**I am looking for a reference which develops the theory of the cotangent complex for complex analytic spaces.** I need this to justify some computations I did assuming some formal properties which hold in the algebraic category. **Here is a list of properties I want** (some might be deducible from others, I haven't checked this carefully):

(1) To each analytic space $X$, assign a well-defined object $\mathbb{L}^\bullet_X$ in the derived category of complexes of $\mathcal O_X$-modules with coherent cohomology sheaves, supported in degrees $\le 0$.

(2) For smooth $X$, we must have a (quasi-)isomorphism $\mathbb L^\bullet_X = \Omega^1_X[0]$, where the right hand side is the holomorphic cotangent sheaf in degree $0$. For each morphism $f:X\to Y$, we have a morphism in the derived category $f^\dagger:Lf^*\mathbb L^\bullet_Y\to\mathbb L^\bullet_X$ (which is the obvious one if $X,Y$ are smooth, i.e., the adjoint of $df$ between the cotangent spaces) and whose cone defines the relative cotangent complex $\mathbb L^\bullet_{X/Y}$ (or $\mathbb L^\bullet_f$), which is an object living on $X$. The assignment $f\mapsto f^\dagger$ respects composition of maps.

(3) For a closed immersion $X\subset Y$ with $Y$ smooth and $\mathscr I$ the ideal sheaf of $X$ in $Y$, we have $h^0(\mathbb L^\bullet_{X/Y})=0$ and $h^{-1}(\mathbb L_{X/Y}^\bullet) = \mathscr I/\mathscr I^2$. Further, the truncation $\tau_{\ge-1}\mathbb L^\bullet_X$ can be identified with $d:\mathscr I/\mathscr I^2\to\Omega^1_Y|_X$ (living in degrees $-1$ and $0$) in a way that is compatible with the exact triangle $Lf^*\mathbb L^\bullet_Y\to\mathbb L^\bullet_X\to\mathbb L^\bullet_{X/Y}\to Lf^*\mathbb L^\bullet_Y[1]$ coming from part (2).

(4) More generally, for any sequence of maps $X\to Y\to Z$, there's an exact sequence $Lf^*\mathbb L^\bullet_{Y/Z}\to\mathbb L^\bullet_{X/Z}\to\mathbb L^\bullet_{X/Y}\to Lf^*\mathbb L^\bullet_{Y/Z}[1]$ where $f:X\to Y$ is the first map in the above.

(5) Crucially, for a morphism $X\to Y$ of algebraic schemes, $\mathbb L^\bullet_{X/Y}$ should be functorially (quasi-)isomorphic to the analytification of the cotangent complex defined in the Stacks Project (tag 08T1).

**I'm actually happy if I just have a reference which treats the corresponding statements for just $\tau_{\ge-1}\mathbb L^\bullet_X$ (called the "Naive cotangent complex" in the Stacks Project). Also, if you can't find any reference which treats this, I'd also find useful a sketch of how to go about proving this in the analytic category.**

Apologies in advance if this is too basic a question, I'm rather new to these concepts.