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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 …

Question. Let $X$ be a scheme. Let $\mathcal{E}$ be a sheaf of $\mathcal{O}_X$-modules. Is there always a quasicoherent sheaf $\mathcal{E}'$ together with a monomorphism $\mathcal{E} \to \mathcal{E}'$ …

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 …

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'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 …

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 …

My position would be: It is a fundamental feature of pullback along geometric morphisms to not preserve exponentials. But, given objects $X$ and $Y$ in a topos, we can do more than just form the expon …

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 …

It seems that Martin has provided the answer which Daniel sought. But the question in the title doesn't appear to be answered yet: Is there a universal property for graded localization? More precisely …

It appears that in the meantime, full proofs have been added to the Stacks Project. Tag 0564. Let $R \to S$ be a finite and finitely presented ring map. Let $M$ be an $S$-module. Then $M$ is finit …

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 …

Is it possible to do algebraic geometry in a synthetic manner that enables rigorous reasoning but is closer to the style of argument employed by classical algebraic geometers? I sure hope so. You …

I think that Section 11 on transfer principles in these notes of mine is what you're looking for. A general machinery abstracts the business of tracking all the $f_i$'s and the required high powers. T …