# what are the epimorphisms in the category of algebraic stacks?

Let $\mathcal{C}$ be the 2-category of algebraic stacks (lets say Deligne-Mumford) over $\textbf{Sch}_{etale}$, and let $\mathcal{C}'$ be the corresponding 1-category we obtain by identifying all 2-isomorphic morphisms.

Let $T,S$ be schemes, and $T\rightarrow S$ be an epimorphism of $\mathcal{C}'$, then for any algebraic stack $\mathcal{M}\in\mathcal{C}'$, the natural map $\mathcal{M}(S)\rightarrow\mathcal{M}(T)$ should be an injection, where here $\mathcal{M}(S) = Hom_{\mathcal{C}}(S,\mathcal{M})$ is just the set of 2-isom classes of morphisms $S\rightarrow\mathcal{M}$, or equivalently the set of isom classes of objects of $\mathcal{M}(S)$.

Certainly if $T\rightarrow S$ is an epimorphism in $\mathcal{C}'$, then it's also an epimorphism in $\textbf{Sch}$.

Now, what I find curious, is that thanks to HeinrichD's comments on:

epimorphisms and 2-isomorphic maps to an algebraic stack

(ie, thanks to the theory of descent and the existence of plenty of objects which are locally trivial but not globally trivial), it seems that the only epimorphisms $T\rightarrow S$ in $\mathcal{C'}$ are the split epimorphisms (which seem trivial to me).

Questions:

1. Do there exist nonsplit epimorphisms in $\mathcal{C}'$ between schemes?
2. Do there exist nonsplit epimorphisms in $\mathcal{C}'$? (not necessarily between schemes?)
3. Are there split epimorphisms which aren't trivial (ie, in which the image of the splitting isn't a direct summand of the domain?)
4. Do the answers to (1),(2),(3) change if we consider $\mathcal{C}$ instead of $\mathcal{C}'$ (and maybe consider 2-epimorphisms instead of epimorphisms?) (I don't even know of a good definition of a 2-epimorphism - the nlab article wasn't very readable)
5. Do the answers to (1),(2),(3) change if we change our definition of algebraic stack? (ie, change the topology, change DM to Artin,...etc)
6. What are some interesting natural examples of other categories which don't have nontrivial epimorphisms?
• Just a comment. You're aware that $\mathrm{Stack}(S,T)=\mathrm{Sch}(S,T)$ if $S,T$ are schemes, right? Did you have a look at Laumon and Moret-Bailly's book? For example, Définition 3.6 and Proposition 3.7. – Qfwfq Oct 18 '16 at 0:08
• (For non- French readers, Définition 3.6 sais that $F:\mathscr{Z}\to\mathscr{X}$ is an epimorphism in Stacks [hence in algebraic stacks] by definition if $F$ is, so to speak, essentially surjective from the "stalks" $\lim_U \mathscr{Z}_U$ of $\mathscr{Z}$ to those of $\mathscr{X}$ (take my notation "$\lim$" heuristically, by analogy with sheaves of sets on a site or topological space). The following Proposition 3.7 asserts the existence of the "stack theoretic image" (notice that it doesn't envolve algebraic stacks, just stacks). – Qfwfq Oct 18 '16 at 0:19

My impression is that your question doesn't really have to do with stacks (algebraic or otherwise).

In the following, calligraphic is for stacks and usual capital for schemes (or set-valued functors, or categories fibered in sets or in rigid groupoids, ...)

1. Consider a nontrivial principal bundle $\pi:X\to Y$ with $X$ and $Y$ schemes, then $\pi$ is a nonsplit epi in schemes. The piece "in $\mathcal{C}'$" is irrelevant, because the example is also valid in $\mathcal{C}'$. Indeed, I think you can factor $f,g:Y \rightrightarrows \mathscr{Z}$ via $Y\rightrightarrows Z\to \mathscr{Z}$ (at least with $Z$ a sheaf of sets, $Z(U):=f(Y(U))\cup g(Y(U))$), and test epicity of $X\to Y$ by $X\to Y \rightrightarrows Z$ instead of $X\to Y \rightrightarrows \mathscr{Z}$. By Yoneda, epis in schemes are the same as epis in sheaves (of sets).

2. see (1.)

3. What's a "direct sum" in schemes? I'm not very fluent in category theory lingo, but I think any geometric vector bundle $\pi:E\to Y$ over a scheme $Y$ would qualify (it always has the zero section, and -mostly- it's not the disjoint union of copies of $Y$). This picture, I'd say, doesn't change if you work in $\mathcal{C}'$.

4. I think $2$-arrows do no harm. Just replace some occurrences of "$=$" with "there exists an isomorphism".

5. Schemes will always be a full subcategory, and an epi $X\to Y$ (split or not) will stay so when regarded as a morphism in your favourite category of stacks.

6. Manifolds, topological spaces, schemes...

Remark: the presence of non-surjective-on-points epis is already captured in affine schemes - stacks will not make it worse.

• I'm confused about your answer to 1 (and 2). Are you saying that any nontrivial principal bundle $X\rightarrow Y$ is an epimorphism? In that case can't you just let $X\rightarrow Y$ be any Galois extension of fields ${\rm Spec}\;L\rightarrow{\rm Spec}\;K$, in which case $Spec\;L\rightarrow Spec K$ certainly doesn't satisfy the epimorphism property when $\mathcal{M}$ is say the moduli stack of elliptic curves, and $Spec K\rightrightarrows\mathcal{M}$ is given by two nonisomorphic curves over $K$ which become isomorphic over $L$... – stupid_question_bot Oct 19 '16 at 2:44
• You're probably right. I tend to think of schemes over $\mathbb{C}$ only! – Qfwfq Oct 20 '16 at 17:06