I am a beginner in derived algebraic geometry and I am trying to develop some visual and geometrical intuition about derived schemes (and stacks), or more precisely about the new geometrical phenomena that they introduce.

It seems to me that (broadly speaking) the new spaces that derived geometry gives rise to are:

  1. (Possibly higher dimensional) Loop spaces. They arise as self-intersections: e.g. see comments of J.Pridham and the answer of DamienC (1) below.
  2. Derived infinitesimal disks/Formal neighbourhoods. Originated by nilpotent extensions. See for example the definition 1.1 in Vezzosi - A note on the cotangent complex in derived algebraic geometry.
  3. QUESTION: What else? (See also an answer of DamienC below (2)). I think that while 1 and 2 are already present in derived schemes other phenomena require derived stacks.

I would like to see more examples that have some geometrical interpretation. There are cases of derived stacks for example in Toen - Higher and derived stacks: A global overview. Such examples include the derived stack of rank $n$ local systems over some topological space (and the derived moduli stack of vector bundles), derived linear stacks, and the derived stack of perfect complexes. However I am unable to obtain a geometrical meaning for this examples.

EDIT: What is the geometrical interpretation of the higher homology groups in (for example) the derived stack of vector bundles over a projective variety $\mathbb{R} \underline{{Vect}_{n}}(X)$?

According to this paper from Toen-Vezzosi some of the motivation for this derived stack comes from the will to build a smooth moduli space (unlike the underived case). When $X=S$, a smooth projective surface, they claim that the tangent space at a point $E$ is: $T_{E} \mathbb{R} \underline{\operatorname{Vect}}_{n}(S) \simeq-H^{2}(S, \underline{E n d}(E))+H^{1}(S, \underline{E n d}(E))-H^{0}(S, \underline{E n d}(E))$. However here the $H^{2}$ term (which is the derived part) seems to come from the fact of $S$ is $2$-dimensional and not from any singularity or self-intersection (which seems strange to me)

If you look at the example of (2), which is quite similar (I think it is the derived stack of local systems), the $H^{2}$ term appears when you take into account the self intersection of $0$ in $\mathbb{A}^{1}$, (i.e. a truly derived structure).

What I am misunderstanding here?

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    $\begingroup$ The difficulty in finding a visualisation is that whereas underived nilpotents come from truncating schemes with more points, the derived structure is entirely formal. One example you might like to mull is the derived fibre product $\{0\}\times^h_{\mathbb{A}^1}\{0\} = \mathrm{Spec } k[x_1]$, for $x_1$ in chain degree $1$ (cochain degree $-1$). The tangent space at the closed point is $k[-1]$, which you should think of as the homotopy kernel of $0 \to k$. $\endgroup$ – Jon Pridham Jan 21 '20 at 18:17
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    $\begingroup$ Thank you for the example! Is that the loop space at $0$, right? I am writing an answer/reflection that is related to this but is too long so I will add it in the main question. $\endgroup$ – Chequez Mar 8 '20 at 20:56
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    $\begingroup$ Yes, in a sense that's the loop space at $0$. $\endgroup$ – Jon Pridham Mar 8 '20 at 21:09
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    $\begingroup$ I edited in a link to @‍DamienC's answers, but saw a comment, not an answer, from @JonPridham. I guessed that that link was the right one, and so edited it in, too. I hope that was correct. $\endgroup$ – LSpice Dec 2 '20 at 18:49
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    $\begingroup$ On a side note, for a later occasion, I think it'd be better to start a new question than significantly changing the question several times after answers have been given. The new question can refer to the previous one for background. But it seems to me that multiple editions makes the whole thread not easy to follow for people who may want to catch up. $\endgroup$ – DamienC Dec 15 '20 at 12:32

I'm not quite sure what kind of answer you're expecting, but here is a geometric example that may help to grasp some intuition.

