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Why not just have arrows in the category of opens represent coverings instead of inclusions?

It seems to me like both conventions (whether presheaves are co/contra and which of the two dual orderings to use to represent spaces as thin categories) are pretty arbitrary, and switching them would simplify notation.

Is it a topology-as-logic issue, where we want to be able to regard meets as conjunctions (a point satisfies $a \wedge b$ when it satisfies both opens $a,b$) etc.?

What, if anything, am I missing?

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    $\begingroup$ You can always avoid talking about contravariant functors by reversing the arrows in the domain category. Nothing is ever gained. You could also reverse the arrows of the codomain category... $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Aug 10, 2013 at 17:17
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    $\begingroup$ We represent the inclusion $U \subset V$ by an arrow $U \to V$ because it literally is a continuous map from $U$ to $V$. The same holds in other Grothendieck topologies (like etale or some four-letter words...) $\endgroup$
    – Dan Petersen
    Commented Aug 10, 2013 at 17:23

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The category of open subsets of a space $X$ is a subcategory of the category of spaces $Y \to X$ over $X$, so the usual direction of arrows is natural in this sense.

Constructing the category of presheaves $\text{Psh}(C)$ on a category $C$ is a covariant construction. Moreover, it is the free cocompletion of $C$. So the category of presheaves over $X$ is a certain completion of a certain subcategory of the category of spaces over $X$, which should give you some indication of the relationship between sheaves and étalé spaces. In somewhat more detail, there is an adjunction between the category of spaces over $X$ and the category of presheaves over $X$, and this adjunction restricts canonically to an equivalence of categories between étalé spaces and sheaves.

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    $\begingroup$ $\mathrm{Psh}(C)$ is the cocompletion when $C$ is small. In general you have to take cofinally small presheaves. $\endgroup$
    – Martin Brandenburg
    Commented Aug 11, 2013 at 9:39
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In this post a presheaf is by definition a contravariant functor from a given category to the category of sets. (See Definition 1.2 in Préfaisceaux by Grothendieck and Verdier.)

If we have to choose between two equally nice functors, one being covariant and the other contravariant, we take the covariant one. (If you disagree with that, dear Reader, this answer won't convince you.)

Now let $\mathcal C$ be a category. The functor $\text{Hom}_{\mathcal C}(X,Y)$ is contravariant in $X$ and covariant in $Y$. So, if you want to attach, in a covariant way, a functor $F(Y)$ to an object $Y$ of $\mathcal C$, the most natural choice is to put $$ F(Y):=\text{Hom}_{\mathcal C}(\ ,Y), $$ and $F(Y)$ is a contravariant functor from $\mathcal C$ to the category of sets.

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I think the real reason is actually a cognitive one: A lot of category theory comes from the investigation of geometric objects.

Now, for the human brain (at least for mine) it is more natural to imagine subspaces than it is to imagine quotients of geometric objects. [My uneducated guess would be that human vision relies on discriminating objects from a background and, thus, subobjects are more natural for us.]

This again lead to an implicit bias towards the category $\mathrm{Spc}/X$ of spaces over $X$ in the development of category theory.

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