# Friday, February 18

:::{.remark}
Last time: abelian categories $\cat{C}$.

1. Existence of kernels, cokernels, and biproducts: $\exists A\times B \iff \exists A \oplus B$.
2. Existence of isomorphisms $\coim \phi \to \im \phi$ for all $\phi\in \cat{C}(A, B)$

:::

:::{.corollary title="?"}
For $A\in \Ab\Cat$, every morphism has a mono-epi factorization:

\begin{tikzcd}
	A && B \\
	\\
	& I
	\arrow[dashed, two heads, from=1-1, to=3-2]
	\arrow[dashed, hook, no head, from=3-2, to=1-3]
	\arrow["f", from=1-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJBIl0sWzIsMCwiQiJdLFsxLDIsIkkiXSxbMCwyLCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifSwiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzIsMSwiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifSwiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn0sImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMCwxLCJmIl1d)

:::

:::{.remark}
The main technical tool: every SES induces a LES in cohomology.
The proof used for $\cat{C} = \Ab\Grp$ works nearly identically in an arbitrary abelian category using either

- *generalized elements*, c/o MacLane, or
- the full Freyd-Mitchell embedding.

MacLane's idea: define a functor 
\[
F: \cat{A} \to \Set_{\pt} \\
A &\mapsto \ts{X\in \cat{A} \st X\injects A}/\sim
,\]
sending $A$ to the set of its subobjects (equivalence classes of monomorphisms), and on morphisms $A \mapsvia{f} B$ sending $X\injects A$ to its image $f(X)\injects B$, so $F(f)(X) = \im_B(X)$.
The point in the pointed set is the subobject $0_A \to A$.
One then proves

- $f = 0 \iff F(f) = 0$,
- $f$ is mono/epi $\iff F(f)$ is mono/epi,
- Thus $F$ is exact.

So one can reduce checking exactness of $f$ (where $\cat{A}$ may not have sets of elements) to checking exactness of $F(f)$, where the source/target are sets.
:::

:::{.theorem title="Freyd-Mitchell"}
For $\cat A \in \Ab\Cat$, there is a fully faithful embedding $\cat{C} \embedsvia{F} \rmod$ for some ring $R$.
Here *full* means that $\hom_{\cat A}(A, B) \cong \hom_{\rmod}(FA, FB)$.
:::

:::{.proof title="Idea"}
\envlist

- Use the Yoneda/functor of points embedding, which is fully faithful:
\[
\cat{A} &\to [\cat{A}, \Set] \\
X &\mapsto h^X(\wait) \da \Mor_{\cat{A}}(X, \wait)
.\]

- Identify $[A, \Set] \homotopic \rmod$ where $R = \Mor_{\cat A}(I, I)$ for $I$ an injective generator of this category, so every object comes from a subobject or quotient of $I$.
  Then every $M = \Mor_{\cat{A}}(I, M)$ becomes an $R\dash$module.

:::

:::{.observation}
Some observations about abelian categories:

- $\Ab\Cat$ is closed under $\opcat{(\wait)}$, i.e. $\cat{A} \in \Ab\Cat \iff \opcat{\cat{A}} \in \Ab\Cat$
- $\cat{A} \in \Ab\Cat \implies \Ch \cat{A} \in \Ab\Cat$.
- If $\cat{I}$ is any index category, $\cat{A}^{\cat I} = [\cat I, \cat A] \in \Ab\Cat$.
  - E.g. $\ZZ$ with $i\to j\iff i\leq j$ yields $\cat{A}^{\ZZ}$ the category of sequences of elements of $\cat{A}$, i.e. $\cdots \to A_{-1}\to A_0 \to A_1\to \cdots$.
  - E.g. for $I = \bullet \to \bullet \from \bullet$, $\cat{A}^{\cat I}$ is the category of pushouts in $\cat{A}$ whose morphisms are commuting diagrams:

\begin{tikzcd}
	{A_1} && {A_2} \\
	&& {B_1} && {B_2} \\
	{A_3} \\
	&& {B_3}
	\arrow[dashed, from=1-3, to=2-5]
	\arrow[dashed, from=1-1, to=2-3]
	\arrow[dashed, from=3-1, to=4-3]
	\arrow[from=2-3, to=2-5]
	\arrow[from=2-3, to=4-3]
	\arrow[from=1-1, to=3-1]
	\arrow[from=1-1, to=1-3]
	\arrow[from=4-3, to=2-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCJBXzEiXSxbMiwwLCJBXzIiXSxbMCwyLCJBXzMiXSxbMiwxLCJCXzEiXSxbNCwxLCJCXzIiXSxbMiwzLCJCXzMiXSxbMSw0LCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMCwzLCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMiw1LCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMyw0XSxbMyw1XSxbMCwyXSxbMCwxXSxbNSwzXV0=)

:::

:::{.remark}
Some additional axioms that hold in $\Ab\Grp$ which we could ask $\cat{A}\in \Ab\Cat$ to have:

- AB3: existence of arbitrary sums $\bigoplus_i A_i$.
- AB4: AB3 and if $A_i\injects B_i$ for all $i$, then \( \bigoplus _i A_i \injects \bigoplus _i B_i \) is again injective.
- The dual of AB4, with products replaced by coproducts and injectives replaced by surjections.
- AB5: AB3 and for all filtered system of subobjects $A_i \subseteq A$ and a subobject $B \subseteq A$,
\[
(\sum A_i) \intersect B \cong \sum (A_i \intersect B)
.\]

- AB6: AB3 and for all filtered systems $B_{i}^j \subseteq B^j \subseteq A$,
\[
\intersect _{j\in J} \qty{ \sum_{i\in I_j} B_i^j } = \sum_{i\in \disjoint I_j}\qty{\intersect _{j\in J} B_i^j }
.\]

- AB: AB6 and AB4$\dual$, the dual conditions for AB4.

The categories $\Sh_X(\Ab\Grp)$ and $\Sh_X(\mods{\OO_X})$ satisfy AB5 and AB3$\dual$
:::