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In the use of the term topos I read about today, $\mathfrak{Top}$ is a 2-category. In the first use I ever read about a topos was just a category of sheaves on a site. I didn't think if it was a 2-category instead of a 1-category though. Is the category of sheaves on a site a 1-category or a 2-category?

sheafpants4
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    The category of sheaves (of sets) on a site is a 1-category. The category of toposes is a 2-category. – Zhen Lin Jun 09 '22 at 22:06

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The category of presheaves on a category C is a functor category with objects being the Functors and arrows being the natural transformations. This is a 1-category.

The category of sheaves on a site (C,J) is a full subcategory of the 1-category of Presheaves on C.

Category_Theorist
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