ScalingStacks

Proof. The functor $-\otimes A\colon\mfld_{n}^{B}\to\mathcal{V}$ carries each contractible manifold to the underlying object of the commutative algebra $A$. For $(A_{i})_{i\in I}$ a finite sequence of commutative algebras in $\mathcal{V}$, the $I$-fold coproduct in $\Alg_{\com}(\mathcal{V})$ is the pointwise tensor product $\underset{i\in I}{\bigotimes}A_{i}$ (see Proposition 3.2.4.7 of [Lu2]). It follows that this functor $-\otimes A$ is symmetric monoidal. From the defining expression of factorization homology as a colimit, there results a natural transformation

$$\int_{-}A\longrightarrow-\otimes A$$

between symmetric monoidal functors $\mfld_{n}^{B}\to\mathcal{V}$, which evaluates as an equivalence on objects of $\disk_{n}^{B}$. Lemma 3.18 grants that the domain of this natural transformation satisfies $\otimes$-excision. Because a collar-gluing $M^{\prime}\underset{M_{0}\times\mathbb{R}}{\bigcup}M^{\prime\prime}\cong M$ determines a pushout of underlying spaces $M^{\prime}\underset{M_{0}}{\coprod}M^{\prime\prime}\simeq M$, the codomain of this natural transformation too satisfies $\otimes$-excision. That the natural transformation evaluates on each $B$-framed $n$-manifold $M$ as an equivalence then follows by induction on a handle decomposition on $M$.

∎

Proof. The two sides are evidently equivalent in the case where $M$ is homeomorphic to $\mathbb{R}^{n}$, so to establish the result it suffices, as usual, to check by induction over a handle decomposition of $M$. Given a handle decomposition $N\underset{S^{k}\times\mathbb{R}^{n-k}}{\bigcup}\mathbb{R}^{n}\cong M$, we have a homotopy pullback diagram of spaces

(8)

which gives rise to a natural map in $R$-homology

$$\mathsf{C}_{\ast}(X^{M},R)\longrightarrow\mathsf{C}_{\ast}(X^{N},R)\underset{\mathsf{C}_{\ast}(X^{S^{k}\times\mathbb{R}^{n-k}},R)}{\bigotimes}\mathsf{C}_{\ast}(X^{\mathbb{R}^{n}},R)$$

from the homology of the mapping spaces to the cotensor product of the comodules $\mathsf{C}_{\ast}(X^{N},R)$ and $\mathsf{C}_{\ast}(X^{\mathbb{R}^{n}},R)$ over the coalgebra $\mathsf{C}_{\ast}(X^{S^{k}\times\mathbb{R}^{n-k}},R)$. This map is an equivalence exactly if the homological Eilenberg–Moore, or Rothenberg–Steenrod, spectral sequence for this homotopy Cartesian diagram converges. By Dwyer [Dw], the convergence of this Eilenberg–Moore spectral sequence is assured if the base $X^{S^{k}\times\mathbb{R}^{n-k}}$ is connected and the action

$$\pi_{1}\bigl(X^{S^{k}\times\mathbb{R}^{n-k}},f\bigr)\circlearrowright\pi_{\ast}\bigl({\sf fiber}_{f}(X^{\mathbb{R}^{n}}\rightarrow X^{S^{k}\times\mathbb{R}^{n-k}})\bigr)$$

is nilpotent for a choice of basepoint $f\in X^{S^{k}\times\mathbb{R}^{n-k}}$. Since $X$ is $n$-connective, for $k<n$ any map $f:S^{k}\rightarrow X$ is nullhomotopic, and therefore the base $X^{S^{k}\times\mathbb{R}^{n-k}}\simeq X^{S^{k}}$ is connected. We can thus take $f$ to be the constant map valued at the basepoint of $X$, and so identify ${\sf fiber}_{f}(X^{\mathbb{R}^{n}}\rightarrow X^{S^{k}\times\mathbb{R}^{n-k}})\simeq\Omega^{k+1}X$. We now show the action of $\pi_{1}\bigl(X^{S^{k}}\bigr)$ on $\pi_{\ast}\Omega^{k+1}X$ is nilpotent. Consider the fiber sequence $\Omega^{k+1}X\to X^{S^{k}}\xrightarrow{{\sf ev}_{\ast}}X$. This fibration admits a section, given by the constant maps. Consequently, there is an identification as a semi-direct product:

