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According to the cobordism hypothesis, if $\mathcal{C}$ is a symmetric monoidal $(\infty,n)$-categories with duals, then framed fully extended TQFTs with target $\mathcal{C}$ are an $\infty$-groupoid, and so a homotopy type. More precisely, by the cobordism hypothesis this groupoid is equivalent to the core $\mathcal{C}^\sim$ of $\mathcal{C}$: the $\infty$-groupoid obtained from $\mathcal{C}$ by discarting all the noninvertible $k$-morphisms in $\mathcal{C}$ for $k\geq 1$. More generally, for $G$-cobordism one gets the $\infty$-groupoid $(\mathcal{C}^\sim)^{hG}$ of $G$-homotopy invariants in $\mathcal{C}^\sim$.

My question is: what are the shapes, i.e., the homotopy types arising this way? In other words, given a (nice) topological space $A$, a nonnegative integer $n$ and a group $G$ over the infinite orthogonal group $O$ (i.e., a sequence of compatible group homomorphisms $G(k)\to O(k)$), are there conditions on $A$, $n$ and $G$ ensuring there exists symmetric monoidal $(\infty,n)$-categories with duals $\mathcal{C}$ such that $(\mathcal{C}^\sim)^{hG}\simeq A$?

The reason behind the question is that for such a triple one could interpret the set $[X,A]$ of homotopy classes of maps from a (nice) topological space $X$ to $A$ as the set of isomorphism classes of $\infty$-functors $\Pi X \to (\mathcal{C}^\sim)^{hG}$ and so (again by the cobordism hypothesis) as isomorphism classes of $G$-TQFTs over $X$ with target $\mathcal{C}$.

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    $\begingroup$ Can’t you just take $C=C^{\simeq}$? This is an infinite loop space, so you may as well deloop it to a spectrum, X. This has an action of O via the J-homomorphism, and you’re asking about all the possible connective spectra you can get by forming X^{hG} for maps BG—>BO. The answer includes every connective spectrum, since we are allowed to let G be trivial, and in general it seems hard to characterize the other possibilities when G is nontrivial $\endgroup$
    – Dylan Wilson
    Commented Dec 17, 2019 at 10:51
  • $\begingroup$ Right, that was indeed a silly question, with only possibly interesting part the one of homotopy fixed points. In full generality that’s too general, but maybe for $G$ in the Whitehead tower of $O$ something can be said. For the framed case: but then doesn’t this say that any space occurring as $E_n$ in some spectrum $E_\bullet$ can occur? Namely $E_n$ would be the 0-th space of the connective spectrum $(E_\bullet[n])_{\geq 0}$, doesn’t it? $\endgroup$
    – domenico fiorenza
    Commented Dec 17, 2019 at 12:26
  • $\begingroup$ @DylanWilson No, in general the core will correspond an $E_\infty$-space but it is usually not grouplike, so is not an infinite loop space (in general). Your proposal does work when the core is grouplike, but that is a strong condition. It is equivalent to saying every fully dualizable object is actually invertible. $\endgroup$
    – Chris Schommer-Pries
    Commented Jun 19, 2020 at 20:09
  • $\begingroup$ Indeed, looking back at this question, the correct thing to be done should be taking the group completion of the core on fully dualizable objects. This should be what is usually denoted by $K(C)$ and should be the construction that gives the connective K-theory associated with a symmetric monoidal $(\infty,n)$-category with duals. For instance, for $C=Vect_{\mathbb{K}}$ this should produce the algebraic K-theory of $\mathbb{K}$. For $\mathbb{K}=\mathbb{C}$ one take a topological variant and consider $Vect^{fin;top}_{\mathbb{C}}$, the topological category of finite dimensional (continues) $\endgroup$
    – domenico fiorenza
    Commented Jun 20, 2020 at 18:21
  • $\begingroup$ ... complex vector spaces (with the Euclidean topology on the Hom-spaces), seen as an $(\infty,1)$-category. In this case the corresponding K-theory should be $ku$ the connective topological complex K-theory spectrum. Similarly, with $\mathbb{K}=\mathbb{R}$ one can obtain $ko$ this way. $\endgroup$
    – domenico fiorenza
    Commented Jun 20, 2020 at 18:24

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