Exercises 5: Equalizers, coequalizer

Equalizers and coequalizers

1.

Equalizer

\[\begin{xy} \xymatrix{ E \ar[r]^m & X \ar@/^/[r]^f \ar@/_/[r]_g & Y\\ F\ar[ru]_n \ar@{.>}[u]^{∃! \; h} & } \end{xy}\]

1.

In the category $Set$:

• \[E = \lbrace x∈ X \mid f(x)=g(x) \rbrace\]
• \[m = E \hookrightarrow X, x ⟼ x\]

2.

$m$ is a monomorphism:

iff for all $f, g$: \(m f = m g ⟹ f = g\)

$m$ is an epimorphism:

iff for all $f, g$: \(f m = g m ⟹ f = g\)

---

In the case of equalizers, let us show that $m$ is mono: let $f’, g’$ be two morphisms s.t.

\[mf' = mg'\]

Then there is a unique $h$ such that

\[mf' = mg' = mh\]

So \(f' = g'\)

3.

It is the colimit of the diagram

\[\begin{xy} \xymatrix{ X \ar@/^/[r]^f \ar@/_/[r]_g & Y\\ } \end{xy}\]

i.e.

\[\begin{xy} \xymatrix{ Q \ar@{.>}[d]_{∃! \; h} & Y \ar[ld]^n \ar[l]^m & X \ar@/^/[l]^f \ar@/_/[l]_g \\ Q' & } \end{xy}\]

(i.e. an equalizer in the dual category)

4.

Same demonstration: the coequalizer $e: Y ⟶ Q$ is an equalizer in the dual category.

By definition of the dual category, monic morphisms are turned into epic ones.

5.

In $Set$:

\[Q ≝ Y/\sim\]

where $\sim$ is the smallest equivalence relation such that

\[y \sim y' ⟺ ∃x∈X; y = f(x) ∧ y' = g(x)\]

Necessary condition:

If $e’ \circ f = e’ \circ g$ and $e’ = h \circ e$, then:

\[h(\bar{y}) = h(e(y)) = e'(y)\]

Conversely:

If $\bar{y} ∈ Q$, by letting \(h(\bar{y}) ≝ e'(y)\)

Then

\[∀z = g(a) ∈\bar{y}, e'(z) = e'(g(a)) = e'(f(a)) = e'(y)\]

6.

\[e: Y ⟶ Q\] \[x \sim y ⟺ e(x) = e(y)\]

As $e$ is surjective, there exists a section $s$ s.t. $e \circ s = id_Y$

For all equivalence classes, we choose a representant $g ≝ s \circ e$, and

\[\begin{xy} \xymatrix{ Y \ar@/^/[r]^{id} \ar@/_/[r]_g & X\ar[r]^e & Q } \end{xy}\]

is a coequalizer.

2. Epi-mono factorization

1.

\[\begin{xy} \xymatrix{ A \ar[rd]^f \\ f(A) \ar[u]^f \ar@{^{(}->}[r]_{id} & B } \end{xy}\]

2.

\[\begin{xy} \xymatrix{ A \ar[r]^{u} \ar[d]_{e} & X \ar[d]^m \\ B \ar[r]_v\ar@{.>}[ru]_{∃ ! \; h} & Y } \end{xy}\] \[mu = ve\]

As $e$ is surjective, there exists a section $s$ s.t. \(e \circ s = id\)

Let \(h ≝ u s\)

It doesn’t depend on the section: if $s’$ is another such section,

\[mus = ves = v = ves' = mus'\]

So

\(us = us'\) since $m$ is monic.

---

For all $a∈ A$,

\[he(a) = u(se(a)) = u(a)\]

Since $u$ is has the same image on preimages of $e$.

And:

\[mu = ve \\ ⟹ mhe = ve \\ ⟹ mh = v\]

since $e$ is epic.

4.

\[\begin{xy} \xymatrix{ X \ar[r]^{e_1} \ar[d]_{u} & U_1 \ar@{.>}[d]_{h} \ar[r]^{m_1} & Y \ar[d]^{v}\\ X' \ar[r]_{e_2} & U_2 \ar[r]^{m_2} & Y' } \end{xy}\]

Uniqueness

\[\begin{xy} \xymatrix{ X \ar[r]^{e_1} \ar[d]_{id} & U_1 \ar@{.>}[d]_{h} \ar[r]^{m_1} & Y \ar[d]^{id}\\ X \ar[r]_{e_2} & U_2 \ar@{.>}@/_/[u]_{\tilde{h}} \ar[r]^{m_2} & Y } \end{xy}\]