# Wednesday, February 16

:::{.definition title="Equalizer and coequalizer"}
The definition of equalizers and coequalizers:

\begin{tikzcd}
	&&&&&& Y \\
	\\
	{K = \cocoeq(f, g)} && A && B && {C = \coeq(f, g)} \\
	\\
	X
	\arrow[from=3-1, to=3-3]
	\arrow["f", shift left=2, from=3-3, to=3-5]
	\arrow[from=3-5, to=3-7]
	\arrow["{\exists !}"', dashed, from=3-7, to=1-7]
	\arrow[from=3-5, to=1-7]
	\arrow[from=5-1, to=3-3]
	\arrow["{\exists !}"', dashed, from=5-1, to=3-1]
	\arrow["g"', shift right=1, from=3-3, to=3-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwyLCJLID0gXFxlcShmLCBnKSJdLFsyLDIsIkEiXSxbNCwyLCJCIl0sWzYsMiwiQyA9IFxcY29lcShmLCBnKSJdLFs2LDAsIlkiXSxbMCw0LCJYIl0sWzAsMV0sWzEsMiwiZiIsMCx7Im9mZnNldCI6LTJ9XSxbMiwzXSxbMyw0LCJcXGV4aXN0cyAhIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzIsNF0sWzUsMV0sWzUsMCwiXFxleGlzdHMgISIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFsxLDIsImciLDIseyJvZmZzZXQiOjF9XV0=)


:::

:::{.remark}
Notes:

- $\ker f \to A \covers^f_0 B\to \coker f$.
- $B\injectsvia{h} X$ is injective iff $A\covers_g^f B \to X$
- $X \surjectsvia{h} A$ is surjective iff $X \mapsvia{h} A \covers^f_g B$
- Iso = mono and epi

:::

:::{.exercise title="?"}
Show that if $\cocoeq(f, g)\to A$ exists then $\cocoeq(f, g) \injects A$ is mono.
:::

:::{.warnings}
Iso implies bijective on underlying sets, but not conversely.

Take the subcategory of $\Top\Ab\Grp$ whose objects are $\RR$ with various topologies, then take $\id: \RR^{\mathrm{disc}}\to \RR^{\mathrm{Euc}}$.
Note that $\ker \id = \coker \id = 0$ but this is not an isomorphism.
The issue: this is an additive category that isn't abelian.
:::

:::{.definition title="Additive categories"}
For $\cat{C} \in \Cat$,

- $\Hom_{\cat C}(A, B) \in \Ab\Grp$
- Composition is distributive, so $f(g+h) = fg+gh$ and $(g+h)f = gf + hf$.

:::

:::{.definition title="Abelian categories"}
For $\cat{C} \in \Cat$,

- Closed under all kernels and cokernels
- Closed under products $\prod A_i$ 
  - Equivalently, closed under coproducts $\bigoplus A_i$, and in fact $A\times B = A \oplus B$ in $\cat{C}$.
- There exists a zero object $0 = \initial = \terminal$ with $\Hom(0, X) = \Hom(X, 0) = 0$.
- Images are uniquely isomorphic to coimages:

\begin{tikzcd}
	0 && {\ker f} && A && B && {\coker f} && 0 \\
	\\
	&&&& I && {I'} \\
	\\
	&&&& 0 && 0
	\arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-1, to=1-3]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-3, to=1-5]
	\arrow["f", from=1-5, to=1-7]
	\arrow[color={rgb,255:red,214;green,153;blue,92}, from=1-7, to=1-9]
	\arrow[color={rgb,255:red,214;green,153;blue,92}, from=1-9, to=1-11]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-5, to=3-5]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, from=3-5, to=5-5]
	\arrow[color={rgb,255:red,214;green,153;blue,92}, from=5-7, to=3-7]
	\arrow[color={rgb,255:red,214;green,153;blue,92}, from=3-7, to=1-7]
	\arrow["{\exists !}", dashed, from=3-5, to=3-7]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=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)

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:::{.remark}
For $\cat C = \Ab\Grp$, $\Hom_{\cat C}$ form abelian groups under pointwise operations.
For morphisms $\cat C = \Sh(X; \Ab\Grp)$ and $f,g\in \cat{C}(\mcf, \mcg)$, writing $f = \ts{f_U}, g = \ts{g_U}$ in components, one can set $f+g = \ts{f_U + g_U}$ to make $\Hom_{\cat C}$ an abelian group.
Images will be isomorphic to coimages in $\cat{C}$ since the induced maps will be isomorphisms on stalks, using that $\Ab\Grp$ is abelian.
:::

:::{.remark}
If $\mca\in \Ab\Cat$, then $\Sh(X; \mca)\in \Ab\Cat$.
:::

:::{.exercise title="?"}
Show that $A_1\times A_2 = A_1 \oplus A_2$ in an abelian category using the universal properties.
:::

:::{.solution}
See course notes.
:::