ScalingStacks

Even if we are interested in studying primarily schemes, the world of derived stacks is a necessary setting in which to calculate homotopically correct quotients, fiber products and mapping spaces. We illustrate this with an important geometric operation on derived stacks: the formation of the derived loop space. This operation is one of our motivations for considering derived stacks in the first place, since the derived loop space of an ordinary scheme or stack is already a nontrivial derived stack. (See [BN1] for a different appearance of derived loop spaces in relation to cyclic homology, ${\mathcal{D}}$-modules and representation theory.)

The free loop space of a derived stack $X$ is the internal hom $\mathcal{L}X=X^{S^{1}}=\Map(S^{1},X)$ of maps from the constant stack given by the circle $S^{1}$. As a derived stack, the loop space may be described explicitly as the collection of pairs of points in $X$ with two paths between them, or in other words, as the derived self-intersection of the diagonal

Let us illustrate this notion in a few examples.

For $X$ a topological space (constant stack), $\mathcal{L}X$ is of course the free loop space of $X$ (again considered as a constant stack).

For $X=BG$ the classifying space of an algebraic group, $\mathcal{L}X=G/G$ is the adjoint quotient of $G$, or in other words, the adjoint group for the universal bundle $EG={\rm pt}\to BG$. Note that this agrees with the underived inertia stack of $BG$.

For $X$ a smooth variety over a field of characteristic zero, the derived self-intersection of the diagonal can be calculated by a Koszul complex to be the spectrum of the complex of differential forms on $X$ (with zero differential) placed in homological degrees. We thus obtain that the loop space of a smooth variety $X$ can be identified with the relative spectrum of the symmetric algebra of the shifted tangent bundle $T_{X}[-1]$.

Similarly, for any $n\geq 0$, we may consider the derived stack of maps $X^{S^{n}}=\Map(S^{n},X)$ from the $n$-sphere $S^{n}$ into $X$. Concretely, the presentation of $S^{n}$ by two cells glued along $S^{n-1}$ leads to an iterated description of $X^{S^{n}}$ as the self-intersection

along two copies of constant maps over $X^{S^{n-1}}$.

More generally, for any topological space $\Sigma$ and stack $X$ we can consider the derived mapping stack $X^{\Sigma}=\Map(\Sigma,X)$. As we will demonstrate, this construction suggests the world of derived algebraic geometry is a natural setting to construct topological $\sigma$-models.