# Friday, January 21

:::{.remark}
Last time: definitions of presheaves and sheaves.
There is an adjunction
\[
\adjunction{(\wait)^+}{\Forget}{\Presh(X)}{\Sh(X)}
.\]

Recall that constant sheaves for $A\in \cat{D}$ are defined as $\ul{A}(\wait) \da \Hom_\Top(\wait, A)$ where $A$ is equipped with the discrete topology.
:::

:::{.exercise title="?"}
What is $\Gamma(\ul{A}, X)$ for $X\da\ts{1/n}_{n\in \ZZ_{\geq 0}} \subseteq \RR$?
So $A(U) \neq A^{\size \pi_0 U}$ in general, since there may not be a notion of connected components for an arbitrary topological space.
:::

:::{.exercise title="?"}
Is it true that for any $X\in \Top$ there is a unique decomposition $X = \disjoint_{i\in I} U_i$ into connected components?

> Hint: form a poset of such decompositions ordered by refinement and apply Zorn's lemma.

:::

:::{.example title="?"}
Consider the following poset with a prescribed topology, and applying some functor $F$:

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![](figures/2022-01-21_10-49-22.png)

For this to be a sheaf, this forces

- $F(\emptyset) = \terminal$
- $F_{12} \cong F_1 \oplus F_2$ by the universal property of $\oplus$ if this is to be a sheaf.
- $F_3$ can be anything mapping to $F_{12}$.

What are the stalks?

- $F_x = F(X)$ for $x=3$, since $X$ is the smallest open set containing $3$.
- $F_{x_i} = F_i$ for $x_i = 1, 2$.

:::

:::{.example title="?"}
Consider now a poset in the order topology:

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![](figures/2022-01-21_10-57-13.png)

Now $F$ is a sheaf iff $F_{124}\cong F_1 \fiberprod{F_4}F_2$ is the fiber product.
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:::{.definition title="Sheaf space"}
A map $\pi: Y\to X \in \Top$ is a **sheaf space** if it is a local homeomorphism, so every $y\in Y$ admits a neighborhood $U_y\ni y$ where $\ro{\pi}{U_y}: U_y\to \pi(U_y)$ is a homeomorphism onto its image.
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:::{.example title="?"}
Some examples:

- $X\times A\to X$ for $A$ discrete.

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![](figures/2022-01-21_11-04-47.png)

:::

:::{.example title="?"}
One possibility: "jumping".
Take $Y \da X\Disjoint_{X\smz} X$ for $X\subseteq \RR$, which is a version of the line with two zeros.
Then $Y\to X$ is a sheaf space, since it is a local homeomorphism.

The other possibility is "skipping":

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![](figures/2022-01-21_11-10-26.png)

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:::{.remark}
These two definitions of sheaf coincide: for new to old, given $Y \mapsvia{\pi} X$ apply $\mathrm{ContSec}_\pi \subseteq \Hom_\Top(X, Y)$.
In the other direction, define $Y\da \Disjoint_{x\in X} F_x$ and prove it is a local homeomorphism.
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:::{.remark}
Next time: direct/inverse image, shriek functors, sheaves of modules.
:::