# Monday, February 14

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

- Categories
- Functors
- Diagram/index categories
  - $\bullet \to \bullet \from \bullet$
  - $\bullet \from \bullet \to \bullet$
  - $\bullet \mapstofrom \bullet$
  - $\NN: \bullet \to \bullet \to \cdots$
  - $\bullet \covers \bullet$
- Sets and posets as categories
- Collections of objects $\cat C$ as functors $F\in [\cat I, \cat C]$ for $\cat I$ an index category
- Products and coproducts (via their universal properties).
  Useful mnemonic diagram:

\begin{tikzcd}
	{\forall P} && {\prod A_i} \\
	\\
	& \cdots & {A_i} & \cdots \\
	\\
	&& {\coprod A_i} && {\forall C}
	\arrow[from=3-2, to=3-3]
	\arrow[from=3-3, to=3-4]
	\arrow[from=1-3, to=3-3]
	\arrow[from=3-3, to=5-3]
	\arrow["{\forall \pi_P}", from=1-1, to=3-3]
	\arrow["\exists"', dashed, from=1-1, to=1-3]
	\arrow["{\forall \iota_C}", from=3-3, to=5-5]
	\arrow["\exists"', dashed, tail reversed, no head, from=5-5, to=5-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNyxbMSwyLCJcXGNkb3RzIl0sWzIsMiwiQV9pIl0sWzMsMiwiXFxjZG90cyJdLFsyLDAsIlxccHJvZCBBX2kiXSxbMiw0LCJcXGNvcHJvZCBBX2kiXSxbMCwwLCJcXGZvcmFsbCBQIl0sWzQsNCwiXFxmb3JhbGwgQyJdLFswLDFdLFsxLDJdLFszLDFdLFsxLDRdLFs1LDEsIlxcZm9yYWxsIFxccGlfUCJdLFs1LDMsIlxcZXhpc3RzIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzEsNiwiXFxmb3JhbGwgXFxpb3RhX0MiXSxbNiw0LCJcXGV4aXN0cyIsMix7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6ImFycm93aGVhZCJ9LCJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dXQ==)

- Algebraic cats over sets (concrete categories) will be closed under products, i.e. $\prod A_i$ will admit the same algebraic structure by taking pointwise operations.
- Examples of (co)products in common categories:
  - $\Set$: direct cartesian product and disjoint union.
  - $\Ab\Grp$: direct cartesian product and direct sum $\oplus$.
  - $\Ring$: $\prod$ and $\tensor_\ZZ$
  - $\Top$: $\prod$ whose underlying set is the cartesian product with the product topology and $\coprod$ as the disjoint union with the union of topologies
    Note the difference between the box and product topologies.
- A diagram in $\cat C$ defined as a functor.
- (co)filtered diagram categories $\cat I$: for any pair $i,j$, $\size \Mor_{\cat I}(i, j) \leq 1$ and there exists a $k$ with $i, j \to k$.
  Reverse arrows for cofiltered.
  - This allows for distinct but isomorphic objects, useful e.g. in $\Vect\slice k$ where abstractly $V\cong V\dual$ but it's useful to distinguish.
- Limits (injective, cones that live above) and colimits (projective, cocones that live below):

\begin{tikzcd}
	{\forall P} && {\cocolim A_i} \\
	\\
	& \cdots & {A_i} & \cdots \\
	\\
	&& {\colim A_i} && {\forall C}
	\arrow[from=3-2, to=3-3]
	\arrow[from=3-3, to=3-4]
	\arrow[from=1-3, to=3-3]
	\arrow[from=3-3, to=5-3]
	\arrow["{\forall \pi_P}", from=1-1, to=3-3]
	\arrow["\exists"', dashed, from=1-1, to=1-3]
	\arrow["{\forall \iota_C}", from=3-3, to=5-5]
	\arrow["\exists"', dashed, tail reversed, no head, from=5-5, to=5-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNyxbMSwyLCJcXGNkb3RzIl0sWzIsMiwiQV9pIl0sWzMsMiwiXFxjZG90cyJdLFsyLDAsIlxcY29jb2xpbSBBX2kiXSxbMiw0LCJcXGNvbGltIEFfaSJdLFswLDAsIlxcZm9yYWxsIFAiXSxbNCw0LCJcXGZvcmFsbCBDIl0sWzAsMV0sWzEsMl0sWzMsMV0sWzEsNF0sWzUsMSwiXFxmb3JhbGwgXFxwaV9QIl0sWzUsMywiXFxleGlzdHMiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMSw2LCJcXGZvcmFsbCBcXGlvdGFfQyJdLFs2LDQsIlxcZXhpc3RzIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiYXJyb3doZWFkIn0sImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9LCJoZWFkIjp7Im5hbWUiOiJub25lIn19fV1d)

- Fiber products/pullbacks and pushouts
- Equalizers/difference kernels $K$ and coequalizers/difference cokernels $C$ fitting into $K \to A_1 \covers A_2\to C$.
- Computing cofiltered colimits in $\Ab\Grp$: for the cofiltered set $\ts{A_i, \phi_{ij}: A_i\to A_j}_{i, j}$, can construct as $\colim A_i = \disjoint A_i/\sim$ here $a_i \sim \phi_{ik}(a_i)$ for any $k$ with $i\to k$.
  - For filtered limits, one generally gets $\cocolim A_i = \bigoplus A_i/\sim$ where $a_i \sim \phi_{ik}(a_i)$
- Example: $\disjoint A_i \in \Ab\Grp$ is not a cofiltered colimit, since the diagram category is discrete.
  - Claim: the underlying set is not $\disjoint A_i$.
- For fixed objects $A\in \cat{C}$, the functors $\Mor_{\cat C}(A, \wait): \cat{C}\to \Set$ and $\Mor_{\cat C}(\wait, A): \cat{C} \to \opcat{\Set}$.
  - More generally the target can be $\Ab\Grp, \CRing$, etc.
:::

:::{.remark}
Next time: additive and abelian categories, why $\Sh(X; \Ab\Grp)$ is an abelian category.
:::