Franklin's Notes


Sheaf

Given a topological space $(I,\Theta)$, a sheaf over $I$ is defined to be a continuous local homeomorphism (and therefore an open map) $p:X\to I$, where $(X,\mathcal T)$ is some other topological space. In fact, given $(I,\Theta)$, we can assemble all of the sheaves over $I$ into a category $\mathsf{Top}(I)$ called the category of sheaves over $I$, so that a morphism between the sheaves $p:X\to I$ and $q:Y\to I$ is defined as a continuous map $k:X\to Y$. Note the similarity to how a comma category is defined.

topology

topos-theory

category-theory

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