higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
A derived loop space is a free loop space object in derived geometry.
Let $T$ be an (∞,1)-algebraic theory and $C \subset T Alg_\infty^{op}$ an (∞,1)-site of formal duals of $\infty$-algebras over $T$. Then the (∞,1)-topos $\mathbf{H} = (\infty,1) Sh(C)$ encodes derived geometry modeled on $T$.
A derived loop space is a free loop space object in such $\mathbf{H}$.
More specifically, if $T$ is an ordinary Lawvere theory, regarded as a 1-truncated $(\infty,1)$-theory, then $T Alg_{\infty}$ are its simplicial algebras. There is a canonical embedding $T Alg^{op} \hookrightarrow T Alg_\infty^{op}$ of the ordinary algebras into the $\infty$-algebras, so that we may regard $X \in T Alg^{op}$ as an object of $\mathbf{H}$. Then the derived loop space of $X$ is its free loop space object computed in $\mathbf{H}$.
The point is that the derived loop space of an ordinary $X \in T Alg^{op}$ in general is a significantly richer object than the free loop space object of $X$ as computed just in the underived $(\inftym1)$-topos $(\infty,1)Sh(T Alg^{op})$. In fact, since $X$ is 0-truncated in $(\infty,1)Sh(T Alg^{op})$, its coincides with its free loop space object there. But the derived loop space does not.
The function complex on the derived loop space $\mathcal{L}X$ is the Hochschild homology complex of $C(X)$. See there for further details.
Also see free loop space object for more informaiton.
loop space object, free loop space object,
formal loop space, derived loop space
The relevance of derived loop spaces was amplified in a series of articles by David Ben-Zvi and David Nadler,
Loop Spaces and Langlands Parameters (arXiv:0706.0322)
Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry (arXiv:0805.0157)
Loop Spaces and Connections (arXiv:1002.3636)
This article uses Toën’s theory of function algebras on ∞-stacks for showing that the function complex on a derived loop space $\mathcal{L}X$ is under mild conditions the Hochschild homology complex of $X$ hence by Hochschild-Kostant-Rosenberg theorem the collection of Kähler differential forms on $X$, and that the functions on $\mathcal{L}X$ that are invariant under the canonical $S^1$-action on $\mathcal{L}X$ are the closed forms. This also gives a geometric interpretation of the old observation by Maxim Kontsevich and others, that the differential and grading on the de Rham complex may be understood as induced from automorphisms of the odd line.
Loop Spaces and Representations (arXiv:1004.5120)