Isomorphisms, Monos, and Epis

A function between sets is injective when distinct inputs go to distinct outputs, and surjective when every element of the codomain is hit. Both definitions mention elements, so neither makes sense in a general category, where objects are opaque. The categorical replacements are cancellation properties: an arrow is monic when it can be cancelled from the left of an equation, epic when it can be cancelled from the right. In $\mathbf{Set}$ these recover injective and surjective exactly. In other categories they behave more subtly, and the gap between monic and epic and isomorphism is genuine.

Monomorphisms

The definition quantifies over all arrows into $A$. Barr & Wells give this quantification a useful reading: an arbitrary arrow $x:X\to A$ is a variable element of $A$ parametrized by $X$, and one may write $m(x)$ for $m\circ x$. In that notation the definition reads exactly like injectivity — if $x\ne y$ then $m(x)\ne m(y)$ — except that the elements now range over arrows.¹

In $\mathbf{Set}$, monic equals injective. If $m$ is injective and $mf=mg$, then for each $x$ in the common domain, $m(f(x))=m(g(x))$ forces $f(x)=g(x)$. Conversely, if $m$ is monic, test against arrows from a one-element set: distinct global elements $x,y:1\to A$ with $mx=my$ would violate monicity, and global elements of a set are exactly its elements.²

The same argument shows injective implies monic in any category of structured sets — $\mathbf{Grp}$, $\mathbf{Mon}$, $\mathbf{Top}$, $\mathbf{Pos}$. The converse direction needs a test object rich enough to distinguish elements, and the terminal object does not always suffice. In $\mathbf{Mon}$ the one-element monoid has exactly one arrow into any monoid (it must send $1$ to $1$), so global elements are useless. The repair: test against $(\mathbb{N},+,0)$ instead. For each $x\in M$ the map $p_x:\mathbb{N}\to M$, $k\mapsto x^k$, is a monoid morphism, and $x\ne y$ gives $p_x\ne p_y$; if $f(x)=f(y)$ then $f\circ p_x=f\circ p_y$, contradicting monicity. So in $\mathbf{Mon}$ the monos coincide with the injective morphisms after all, though the proof had to find the right probe.³

Two closure facts, used constantly in diagram chases:

Epimorphisms

Epimorphisms are the formal dual: monos in ${\mathcal{A}}^{\mathrm{op}}$.

In $\mathbf{Set}$, epic equals surjective. Surjectivity gives cancellation directly. For the converse, suppose $e:A\to B$ misses $t\in B$; define $f,g:B\to \{0,1\}$ agreeing everywhere except that $f(t)=0$ and $g(t)=1$. Then $fe=ge$ but $f\ne g$, so $e$ is not epic.⁴

Outside $\mathbf{Set}$, the correspondence with surjectivity fails. The standard counterexample:

The inclusion is injective, hence also monic. It is not an isomorphism. So:

• monic + epic does not imply isomorphism. An arrow that is both is called a bimorphism, and a category in which every bimorphism is an isomorphism is called balanced. $\mathbf{Set}$ and $\mathbf{Grp}$ are balanced; $\mathbf{Mon}$, $\mathbf{Ring}$ (where $\mathbb{Z}\hookrightarrow \mathbb{Q}$ is epic), and $\mathbf{Top}$ (where any continuous bijection is a bimorphism) are not.⁵
• In a preorder viewed as a category, every arrow is vacuously both monic and epic — there are no parallel pairs to separate — so every arrow is a bimorphism, while the only isomorphisms in a poset are identities. Thin categories are as unbalanced as possible.

Worked example ($\mathbb{Z}\hookrightarrow \mathbb{Q}$ is a non-surjective epi in $\mathbf{Ring}$). A ring homomorphism $h:\mathbb{Q}\to S$ is fixed by its restriction to $\mathbb{Z}$. For a nonzero integer $b$, applying $h$ to $b\cdot \frac{1}{b}=1$ gives $h(b)h(\frac{1}{b})=h(1)=1$, so $h(\frac{1}{b})$ is the inverse of $h(b)$ in $S$. Inverses are unique, so $h(\frac{1}{b})$ is determined by $h(b)$, and $h(b)$ by $h{|}_{\mathbb{Z}}$. Then $h(\frac{a}{b})=h(a)h(\frac{1}{b})$ is fixed on all of $\mathbb{Q}$. Two ring maps out of $\mathbb{Q}$ that agree on $\mathbb{Z}$ therefore agree everywhere, giving the right-cancellation $fi=gi⟹f=g$ for the inclusion $i$. So $i$ is epic; it is injective and not surjective, hence a bimorphism that is no isomorphism, and $\mathbf{Ring}$ is not balanced.

