ScalingStacks

Example 3.28. Let $\mathcal{V}^{\oplus}$ be either the $\oo$-category of chain complexes or of spectra equipped with the direct sum monoidal structure. Since every object $V$ therein has an essentially unique morphism $V\oplus V\rightarrow V$, there is an equivalence $\Alg_{\disk_{n}^{\sf fr}}(\mathcal{V}^{\oplus})\simeq\mathcal{V}$. The factorization homology of a framed $n$-manifold $M$ with coefficients in $V$ is then equivalent to $\int_{M}V\simeq\mathsf{C}_{\ast}(M,V)$, or $\Sigma^{\infty}_{\ast}M\otimes V$ for spectra, the stabilization of $M$ smashed with $V$. There is a natural functor $\varinjlim\mfld_{n}^{\fr}\rightarrow\spaces^{\sf fin}$, and this functor is an equivalence because: it is fully faithful since $\varinjlim_{k}\Emb^{\fr}(M\times\mathbb{R}^{k},N\times\mathbb{R}^{k})\rightarrow\Map(M\times\mathbb{R}^{\infty},N\times\mathbb{R}^{\infty})\simeq\Map(M,N)$ is a weak homotopy equivalence for every $M$ and $N$; it is essentially surjective since every finite CW complex $X$ can be embedded into $\mathbb{R}^{m}$ for $m$ sufficiently large, and thus it is homotopy equivalent to a framed $n$-manifold, namely an open regular neighborhood of the embedding. Theorem 3.24 thereby specializes to the formulation of the Eilenberg–Steenrod axioms given in the introduction. If one sets $\mathcal{V}$ to be the opposite ${\sf Ch}^{\op}$, then one likewise recovers the Eilenberg–Steenrod axioms for cohomology.