# On 2-Category Of Toposes

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Warning: Incomplete

Toposes, geometric morphisms, and natural transforamtions form a 2-category denoted by $\mathfrak{Top}$. Note that for each pair of toposes $D$ and $E$, $\mathbf{Geom}(D,E)$ is a large though locally small category.

#### Example.

Suppose $X$ is a (non-$T_1$) topological space We define the following (non-trivial) partial order on points of $X$. $$x \leq x’ \ \ \text{iff every neighbourhood of} \ \ x \ \ \text{contains} \ \ x’$$

We can extend this order to all maps between topological spaces. Suppose $f,g: X \rightrightarrows Y$ are (continuous) maps. Define

$$\label{def-order on maps} f \leq g \ \ \text{iff} \ \ f(x) \leq g(x) \ \ \text{for every} \ \ x \in X$$

A ramification of above defintion, which is straightforward to see, is that $f \leq g$ if and only if $f^{-1} (V) \subseteq g^{-1}(V)$ for evey open set $V$ of $Y$. Notice that $f$ and $g$ give us $f_{\ast},g_{\ast}: Sh(X) \rightrightarrows Sh(Y)$ which are pushfoward geometric morphisms along $f$ and $g$ respectively.

Thus, if $f \leq g$, then for any sheaf $P$ on $X$, and any open $V$ of $Y$, we get the restriction $g_{\ast}(P)(V)=P(g^{-1}(V)) \rightarrow P(f^{-1}(V))= f_{\ast}(P)(V)$. This yields a natural transforamtion $f \Rightarrow g$ in $\geom(Sh(X),Sh(Y))$.

#### Remark.

From the construction above we obtain a functor $\mathbf{Top}(X,Y) \rightarrow \mathbf{Geom}(Sh(X),Sh(Y))$.

As a consequence, it is generally too much to expect diagrams in Top to commute on the nose”, i.e up to equality { commuting is usually only up to isomorphism. In broad terms, this is because in a category equality between ob jects is less important that isomorphism

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