Epimorphism

definition example

An arrow $f:a\to b$ is an epimorphism (“epi”, drawn $a\twoheadrightarrow b$) if for every object $c$ and every pair $g_1,g_2:b\to c$,

$$g_1\circ f=g_2\circ f⟹g_1=g_2.$$

Equivalently, pre-composition $(-\circ f):\mathcal{C}(b,c)\to \mathcal{C}(a,c)$ is injective for every $c$. To show $f$ is not epi, find $c$ and two different $g_1,g_2$ that agree after precomposing with $f$.

Sources: DaoFP §2.5 (“Epimorphisms”), Exercise 2.5.1; 7 Sketches §1.4.2; Kittenlab Lecture 2; CTfS Definition 2.7.5.3, Proposition 2.7.5.4, Exercise 2.7.5.6

Intuition (DaoFP). Mappings out of an object define its properties: think of elements of a finite target $c$ as colours painting $b$. If $f$ is not epi, its image may cover only the part of $b$ painted alike by $g_1$ and $g_2$, so the two agree on $a$ although they differ on $b$. “Of course, in an actual category there is no peeking inside objects.”

• In $\mathbf{Set}$ epis are exactly the surjections (even :: Int -> Bool covers all of Bool). Any arrow to the Terminal Object is epi (DaoFP Exercise 2.5.1).
• Proof that epi ⇒ surjective in $\mathbf{Set}$ (CTfS Proposition 2.7.5.4) uses the Subobject Classifier $\Omega =\{True,False\}$ as the “test object”: if $y_0\in Y$ were missed by $f$, the characteristic functions of $\{y_0\}$ and of $\varnothing$ would be two different maps $Y\to \Omega$ that agree after precomposing with $f$. Pushouts preserve epimorphisms, dually to pullbacks preserving monos (CTfS Exercise 2.7.5.6).
• Epi is dual to Monomorphism: an epi in $\mathcal{C}$ is a mono in ${\mathcal{C}}^{\mathrm{op}}$. A retraction is always epi.
• Surjections out of $A$ = partitions of $A$; the Epi-Mono Factorization $A\twoheadrightarrow \mathrm{im}(f)\hookrightarrow B$ underlies Pushforward and Pullback of Partitions.
• Epi + mono need not be iso (DaoFP; e.g. dense inclusions in $\mathbf{Top}$).
• Via pushouts (7 Sketches Definition 7.5): $f:A\to B$ is epi iff the square with $f$ twice and ${\mathrm{id}}_B$ twice is a Pushout (the cokernel pair of $f$ is trivial) — the exact dual of the pullback characterization of monos. In a Topos ”$f$ is epi” is expressed by the internal formula $\forall (t:T).\exists (s:S).f(s)=t$ (Internal Language of a Topos, Example 7.74).

JuliaLeanHaskell

Docs: FinSets · C-set morphisms — Kittenlab Lecture 2

using Catlab
is_epic(FinFunction([1, 2, 2], 2))    # true: surjective
is_epic(FinFunction([1, 1, 1], 2))    # false

#check CategoryTheory.Epi            -- class Epi f : ∀ g h, f ≫ g = f ≫ h → g = h
#check @CategoryTheory.epi_iff_surjective

even' :: Int -> Bool                  -- an epimorphism in Hask
even' n = n `mod` 2 == 0