Remark 4.7 (0N51) · ScalingStacks

0N51

Remark 4.7. The factorization homology $\int_{M}\Omega^{n}_{B}X$ is built from configuration spaces of disks in $M$ with labels defined by $X\to B$ , and the preceding result thereby has roots in the configuration space models of mapping spaces dating to the work of Segal, May, McDuff and others in the 1970s, see [Se1], [Ma], [Mc], and [Bö]. Factorization homology is not a generalization of the classical configuration spaces with labels, as described in [Bö], because the configuration space with labels in $X$ models a mapping space with target the $n$ -fold suspension of $X$ , rather than into $X$ itself. Instead, factorization homology generalizes the configuration spaces with summable or amalgamated labels of Salvatore [Sa] and Segal [Se3].

0N52

Proof of Theorem 4.4. Corollary 4.6 identifies the functor $\Gamma_{\sf c}\colon\spaces_{B}^{\geq n}\to\mathbf{H}(\mfld_{n}^{B},\Space)$ as the composition $\Gamma_{\sf c}\colon\spaces_{B}^{\geq n}\xrightarrow{\Omega^{n}_{B}}\disk_{n}^{B}\xrightarrow{\int}\mathbf{H}(\mfld_{n}^{B},\Space)$ . Theorem 3.24 gives that $\int$ is fully faithful, so it remains to argue that $\Omega_{B}^{n}$ is fully faithful with essential image the group-like $\disk_{n}^{B}$ -algebras in spaces. This is immediate because, for instance, $\spaces_{/B}$ is an $\infty$ -topos (Theorem 5.1.3.6 of [Lu2]).

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