of $R_2 L_1$ as the maps whose $L_2\cap R_1$-part is an isomorphism. However, Theorem 2 does still hold — a map is an acyclic cofibration iff it is both a cofibration and a weak equivalence.

Is there something deep going on here? Is it good for anything? I don’t know. But it reminds me of the theory of algebraic weak factorization systems structures, and the difficulty in finding a corresponding notion of “algebraic model category.” An inclusion between two WFS (or a map, in the algebraic case) determines a model structure *if* and only if the class of weak equivalences, defined as the maps that factor as an acyclic cofibration followed by an acyclic fibration, has the 2-out-of-3 property. But in practice, starting from that definition the 2-out-of-3 property is usually quite difficult to verify. Could the perspective of ternary factorization systems be helpful?

## Re: Ternary factorization systems

Maybe one place nForum doesn’t function as well as the Café is in linking discussions to older discussions. E.g., from the post we get sent here, but that doesn’t lead us back to an earlier discussion.

Maybe there’s little there, but you never know.