$$\pi_{1}\bigl(X^{S^{k}}\bigr)\cong\pi_{1}X\ltimes\pi_{1}\Omega^{k}X\cong\pi_{1}X\ltimes\pi_{0}\Omega^{k+1}X~.$$

Through this identification, the action of $\pi_{1}\bigl(X^{S^{k}}\bigr)$ on $\pi_{\ast}\Omega^{k+1}X$ is the unique action that extends the standard actions of $\pi_{1}X$ and of $\pi_{0}\Omega^{k+1}X$ on $\pi_{\ast}\Omega^{k+1}X$. By assumption, the action of $\pi_{1}X$ on $\pi_{\ast+k+1}X\cong\pi_{\ast}\Omega^{k+1}X$ is nilpotent. In the case that $k=0$, the same assumption grants that the action of $\pi_{0}\Omega^{k+1}X$ on $\pi_{\ast}\Omega^{k+1}X$ is nilpotent. In the case that $k>0$, the action of $\pi_{0}\Omega^{k+1}X$ on $\pi_{\ast}\Omega^{k+1}X$ is automatically nilpotent due to commutativity. Nilpotence of the action of $\pi_{1}\bigl(X^{S^{k}}\bigr)$ on $\pi_{\ast}\Omega^{k+1}X$ follows. Consequently, the natural map in $R$-homology above is an equivalence.

The remainder of this argument is checking that we have imposed sufficient finiteness conditions to ensure the convergence in cohomology as well as homology. Dualizing, we obtain an equivalence

$$\Bigl(\mathsf{C}_{\ast}(X^{N},R)\underset{\mathsf{C}_{\ast}(X^{S^{k}\times\mathbb{R}^{n-k}},R)}{\bigotimes}\mathsf{C}_{\ast}(X^{\mathbb{R}^{n}},R)\Bigr)^{\vee}\overset{\sim}{\longrightarrow}\mathsf{C}^{\ast}(X^{M},R)~.$$

Since $\pi_{n}X$ is finite, the mapping space $X^{K}$ has finitely many components for any $n$-dimensional finite CW complex $K$. Because the source spaces, $M$, $N$, $S^{k}\times\mathbb{R}^{n-k}$, and $\mathbb{R}^{n}$, all have have the homotopy types of finite $n$-dimensional CW complexes, we obtain that all these mapping spaces have finitely many components. Since they are additionally finite CW complexes and $X$ is finite type, the homology groups of the mapping spaces $\mathsf{H}_{i}(X^{K},R)$ are finite rank over $R$, and therefore $\mathsf{C}_{*}(X^{K},R)$ is its own double dual: the map $\mathsf{C}_{*}(X^{K},R)\rightarrow\mathsf{C}^{*}(X^{K},R)^{\vee}$ is an equivalence. Likewise, there is an equivalence between the dual of the tensor product and the cotensor product

$$\Bigl(\mathsf{C}^{\ast}(X^{N},R)\underset{\mathsf{C}^{\ast}(X^{S^{k}\times\mathbb{R}^{n-k}},R)}{\bigotimes}\mathsf{C}^{\ast}(X^{\mathbb{R}^{n}},R)\Bigr)^{\vee}\simeq\mathsf{C}_{\ast}(X^{N},R)\underset{\mathsf{C}_{\ast}(X^{S^{k}\times\mathbb{R}^{n-k}},R)}{\bigotimes}\mathsf{C}_{\ast}(X^{\mathbb{R}^{n}},R)\simeq\mathsf{C}_{\ast}(X^{M},R)$$

– this can be seen by commuting duality with the colimit to obtain a limit of a cosimplicial object, then comparing termwise. Continuing, one then concludes the equivalence

$$\mathsf{C}^{\ast}(X^{M},R)\simeq\mathsf{C}^{\ast}(X^{N},R)\underset{\mathsf{C}^{\ast}(X^{S^{k}\times\mathbb{R}^{n-k}},R)}{\bigotimes}\mathsf{C}^{\ast}(X^{\mathbb{R}^{n}},R)~,$$

thereby finishing the proof.

