# Friday, February 11

:::{.remark}
Recall the definitions of:

- Cochain complexes, 
- Boundaries, 
- Cycles, 
- Homology as cycles mod boundaries,
- Morphisms of chain complexes
- Chain homotopies
- Nullhomotopic morphisms
- Homotopic morphisms of chain complexes
- Short exact sequences of complexes:

\begin{tikzcd}
	&& 0 && 0 && 0 \\
	\\
	\cdots && {A^{n+1}} && {A^{n}} && {A^{n-1}} && \cdots \\
	\\
	\cdots && {C^{n}} && {B^{n}} && {B^{n-1}} && \cdots \\
	\\
	\cdots && {C^{n-1}} && {C^{n}} && {C^{n-1}} && \cdots \\
	\\
	&& 0 && 0 && 0
	\arrow[from=3-1, to=3-3]
	\arrow[from=3-3, to=3-5]
	\arrow[from=3-5, to=3-7]
	\arrow[from=3-7, to=3-9]
	\arrow[from=5-1, to=5-3]
	\arrow[from=5-3, to=5-5]
	\arrow[from=5-5, to=5-7]
	\arrow[from=5-7, to=5-9]
	\arrow[hook, from=1-3, to=3-3]
	\arrow[hook, from=1-5, to=3-5]
	\arrow[hook, from=1-7, to=3-7]
	\arrow[hook, from=3-3, to=5-3]
	\arrow[hook, from=3-5, to=5-5]
	\arrow[hook, from=3-7, to=5-7]
	\arrow[two heads, from=5-3, to=7-3]
	\arrow[two heads, from=5-5, to=7-5]
	\arrow[two heads, from=5-7, to=7-7]
	\arrow[from=7-1, to=7-3]
	\arrow[from=7-3, to=7-5]
	\arrow[from=7-5, to=7-7]
	\arrow[from=7-7, to=7-9]
	\arrow[two heads, from=7-7, to=9-7]
	\arrow[two heads, from=7-5, to=9-5]
	\arrow[two heads, from=7-3, to=9-3]
\end{tikzcd}

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

- Small categories
  - Sets of objects, sets of morphisms, a pairing $\Mor(A, B)\times \Mor(B, C)\to \Mor(A, C)$.
- Universes

:::

:::{.exercise title="?"}
Show that a morphism of chain complexes induces a morphism on homology.
:::

:::{.exercise title="?"}
Show that $f\homotopic g \implies \cxH(f) = \cxH(g)$, i.e. homotopic chain morphisms induce equal maps on homology.

> Hint: reduce to showing that $f$ nullhomotopic implies $\cxH(f) = 0$.

:::

:::{.exercise title="Show a SES induces a LES in homology"}
Show that a SES of complexes induces a LES in homology.
Write a formula for the connecting morphism, and do the check that everything is well-defined!
Use the grid diagram from above.
:::

:::{.example title="?"}
Examples of categories:

- $\Set$
- $\modsleft{R}$
- $\modsright{R}$
- $\Top$
- $\CRing$, assumed to be unital
- $\Sch\slice k$
- $\Alg\Var\slice k$ for $k=\kbar$
- $\Sh(X; \zmod)$
- $\mods{\OO_X}$
- $\Top\Ab\Grp$
- $G\actson\rmod$

Note that many of these are not abelian, since they are not even additive, or e.g. are not closed under kernels.
:::