Warning 6.96.
The terminology on “coherence” can be somewhat confusing in the literature. For example, in SGA the word “coherent” is used in the following three different ways:
There is a notion for an object \(X \in T\) to be coherent; it just means that it is qcqs.
There is a notion for a morphism \(X \to Y\) to be coherent. Confusingly, this is not equivalent to the condition that the morphism \(X \to Y\) is qcqs when regarded as an object in \(T_{/Y}\). Since we would like to stick to the general convention that notions for maps are simply those for the associated object in the slice topos, we will follow Lurie to refer to this other definition by adding the adverb relatively, giving notions like “relatively quasi-compact” and “relatively qcqs”.
There is a notion for a topos to be coherent. This is the “correct” notion from the viewpoint of general topos theory. Marc mentioned he finds it quite impressive that Grothendieck managed to find the correct definition of coherence already so early on.