Let me expand on some additional curiosities, examples and obsevations about Factorization Homology:

1 - Actions of the Mapping Class Group

Take \((\mathcal{S},\otimes)\) to be a target symmetrix monoidal \((\infty,1)\)-category, and consider \(A\colon \mathcal{D}\mathsf{isk}_2^{\mathsf{or}}\rightarrow \mathcal{S}\) to be an (oriented) \(\mathbb{E}_2\)-algebra in \(\mathcal{S}\). Then consider, for any genus \(g\) oriented surface \(\Sigma_{g}\), the Lie group \(\mathsf{Diff}^{+}(\Sigma_g)\), of orientation-preserving diffeomorphisms of \(\Sigma_g\). Not that this corresponds to the automorphisms of \(\Sigma_g\) in \(\mathsf{M}\mathsf{fld}_{2}^{\mathsf{or}}\), and therfore factorization homology furnishes a map \(\mathsf{Diff}^{+}(\Sigma_g)=\mathsf{hom}_{\mathcal{M}\mathsf{fld}_2^{\mathsf{or}}}(\Sigma_g,\Sigma_g) \rightarrow \mathsf{hom}_{\mathcal{S}}\Big(\int_{\Sigma_{g}}A,\int_{\Sigma_{g}}A\Big)\) Taking connected components, we get an action of the mapping class group on \(\pi_0\) of the object on the right hand side, and in the case that \(\mathcal{S}^{\otimes}=\mathsf{LinCat}_{\mathbb{k}}^{\times}\), this recovers well-known actions on certain invariants of quantum groups.

2 - Pushforward Property

Factorization homology enjoys a Fubini-like property. This is often called the pushforward property. In its “simpler” guise it states that, for manifolds \(M^{m},N^{n}\) (with superscript indicating dimension), we get the following equivalence

\[\int_{M\times N}A \simeq \int_{M}\bigg(\int_{N\times \mathbb{R}^m}A\bigg)\]

To make sense of it, one should note that \(\int_{N\times \mathbb{R}^m}A\) is an \(\mathbb{E}_m\)-algebra, much like we saw that \(\int_{M\times \mathbb{R}} A\) was an \(\mathbb{E}_1\)-algebra. In its fancier, and evidently more general form, it says that given a map \(f\colon M^m \rightarrow N^n\) which fibers over the interior and boundary of \(N\), and for \(A\) an \(\mathbb{E}_m\)-algebra, we get an equivalence

\[\int_{N}f_{\ast}A \simeq \int_{M}A.\]

Where \((f_\ast A)(U):=\int_{f^{-1}(U)}A\). To recover the Fubini-looking identity we choose \(\pi_{M}\colon M\times N \rightarrow M\), in this case \(f^{-1}(\mathbb{D}^m)=\mathbb{D}^m\times N\), as expected.

3 - Factorization homology as ordinary homology

Factorization homology, in its version using \(\mathsf{Ch}_{\mathbb{Z}}^{\oplus}\) as a target symmetric monoidal category, recovers “usual” homology of CW-complexes à la Eilenberg-Steenrod. Note that in this case \(\oplus\) corresponds to the coproduct in the category, and as such forces any \(\mathbb{E}_n\)-algebra \(V\oplus V \rightarrow V\) to be simple addition of chains.

We define the following chain complex

\[\mathsf{C}_\ast(-;V):=\int_{-}V\]

and we call it “formal singular chains with coefficients in \(V\)”. Consider the \(\otimes\)-excision axiom:

\[\mathsf{C}_\ast(X_{1}\underset{X_1 \cap X_2}{\sqcup} X_2;V)\simeq \mathsf{C}_\ast(X_1;V)\underset{\mathsf{C}_\ast (X_1 \cap X_2;V)}{\bigotimes} \mathsf{C}_\ast(X_2;V)\]

In this case, the module structure has to be also given by addition i.e. this is the right \(\mathsf{C}_\ast(X_1\cap X_2;V)\)-module structure map

\[\mathsf{C}_\ast(X_1;V)\oplus \mathsf{C}_\ast(X_1\cap X_2;V)\xrightarrow{(-+-)} \mathsf{C}_\ast(X_1;V).\]

This means that the relative tensor product of modules is given by the coequalizer

\[\mathsf{coeq}\Big(\mathsf{C}_\ast(X_1;V)\oplus \mathsf{C}_\ast(X_1\cap X_2;V) \oplus \mathsf{C}_\ast(X_2;V) \substack{\longrightarrow\\[-1em] \longrightarrow \\} \mathsf{C}_\ast(X_1;V)\oplus \mathsf{C}_\ast(X_2;V)\Big)\]

where the top arrow is the right module structure map, and the bottom one the left-module structure map. We can recast in more familiar terms as the cokernel

\(\mathsf{coker}\Big(\mathsf{C}_\ast(X_1\cap X_2;V)\xrightarrow{(x,-x)} \mathsf{C}_\ast(X_1;V)\oplus \mathsf{C}_\ast(X_2;V)\Big)\). This is an injective map, and so all put together we get a short exact sequence of chain complexes

\[0 \rightarrow \mathsf{C}_\ast(X_1\cap X_2;V) \xrightarrow{(x,-x)} \mathsf{C}_\ast(X_1;V)\oplus \mathsf{C}_\ast(X_2;V) \xrightarrow \mathsf{C}_\ast(X_1\cup X_2;V)\xrightarrow 0\]

and this can be seen as the Mayer-Vietoris sequence which “universally” characterizes singular homology. From this, we can then conclude that

\[\int_X V\]

are actual singular chains in \(X\) with coefficients in \(V\), and \(\otimes\)-excision translates to Mayer-Vietoris.

4 - Where to look for \(\mathbb{E}_n\)-algebras?

  1. Fix a base ring \(R\), letting the target symmetric monoidal \((\infty,1)\)-category be \(\mathcal{C}\mathsf{hain}_{R}^{\otimes_{R}^{\mathbb{L}}}\) whose objects are chain complexes of \(R\)-modules, and whose symmetric monoidal structure is given by the derived tensor product \(\otimes_R ^{\mathbb{L}}\). A commutative \(R\)-algebrea is an \(\mathbb{E}_n\)-algebra in \(\mathcal{C}\mathsf{hain}_{R}^{\otimes_{R}^{\mathbb{L}}}\) for any \(n\). More generally, a cdga (a commutative dg algebra) is an \(\mathbb{E}_n\)-alegbra for all \(n\).

  2. The Hochschild cochain complex of any associative algebra (or dg algebrea, or \(A_{\infty}\)-algebra) is an exmaple of an \(\mathbb{E}_2\)-algebra in \(\mathcal{C}\mathsf{hain}_{\mathbb{k}}^{\otimes_\mathbb{k}}\).

  3. \(n\)-fold loop spaces are \(\mathbb{E}_n\)-algebras in \(\mathcal{T}\mathsf{op}^{\times}.\)

  4. The free \(\mathbb{E}_n\)-algebra on one generator (in \(\mathcal{T}\mathsf{op}^{\times}\)) is \(\mathsf{Conf}(\mathbb{R}^n)\) - the unordered configuration space of \(\mathbb{R}^n\).

  5. \(\mathbb{E}_n\)-algberas are also often constructed as deformations of commutative or cocommutative objects. The braided monoidal category of representations of the quantum group \(\mathsf{U}_q(\mathfrak{g})\) is obtained by deforming a cocommutative Hopf algebra (the universal enveloping algebra of \(\mathfrak{g}\)).

References:

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