# Monday, March 14

:::{.remark}
Call the definition of the derived functor $\RR F$ for a left-exact functor $F\in\Ab\Cat(\cat C, \cat D)$ where $\cat A$ has enough injectives.
These satisfy $\RR^0 F = F$, and for a SES $0\to A\to B\to C\to 0$ there is an induced LES $\RR F A\to \RR F B \to \RR F C \to \RR F A[1]$ which is functorial in the triple $(A,B,C)$.
Next: Grothendieck's universality theorem.
:::

:::{.definition title="$\delta\dash$functor"}
A $\delta\dash$functor is a sequence of functors $\ts{S^i: \cat A\to \cat B}_{i\geq 0}$ such that for all SESs $0\to A\to B\to C\to 0$ there is a (not necessarily exact) complex:

\begin{tikzcd}
	0 \\
	A && B && C \\
	\\
	{S^1A} && {S^1B} && {S^1 C} \\
	\\
	{S^2A} && {S^2B} && \cdots
	\arrow[from=1-1, to=2-1]
	\arrow[from=2-1, to=2-3]
	\arrow[from=2-3, to=2-5]
	\arrow[from=2-5, to=4-1]
	\arrow[from=4-1, to=4-3]
	\arrow[from=4-3, to=4-5]
	\arrow[from=4-5, to=6-1]
	\arrow[from=6-1, to=6-3]
	\arrow[from=6-3, to=6-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMTAsWzAsMSwiQSJdLFsyLDEsIkIiXSxbNCwxLCJDIl0sWzAsMywiU14xQSJdLFswLDUsIlNeMkEiXSxbMiwzLCJTXjFCIl0sWzQsMywiU14xIEMiXSxbMiw1LCJTXjJCIl0sWzQsNSwiXFxjZG90cyJdLFswLDAsIjAiXSxbOSwwXSxbMCwxXSxbMSwyXSxbMiwzXSxbMyw1XSxbNSw2XSxbNiw0XSxbNCw3XSxbNyw4XV0=)

A **morphism** of $\delta\dash$functors is a collection $\ts{f^i: S^i\to T^i}_{i\geq 0}$ such that for all such SESs, there is a commutative diagram:

\begin{tikzcd}
	{S^iA} && {S^iB} && {S^iC} && {S^{i+1}A} \\
	\\
	{T^iA} && {T^iB} && {T^iC} && {T^{i+1}A}
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-3, to=1-5]
	\arrow["{\phi_s^i}", color={rgb,255:red,92;green,92;blue,214}, from=1-5, to=1-7]
	\arrow[from=3-1, to=3-3]
	\arrow[from=3-3, to=3-5]
	\arrow["{\phi_t^i}", color={rgb,255:red,92;green,92;blue,214}, from=3-5, to=3-7]
	\arrow[from=1-1, to=3-1]
	\arrow[from=1-3, to=3-3]
	\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=1-7, to=3-7]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwwLCJTXmlBIl0sWzIsMCwiU15pQiJdLFs0LDAsIlNeaUMiXSxbMCwyLCJUXmlBIl0sWzIsMiwiVF5pQiJdLFs0LDIsIlReaUMiXSxbNiwwLCJTXntpKzF9QSJdLFs2LDIsIlRee2krMX1BIl0sWzAsMV0sWzEsMl0sWzIsNiwiXFxwaGlfc15pIiwwLHsiY29sb3VyIjpbMjQwLDYwLDYwXX0sWzI0MCw2MCw2MCwxXV0sWzMsNF0sWzQsNV0sWzUsNywiXFxwaGlfdF5pIiwwLHsiY29sb3VyIjpbMjQwLDYwLDYwXX0sWzI0MCw2MCw2MCwxXV0sWzAsM10sWzEsNF0sWzIsNSwiIiwxLHsiY29sb3VyIjpbMjQwLDYwLDYwXX1dLFs2LDcsIiIsMSx7ImNvbG91ciI6WzI0MCw2MCw2MF19XV0=)

Note that the first 2 square are commutative by functoriality, and the content here is that the map commutes with the connecting morphisms.
:::

:::{.definition title="Effaceable functors"}
An additive functor $G: \cat{A}\to \cat{B}$ is **effaceable** iff for all $A\in \cat{A}$ there is a monomorphism $A\injectsvia{f} M$ such that $GA \mapsvia{Gf} GM$ is the zero map.
:::

:::{.slogan}
Effaceable functors are those which erase some monomorphism.
:::

:::{.definition title="Universal delta functors"}
A delta functor $(S_i, \phi_S)$ is *exact* iff the induced complex is a LES, and is **universal** iff for any other delta functor $(T_i, \phi_T)$ and any natural transformation $\eta: S^0\to T^0$, there is a unique morphism $(S_i, \phi_S) \to (T_i, \phi_T)$ extending $\eta$.
:::

:::{.theorem title="Grothendieck, Tohoku: exact fully effaceable functors are universal"}
Suppose $\ts{S^i F, \phi}_{i\geq 0}$ is an exact delta functor and that the $S^i$ are effaceable for all $i$.
Then it is a universal $\delta$ functor.
:::

:::{.corollary title="?"}
When $F\in \Ab\Cat(\cat A, \cat B)$ where $\cat A$ has enough injectives, $(\RR^i F, \phi)$ is universal and there is a unique such delta functor with $\RR^0 F = F$.
:::

:::{.proof title="of corollary"}
Embed $A\embeds I$ into an injective object, which is $F\dash$acyclic, and thus $\RR^i F A \mapsvia{0} \RR^i F I = 0$.
:::

:::{.proof title="of theorem"}
Proceed by induction.
Let $0\to A \to M \to Q \to 0$ be arbitrary, and use a diagram chase to define a map $f^i(\iota)$:

\begin{tikzcd}
	{S^i M} && {S^i Q} && {S^{i+1}A} && {S^{i+1}M} \\
	\\
	{T^i M} && {T^i Q} && {T^{i+1}A} && {T^{i+1}B}
	\arrow[from=3-1, to=3-3]
	\arrow["{\phi_T^i}", from=3-3, to=3-5]
	\arrow[from=3-5, to=3-7]
	\arrow[from=1-1, to=1-3]
	\arrow["{\phi_S^i}", two heads, from=1-3, to=1-5]
	\arrow["0", from=1-5, to=1-7]
	\arrow[from=1-1, to=3-1]
	\arrow[from=1-3, to=3-3]
	\arrow["{\exists?}"', dashed, from=1-5, to=3-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwwLCJTXmkgTSJdLFsyLDAsIlNeaSBRIl0sWzQsMCwiU157aSsxfUEiXSxbNiwwLCJTXntpKzF9TSJdLFs0LDIsIlRee2krMX1BIl0sWzYsMiwiVF57aSsxfUIiXSxbMiwyLCJUXmkgUSJdLFswLDIsIlReaSBNIl0sWzcsNl0sWzYsNCwiXFxwaGlfVF5pIl0sWzQsNV0sWzAsMV0sWzEsMiwiXFxwaGlfU15pIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzIsMywiMCJdLFswLDddLFsxLDZdLFsyLDQsIlxcZXhpc3RzPyIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==)

One needs to show:

1. $f^i(\iota)$ does not depend on $\iota$
2. It is a ? for all $A\to B$
3. This map commutes with $\phi_S, \phi_T$.

:::