# Friday, February 25

## Adjoint Functors, Exactness

:::{.remark}
Consider the setup:
\[
\adjunction F G { \cat{A}}{ \cat{B} }
.\]
We say $F$ is a **left adjoint** and $G$ is a **right adjoint**, so $F$ *has* a right adjoint and $G$ *has* a left adjoint, if there are natural isomorphisms
\[
[FA, B]_{\cat B} \iso [A, GB]_{\cat A}
,\]
i.e. there is a natural isomorphism of functors $[A, G(\wait)] \iso [FA, (\wait)]$.
For a fixed object $B$, there is a natural transformation $\eps_B: FG\to \id_B$ which we call the **counit** and $\eta_A: \id_A\to GF$ called the **unit**:

\begin{tikzcd}
	A &&&& FA \\
	\\
	GB &&&& FGB && B
	\arrow["{\exists \eps_B}", from=3-5, to=3-7]
	\arrow[""{name=0, anchor=center, inner sep=0}, from=1-1, to=3-1]
	\arrow[""{name=1, anchor=center, inner sep=0}, from=1-5, to=3-5]
	\arrow[from=1-5, to=3-7]
	\arrow["F", shorten <=26pt, shorten >=26pt, Rightarrow, from=0, to=1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCJBIl0sWzAsMiwiR0IiXSxbNCwwLCJGQSJdLFs0LDIsIkZHQiJdLFs2LDIsIkIiXSxbMyw0LCJcXGV4aXN0cyBcXGVwc19CIl0sWzAsMV0sWzIsM10sWzIsNF0sWzYsNywiRiIsMCx7InNob3J0ZW4iOnsic291cmNlIjoyMCwidGFyZ2V0IjoyMH19XV0=)


:::

:::{.theorem title="?"}
If $\cat A, \cat B \in \Ab\Cat$, then

- If $F$ is a right adjoint, $F$ is left exact.
- If $G$ is a left adjoint, $G$ is right exact.

:::

:::{.proof title="?"}
Note that the following lift exists iff $\ker(A\to A'') = (A'\to A)$:

\begin{tikzcd}
	0 && {A'} && A && {A''} && 0 \\
	\\
	&&&& X
	\arrow[from=3-5, to=1-5]
	\arrow["0"', from=3-5, to=1-7]
	\arrow[from=1-1, to=1-3]
	\arrow["i"', from=1-3, to=1-5]
	\arrow["p"', from=1-5, to=1-7]
	\arrow[from=1-7, to=1-9]
	\arrow["\exists", dashed, from=3-5, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCIwIl0sWzIsMCwiQSciXSxbNCwwLCJBIl0sWzYsMCwiQScnIl0sWzgsMCwiMCJdLFs0LDIsIlgiXSxbNSwyXSxbNSwzLCIwIiwyXSxbMCwxXSxbMSwyLCJpIiwyXSxbMiwzLCJwIiwyXSxbMyw0XSxbNSwxLCJcXGV4aXN0cyIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==)

Given $0\to B'\to B\to B''$, we want to show $0\to GB'\to GB\to GB''$ is exact.
Given $A\to B''$ factoring through zero, we can use adjointness to flip diagrams:

\begin{tikzcd}
	0 && {GB'} && GB && {GB''} && 0 \\
	\\
	&&&& A \\
	\\
	0 && {B'} && B && {B''} && 0 \\
	\\
	&&&& FA
	\arrow[from=3-5, to=1-5]
	\arrow[""{name=0, anchor=center, inner sep=0}, "0"', from=3-5, to=1-7]
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-3, to=1-5]
	\arrow[from=1-5, to=1-7]
	\arrow[from=1-7, to=1-9]
	\arrow["{\therefore \exists}", dashed, from=3-5, to=1-3]
	\arrow[from=5-1, to=5-3]
	\arrow[from=5-3, to=5-5]
	\arrow[""{name=1, anchor=center, inner sep=0}, from=5-5, to=5-7]
	\arrow[from=5-7, to=5-9]
	\arrow[from=7-5, to=5-5]
	\arrow["0"', from=7-5, to=5-7]
	\arrow["\exists", dashed, from=7-5, to=5-3]
	\arrow["{F(\wait)}"', shorten <=13pt, shorten >=13pt, Rightarrow, from=1, to=0]
\end{tikzcd}

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

:::

:::{.example title="?"}
There is an adjunction between global sections and constant sheaves:
\[
\adjunction{ \Globsec{X; \wait} }{ \constantsheaf{(\wait) }  }{\Sh(X; \Ab\Grp) }{\Ab\Grp}
.\]

One can define the map explicitly:
\[
[A, \globsec{X; \mcf} ]_{\Ab\Grp} &\to [\ul{A}, \mcf]_{\Sh(X; \Ab\Grp)} \\
(a\mapsto s_a) &\mapsto (a_U \mapsto \ro{s_a}{U} )
.\]
It suffices to check this locally. 
Use that $\Globsec{X; \ul{A}}$ contains a copy of $A$ to define the reverse map, and check they are mutually inverse.
:::

:::{.example title="?"}
For $f\in [X, Y]_{\Top}$, there is an induced adjunction
\[
\adjunction{f_*}{f\inv}{\Sh(X; \Ab\Grp)}{\Sh(Y; \Ab\Grp)}
.\]
Thus $f_*$ is left exact.
:::

:::{.exercise title="?"}
Define the map
\[
[\mcg, f_* \mcf]_{\Sh_Y} \to [f\inv \mcg, \mcf]_{\Sh_X}
.\]
:::

:::{.remark}
Note that $f_*$ is fully exact, as we knew before by checking on stalks.
Also note that $\st_x$ for $\mcf\in \Sh(X)$ is $f\inv \mcf$ for $f:\ts{x} \injects X$.
:::

:::{.example title="?"}
\[
\adjunction{(\wait)^+}{\Forget}{\presh(X; \Ab\Grp)}{\Sh(X; \Ab\Grp)}
,\]
so sheafification is right exact and the forgetful functor is left exact.
In fact, $(\wait)^+$ is fully exact since it preserves stalks.
:::

:::{.example title="?"}
For $j\in [U, X]_{\Top}$ with $U$ open in $X$,
\[
\adjunction{j_!}{j\inv}{\Sh(U; \Ab\Grp) }{\Sh(X; \Ab\Grp) }
.\]
In general there is a SES
\[
0 \to j_! \ro{\mcf}{U} \to \mcf \to i_* \ro{\mcf}{X\sm U} \to 0
.\]
:::

:::{.example title="from algebra"}
\[
\adjunction{(\wait)\tensor_R (\wait) }{[\wait, \wait]_{\rmod}}{\rmod}{\rmod}
,\]
so tensoring is right exact when an object is fixed.
Note the isomorphism
\[
[A\tensor_R B]_{\rmod}\iso [A, [B,C]_{\rmod}]_{\rmod}
.\]
:::