Sheaves in the Alexandroff Topology

In the last blog post we defined the Sierpinski locale $\mathbb{S}$ by $\Omega(\mathbb{S}) := \{\text{Down sets of } \{0, 1\}\}$. This is a special case of the Alexandroff topology associated to an ordered set. Given an ordered set $P$, we call $\widehat{P} = \{ \text{Down sets of } P\}$ the Alexandrov topology associated to $P$. It is easy to check that it is a topology on the underlying set of $P$. Note that in the nlab link above they use upper sets, i.e. down sets in the opposite ordering. In this blog post, we are going to describe sheaves in the Alexandroff topology.

Sheaves

The Yoneda map $$ y \colon P \to \widehat{P}, \qquad p \mapsto p \downset $$ satisfies $$ p \le q \iff p\downset \subset q\downset, $$ in particular, $y$ is injective.

Note that if $\sheaf{F}$ is a sheaf on $\widehat{P}$, then it is in particular a presheaf, i.e. a functor $(\widehat{P})^{\op} \to \Set$ where we regard the ordered set $\widehat{P}$ as a category in the usual way. We get a functor $$ \Sh(\widehat{P}) \to \PSh(P), \qquad \sheaf{F} \mapsto \sheaf{F}_P := \sheaf{F} \circ y. $$

It is the aim of this post to show that this functor is an equivalence of categories. The inverse functor is given by $$ \PSh(P) \to \Sh(\widehat{P}), \qquad \sheaf{G} \mapsto \widehat{\sheaf{G}}, $$ where $$ \widehat{\sheaf{G}}(U) = \lim_{p \in U} \sheaf{G}(p) = \{(s_p)_{p \in U} \in \prod_{p \in U} \sheaf{G}(p) \mid s_p = s_q|_p \text{ for all }p \le q \in U\}. $$

Lemma. The presheaf $\widehat{\sheaf{G}}$ defined above is indeed a sheaf.

Proof. Let $U \in \widehat{P}$ and let $U = \bigcup_{i \in I} U_i$ be an open covering of $U$. Let $\{s^{(i)}\}_{i \in I}$ be a compatible family of sections $s^{(i)} \in \widehat{\sheaf{G}}(U_i)$. We want to show that there exists a unique section $s \in \widehat{\sheaf{G}}(U)$ such that $s|_{U_i} = s^{(i)}$ for all $i \in I$.

To show uniqueness, assume that $s, s' \in \widehat{\sheaf{G}}(U)$ both satisfy this property. Then given $p \in U$ there exists $i \in I$ such that $p \in U_ i$. Then we have $s_p = (s|_{U_i})_p = s^{(i)}_p = (s'|_{U_i})_p = s'_p$. Since $p \in U$ was arbitrary, we have $s = s'$ which shows the uniqueness.

In order to show existence, we construct $s$ with the required properties. Let $p \in U$. There exists $i \in I$ such that $p \in U_i$. We claim that the definition $s_p := s^{(i)}_p$ does not depend on the choice of $i$. In other words, we have to show that if $j \in J$ also satisfies $p \in U_j$, then $s_p^{(i)} = s_p^{(j)}$. This is true because of the compatibility of the family $\{s^{(i)}\}_{i \in I}$.

Next we check that the family $(s_p)_{p \in U}$ thus defined is indeed an element of $\widehat{\sheaf{G}}(U)$. So let $p \le q \in U$. Pick $i \in I$ such that $q \in U_i$. Then we also have $p \in U_i$ because $U_i$ is a down set. Hence we have $s_p = s^{(i)}_p = s^{(i)}_q|_p = s_q|_p$.

We have $s|_{U_i} = s^{(i)}$ by construction. ∎

Lemma. Given $\sheaf{F} \in \Sh(\widehat{P})$ we have an isomorphism $\sheaf{F} \isoto \widehat{\sheaf{F}_P}$ given on an open subset $U \in \widehat{P}$ by $$ \sheaf{F}(U) \isoto \widehat{\sheaf{F}_P}(U) = \lim_{p \in U} \sheaf{F}(p\downset), \qquad s \mapsto (s|_{p\downset})_{p \in U}. $$

Proof. This is because $\sheaf{F}$ is a sheaf and $U = \bigcup_{p \in U} p\downset$ is an open covering. ∎

Lemma. Given $\sheaf{G} \in \PSh(P)$ we have an isomorphism $(\widehat{\sheaf{G}})_P \isoto \sheaf{G}$ given pointwise by $$ (\widehat{\sheaf{G}})_P = \lim_{q \in p\downset} \sheaf{G}(q) \isoto \sheaf{G}(p), \qquad (s_q)_{q \in p\downset} \mapsto s_p. $$ Proof. This is because $p \in p\downset$ is a terminal object of the index category in the limit. ∎

All in all we have shown:

Theorem. The functor $$ \Sh(\widehat{P}) \to \PSh(P), \qquad \sheaf{F} \mapsto \sheaf{F}_P := \sheaf{F} \circ y $$ is an equivalence of categories. ∎

Sierpinski Space

Let $\mathbb{S}$ be the Sierpinski locale with the universal open $U_0 \in \Omega(\mathbb{S})$ and the top open $\top \in \Omega(\mathbb{S})$. Note that regarding $\Omega(\mathbb{S}) = \widehat{\{0, 1\}}$, we have $U_0 = 0\downset$ and $\top = 1\downset$.

Corollary. The functor $$ \Sh(\mathbb{S}) \to \Set^{\to}, \qquad \sheaf{F} \mapsto (\sheaf{F}(\top) \to \sheaf{F}(U_0)) $$ is an equivalence of categories. ∎