nLab derived loop space



Higher geometry

Mapping space



A derived loop space is a free loop space object in derived geometry.


Let TT be an (∞,1)-algebraic theory and CTAlg opC \subset T Alg_\infty^{op} an (∞,1)-site of formal duals of \infty-algebras over TT. Then the (∞,1)-topos H=(,1)Sh(C)\mathbf{H} = (\infty,1) Sh(C) encodes derived geometry modeled on TT.

A derived loop space is a free loop space object in such H\mathbf{H}.

More specifically, if TT is an ordinary Lawvere theory, regarded as a 1-truncated (,1)(\infty,1)-theory, then TAlg T Alg_{\infty} are its simplicial algebras. There is a canonical embedding TAlg opTAlg opT Alg^{op} \hookrightarrow T Alg_\infty^{op} of the ordinary algebras into the \infty-algebras, so that we may regard XTAlg opX \in T Alg^{op} as an object of H\mathbf{H}. Then the derived loop space of XX is its free loop space object computed in H\mathbf{H}.

The point is that the derived loop space of an ordinary XTAlg opX \in T Alg^{op} in general is a significantly richer object than the free loop space object of XX as computed just in the underived (,1)(\infty,1)-topos (,1)Sh(TAlg op)(\infty,1)Sh(T Alg^{op}). In fact, since XX is 0-truncated in (,1)Sh(TAlg op)(\infty,1)Sh(T Alg^{op}), it coincides with its free loop space object there, but the derived loop space does not.

Function complexes on derived loop spaces: Hochschild homology

The function complex on the derived loop space X\mathcal{L}X is the Hochschild homology complex of C(X)C(X). See there for further details. In particular see the section Hochschild cohomology – As function algebra on the derived loop space.

Also see free loop space object for more information.


The relevance of derived loop spaces was amplified in a series of articles by David Ben-Zvi and David Nadler,

Last revised on June 26, 2021 at 11:39:20. See the history of this page for a list of all contributions to it.