Hello,

I hope this is the right board for this type of question. I am currently reading "Some aspects of topological Galois theory" by Olivia and Laurent and I am struggling to understand the combinatorial description of the atomic completion of a category in a simple case--I believe I am missing something obvious.

Let's say I am starting from an essentially small category $\mathcal C$ whose opposite satisfies the amalgamation property, and which also has all finite products. If I am reading Lemma 6.4 correctly, any two family of arrows are considered equivalent in this case, since every family of arrows with common domain factors through the product of its codomains. So, the set of components of $c_1 \times \dotsc \times c_n$ is a singleton. This seems in line with Lemma 6.11 further down in the paper (in the very special case of multi-products that are actually products).

Now, in Definition 6.5, an equivalence relation on a given object $c$ is defined as a subset of the set of components of $c \times c$. In our case, there are only two such subsets, namely $\emptyset$ and the singleton whose sole object is the set of all pairs of morphisms with common domain and with codomain $c$. The first one doesn't contain the diagonal, so we have only one equivalence relation on $c$, its diagonal.

Is that correct? My category cannot be atomically complete, as I did not make any assumption implying that all its morphisms are strict epimorphisms. And yet, having no non-trivial equivalence relation seems to imply that there are no "imaginaries", since $\mathcal C^{\text{at}}$ is constructed using formal quotients of objects of $\mathcal C$ by their equivalence relations.

@tjaj

Hello, if we start from a category C with finite products and a well-defined atomic topology, then the associated topos E is the topos of sets. Indeed, in E, all finite products of objects coming from C are atomic. So, for any such object X, the canonical morphism X --> 1 to the terminal object 1 of E is a monorphism as the product XxX object coincides with the diagonal X. It is also an epimorphism as, for any such objects X and Y, the morphism XxY --> X is epimorphic. So any such object X is isomorphic to the terminal object of E, and E is the topos of sets.

Thank you very much for your answer Laurent!

Embarrassingly, what I was missing was the fact that sheafification (for the atomic topology) of representable functors necessarily produces atoms. I found a proof of that in Olivia’s paper on atomic sites—then your argument was very clear.