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).
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 $i\in I$:
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”).
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.
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.
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).
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.
If $(E,M)$ is a factorization structure for sinks, then $M$ consists of monomorphisms.
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.
Created on September 10, 2011 at 18:29:31. See the history of this page for a list of all contributions to it.