# Tuesday, February 01

:::{.definition title="Additive categories"}
A category $\cat C$ is **additive** iff

- $\cat C$ has zero object
- There exists a binary operation $+: \Hom(A, B)\cartpower{2}\to \Hom(A, B)$ for all $A, B\in \cat{C}$ making $\Hom(A ,B)$ an abelian group.
- Distributivity with respect to composition: $(g_1 + g_2)f = g_1f + g_2 f$
- For any collection $\ts{A_1,\cdots, A_n}$, there exists an object $A$, projections $p_j: A\to A_j$ with sections $i_k: A_k\to A$ with $p_j i_j = \id_A$, $p_j i_k = 0$ for $j\neq k$, and $\sum i_j p_j = \id_A$.

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:::{.definition title="Monomorphisms and epimorphisms"}
A morphism: $k:K\to A$ is **monic** iff whenever $g_1, g_2: L\to K$, $kg_1 = kg_2 \implies g_1 = g_2$:

\begin{tikzcd}
	L && K && A
	\arrow["k", from=1-3, to=1-5]
	\arrow["{g_1}", shift left=3, from=1-1, to=1-3]
	\arrow["{g_2}", shift right=3, from=1-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJMIl0sWzIsMCwiSyJdLFs0LDAsIkEiXSxbMSwyLCJrIl0sWzAsMSwiZ18xIiwwLHsib2Zmc2V0IjotM31dLFswLDEsImdfMiIsMCx7Im9mZnNldCI6M31dXQ==)

Define $k$ to be **epic** by reversing the arrows.
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:::{.definition title="Kernel"}
Assume $\cat C$ has a zero object.
Then for $f:A\to B$, the *morphism* $k: K\to A$ is the **kernel** of $f$ iff

- $k$ is monic
- $fk=0$
- For any $g:G\to A$ with $fg=0$, there exists a $g'$ with $g=kg'$.

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:::{.example title="?"}
For $f\in \rmod(A, B)$, take $k: \ker f\injects A$.
If $g\in \cat{C}(G, A)$ with $f(g(x)) = 0$ for all $x\in G$, then $\im g \subseteq \ker f$ and we can factor $g$ as $G \mapsvia{g'} \ker f \injectsvia{k} A$.
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:::{.definition title="Cokernel"}
For $f: A\to B$, a morphism $c: B\to C$ is a **cokernel of $f$** iff 

- $c$ is epic,
- $cf=0$
- For any $h\in \cat{C}(B, H)$ with $hf=0$, there is a lift $h': C\to G$ with $h=h'c$.

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:::{.example title="?"}
For $\cat{C} = \rmod$ and $f\in \rmod(A, B)$, set $c: B\to B/\im f$.
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:::{.exercise title="?"}
Show that kernels are unique.
Sketch:

- Set $k:K\to A$, $k': K'\to A$.
- Factor $k=k' u_1$ and $k' = ku_2$.
- Then $k\id = k(u_2 u_1) \implies \id = u_2 u_1$, similarly $u_1u_2=\id$.

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:::{.definition title="Abelian categories"}
$\cat{C}$ is **abelian** iff $\cat{C}$ is additive and

- A5: Every morphism admits kernels and cokernels.
- A6: Every monic is the kernel of its cokernel, and every epic is the cokernel of its kernel.
- A7: Every morphism $f$ factors as $f=me$ with $m$ monic and $e$ epic.

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:::{.example title="?"}
For $f\in \rmod(A, B)$,

- A5: Take $k: \ker f\injects A$ and $c: B\surjects B/\im f$
- A6: For $m: A\injects B$ monic, consider the composition $A\injects B \mapsvia{\coker m} B/A$ and check $A\cong \ker(\coker m)$.
- A7: Use the 1st isomorphism theorem:

\begin{tikzcd}
	A &&&& B \\
	\\
	& {A/\ker f} && {\im f}
	\arrow["f", from=1-1, to=1-5]
	\arrow["i"', two heads, from=1-1, to=3-2]
	\arrow["{\text{1st iso}}"', from=3-2, to=3-4]
	\arrow["m"', hook, from=3-4, to=1-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJBIl0sWzQsMCwiQiJdLFszLDIsIlxcaW0gZiJdLFsxLDIsIkEvXFxrZXIgZiJdLFswLDEsImYiXSxbMCwzLCJpIiwyLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzMsMiwiXFx0ZXh0ezFzdCBpc299IiwyXSxbMiwxLCJtIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XV0=)