Sections and retractions

A one-sided inverse is an explicit witness certifying cancellation.

Applying $r$ to both sides of $sx=sy$ gives $x=y$, so every section is monic; dually, every retraction is epic. The converses fail:

• Epis that do not split. In $\mathbf{Set}$, every surjection splits — for each $b\in B$ choose a preimage; this is precisely the axiom of choice. In $\mathbf{Mon}$ even surjective epis can fail to split: the quotient ${\mathbb{Z}}_4\to {\mathbb{Z}}_2$ has no monoid section.⁶
• Monos that do not split. In $\mathbf{Set}$, any map out of $\varnothing$ is monic, but only $1_{\varnothing }$ has a left inverse. Every mono with nonempty source splits in $\mathbf{Set}$; in $\mathbf{Grp}$, the inclusion of a subgroup usually does not.
• In a poset, the only split monos or split epis are identities: $rs=1$ forces $B\le A\le B$, hence $A=B$.

An isomorphism is an arrow with a two-sided inverse, so every isomorphism is both a split mono and a split epi. A weaker hypothesis already forces an isomorphism:

The full hierarchy, with all containments strict in general:

Hom-set characterizations

All five classes can be phrased uniformly through the induced maps on hom-sets. An arrow $f:B\to C$ induces, for each object $A$, a post-composition function $\mathrm{Hom}(A,f):\mathrm{Hom}(A,B)\to \mathrm{Hom}(A,C)$ sending $g\mapsto f\circ g$, and for each $D$ a pre-composition function $\mathrm{Hom}(f,D):\mathrm{Hom}(C,D)\to \mathrm{Hom}(B,D)$ sending $h\mapsto h\circ f$ (note the reversal).

The first two lines are the definitions restated. The third is worth unwinding once: surjectivity of $\mathrm{Hom}(f,D)$ at $D=B$ produces an $r$ with $rf=1_B$. This proposition is an early sighting of a theme that dominates the Yoneda lemma: every property of an arrow is encoded in what composition with it does to hom-sets.

Subobjects

Monomorphisms generalize injections, and injections generalize subset inclusions, so monos are the raw material for the categorical notion of subthing. The subtlety is that many different monos into $B$ carve out the same piece of $B$, and the definition must quotient that redundancy away.

Say a mono $f_0:C_0\to B$ factors through a mono $f_1:C_1\to B$ if $f_0=f_1\circ h$ for some $h$. Write $f_0\sim f_1$ when each factors through the other; then the connecting arrows are unique and are mutually inverse isomorphisms, and $\sim$ is an equivalence relation on monos with target $B$.⁸

In $\mathbf{Set}$ this recovers subsets, in a precise sense: every subobject of a set $S$ contains exactly one genuine inclusion $I\subseteq S$ (the common image of all its members), and distinct subsets give distinct subobjects. Subobjects are not literally subsets — no categorical definition could be, since the same element in both violates the typing discipline — but the two systems of data determine each other.

The categorical attitude that results is strict typing with explicit coercions: $\mathbb{Z}$ is a subobject of $\mathbb{R}$ not because an integer is a real number but because there is a canonical mono $\mathbb{Z}\to \mathbb{R}$, regarded as an inclusion even when the underlying construction (Dedekind cuts, Cauchy sequences) changes what the elements really are. Barr & Wells press the computing analogy: converting an INT to a REAL may change the representation in memory; what matters is that the conversion map is monic and canonical. The mono is the type conversion.

Duality

Nearly every definition here came in mirror pairs, and only one member of each pair was ever proved: epi is mono in ${\mathcal{A}}^{\mathrm{op}}$, a retraction is a section in ${\mathcal{A}}^{\mathrm{op}}$, and the composition and factor lemmas dualize word for word (if $gf$ is epic, so is $g$). This is the duality principle from the first lesson: one proof, two theorems. Initial and terminal objects, products and coproducts, and equalizers and coequalizers are all dual pairs, and monos return there as the arrows that equalizers produce.