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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)?

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    $\begingroup$ 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? $\endgroup$ – Yemon Choi May 23 '19 at 13:15
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    $\begingroup$ For your final question, see HAG2 Prop 1.3.4.2, but beware that indexing conventions differ slightly between different arXiv versions. $\endgroup$ – Jon Pridham May 23 '19 at 13:51
  • $\begingroup$ In the title, you should spell out DAG, which also stands for (the much more common use of DAG) Directed Acyclic Graph). $\endgroup$ – Mark L. Stone May 23 '19 at 22:11
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    $\begingroup$ @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. $\endgroup$ – user140765 Jun 2 '19 at 13:16
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Have a look at HAG II

There, the definition of an Artin n-stack is the following:

Definition 2.1.1.3.An Artin n-stack is an n-truncated stack which is m-geometric for some integer m.

I think the question is what it means to be m-geometric. There is a definition that goes through atlases (see Definition 1.3.3.1 in HAG II), and there is another characterization using Smooth Segal groupoids (see Proposition 1.3.4.2 in HAG II, which basically answers your question: Artin means exactly the same thing in the two texts, and means that it is m-geometric for some m).

There's also a non-inductive characterization of m-geometricity via smooth m-hypergroupoids (see e.g. this note of Pridham).

One can then rather easily show that every m-geometric stack is m-truncated.

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