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:::{.remark}
Some notes:

- Recall the definition the category of chain complexes $\Ch(\cat C)$ over an abelian category: $d_i d_{i+1} = 0$, so $\im d_i \subseteq \ker d_{i+1}$.
- Every exact sequence is an acyclic complex.
- $\cat{C}\embeds \Ch(\cat C)$ by $M\mapsto \cdots \to 0 \to M \to 0 \to \cdots$.
  Note that this isn't an acyclic complex.
- Morphisms between complexes: chain maps, just levelwise maps forming commutative squares, i.e. maps commuting with the differentials.
- $\Ch(\cat C)$ is additive: given $\alpha_\bullet, \beta_\bullet\in \Ch\cat{C}( (A, d), (B, \delta) )$, check that $(\alpha_{i-1} + \beta_{i-1})d_i = \delta_i (\alpha_i + \beta_i)$.
- There are direct sums: $(A \oplus B)_i \da A_i \oplus B_i$ with $d \da d_A + d_B$.
- Define cycles as $Z_i \da \ker\qty{ C_i \mapsvia{d_i} C_{i-1}}$ for $C_\bullet \in \Ch(\cat C)$, and boundaries $B_i \da \im\qty{C_{i+1} \mapsvia{d_{i+1}} C_i} \subseteq \ker d_i$.
- Define $H_i(C_\bullet )\da Z_i/B_i$.
- Show that chain morphisms induce morphisms on homology:
  - Let $\alpha\in \Ch(\cat C)(C, C')$, then $\alpha_i(Z_i) \subseteq Z_i'$.
  - Check $d_2(a_i(Z_i)) = a_{i-1} d_i(Z_i) = 0$.
  - Factor $Z_i \mapsvia{\alpha_i} Z_i' \surjects Z_i'/B_i'$.
  - Show that $x\in B_i$ maps lands in $B_i'$ to get well-defined map on $H_i$.
  - Use $\alpha(B_i) \subseteq Z_i'$, so pull back $x\in B_i$ to $y\in C_{i+1}$.
  - Check $d_{i+1}(y) = x$, so $\alpha(d_{i+1}(y)) = \alpha(x)$.
  - The LHS is $d_{i+1}'(\alpha_{i+1}(y))$, so $\alpha_i(x) in \im d_{i+1}' = B_{i+1}'$
- Chain homotopies: for $\alpha, \beta\in \Ch(\cat C)(C, C')$, write $\alpha \homotopic \beta$ iff there exists $\ts{s_i: C_i \to C_{i+1}' }$ with $\alpha_i - \beta_i = d_{i+1}' s_i + s_{i-1} d_i$.

\begin{tikzcd}
	\cdots && {C_{i+1}} && {C_{i}} && {C_{i-1}} && \cdots \\
	\\
	\cdots && {C_{i+1}'} && {C_{i}'} && {C_{i-1}'} && \cdots
	\arrow[from=1-1, to=1-3]
	\arrow["{d_{i+1}}", from=1-3, to=1-5]
	\arrow["{d_{i+1}}", color={rgb,255:red,92;green,92;blue,214}, from=1-5, to=1-7]
	\arrow[from=1-7, to=1-9]
	\arrow[from=3-1, to=3-3]
	\arrow["{d_{i}'}", from=3-5, to=3-7]
	\arrow["{d_{i+1}'}", color={rgb,255:red,214;green,92;blue,92}, from=3-3, to=3-5]
	\arrow[from=3-7, to=3-9]
	\arrow[from=1-3, to=3-3]
	\arrow["{\alpha_i-\beta_i}"{description}, color={rgb,255:red,92;green,92;blue,214}, from=1-5, to=3-5]
	\arrow[from=1-7, to=3-7]
	\arrow["{s_i}"{description}, color={rgb,255:red,214;green,92;blue,92}, from=1-5, to=3-3]
	\arrow["{s_{i-1}}"{description}, color={rgb,255:red,214;green,153;blue,92}, from=1-7, to=3-5]
\end{tikzcd}

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

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