# Friday, February 19

:::{.remark}
We looked at $B\in \bimod{R}{S}$ and showed $\wait \tensor_R B: \mods{R} \to \mods{S}$ is left adjoint to hom, and has left-derived functors $\Tor_n^R(\wait, B) \da L_n(\wait \tensor_R B)$.
\[
\adjunction{\wait\tensor_R B}{ \Hom_{{S}}(B, \wait) }{\modsleft{R}}{\modsleft{S}}
.\]

Note that $\Tor_0^R(A, B) \cong A\tensor_R B$.
:::

:::{.remark}
$A\tensor_R \wait$ is also right exact, and it turns out that 
\[
L_n(A\tensor_R \wait)(B) \cong L_n(\wait \tensor_R B)(A)
.\]
So unambiguously denote either of this left derived functors as $\Tor_n(A, B)$.
:::

## Limits and Colimits

:::{.definition title="Functor Category"}
Given categories $\cat{I}, \cat{A}$, define a **functor category** $\cat{A}^{\cat{I}}$ by

- $\Ob( \cat{A}^{\cat{I}} )$: functors $A: \cat{I} \to \cat{A}$.

- $\Mor(\cat{A} ^{\cat{I} })$: natural transformations $\eta:A\to B$ between functors.

$\cat{I}$ is thought of as an index category, and we'll write $A_i \da A(i) \in \cat{A}$ for $i\in \cat{I}$.
If \( \alpha:i \to j \) is a morphism in $I$, then denote \( A(\alpha) \da \alpha_* \), which is the morphism defined by the following:

\begin{tikzcd}
	{A_i} &&& {A_j} \\
	\\
	{B_i} &&& {B_j}
	\arrow["{\alpha_*}", from=1-1, to=1-4]
	\arrow["{\alpha_*}", from=3-1, to=3-4]
	\arrow["{\eta_i}"{description}, from=1-1, to=3-1]
	\arrow["{\eta_j}"{description}, from=1-4, to=3-4]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJBX2kiXSxbMywwLCJBX2oiXSxbMCwyLCJCX2kiXSxbMywyLCJCX2oiXSxbMCwxLCJcXGFscGhhXyoiXSxbMiwzLCJcXGJldGFfKiJdLFswLDIsIlxcZXRhX2kiLDFdLFsxLDMsIlxcZXRhX2oiLDFdXQ==)


Composition is defined by $A \mapsvia{\eta} B \mapsvia{\zeta} C$ is given by \( (\zeta_\eta)_i = \zeta_i \circ \eta_i \).
We need the collection of morphisms to be sets, so we'll require $\cat{I}$ to be a *small category* (i.e. the class of objects forms a set).
:::

:::{.example title="Poset Category"}
Take $(I, \leq)$ a poset (which is reflexive, antisymmetric, transitive, but not every two elements are comparable), define a category by

- $\Ob(\cat{I}) = I$

- $\abs{ \Hom_{\cat{I}}(i, j) } \leq 1$, and $i\to j \iff i\leq j$

Note that if $i\not\leq j$, then $\Hom_{\cat{I}}(i, j) = \emptyset$.
:::

:::{.remark}
Both $\cat{A}, \cat{A}^{\cat{I}}$ are small, so we can consider functors between them.
:::

:::{.definition title="Diagonal Functor"}
The **diagonal functor** is defined as $\Delta: \cat{A} \to \cat{A}^{\cat{I}}$ where for \( B \in \cat{A} \) the functor $\Delta(B)$ is the constant functor, i.e. $\Delta(B)_i = B$ for all $i\in \cat{I}$.
All morphism are sent to the identity, i.e. $i \mapsvia{\alpha} j \mapsvia{\Delta(B)} B \mapsvia{\one_B} B$.
:::

\todo[inline]{Work out how morphisms work here with respect to natural transformations.}

