I’m writing this post to deconfuse myself and productively order my notes on this topic, as well as popularizing a topic in category theory that doesn’t get much attention outside the categorical algebra literature (or at least, that’s how it seems from where I’m sitting!).
Now, let’s start with a definition:
Definition 1. A factorization system on a category \cal X is a pair of subcategories (\cal L, R) (the left class and the right class) such that:
- Both \cal L and \cal R contain all isomorphisms,
- “\cal X =\cal L ;\cal R”: Every morphism in \cal X factors as a morphism of \cal L followed by a morphism in \cal R and this factorization is unique up to unique isomorphism:
We denote left morphisms as \twoheadrightarrow and right ones as \rightarrowtail. People often call the right class as \cal E and the left class as \cal M, though epi-mono do not form a factorization system in general (they do in balanced categories, like pretopoi). But having in mind surjections and injections in \bf Set is a good enough intuition for factorization systems, and in fact the middle object in the factorization of a morphism is usually called the image:
More examples can be found in Joyal’s catlab.
As often happens in category theory, while we defined the factorization of a morphism as something that just happens to exists, factorization can be given as an actual functor, specifically a section of the composition functor \_;\_ : \cal C^{\to\to} \to C^\to. It sends an arrow to its factorization. Conversely, given such a functor one can obtain a factorization system (\cal L is the subcategory of morphisms whose factorizations look like \to =, and viceversa for \cal R). In fact, a section of the identity 1:\cal C \to C^\to, sending a morphism to its image, suffices, provided it satisfies the axioms of a (normal) pseudoalgebra for the 2-monad (-)^\to : \bf Cat \to Cat. This is cool! It means factorization systems are an algebraic structure on categories, and makes them easily generalizable to other ambient 2-categories (notice, (-)^\to is a 2-monad on any 2-category with \to-powers!)
Another cool fact about factorization systems, which justifies writing \cal X = L;R, is that they literally present \cal X as a ‘composition’ of two categories when we think of them as monad in spans. This is an old idea of Rosebrugh and Wood, and has some caveats.
I’m starting to accumulate facts that I couldn’t prove without the most important property of factorization systems:
Proposition 2. The left and right class are orthogonal (denoted \cal L \perp R), meaning every square as below (left side is left, right side is right) has a unique diagonal fill-in:
Proof. We factor the top and bottom morphism, and get a unique isomorphism between the respective images. The red composite is the sought diagonal fill-in, and it is unique because each of its component is unique:
\square
This property means that we have simultaneous extensions along left morphisms and lifts along right ones. Note, however, that in general we can’t just extend along a left morphism or just lift along a right one, since we need a full square to invoke orthogonality!
Orthogonality is so important that factorization systems are often called orthogonal factorization systems. In fact, orthogonality can even replace uniqueness of factorization in the definition: a pair of subcategories that can factor every morphism (possibly non-uniquely) and are orthogonal is automatically a factorization system.
Orthogonality can be used to prove all the facts I mentioned above. For instance, one can show \cal L \cap R is all and only the isomorphisms of \cal X, a fact we evince by contemplating the following square built for f \in \cal L \cap R:
This explains why epi-mono is factorization system iff the category is balanced: not all epic and monic morphisms are isos in general!It also shows a way to overcome such a limitation. Namely, if one has a class of morphisms \cal R they really like (say, monomorphisms), they can match it with ^\perp\cal R = \{\ell \in \cal X^\to \mid \forall r \in R, \ell \perp r\} to get a factorization system, with the caveat that they might need to replace \cal R with \cal L^\perp after the fact (but then that’s it—I’m describing a closure operation on pairs of classes of morphisms, see [CHK]).
I hope the (-)^\perp and ^\perp(-) notations are self-describing: the first means ‘all things that have the diagonal-fill in property on squares where morphisms in the argument appear on the left side’, and dually for the other one.
So, for instance, (epi, mono) isn’t a factorization system but in a regular category (strong epi, mono) is, where a strong epi is, by definition, something left ortoghonal to all monomorphisms!
Finally, orthogonality is so cool it can stand by itself. Thus one might have \cal L = {^\perp R} and \cal R = L^\perp without (\cal L, R) forming a factorization system! This is called a prefactorization system, and it’s a factorization system without the existence part of the factorization.
Reflections
One of the nicest consequences of having a factorization system is to get a reflective subcategory. I believe this is one of the most beautiful theorems in category theory!
Let’s start with a category \cal X with a terminal object 1 and a factorization system (\cal L, R). Then we can define the full subcategory \cal R/1 \subseteq X given by those objects whose terminal map !_X:X \rightarrowtail 1 is in the right class. These are called fibrant, generalizing a terminology from model categories.
The archetypal example is subterminal objects, i.e. those objects for which the terminal map is mono, which are thus fibrant for the (strong epi, mono) factorization system.
What’s interesting is, even though an object isn’t fibrant, we can always factor its terminal map !_X:X \to 1 as X \twoheadrightarrow \mathrm{im} !_X \rightarrowtail 1, thus yielding another object rX := \mathrm{im} !_X which is fibrant. This is called the fibrant replacement of X, again abusing terminology from model categories, or its reflection, foreshadowing the result I’m about to expound.
It’s easy to see that, by orthogonality, fibrant replacement is functorial, and in fact, left adjoint to the inclusion of fibrant objects:
In particular, this adjunction is a reflection meaning the counit is the identity. This can be noticed either by abstract nonsense (the right adjoint is fully faithful) or by concrete nonsense (clearly rX = X if X is already fibrant).
On other hand, the unit is defined as a byproduct of the factorization we used to construct the fibrant replacement, and thus we define \rho_X : X \twoheadrightarrow rX to be it. Notice all its components are, by definition, left maps.
We can then prove that r is left adjoint by proving that \rho : 1_{\cal X} \Rightarrow r is a universal arrow, that is, for every object X:\cal X and map f:X \to X' with X' fibrant there is a unique factorization of f through \rho_X:
and we can get such a map by invoking ortoghonality for the following square (!_{X'} is right because we assumed X' to be fibrant):
Done! Isn’t that beautiful?
In [CHK], a lot of attention is devoted to obtaining a converse to this fact, i.e. obtain a factorization system from a reflective subcategory. In general, this cannot be done (one only gets a prefactorization system), and there is only a Galois connection (which is, amusingly, a reflection again!) between the poset of reflective subcategories of \cal X and the poset of factorization systems.
They thus prove two theorems. One characterizes those categories for which the Galois connection is actually an equivalence—these are the ‘finitely well-complete’ ones, which is a condition slightly weaker than finitely complete and well-powered. The other theorem characterizes the fixed points of the Galois connection, hence those factorization systems that do arise from a reflective subcategory, and these are the ones for which left left class satisfies the left cancellation property:
(It’s a fact of life that all factorization systems have the left class satisfy the right cancellation property, which is exactly like the above but with g and f swapped.)
References
The results and proofs above are not mine, but come from the following sources:
- Joyal’s catlab,
- [CHK] Cassidy, Hébert, Kelly, Reflective subcategories, localizations and factorization systems, 1985
- [RT] Rosicky and Tholen, Factorization, fibration and torsion, 2007