In differential geometry, when an intersection is badly behaved (e.g. it doesn't have the expected dimension) one can geometrically perturbe one of the two factors. For instance, if you are intersecting tow submanifolds $X,Y\subset Z$, and if $X$ is locally given as the zero of some functions $X\overset{\text{loc}}{=}\{f_1=\dotsb=f_k=0\}$, you may want to introduce a deformation $X_{t_1,\dotsc,t_k}$ of $X$ defined as $$ X_{t_1,\dotsc,t_k}\overset{\text{loc}}{=}\{f_1=t_1,\dotsc,f_k=t_k\}. $$ One of the main idea of derived geometry is to replace these deformation/perturbation parameters $t_i$'s by a homological perturbation: $$ X_{t_1,\dotsc,t_k}\overset{\text{loc}}{=}\{f_1=d\tau_1,\dotsc,f_k=d\tau_k\}, $$ with $\operatorname{deg}(\tau_i)=-1$ (my degree convention is cohomological).

Homological perturbation has two advantages above geometric perturbations:

  • it can be made functorial.

  • it exists even in the (quite non-flexible) algebraic setting, where geometric perturbation may not exist.

Let's try to apply informally the above reasoning to the case discussed in Jon Pridham's comment: consider $X=Y=\{x=0\}$ inside $Z=\mathbb{A}^1=\operatorname{Spec}(k[x])$. You deform $\{x=0\}$ to $\{x=d\tau\}$ and then proceed with the intersection of $\{x=d\tau\}$ with $\{x=0\}$, and get $\{d\tau=0\}$, which is just a ($k$-)point (it is $0$ in $\mathbb{A}^1$) together with a self-homotopy (given by $\tau$). This is indeed the "space" of derived loops in the affine line that are based at $0$.

I apologize for self-promoting, but you can read an informal account of how to view derived self-intersections as some kind of based loop spaces in the introduction of Calaque, Căldăraru, and Tu - On the Lie algebroid of a derived self-intersection.

  • $\begingroup$ Thank you for the answer. It is quite concrete an clarifying. The intro of the paper is useful too. However, I am unsure about higher homotopical groups (i.e. $deg(\tau_{i})<-2$). In which situations you can have higher (or abritary high?) homotopical perturbations (you dont need higher dimensional singularities for that, right?)? On the other side if are all the derived nilpotents (of any derived scheme/stack) of homotopical nature? If so, the relevant geometry/topology would be homotopy classes (i.e. maps from the topological space of our scheme/stack to some n-spheres, ) $\endgroup$ – Chequez Mar 12 '20 at 1:56
  • $\begingroup$ Also, thinking about my last sentence it seems to me that broadly speaking the new spaces that derived geometry gives rise to are self intersections spaces (i.e. formal loop and higher dimensional formal loop spaces) and "derived thickenings" by nilpotent extensions (yielding formal disks/higher dimensional formal disks). Does this make any sense? $\endgroup$ – Chequez Mar 12 '20 at 2:04
  • $\begingroup$ In my first comment I meant "$deg(\tau_{i})\leq2$ " and "On the other side, are all the derived nilpotents (in any derived scheme/stack) of homotopical nature?" $\endgroup$ – Chequez Mar 12 '20 at 2:17

$\DeclareMathOperator\Map{Map}\DeclareMathOperator\ad{ad}$This is an attempt to answer the third question: what else?

Let $X$ be a compact space and let $G$ be an affine algebraic group. One can contemplate the following (underived) higher stacks:

  • $BG$: the classifying stack of $G$-torsors.

  • $X_B$: the constant stack associated to $X$.

One can consider the higher underived mapping stack $\Map(X_B,BG)$, which is nothing but the ordinary (ie non-derived) stack of $G$-local systems on $X$. Its tangent complex has amplitude $[-1,0]$:

  • in degree $-1$, at a $k$-point $P$ ($P$ is a $G$-local system), its cohomology is $H^0(X,\ad(P))$, where $\ad(P)$ is the linear local system associated with $P$ and the adjoint $G$-representation $\mathfrak{g}$: $\ad(P)=P\times_G\mathfrak{g}$.

