Why You Should Learn About Factorization Homology

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

1 - Actions of the Mapping Class Group

Take $(\mathcal{S},\otimes)$ to be a target symmetric 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 $\Sigma_g$. Note that this corresponds to the automorphisms of $\Sigma_g$ in $\mathsf{M}\mathsf{fld}_{2}^{\mathsf{or}}$, and therefore factorization homology furnishes a map

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

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

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

and we call it "formal singular chains with coefficients in $V$". Consider the $\otimes$-excision axiom:

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

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

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

. This is an injective map, and so all put together we get a short exact sequence of chain complexes

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

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 $R$-algebra 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 $\mathbb{E}_n$-algebra for all $n$.

2. The Hochschild cochain complex of any associative algebra (or dg algebra, or $A_{\infty}$-algebra) is an example 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. $\mathbb{E}_n$-algebras 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 $\mathfrak{g}$).