## Franklin's Notes

### Spatial topos

Given a topological space $(I,\Theta)$, it turns out that the category of sheaves $\mathsf{Top}(I)$ is always a topos , and is called a spatial topos. We have the following data for the subobject classifier of this topos:

• The terminal object is the open map $1_{I}:I\to I$.

• The codomain of the subobject classifier is given by the projection map $\Omega:\hat I\to I$ of the sheaf of germs of elements of $I$ back onto $I$.

• Since a germ measures "the degree to which an open set contains a point", the true arrow $\top:1_I\to \Omega$ is obtained from the map $I\to \hat I$ sending each $i\mapsto [I]_i$, i.e. sending each $i\in I$ to the germ of open sets containing $i$.

• Given a monic $\mathsf{Top}(I)$-arrow $k:\alpha\to\beta$ between the sheaves $\alpha:a\to I$ and $\beta:b\to I$, we define $\chi_k:\beta\to\Omega$ using the function $b\to \hat I$ that sends each $x\in b$ to $\langle \beta(x), [\alpha(k(a)\cup s)]_{\beta(x)}\rangle$ where $s$ is an arbitrary neighborhood of $b$ on which $\beta$ restricts to a homeomorphism (since it is a sheaf); that is, we send each $x\in b$ to the germ measuring "the extent to which" $\beta(x)$ is contained in the image of $\alpha\circ k$ restricted to some sufficiently small neighborhood. In this way, we use open maps into $I$ as ways of translating points and subspaces of other topological spaces into points and subspaces of $I$, so that their "level of containment" can be measured using the germ structure of $I$.

Notice that we can define a partial ordering on the germs $[U]{\beta(x)}$ by letting $[U]{\beta(x)}\sqsubset [V]{\beta(x)}$ if and only if there is a neighborhood $W$ of $q(x)$ in which the restriction of $U$ is contained in the restriction of $V$, or such that $W\cap U\subset W\cap V$. That is, we say that $[U]{\beta(x)}\sqsubset [V]_{\beta(x)}$ if "points of $U$ sufficiently close to $q(x)$ are contained in $V$".