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We have a canonical map in one direction, namely $f^*(W(p)) \to W(f^*(p))$, but this map can fail to be an isomorphism. Here is an explicit counterexample. Let $X$ be the set of countably-brancing tre …

Great question! One example is Zorn's lemma. Assuming ZL holds in the metatheory, ZL also holds in toposes of sheaves over locales, so in particular in toposes of sheaves over topological spaces. Howe …

Already elementary toposes can fail to have enough injectives: Let $M$ be the model of ZF by Andreas Blass in his 1979 paper Injectivity, Projectivity, and the Axiom of Choice. The category of $M$-set …

I'm not precisely sure what you're looking for. Here is an example for the external interpretation of an apartness relation: Recall that the object of Dedekind reals $\mathbb{R}$ in a sheaf topos $\m …

Many of these toposes admit descriptions as internal classifying toposes, hence indeed enjoy useful universal properties. Here is a selection of such descriptions: Constructing the big Zariski topos …

I'll gather all the snippets from the various comments here (and mark this post as community wiki). The answer to the question "Is the theory of vector bundles just linear algebra done in a suitable …

The internal language of the effective topos can be understood with requiring barely any technology and is lots of fun! For instance, Andrej Bauer observed that if you construct the effective topos us …

An intuition that I find useful is the following. Recall that a presheaf $F : \mathcal{C}^\mathrm{op} \to \mathrm{Set}$ can be seen as "gluing specification": If $G : \mathcal{C} \to \mathcal{D}$ is …

To amplify Inna's answer, your sheaf $\mathcal{F}$ on $X$ is even a sheaf for the Lawvere–Tierney topology on $\mathrm{Sh}(X)$ given by the double-negation modality. In other words, your sheaf is the …