# Confusion about DAG terminology

This question refers to higher and derived algebraic geometry as developed by Toen-Vezzosi, not by Lurie. I have seen two expository documents by Toen.

In the first text, there is a definition:

A derived stack is a derived Artin $$n$$-stack if it is an $$m$$-geometric derived stack for some $$m$$, and if $$t_0(F)$$ is an (Artin) $$n$$-stack. A derived Artin stack is an Artin $$n$$-stack for some $$n$$.

In the second text, there is a definition:

(1) A derived stack is a derived $$n$$-Artin stack if it is of the form $$|X_∗|$$ for some smooth groupoid of derived $$(n − 1)$$-Artin stacks $$X_∗$$.

(2) A morphism between derived $$n$$-Artin stacks $$f:X\rightarrow Y$$ is smooth if there exists smooth groupoid of derived $$(n − 1)$$-Artin stacks $$X_∗$$ and $$Y_∗$$, a morphism of groupoid objects $$f_∗:X_*\rightarrow Y_∗$$ with $$f_0:X_0\rightarrow Y_0$$ smooth, and such that $$|f_∗|$$ is equivalent to $$f$$.

A derived stack is a derived Artin stack if it is a derived $$n$$-Artin stack for some $$n$$.

In the first text there is also an admonishment:

The reader is warned that there is a small discrepancy for the indices in the notions of $$n$$-Artin stack and Artin $$n$$-stack. For example a scheme is always an Artin $$0$$-stack, but is only a $$1$$-Artin stack. It is a $$0$$-Artin stack if and only if its diagonal is an affine morphism...

Does the term "derived Artin stack" mean exactly the same thing in the first text and in the second text? If the answer is positive, please provide a detailed proof.

Is there any constraint on the two indices (e.g. given a derived Artin stack, minimal $$n$$ in the second definition is a bounded linearly in terms of minimal $$n$$ in the first definition)?

• This doesn't address your question, but given the different unregistered accounts you keep opening mathoverflow.net/users/140276/m-for-motive mathoverflow.net/users/140971/m-for-motive mathoverflow.net/users/140978/m-for-motive mathoverflow.net/users/140961/m-for-motive mathoverflow.net/users/140893/m-for-motive would it not make sense to register a single account? May 23, 2019 at 13:15
• For your final question, see HAG2 Prop 1.3.4.2, but beware that indexing conventions differ slightly between different arXiv versions. May 23, 2019 at 13:51
• In the title, you should spell out DAG, which also stands for (the much more common use of DAG) Directed Acyclic Graph). May 23, 2019 at 22:11
• @MarkL.Stone Interesting, never heard about Directed Acyclic Graphs before (and if I had to guess which one is more common I would bet on derived algebraic geometry). Guess people have different tastes.
– user140765
Jun 2, 2019 at 13:16