:::{.definition title="Colimit"}
The **colimit** of a functor $A: \cat{I} \to \cat{A}$ is an object $C\in \cat{A}$ which we'll denote $\colim_{i\in \cat{I}} A_i$, along with a natural transformation $\eta:A\to \Delta(C)$ which is universal among natural transformations of the form $\theta: A\to \Delta(B)$ for $B\in \cat{A}$.
The unique map in the universal property is from $C\to B$, and we have the following situation:

\begin{tikzcd}
	{\cat{I}} \\
	i && {A_i} && C \\
	\\
	j && {A_j} && C \\
	\\
	&& {A_i} \\
	&&&& C && B \\
	&& {A_j}
	\arrow["\alpha", from=2-1, to=4-1]
	\arrow["{\alpha_*}", from=2-3, to=4-3]
	\arrow["{\eta_i}"', from=2-3, to=2-5]
	\arrow["{\eta_j}", from=4-3, to=4-5]
	\arrow[equals, from=2-5, to=4-5]
	\arrow["{\eta_i}"', from=6-3, to=7-5]
	\arrow["{\eta_j}", from=8-3, to=7-5]
	\arrow["{\alpha_*}", from=6-3, to=8-3]
	\arrow["{\theta_i}", curve={height=-12pt}, from=6-3, to=7-7]
	\arrow["{\theta_j}", curve={height=12pt}, from=8-3, to=7-7]
	\arrow["{\exists ! \gamma}", dashed, from=7-5, to=7-7]
\end{tikzcd}

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

:::

:::{.example title="?"}
Let $(I, \leq)$ be a poset and take $\cat{I}$ its poset category.
Then there are morphisms $i\to j \iff i\leq j$, and we have a diagram

\begin{tikzcd}
	{A_i} \\
	&& C && D \\
	{A_j} \\
	{}
	\arrow["{\eta_i}", from=1-1, to=2-3]
	\arrow["{\eta_j}"', from=3-1, to=2-3]
	\arrow["{\theta_j}", curve={height=12pt}, from=3-1, to=2-5]
	\arrow["{\theta_i}"', curve={height=-12pt}, from=1-1, to=2-5]
	\arrow["{\exists ! \gamma}"{description}, dashed, from=2-3, to=2-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCJBX2kiXSxbMCwzXSxbMCwyLCJBX2oiXSxbMiwxLCJDIl0sWzQsMSwiRCJdLFswLDMsIlxcZXRhX2kiXSxbMiwzLCJcXGV0YV9qIiwyXSxbMiw0LCJcXHRoZXRhX2oiLDAseyJjdXJ2ZSI6Mn1dLFswLDQsIlxcdGhldGFfaSIsMix7ImN1cnZlIjotMn1dLFszLDQsIlxcZXhpc3RzICEgXFxnYW1tYSIsMSx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==)


This is the **direct limit**.
Note that for a poset of category of subsets, this ends up being the union.
:::

:::{.example title="?"}
Let $\Ob(\cat{I}) = \ts{ 1, 2 }$, and take two maps, one of which we'll label by "0":

\begin{tikzcd}
	1 \\
	\\
	2
	\arrow[shift right=5, from=1-1, to=3-1]
	\arrow["0"{description}, shift left=5, from=1-1, to=3-1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMixbMCwwLCIxIl0sWzAsMiwiMiJdLFswLDEsIiIsMCx7Im9mZnNldCI6NX1dLFswLDEsIjAiLDEseyJvZmZzZXQiOi01fV1d)


Suppose now that \( \cat{A} \) is an abelian category, and suppose we're given a morphism \( A_1 \mapsvia{f} A_2 \) in \( \cat{A} \).
Define \( A\in \cat{A}^{\cat{I}} \), and define a functor

\begin{tikzcd}
	1 && {A_1} \\
	&&&&&& B \\
	2 && {A_2}
	\arrow[shift right=2, from=1-1, to=3-1]
	\arrow["0", shift left=2, from=1-1, to=3-1]
	\arrow["0", shift left=2, from=1-3, to=3-3]
	\arrow["f"', shift right=2, from=1-3, to=3-3]
	\arrow["{\theta_1}", curve={height=-12pt}, from=1-3, to=2-7]
	\arrow["{\theta_2}", curve={height=12pt}, from=3-3, to=2-7]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCIxIl0sWzAsMiwiMiJdLFsyLDAsIkFfMSJdLFsyLDIsIkFfMiJdLFs2LDEsIkIiXSxbMCwxLCIiLDAseyJvZmZzZXQiOjJ9XSxbMCwxLCIwIiwwLHsib2Zmc2V0IjotMn1dLFsyLDMsIjAiLDAseyJvZmZzZXQiOi0yfV0sWzIsMywiZiIsMix7Im9mZnNldCI6Mn1dLFsyLDQsIlxcdGhldGFfMSIsMCx7ImN1cnZlIjotMn1dLFszLDQsIlxcdGhldGFfMiIsMCx7ImN1cnZlIjoyfV1d)

By commutativity,

- $\theta_2 \circ 0 = \theta_1 \implies \theta_1 = 0$

- $\theta_2 \circ f = \theta_1 = 0$.

