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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 …

Since the axioms describing what a valuation ring can be put as what's called geometric sequents [*], by the fundamental theorem on classifying toposes, there is a topos $T_{val}$ with precisely the u …

I assume that you mean that $\mathcal{F}$ is a sheaf of rings. What's internally an ideal of $\mathcal{F}$ is externally simply a sheaf of ideals. In case that the topos in question is the little Za …

First note that a morphism $\operatorname{Spec}(A) \to \mathbb{P}^1$ is just given by an element of the "classical projective space" $\mathbb{P}^1(A) = \{ [a:b] \,|\, \text{$a$ is invertible or $b$ is …

Let $\mathbb{T}$ be a geometric theory. Let $\mathrm{Set}[\mathbb{T}]$ be its classifying topos, such that geometric morphisms from any (cocomplete) topos $\mathcal{E}$ into $\mathrm{Set}[\mathbb{T}]$ …

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 …

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 …

Unfortunately I don't know an interesting intrinsically formulated sufficient criterion for a locally ringed topos to be the little Zariski topos of a scheme. This is an extremely interesting question …

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 …

Let's simplify and consider the presheaf topos. I asked the same question over at the nForum a while back. There Marc Hoyois reminded me of the following quite general fact: The category of topos-the …

This question was previously asked over at math.SE. Let $X$ be a scheme. Let $\mathcal{E}$ be a sheaf of sets on $X$. Then we can define $\mathcal{O}_X\langle\mathcal{E}\rangle$, the free module over …

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 …

Since I don't know precisely which parts of Lawvere's article you have difficulties with, this answer is a bit a long and tries to give a bit of context. If you want me to be more specific at some poi …

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 …

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 …