I am thinking about higher Artin stacks in the sense of Simpson, concretely I would like to calculate the dimension and compare these two cases:
- $\mathfrak{X}_{1}=$ Higher linear stack classifying (over some affine scheme $X$) positively graded cochain complexes $E$ of length $2$ of vector spaces (being $E$ perfect on $X$)
- $\mathfrak{X}_{2}=$ Moduli stack of $2$-vector bundles over some affine scheme $X$
Let us assume for simplicity (smoothness) that $X$ is a reduced curve.
The tricky thing here is that the categorification of a vector space into a $2$-vector space is somewhat ambiguous: they exist two main versions: Kapranov–Voevodsky and Baez–Crans 2-vector spaces
Baez-Crans $2$-vector space is said to be equivalent to a $2$-term chain complex of vector spaces $E:V_{2}\overset{d_{2}}{\rightarrow} V_{1}$, so we would be in the case $\mathfrak{X}_{1}$.
QUESTION 1.1: In a $2$-stack $\mathfrak{X}_{1}$ of this form what would be the $1$-(iso)morphims and the $2$-(iso)morphisms? Take two complexes $E_{a}:V_{2a}\overset{d_{2a}}{\rightarrow} V_{1a}$ and ake $E_{b}:V_{2b}\overset{d_{2b}}{\rightarrow} V_{1b}$. I guess the $1$-morphisms would be the same that in $1$-stacks (linear transformtions applied on each $V_{i\alpha}$) and the $2$-morphisms are the chain maps $f:E_{a}{\rightarrow}E_{b}$, right?
QUESTION 1.2: What is the stacky dimension of $\mathfrak{X}_{1}$ at a point $x$ in such a case? I expect it to be something like $\dim_{x}(\mathfrak{X}_{1})=\dim_{x}(H^{-1}(X,End(E)))+\dim_{x}(H^{0}(X,End(E)))+\dim_{x}(H^{1}(X,End(E)))$ where the first two terms are related to $V_{2}$ and $V_{1}$ respectively, But I am not sure about this expression and specially about which set of automorphisms and/or deformations count each term.
The Kapranov-Voevodsky case would be somehow a wilder space (I expect its stacky dimension to be bigger). For a quite explicit formulation of it I have followed this paper. A $2$-vector space $Vect^{k}$ is there a category having $k$-tuples of vector spaces as objects and $k$-tuples of linear maps as morphisms. As we are interested in the moduli $2$-stack $\mathfrak{X}_{2}$ we have also to take account of $2$-morphims $T$ : $Vect^{k} \rightarrow$ $Vect^{l}$ which are $k \times l$ matrices of vector spaces,defined as $$ \left(\begin{array}{ccc} T_{1,1} & \ldots & T_{1, k} \\ \vdots & & \vdots \\ T_{l, 1} & \ldots & T_{l, k} \end{array}\right)\left(\begin{array}{c} V_{1} \\ \vdots \\ V_{k} \end{array}\right)=\left(\begin{array}{c} \oplus_{i=1}^{k} T_{1, i} \otimes V_{i} \\ \vdots \\ \oplus_{i=1}^{k} T_{l, i} \otimes V_{i} \end{array}\right) $$
For simplicity we can choose all the $V_{1,...,k}$ equdimensional, say of dimension $n$. It is easy to see that the right hand side of that equation is a $l$-tuple of ($k \times n$)-dimensional vector spaces (not just $n$) so for a sensible $2$-stack we should fix both $n$ (dimension of the $1$-vector spaces involved), and $k$ and $l$ (how long are the tuples of vector spaces that we combine in a $2$-morphism) whereas in a usual $1$-stack we would fix $n$.
QUESTION 2: What is the stacky dimension $\mathfrak{X}_{2}$ at a point $x$ in such a case? I expect that the part coming from the automorphims of the $2$-vector space to be $\left (k\times n \right )\times \left (l\times n \right )$-dimensional but I am unsure about the form of the formula in terms of the homology groups of the cotangent complex.
