To address the question of whether chromatic spectra like Brown-Peterson (), Johnson-Wilson (), and Morava -theory () can be given a universal characterization using the geometric realization functor's colimit preservation, we proceed as follows:

1. Cobordism Hypothesis and Geometric Realization

The cobordism hypothesis states that the framed bordism ca...

Short answer.

• For ordinary tangential structures (X, ζ) the cobordism hypothesis does not give anything “new”: one has a canonical identification |Bordn(X,ζ)| ≃ Ω∞−n Mζ, i.e. the geometric realization of the n-dimensional (∞,n)-bordism category is the (∞−n)-fold loop space of the Thom spectrum Mζ. This is the same identification one gets from Pontryagin–Thom/“GMTW” results; it is not a diffe...

Let Bord be any of the symmetric‐monoidal (∞,n)-categories that are constructed in §4 of Lurie’s “On the Classification of Topological Field Theories”.
Write

|–| : SymMon_(∞,n) → Alg_{E_∞}(Gpd_∞) ≃ CAlg(Sp)

for the geometric realisation functor (it is left adjoint to the inclusion of E_∞-spaces into symmetric–monoidal (∞,n)-categories, hence preserves all homotopy colimits).

1. ...

Here is what is really going on: nothing mystical is hiding in the cobordism‐hypothesis beyond the well–known Baas–Sullivan picture of Thom–spectra as colimits of “bordism with singularities” and the fact that geometric realization [ | - |;:;\SymMon_{(\infty,n)};\longrightarrow;\Alg_{\E_\infty} ] preserves all homotopy colimits. In particular one recovers at once all of the usual push–o...

Let me recall the two ingredients that are relevant here.

(1) For every tangential structure
ξ : X → BO(n)
Lurie defines a symmetric–monoidal bordism ((\infty ,n))-category
Bord({n}^{,\xi }). His Cobordism Hypothesis implies that Bord({n}^{,\xi }) is the free symmetric–monoidal ((\infty ,n))-category with duals generated by a ξ–framed point. ...

This is an excellent question that connects two deep and powerful areas of modern algebraic topology. Your intuition is very much on the right track. The cobordism hypothesis and its machinery do provide a universal characterization for these spectra, which is conceptually different from their construction as Thom spectra.

Let's break it down.

The Universal Property of Bordism Categories

Fi...

Chromatic Spectra and the Cobordism Hypothesis

Your question probes deep connections between cobordism theory, chromatic homotopy theory, and higher categorical structures.

On Universal Characterizations

The key insight of the cobordism hypothesis is that is the free symmetric monoidal -category with duals generated by a point. When we apply geomet...

The context is the cobordism hypothesis, which states that the bordism category with framing, denoted $\mathbf...

This is a fascinating question that touches on deep connections between chromatic homotopy theory, cobordism categories, and the cobordism hypothesis. Let me address your questions systematically.

The Main Question

Your core question asks whether chromatic spectra like BP, E(n), and K(n) can be given universal characterizations using the fact that geometric realization preserves colimits. Thi...