∎

Proof. The following equivalences

$$\int_{M}\free_{n}(V)\simeq\colim_{U\in\disk^{\partial}_{n/M}}\coprod_{i\geq 0}\conf_{i}(U,\partial U)\underset{\Sigma_{i}}{\otimes}V^{\otimes i}\simeq\coprod_{i\geq 0}\colim_{U\in\disk^{\partial}_{n/M}}\conf_{i}(U,\partial U)\underset{\Sigma_{i}}{\otimes}V^{\otimes i}$$

follow from the commutativity of colimits. To conclude the result it therefore suffices to show that for each $i$ the canonical morphism

$$\colim_{U\in\disk^{\partial}_{n/M}}\conf_{i}(U,\partial U)\underset{\Sigma_{i}}{\otimes}V^{\otimes i}\longrightarrow\conf_{i}(M,\partial M)\underset{\Sigma_{i}}{\otimes}V^{\otimes i}$$

in $\mathcal{V}$ is an equivalence. By the assumed distributivity in the $\otimes$-presentability condition, this follows if the natural $\Sigma_{i}$-equivariant map of $\Sigma_{i}$-spaces

(9)

$$\colim_{U\in\disk^{\partial}_{n/M}}\conf_{i}(U,\partial U)\longrightarrow\conf_{i}(M,\partial M)$$

is an equivalence, which we now show.

We first consider the case that the boundary of $M$ is empty, so that the natural $\Sigma_{i}$-equivariant map $\conf_{i}(M)\xrightarrow{\cong}\conf_{i}(M,\partial M)$ is a homeomorphism, and the natural functor $\disk_{n/M}\xrightarrow{\simeq}\disk^{\partial}_{n/M}$ is an equivalence of $\infty$-categories. In this case we are to show that the $\Sigma_{i}$-equivariant map of $\Sigma_{i}$-spaces

$$\colim_{U\in\disk_{n/M}}\conf_{i}(U)\longrightarrow\conf_{i}(M)$$

is an equivalence. After Proposition 2.19, it is enough to show that the $\Sigma_{i}$-equivariant map of $\Sigma_{i}$-topological spaces

(10)

$$\underset{U\in{\sf Disk}_{n/M}}{\colim}\conf_{i}(U)\longrightarrow\conf_{i}(M)$$

is a weak homotopy equivalence from the homotopy colimit. Each map $\conf_{i}(U)\to\conf_{i}(M)$ comprising this homotopy colimit is an open embedding. Also, for each element $\{1,\dots,i\}\xrightarrow{c}M$ of $\conf_{i}(M)$, choosing mutually disjoint Euclidean neighborhoods about each $c(j)\in M$ demonstrates that $c$ lies in the image of at least one such open embedding. Therefore this augmented diagram is an open cover of $\conf_{i}(M)$. This open cover of $M$ has the property that each finite intersection of its terms is covered by terms contained in this finite intersection. This is to say that this open cover of $\conf_{i}(M)$ is in fact a hypercover. That the map (10) is a weak homotopy equivalence follows from Corollary 1.6 of [DI].

Now suppose $\partial M$ is not empty. Fix a collar-neighborhood $\partial M\times\mathbb{R}_{\geq 0}\hookrightarrow M$. Such a collar-neighborhood determines the top horizontal arrow in the diagram of topological spaces

which commutes up to homotopy – here, the homotopy colimit is indexed by the opposite of the poset of non-empty subsets of $\{1,\dots,i\}$. This collar-neighborhood also gives that this diagram is a weak homotopy pushout. The result for this case of non-empty boundary thus follows from the previous case of empty boundary applied to $\partial M$ and to $\mathring{M}$, using that homotopy colimits commute with one another.

