# When is a substack closed?

For this question we will consider the Zariski site of affine schemes and a stack $\mathcal{M}$ over it. I don't know what a substack is, but I have a guess. The stack $\mathcal{M}$ has an underlying path-component sheaf $\pi_0\mathcal{M}$, and we have a map $\mathcal{M}\rightarrow \pi_0\mathcal{M}$. A stack $\mathcal{N}\rightarrow \mathcal{M}$ is substack if the induced map of sheaves $\pi_0\mathcal{N}\rightarrow\pi_0\mathcal{M}$ is an inclusion, and the resulting square is $2$-categorical pullback. I am not exactly sure this definition is right, but at least the notion is "homotopic". So my first question is:

$1$. Is this a definition of a substack?

I also have a guess of what a closed substack is supposed to be (of course, assuming that the definition of a substack is more or less what I wrote above). We can first define it for sheaves. A map of sheaves $F\rightarrow G$ is a closed immersion of sheaves, if any pullback along a map $\mathrm{Spec}(R)\rightarrow G$ is a map of form $\mathrm{Spec}(R/I)\rightarrow \mathrm{Spec}(R)$. For the case of stacks, we say $\mathcal{N}\rightarrow\mathcal{M}$ is a closed immersion if it is a substack, such that the induced map $\pi_0\mathcal{N}\rightarrow\pi_0\mathcal{M}$ is a closed immersion. My second question:

$2$. Is this the correct definition of a closed substack?

• A substack $Y \subset X$ is a stack $Y$ equipped with a map to $X$ such that for every affine $A$, $Y(A) \to X(A)$ is the inclusion of a full, replete subcategory (replete: every object isomorphic to one in the image is in the image). At least for topological stacks, the property of a map being a closed embedding is local on the target and invariant under base change, so we can define a representable map of stacks to be closed if it is so locally. – David Roberts Feb 27 '13 at 23:25

## 2 Answers

No need to guess, just look it up. http://ens.math.univ-montp2.fr/~toen/cours8.pdf Definition 1.1.

Here is a definition in terms of presentation of the stack: Definition 1.5 in Angelo Vistoli's paper "Intersection theory on algebraic stacks and on their moduli spaces".