So suppose there was a colimit $C$, then it'd fit into this diagram as follows:

\begin{tikzcd}
	1 && {A_1} \\
	&&&& C && B \\
	2 && {A_2}
	\arrow[shift right=2, from=1-1, to=3-1]
	\arrow["0", shift left=2, from=1-1, to=3-1]
	\arrow["0", shift left=2, from=1-3, to=3-3]
	\arrow["f"', shift right=2, from=1-3, to=3-3]
	\arrow["{\theta_1}", curve={height=-12pt}, from=1-3, to=2-7]
	\arrow["{\theta_2}", curve={height=12pt}, from=3-3, to=2-7]
	\arrow["0", from=1-3, to=2-5]
	\arrow["p", from=3-3, to=2-5]
	\arrow[dashed, from=2-5, to=2-7]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCIxIl0sWzAsMiwiMiJdLFsyLDAsIkFfMSJdLFsyLDIsIkFfMiJdLFs2LDEsIkIiXSxbNCwxLCJDIl0sWzAsMSwiIiwwLHsib2Zmc2V0IjoyfV0sWzAsMSwiMCIsMCx7Im9mZnNldCI6LTJ9XSxbMiwzLCIwIiwwLHsib2Zmc2V0IjotMn1dLFsyLDMsImYiLDIseyJvZmZzZXQiOjJ9XSxbMiw0LCJcXHRoZXRhXzEiLDAseyJjdXJ2ZSI6LTJ9XSxbMyw0LCJcXHRoZXRhXzIiLDAseyJjdXJ2ZSI6Mn1dLFsyLDUsIjAiXSxbMyw1LCJwIl0sWzUsNCwiIiwxLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d)


Note that $C$ is precisely the cokernel of $f$!
:::

:::{.remark}
Think about this last diagram: what happens when you mod out by larger modules?
:::

:::{.exercise title="Colimits always exist"}
Suppose $I$ is a discrete category, i.e. $\Hom(i, j) = \emptyset$ unless $i=j$, in which case $\Hom(i, i) = \ts{ \one_i }$.
Supposing that \( A: I \to \cat{A}  \), show that \( \colim_{i\in \cat{I}} = \coprod_{i} A_i \).
:::

:::{.definition title="?"}
A category \( \cat{A} \) is **cocomplete** if every colimit $\colim_{i\in \cat{I}} A_i$ exists for every \( A\in \cat{A}^{\cat{I}} \) and all small categories $\cat{I}$.
:::

:::{.exercise title="Taking colimits defines a functor for cocomplete categories"}
Show that when \( \cat{A} \) is cocomplete, \( \colim: \cat{A}^{\cat{I}} \to \cat{A} \) defines a functor.
:::

:::{.exercise title="Weibel 2.6.4"}
Show that the functor $\colim$ is left-adjoint to the diagonal functor \( \Delta \), so there is an adjunction
\[
\adjunction{\colim}{\diagonal}{\cat{A}^{\cat{I}}  }{\cat{A} }
.\]

Thus when \( \cat{A} \) is abelian and $\colim$ exists, it is right-exact (since left-adjoints are always right-exact).
Note that it's not exact in general.
:::

:::{.proposition title="Cocomplete iff all coproducts exist"}
For any abelian category \( \cat{A} \), the following are equivalent: 

1. $\coprod A_i$ exists in \( \cat{A} \) for every set \( \ts{ A_i } \) of objects in \( \cat{A} \) (*set-indexed coproducts*).

2. \( \cat{A} \) is cocomplete.

:::

:::{.remark}
We'll prove this next time, note that $2\implies 1$ since coproducts are special cases of limits.
:::