nLab factorization structure

Idea

A factorization structure is a strengthening of an orthogonal factorization system which allows us to factor, not only single morphisms, but arbitrary sinks (or cosinks).

Definition

Let $\{e_i \colon X_i \to Y\}_{i\in I}$ be a sink and $m\colon A \to B$ a morphism in some category. We say that $\{e_i\}$ is orthogonal to $m$ if for any sink $\{f_i \colon X_i \to A\}_{i\in I}$ and morphism $g\colon Y\to B$ such that $m f_i = g e_i$ for all $i\in I$:

$$\array{X_i & \overset{f_i}{\to} & A\\ ^{e_i}\downarrow && \downarrow^m\\ Y & \underset{g}{\to} & B}$$

there exists a unique arrow $h\colon Y\to A$ such that $h e_i = f_i$ for all $i$ and $m h = g$. Clearly, if ${|I|}=1$ this reduces to the usual notion of orthogonality for morphisms.

Let $(E,M)$ be an (orthogonal) factorization system; we say it extends to a factorization structure for sinks if every sink $\{f_i \colon X_i \to Y\}_{i\in I}$ can be factored as $f_i = m e_i$, where $m\in M$ and the sink $\{e_i \colon X_i \to Z\}_{i\in I}$ is orthogonal to $M$. Note there is no restriction on the sinks involved to be small.

Some authors write $\mathbf{E}$ for the collection of sinks orthogonal to $M$, and say that $(\mathbf{E},M)$ is a factorization structure for sinks, or that the category $C$ is an “$(\mathbf{E},M)$-category”. However, since $\mathbf{E}$ is uniquely determined by $(E,M)$, and $E$ is precisely the 1-ary sinks in $\mathbf{E}$, there is little harm in saying that $(E,M)$ is a factorization structure for sinks.

The dual notion is a factorization structure for cosinks (“sources”).

On size and foundations

Observe that if $C$ is large, then the collection $\mathbf{E}$ contains proper classes as elements. Therefore, in some foundations such as ZF, it is not definable as a single thing. In NBG one may treat it as a “hyperclass” defined by a first-order formula, in the same way that one treats classes in ZF. If we use a Grothendieck universe to define smallness, of course, there is no problem.

Constructions

Many well-known factorization systems, and ways to construct factorization systems, extend to factorization structures for sinks and/or cosinks.

• This theorem implies that if a category is M-complete for a class $M$ of morphisms which consists of monomorphisms and is closed under composition and pullback, then $M$ is the right class in a factorization structure for sinks.

• If $p\colon A\to B$ is a Grothendieck fibration, then factorization structures for sinks can be lifted from $B$ to $A$. Dually, if $p$ is an opfibration, we can lift factorization structures for cosinks. For the “mismatched” types of lifting, we require more: $p$ must be a topological functor.

Examples

• In Set, the factorization system (epi, mono) extends to both a factorization structure for sinks and one for cosinks. The epi-sinks are those that are jointly epimorphic, and the mono-sinks are those that are jointly monomorphic.

• By lifting (epi,mono) to Top (or any other topological category over $Set$), we obtain two factorization structures for sinks: (jointly surjective, subspace inclusions) and (final topologies, injections). Dually, we have two factorization structures for cosinks: (surjections, initial topologies) and (quotient maps, jointly injective).

Properties

$M$ consists of monics

Recall that a factorization system $(E,M)$ is called proper if $E\subseteq Epi$ and $M\subseteq Mono$. In the case of a factorization structure for sinks, the second of these is automatic.

Theorem

If $(E,M)$ is a factorization structure for sinks, then $M$ consists of monomorphisms.

Proof

Let $m\colon A\to B$ be in $M$, and suppose that $m r = m s$ for some $r,s\colon X\to A$, but $r\neq s$. Consider the sink $\{m r: X \to B \}_{f\in Mor(C)}$ consisting of one copy of $m r$ ($= m s$) for each arrow of the ambient category $C$, and factor this sink as an $E$-sink $\{e_f\colon X \to Y\}_{f\in Mor(C)}$ followed by an $M$-morphism $n\colon Y\to B$.

Now there are at least $2^{|Mor(C)|}$ different sinks $\{g_f\colon X \to A\}_{f\in Mor(C)}$ such that $m g_f = n e_f$, since we may take each $g_f$ to be either $r$ or $s$. Therefore, by orthogonality, there are at least $2^{|Mor(C)|}$ different morphisms $Y\to A$, a contradiction to Cantor's theorem, since there can be at most $Mor(C)$.

This proof is of course quite reminiscent of Freyd’s theorem that any complete small category is a preorder. In fact, Freyd’s theorem is a consequence of this one (or at least of its proof). For given a complete small category, there is a factorization structure acting at least on small sinks, where $M$ is the class of all morphisms and $E$ the class of families of injections into coproducts. (Any complete small category is also cocomplete, by the adjoint functor theorem.) Therefore, all morphisms in a complete small category are monic, including the unique maps to the terminal object; hence the category is a preorder.

Since Freyd’s theorem can fail in constructive mathematics, we should expect the use of excluded middle to also be essential in proving the above property of factorization structures.