  • in degree $0$, at a $k$-point $P$, its cohomology is $H^1(X,\ad(P))$.

The infinitesimal theory $\Map(X_B,BG)$ doesn't capture anything about higher cohomology groups $H^{*\geq 2}(X,\ad(P))$.

If you're looking at the derived mapping stack $\mathbb{R}{\Map}(X_B,BG)$ instead, then its tangent complex at a $k$-point $P$ is the full de Rham cohomology $H^{*+1}(X,\ad(P))$.

Why is this so? The point is that the underived stack $\Map(X_B,BG)$ doesn't see anything else than the fundamental groupoid of $X$: this is because $BG$ is a $1$-truncated Artin stack. But if we allow families of $G$-local systems parametrized by geometric objects intrinsically carrying homotopical information (affine derived schemes), then we get back the missing information. For instance, it's a good exercise to check that if $Y$ is the derived self-intersection of $0$ in the affine line that was mentionned in previous answers (i.e. $Y=\operatorname{Spec}(k[\tau])$, $\operatorname{deg}(\tau)=-1$), then a $Y$-point in $\mathbb{R}{\Map}(X_B,BG)$ is the datum of a $k$-point $P$ and a class in $H^2(X,\ad(P))$.

  • $\begingroup$ Did you really mean $\mathbb R{\operatorname{Map}}$ for the derived functor? I thought it was usually denoted $\mathbf R{\operatorname{Map}}$. $\endgroup$ – LSpice Dec 2 '20 at 19:34
  • $\begingroup$ Thanks for your answer Damien, I am still a bit unsure about how to intepret geometrically the class in $H^{2}(X,ad(P))$ (in the example of $Y=$self-intersection of $0$ in $\mathbb{A}^{1}$). While the $H^{0}$ and $H^{1}$ should correspond to the usual automorphisms and deformations of the underived stack I guess that the $H^{2}$ should take into account the derived deformations (coming from the deformations of the derived loop space at $0$?). However $H^{2}(X,ad(P))$ should correspond to a class of $2$-cycles/holes (dependent on sections of a bundle/local system/whatever) but (...) $\endgroup$ – Chequez Dec 7 '20 at 7:43
  • $\begingroup$ seems strange to me that that set of $2$-dimensional objects parametrizes the deformations of the derived $1$-dimensional loop space. Does higher dimensional loop spaces work in the same way (i.e. if $Y$ was the self int. of 0 in $\mathrm{A}^{n}$ you would need classes in $\left.H^{i}(X, a d(P)), i=2, \ldots, n+1\right)$? What happens if $Y$ is a more complicated object, like the derived self-intersection of a closed subvariety (say of $\operatorname{dim}=m$ ) of some other variety (say of $\operatorname{dim}=l$ )? We can know something about the $H^{i}$ 's needed in that case? $\endgroup$ – Chequez Dec 7 '20 at 7:46
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    $\begingroup$ If you replace $S$ with a closed topological surface, and the stack of vector bundles with the one of local systems, things fit nicely together. Say $S$ is the gluing (push-out) of two surfaces $S_{\pm}$ with same boundary boundary $C$. Then $Loc(S)$ is equivalent to the derived intersectin of $Loc(S_+)$ and $Loc(S_-)$ in $Loc(C)$. $\endgroup$ – DamienC Dec 7 '20 at 16:57
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    $\begingroup$ To answer the comments that starts as "seems strange to me". You'd rather want to look at $Y_n$-points for $Y_n=Spec(k[x_n])$ with $x_n$ sitting in degree $-n$. They'll give you a $G$-local system $P$ and a class in $H^{n+1}(X,ad(P))$. The degree is not related to the codimension, but rather to how many times you self-intersect $0$: $Y_1$ is the self-intersection of $0$ in the affine line $Y_0$, $Y_2$ is the self intesection of $0$ in $Y_1$, ... $Y_{n+1}$ is the self intesection of $0$ in $Y_n$. $\endgroup$ – DamienC Dec 7 '20 at 17:59

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