ScalingStacks

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 $\Sigma$.

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