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## Epi-mono factorizations

In many familiar categories, a morphism $f : a \to b$ admits a canonical factorization, which we will write

$a \xrightarrow{e} c \xrightarrow{m} b$,

as the composite of some kind of epimorphism $e$ and some kind of monomorphism $m$. Here we should think of $c$ as something like the image of $f$. This is most familiar, for example, in the case of $\text{Set}, \text{Grp}, \text{Ring}$, and other algebraic categories, where $c$ is the set-theoretic image of $f$ in the usual sense.

Today we will discuss some general properties of factorizations of a morphism into an epimorphism followed by a monomorphism, or epi-mono factorizations. The failure of such factorizations to be unique turns out to be closely related to the failure of epimorphisms or monomorphisms to be regular.

The category of factorizations

Define the category of epi-mono factorizations of a morphism $f$ to be the category whose objects are epi-mono factorizations

$\displaystyle a \xrightarrow{e} c \xrightarrow{m} b$

of $f$ (so $e$ is an epimorphism, $m$ is a monomorphism, and $m \circ e = f$) and whose morphisms $(e_1, c_1, m_1) \to (e_2, c_2, m_2)$ are morphisms $g : c_1 \to c_2$ making the diagram

commute; that is, such that $e_2 = g \circ e_1$ and $m_1 = m_2 \circ g$. These two properties are already enough to conclude the following:

1. $g$ is an epimorphism (since $e_2$ is an epimorphism).
2. $g$ is a monomorphism (since $m_1$ is a monomorphism).
3. If $g$ exists, it is unique (since $e_1$ is an epimorphism, or alternately since $m_2$ is a monomorphism).

Thus the category of epi-mono factorizations of a morphism is a preorder. Moreover, the morphisms $g$ in the category are both monomorphisms and epimorphisms. Call such a morphism a fake isomorphism if it is not an isomorphism (this terminology is nonstandard).

If we are working in a category with no fake isomorphisms, such as $\text{Set}$ or $\text{Grp}$, then any two epi-mono factorizations which are related by a morphism are isomorphic via a unique isomorphism. (This doesn’t rule out the possibility that there are two epi-mono factorizations which are not related by any morphisms at all.) However, because there are categories with fake isomorphisms, we do not expect uniqueness of epi-mono factorizations in general.

Example. In $\text{CRing}$, let $D$ be an integral domain and let $f : D \to \text{Frac}(D)$ be the inclusion of $D$ into its field of fractions. If $D$ is not a field, then $f$ is a fake isomorphism; moreover, the category of epi-mono factorizations of $f$ is equivalent to the poset of subrings of $\text{Frac}(D)$ containing $D$, or equivalently the poset of localizations of $D$.

Example. In $\text{Top}$, any continuous bijection $f : X \to Y$ which does not have a continuous inverse is a fake isomorphism. Without loss of generality, we may take $X$ and $Y$ to have the same underlying set; then we are just talking about a pair of topologies on $X$ one of which is strictly finer than the other. The category of epi-mono factorizations of $f$ is then equivalent to the poset of topologies intermediate between these two topologies.

More generally, if $f : a \to b$ is a fake isomorphism, then it admits two nonisomorphic factorizations

$\displaystyle a \xrightarrow{\text{id}_a} a \xrightarrow{f} b, a \xrightarrow{f} b \xrightarrow{\text{id}_b} b$.

So the problem of non-uniqueness of epi-mono factorizations is closely related to the problem of existence of fake isomorphisms. Furthermore, previously we showed that a morphism which is either both a monomorphism and a regular epimorphism or which is both a regular monomorphism and an epimorphism is necessarily an isomorphism. It follows conversely that the existence of fake isomorphisms indicates the existence of epimorphisms or monomorphisms which are not regular.

Besides uniqueness, in full generality it is also necessary to worry about existence. For example, consider the free category on an idempotent. This is a category with a single object $\bullet$ and a single non-identity morphism $f : \bullet \to \bullet$ satisfying $f^2 = f$. Then $f$ is neither a monomorphism nor an epimorphism, since the above identity shows that it is neither left nor right cancellable, and since the only possible factorizations of $f$ are as

$\displaystyle f = f \circ f = f \circ \text{id} = \text{id} \circ f$

it follows that $f$ does not admit an epi-mono factorization.

Connectivity

Above we observed that one issue with the category of epi-mono factorizations is that it may fail to be connected: that is, there may be two epi-mono factorizations that are not related by any chain of morphisms, hence even if there were no fake isomorphisms we would still not be able to conclude that epi-mono factorizations are unique.

However, mild categorical hypotheses guarantee that this is not an issue.

Theorem: Suppose that a category $C$ has either pushouts or pullbacks. Moreover, suppose that $C$ has no fake isomorphisms (e.g. because all monomorphisms are regular or because all epimorphisms are regular). Then epi-mono factorizations in $C$ are unique (up to unique isomorphism).

Proof. The second hypothesis and the conclusion are categorically self-dual but the first hypothesis is not, so it suffices to prove the statement under the assumption that $C$ has pushouts. If $a \xrightarrow{e_1} c_1 \xrightarrow{m_1} b$ and $a \xrightarrow{e_2} c_2 \xrightarrow{m_2} b$ are two epi-mono factorizations of a morphism $f : a \to b$, consider the pushout $c_1 \sqcup_a c_2$ together with the inclusions $i_1, i_2 : c_1, c_2 \to c_1 \sqcup_a c_2$ and induced map $g : c_1 \sqcup_a c_2 \to b$:

We claim that $i_1 \circ e_1 = i_2 \circ e_2$ is an epimorphism. To see this, suppose $p, q : c_1 \sqcup c_2 \to d$ are two other morphisms such that

$\displaystyle p \circ i_1 \circ e_1 = p \circ i_2 \circ e_2 = q \circ i_1 \circ e_1 = q \circ i_2 \circ e_2$.

Since $e_1, e_2$ are epimorphisms, it follows that $p \circ i_1 = q \circ i_1$ and $p \circ i_2 = q \circ i_2$. Hence $p \circ i_1, p \circ i_2$ and $q \circ i_1, q \circ i_2$ describe the same commutative square, from which it follows by the universal property of the pushout that they factor through the same morphism $c_1 \sqcup c_2 \to d$, namely $p = q$.

It follows that $i_1, i_2$ are both epimorphisms. On the other hand, since $g \circ i_1 = m_1$ and $g \circ i_2 = m_2$ are monomorphisms, it follows that $i_1, i_2$ are both monomorphisms. Since $C$ has no fake isomorphisms, it follows that $i_1, i_2$ are both isomorphisms, hence $g$ is a monomorphism and $c_1 \sqcup_a c_2$ determines an epi-mono factorization which is isomorphic to both $c_1$ and $c_2$. The conclusion follows. $\Box$

Corollary: Suppose $C$ is a category with either pushouts or pullbacks. Then epi-mono factorizations in $C$ are unique if and only if $C$ has no fake isomorphisms.

Note that the corollary does not say anything about existence.