Isomorphisms, Monos & Epis

Injective and surjective, rebuilt out of arrows — and the categories where they come apart.

The idea

Monomorphisms, epimorphisms and isomorphisms are the categorical replacements for injective, surjective and invertible functions, each stated with nothing but arrows, composition and identities.

Replacements are needed because the familiar definitions point at elements — a function is injective when it never sends two different inputs to the same place — and the objects of a category need not have any. Arrows stand in for the elements: an element of a set $A$ is the same thing as a function from a one-point set into $A$, so "$f$ sends two different elements to one place" becomes "two different arrows into $A$ become equal once composed with $f

Rewriting both notions that way, and adding the arrow version of invertible, gives three definitions. An arrow $f \colon A \to B$ is a monomorphism when it cancels on the left: $f \circ g = f \circ h$ forces $g = h$, for every pair of arrows $g, h \colon T \to A$ out of any object $T$. It is an epimorphism when it cancels on the right: $g \circ f = h \circ f$ forces $g = h$. And it is an isomorphism when some arrow $g \colon B \to A$ satisfies $g \circ f = 1_{A}$ and $f \circ g = 1_{B}$.

In $\mathbf{Set}$ these recover injective, surjective and bijective exactly. In other categories they come apart, and the definitions, not the intuitions behind them, are what continue to hold.

Ways to work on it

• Walkthrough. Learn the cancellation properties that generalize injective and surjective to arbitrary categories.
• Proof. Why mono means exactly injective in Set, using the one-point set to turn elements into arrows.
• Practice. Classify familiar maps as monic, epic or invertible across several categories.
• Hardest. Decide whether epimorphisms must be surjective in a category of rings.

Not sure where to start? Take the ten-question placement test.