∎

Proof. We first consider the case that $\mathcal{V}=\bigl({\sf Spectra},\wedge\bigr)$ and $V=\Sigma^{\infty}X$ is the suspension spectrum of a pointed connected space $X$. There is a natural equivalence of spaces

$$\int_{S^{m}\times\mathbb{R}^{n-m}}\Omega^{n}\Sigma^{n}X~\underset{\rm Cor~\ref{non-abel}}{\simeq}~(\Omega^{n-m}\Sigma^{n}X)^{S^{m}}~\simeq~\Omega^{n}\Sigma^{n}X\times\Omega^{n-m}\Sigma^{n}X$$

where the first equivalence is by nonabelian Poincaré duality and the second is by the standard trivialization of each fiber sequence $\Map_{*}(K,G)\rightarrow G^{K}\rightarrow G$ whose base is equipped with the structure of a group-like $\mathcal{E}_{1}$-space. (It is this second step that requires the strict inequality $m<n$.) Passing to suspension spectra, Proposition 5.5 begets the further equivalence

$$\bigvee_{k\geq 0}\conf_{k}(S^{m}\times\mathbb{R}^{n-m})\underset{\Sigma_{k}}{\otimes}\Sigma^{\oo}X^{\otimes k}\simeq\Bigl(\bigvee_{i\geq 0}\conf_{i}(\mathbb{R}^{n})\underset{\Sigma_{i}}{\otimes}\Sigma^{\oo}X^{\otimes i}\Bigr)\otimes\Bigl(\bigvee_{j\geq 0}\conf_{j}(\mathbb{R}^{n-m})\underset{\Sigma_{j}}{\otimes}\Sigma^{\oo}(\Sigma^{m}X)^{\otimes j}\Bigr)~.$$

Collecting coefficients of terms which are homogeneous in $X$ determines a $\Sigma_{k}$-equivariant stable homotopy equivalence

$$\Sigma^{\infty}_{\ast}\conf_{k}(S^{m}\times\mathbb{R}^{n-m})\simeq\coprod_{i+j=k}\Bigl(\Sigma_{k}\underset{\Sigma_{i}}{\times}\Sigma^{\infty}_{\ast}\conf_{i}(\mathbb{R}^{n})\Bigr)\otimes\Bigr(\Sigma_{k}\underset{\Sigma_{j}}{\times}\Sigma^{\infty}_{\ast}\bigl(\Sigma^{mj}_{+}\conf_{j}(\mathbb{R}^{n-m})\bigr)\Bigl)$$

where $\Sigma_{k}\underset{\Sigma_{l}}{\times}-$ is induction from $\Sigma_{l}$-spectra to $\Sigma_{k}$-spectra.

Now consider the general case for $\mathcal{V}$, according to the hypothesis. After Proposition 5.5, both sides of the equivalence split as coproducts in homogeneous terms $V^{\otimes k}$, so it suffices to show that the coefficients of these terms are degreewise equivalent. Inspecting, we thus seek an equivalence in $\mathcal{V}$:

$$\conf_{k}(S^{m}\times\mathbb{R}^{n-m})\underset{\Sigma_{k}}{\otimes}V^{\otimes k}~\simeq~\bigoplus_{i+j=k}\Bigl(\conf_{i}(\mathbb{R}^{n})\underset{\Sigma_{i}}{\otimes}V^{\otimes i}\Bigr)\otimes\Bigr(\conf_{j}(\mathbb{R}^{n-m})\underset{\Sigma_{j}}{\otimes}(\Sigma^{m}V)^{\otimes j}\Bigl)~.$$

This equivalence follows from the conclusion of the previous paragraph upon tensoring with $V^{\otimes k}$ and taking balanced $\Sigma_{k}$-coinvariants.

