Subsection (0N5R) · ScalingStacks

Notation

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$\spaces$ is the $\infty$ -category of spaces. This $\infty$ -category has numerous constructions and characterizations: as the ${\sf Kan}$ -enriched category of Kan complexes; as the free small colimit completion of the terminal $\infty$ -category $\ast$ ; and as $\infty$ -groupoids.
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$\ddisk_{n}$ and $\dmfld_{n}$ are ordinary categories of manifolds with embeddings; $\disk_{n}$ and $\mfld_{n}$ are topological categories of manifolds with embeddings, where the spaces of embeddings carry the compact-open topology. See Definition 2.18 versus Definition 2.1.
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$\Top(n)$ is the topological group of homeomorphisms of $\mathbb{R}^{n}$ , endowed with the compact-open topology.
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After Definition 2.7, we fix a space $B$ with a map $B\rightarrow\BTop(n)$ and consider $B$ -framed $n$ -manifolds. Any occurrence of $B$ thereafter refers to this choice.
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We use $\colim$ and $\limit$ to denote colimits and limits in $\oo$ -categories, which correspond to homotopy colimits and limits in topological categories or model categories. (In the one or two places where we use a point-set colimit, we employ unmistakeably jarring notation to distinguish the two.)
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For $k$ a ring we use the notation ${\sf Mod}_{k}$ for the $\infty$ -category of $k$ -modules. This is an $\infty$ -category associated to the differential graded category of chain complexes over $k$ . (See §1.3 of [Lu2] for a thorough account.) We will sometimes use the notation ${\sf Ch}_{k}$ for this $\infty$ -category, and should $k$ be the integers we drop it from the subscript.
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$\mathcal{E}_{n}$ will stand for the topological operad of little $n$ -cubes, as defined in [BV].
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$\mathsf{C}_{\ast}(X)$ is the singular chains on a topological space $X$ .
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${\sf Sym}(V)$ is the free commutative algebra on an object $V$ of a symmetric monoidal $\infty$ -category. In a symmetric monoidal $\oo$ -category, commutative algebras are equivalent to $\mathcal{E}_{\oo}$ -algebras, so ${\sf Sym}(V)$ is also the free $\mathcal{E}_{\oo}$ -algebra on $V$ .
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For $X\in\mathcal{X}$ an object of an $\infty$ -category, we notate $\mathcal{X}_{/X}$ and $\mathcal{X}^{X/}$ for the over- and under- $\infty$ -categories. For $(A\to X)$ and $(B\to X)$ two objects of $\mathcal{X}_{/X}$ , we may denote the space of morphisms from the first to the second as $\Map_{/X}(A,B)$ .

Acknowledgments 1.3. JF foremost thanks Kevin Costello for many conversations on this subject, which have motivated and clarified this work, from Theorem 3.24 to the computations of the following sections, and without which JF likely would not have pursued it. Our joint works with Hiro Lee Tanaka build and improve on many of the ideas here, and we thank him for his collaboration. JF first learned the basic idea of factorization homology in conversations with Jacob Lurie and Dennis Gaitsgory in 2007, and we have both benefitted greatly from their generosity in sharing many other insights in these intervening years. JF thanks Sasha Beilinson and Mike Hopkins for their great influence which has shaped his thoughts on this subject. We also thank Grégory Ginot and Owen Gwilliam for helpful conversations and Pranav Pandit for comments on an earlier draft of this paper. We thank Amabel Wilson and Theo Johnson–Freyd for correcting the hypotheses in the statement of Proposition 5.3. Finally, we thank the anonymous referee whose careful feedback has considerably improved this article.

David Ayala, John Francis

Original source: arXiv:1206.5522v6