Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

This is a follow up to this MO question: Free symmetric monoidal $(\infty,n)$-categories with duals

Freed-Hopkins-Lurie-Teleman define a classical field theory as a symmetric monoidal functor $I$ from $n$-cobordism to the symmetric monoidal $n$-category $Fam_n(\mathcal{C})$ of $n$-families over a fixed symmetric monoidal $n$-category. This is required to lift a certain "geometric background", i.e., a fixed functor from $n$-cobordism to the $n$-category $Fam_n$.

If $\mathcal{C}$ in an $(\infty,n)$ symmetric monoidal $n$-category with duals, then it is likely that $Fam_n(\mathcal{C})$ has duals, too (I have so far checked this only for $n=1$; in this particular case, the dual of a functor $X\to \mathcal{C}$ from a finite groupoid $X$ to $\mathcal{C}$ is nothing but the dual representation of $X$; and I expect this will still be true for higher $n$'s). Then, by the cobordism hypothesis, the datum of $I$ would be reduced to the choice of a fully dualizable object $I(*)$ in $Fam_n(\mathcal{C})$, i.e., of a functor $X\to \mathcal{C}$ "with a few good properties". The object $I(*)$ is explicited a couple of times in the paper (namely in the cases $n=1$ and $n=2$), but its fully dualizability is never discussed, nor seems to be claimed that $I(*)$ already contains all the information of $I$. For $n=3$, instead, the natural 3-representation of the groupoid $*//G$ which would starightforwardly generalize the $n=1$ and $2$ cases is not considered explicitely, and only its integrated (or quantized) version apperas.

Therefore, here is the question:

Is it correct that a classical field theory $I$ in the sense of Freed-Hopkins-Lurie-Teleman are equivalent to the datum of a fully dualizable object $I(*):X\to \mathcal{C}$ in $Fam_n(\mathcal{C})$ as claimed above? (with the "background geometry" given by the groupoid $X$)

share|improve this question

1 Answer 1

up vote 2 down vote accepted

All of the objects in these iterated span categories are fully dualizable. See this paper by Rune Haugseng.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.