∎

Proof. Via Example 3.10, each augmented associative algebra $A\to\uno$ in $\mathcal{V}$ determines a symmetric monoidal functor $A\colon\disk^{\partial,\sf or}_{1}\to\mathcal{V}$. Applying $\otimes$-excision in this simplest case of the collar-gluing $[-1,1)\underset{(-1,1)}{\bigcup}(-1,1]\cong[-1,1]$, we have that the bar construction ${\sf Bar}(A)$ is identifiable as the factorization homology over the closed 1-disk:

$${\sf Bar}(A)~\simeq~\int_{\mathbb{D}^{1}}A~.$$

Proposition 5.5 gives the first and last of the following identifications

	$\displaystyle\underset{\mathbb{D}^{1}\times\mathbb{R}^{n-1}}{\int}\free_{n}(V)$	$\displaystyle\simeq$	$\displaystyle\coprod_{i\geq 0}\conf_{i}(\mathbb{D}^{1}\times\mathbb{R}^{n-1},\partial\mathbb{D}^{1}\times\mathbb{R}^{n-1})\underset{\Sigma_{i}}{\otimes}V^{\otimes i}$
		$\displaystyle\simeq$	$\displaystyle\coprod_{i\geq 0}\Sigma^{i}\conf_{i}(\mathbb{R}^{n-1})\underset{\Sigma_{i}}{\otimes}V^{\otimes i}$
		$\displaystyle\simeq$	$\displaystyle\coprod_{i\geq 0}\conf_{i}(\mathbb{R}^{n-1})\underset{\Sigma_{i}}{\otimes}(\Sigma V)^{\otimes i}$
		$\displaystyle\simeq$	$\displaystyle\free_{n-1}(\Sigma V)~.$

The second identification follows from the $\Sigma_{i}$-equivariant equivalence of spaces

$$\conf_{i}(\mathbb{D}^{1}\times M,\partial\mathbb{D}^{1}\times M)\simeq\mathbb{D}^{i}\times\conf_{i}(M)\big/\partial\mathbb{D}^{i}\times\conf_{i}(M)\simeq\Sigma^{i}\conf_{i}(M)$$

in the case $M=\mathbb{R}^{n-1}$. The third equivalence is a coproduct of a composite of two equivalences: $\Sigma^{i}X\otimes V^{\otimes i}\simeq X\otimes\Sigma^{i}(V^{\otimes i})\simeq X\otimes(\Sigma V)^{\otimes i}$. The first of these equivalences uses that tensoring with spaces preserves colimits among spaces – an assertion which is direct from definitions. The second of these equivalences directly uses the assumption that the symmetric monoidal structure of $\mathcal{V}$ distributes over colimits.

∎

Proof. We first show that the bar construction defines a functor ${\sf Bar}\colon\Alg_{\disk_{n}^{\fr}}^{\sf aug}(\mathcal{V})\rightarrow\Alg_{\disk_{n-1}^{\fr}}^{\sf aug}(\mathcal{V})$. This is so in as much as ${\sf Bar}\simeq\int_{\mathbb{D}^{1}\times\mathbb{R}^{n-1}}$ is the object in $\mathcal{V}$ underlying the augmented $\mathcal{E}_{n-1}$-algebra $\disk^{\sf fr}_{n-1}\xrightarrow{\mathbb{D}^{1}\times-}\mfld^{\partial,\sf fr}_{n}\xrightarrow{\int A}\mathcal{V}$. We will now argue that this functor carries colimit diagrams to colimit diagrams.

Proposition 5.9 gives a commutative diagram among $\infty$-categories:

As a consequence, the functor ${\sf Bar}$ preserves coproducts of free $\mathcal{E}_{n}$-algebras. We next argue that ${\sf Bar}$ preserves sifted colimits. Using that the symmetric monoidal structure of $\mathcal{V}$ distributes over colimits, it is enough to argue that factorization homology $\int_{M}\colon\Alg^{\sf aug}_{\disk^{\sf fr}_{n}}(\mathcal{V})\to\mathcal{V}$ carries sifted colimit diagrams to colimit diagrams, for each framed $n$-manifold $M$ possibly with boundary.

