I apologize in advance, if some of the things I've written are incorrect.

The cobordism hypothesis states that $\mathbf{Bord}^\mathrm{fr}_n$ is the free symmetric monoidal $(\infty,n)$-category with duals generated by a point. There is a geometric realization functor $|-|:\text{Cat}_{(\infty,n)}\rightarrow \text{Gpd}_\infty$, where $\text{Cat}_{(\infty,n)}$ is the $\infty$-category (I mean a quasicategory, but you can think of it as a model category, or whatever you like) of (small) $(\infty,n)$-categories, and $\text{Gpd}_\infty$ is $\infty$-category of $\infty$-groupoids, otherwise known as spaces. This functor is a left adjoint (in the homotopical sense) to the inclusion functor $\text{Gpd}_\infty\hookrightarrow \text{Cat}_{(\infty,n)}$. Therefore, $|-|$ preserves (homotopy) colimits. If we "restrict" ourselves to the symmetric monodical $(\infty,n)$-categories, we would get a functor $|-|:\text{SymMon}_{(\infty,n)}\rightarrow \text{Alg}_{\mathbb{E}_\infty}$, where $\text{SymMon}_{(\infty,n)}$ is the $\infty$-category of symmetric monoidal $(\infty,n)$-categories and $\text{Alg}_{\mathbb{E}_\infty}$ is the $\infty$-category of infinite loop spaces. This should also preserve colimits. It manifests itself in the fact that $|\mathbf{Bord}^\text{fr}_n|$ is the infinite loop space of the sphere spectrum.

There is a universal characterization of the bordism $(\infty,n)$-category of manifolds with singularities [Theorem 4.3.11 in Lurie's paper]. The type of singularities considered are those of Baas-Sullivan theory, which produce a wide array of chromatic spectra, such as the Brown-Peterson spectrum $\text{BP}$, the Johnson-Wilson spectra $\text{E}(n)$, and, of course, the Morava K-theory spectra $\text{K}(n)$. I am wondering:

Is it possible to give these spectra a universal characterization using the fact that geometric realization preserves colimits?

EDIT: When I think about it, I am not sure whether there is a universal description of $|\mathbf{Bord}^{(X,\zeta)}_n|$, where $X$ is topological space, $\zeta$ a vector bundle on it of dimension $n$, other than the case when $X$ is point. We can write it down. It is simply $\Omega^{\infty-n}M\zeta$, where $M\zeta$ is the Thom spectrum of $\zeta$.

Do we get a different characterization of these infinite loopspaces from the cobordism hypothesis?


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