So let $A\colon J\to\Alg_{\disk^{\sf fr}_{n}}^{\sf aug}(\mathcal{V})$ be a diagram of augmented $\mathcal{E}_{n}$-algebras in $\mathcal{V}$, indexed by a sifted $\infty$-category $J$. The canonical arrow $\underset{j\in J}{\colim}~{\sf Bar}(A_{j})\longrightarrow{\sf Bar}(\underset{j\in J}{\colim}A_{j})$ in $\mathcal{V}$ is a composite

$$\colim_{j\in J}\colim_{U\in\disk^{\partial,\sf fr}_{n/M}}A_{j}(U)~\simeq~\colim_{U\in\disk^{\partial,\sf fr}_{n/(M}}\colim_{j\in J}A_{j}(U)\longrightarrow\colim_{U\in\disk^{\partial,\sf fr}_{n/M}}(\colim_{j\in J}A_{j})(U)$$

where the outer objects are in terms of the defining expression for factorization homology, the left equivalence is through commuting colimits, and the right arrow is a colimit of canonical arrows. Again using that the symmetric monoidal structure of $\mathcal{V}$ distributes over colimits, each arrow $\underset{j\in J}{\colim}A_{j}(U)\to(\underset{j\in J}{\colim}A_{j})(U)$ is an equivalence if and only if it is for $U$ connected. This is the case provided the forgetful functor ${\sf ev}_{\mathbb{R}^{n}}\colon\Alg_{n}^{\sf aug}(\mathcal{V})\to\mathcal{V}$ preserves sifted colimits. This assertion is Proposition 3.2.3.1 of [Lu2].

Continuing, we conclude that ${\sf Bar}$ preserves all coproducts, since any coproduct is a geometric realization of coproducts of free algebras (this is a consequence of the $\infty$-categorical Barr–Beck Theorem 4.7.4.5 of [Lu2]; see §4.7 thereof for a general discussion). Now, coproducts and geometric realizations generate all colimits, and we conclude that ${\sf Bar}$ is a colimit preserving functor from $n$-disk algebras to $(n-1)$-disk algebras.

To complete the proof, both of the $\infty$-categories in the adjunction are presentable (see Corollary 3.2.3.3 of [Lu2]). The adjoint functor theorem (Corollary 5.5.2.9 of [Lu1]) can thus be applied to conclude that ${\sf Bar}$ is a left adjoint. The diagram above is therefore a commutative diagram of left adjoints, and therefore their right adjoints commute. Consequently, ${\sf Bar}$ has a right adjoint which, at the level of objects of $\mathcal{V}$, agrees with based loops $\Omega$, which is right adjoint to suspension $\Sigma$.

∎

Proof. Lie algebra chains defines a functor between $\infty$-categories $\mathsf{C}^{\Lie}_{\ast}:\Alg_{\Lie}({\sf Mod}_{k})\rightarrow{\sf Mod}_{k}$. This functor carries finite products of Lie algebras to finite tensor products of $k$-modules, which is to say that it is symmetric monoidal. Furthermore, this functor preserves geometric realizations, so Lemma 3.25 applies to give a natural identification: $\int_{M}\mathsf{C}_{\ast}^{\Lie}\bigl(\Mapc(\mathbb{R}^{n},\mathfrak{g})\bigr)\simeq\mathsf{C}_{\ast}^{\Lie}\bigl(\int_{M}\Mapc(\mathbb{R}^{n},\mathfrak{g})\bigr)$. The equivalence $\int_{M}\Mapc(\mathbb{R}^{n},\mathfrak{g})\simeq\Mapc(M,\mathfrak{g})$ now follows from the argument of nonabelian Poincaré duality (Corollary 4.6), only here we apply it in the usual abelian setting.⁶⁶ 6 See [AF2] for a complete account, where nonabelian Poincaré duality is an instance of a version of Poincaré/Koszul duality for Cartesian-presentable $\infty$-categories.

∎