Extended TFT with coefficients in spans in any $\infty$-topos

Question

In the TFT classification article by Jacob Lurie (arXiv:0905.0465) the (∞,n)-category of correspondences (there: $Fam_n$) plays a key role, whose $k$-morphisms are $k$-fold spans of $\infty$-groupoids sliced over a suitable coefficient object $\mathcal{C}$.

Motivated by some notes we had made on "Local prequantum field theory", recently Rune Haugseng considered in detail (arXiv:1409.0837) the evident generalization of this where the collection of $\infty$-groupoids is replaced by any other $\infty$-topos. The key properties of the construction are preserved under this generalization, in particular this is still an (∞,n)-category with duals.

It seems plausible that, similarly, essentially all the relevant statements regarding topological field theories with coefficients in such correspondences in any $\infty$-topos will still hold. I am specifically interested in a sanity check of this for the following statement:

First of all, combining prop. 3.2.8 in arXiv:0905.0465 with theorem 2.4.18 there and applying it to the special case that $\mathcal{C}$ is an $\infty$-groupoid (with duals), yields the neat statement that unoriented "local prequantum field theories with phases in $\mathcal{C}$"

$$ Z_L \;\colon\; Bord_n^\sqcup \longrightarrow Fam_n(\mathcal{C})^\otimes $$

are equivalent to the choice of an $\infty$-groupoid $F$ equipped with $O(n)$-$\infty$-action and with an $O(n)$-equivariant map

$$ L \;\colon\; F \longrightarrow \mathcal{C} \,, $$

where $\mathcal{C}$ is equipped with the canonical $O(n)$-action induced via theorem 2.4.6 in the above text. An equivalent way to say this is that $L$ induces a horizontal map fitting into

$$ \array{ F/\!/O(n) && \stackrel{L/\!/O(n)}{\longrightarrow} && \mathcal{C}/\!/O(n) \\ & \searrow && \swarrow_{canonical} \\ && B O(n) } $$

(where the double slash denotes homotopy quotients).

This makes it very manifest that this statement has an immediate analogue with all $\infty$-groupoids replaced by objects of any $\infty$-topos $\mathbf{H}$ (with $B O(n)$ regarded as the inverse image under the terminal geometric morphism of that $\infty$-topos of the homotopy type of the usual classifying space). Now one considers $Fam_n^{\mathbf{H}}(\mathcal{C})$ being the $(\infty,n)$-category of $n$-fold $\mathcal{C}$-phased correspondences, now all inside $\mathbf{H}$, and so forth. And $O(n)$-$\infty$-actions on $F \in \mathbf{H}$ are equivalent to homotopy fiber sequences in $\mathbf{H}$ of the form $F \to F/\!/O(n) \to B O(n)$. But let me maybe restrict attention to the case that $\mathcal{C}$ is still a bare $\infty$-groupoid (similarly embedded into $\mathbf{H}$ under the terminal inverse image).

Is it then still true for general $\mathbf{H}$ that monoidal $(\infty,n)$-functors

$$ Bord_n^\sqcup \longrightarrow Fam_n^{\mathbf{H}}(\mathcal{C})^\otimes $$

are equivalent to diagrams of the form

$$ \array{ F/\!/O(n) && \stackrel{L/\!/O(n)}{\longrightarrow} && \mathcal{C}/\!/O(n) \\ & \searrow && \swarrow_{{canonical}} \\ && B O(n) } $$

in $\mathbf{H}$?

(In case it matters, I am happy to assume that the terminal inverse image of $\mathbf{H}$ is fully faithful.)

Answer 1

Here an argument using the assumption that $\mathbf{H}$ has an $\infty$-site $\mathcal{S}$ of definition all whose objects are étale contractible.

The proof of prop. 3.2.8 arXiv:0905.0465 shows that for $\mathbf{H} = \infty Grpd$ the equivalence in question is natural in the choice of the unoriented bulk field theory $F/\!/O(n)$.

Let then $$ \mathbf{H} \stackrel{\longleftarrow}{\hookrightarrow} Func(\mathcal{S}^{op},\infty Grpd) $$

be the reflection exhibiting $\mathcal{S}$ as an $\infty$-site of definition.

Given $F \in \mathbf{H}$ equipped with $O(n)$-$\infty$-action (throughout $O(n)$ denotes the homotopy type of the topological group $O(n)$, regarded as a group object in constant $\infty$-stacks) write for each $U \in \mathcal{S}$

$$ F_U := \mathbf{H}(U,F) \in \infty Grpd $$

for its value on $U$. Observe that setting

$$ F_U/\!/O(n) := \mathbf{H}(U,F/\!/O(n)) \in \infty Grpd $$

exhibits an $O(n)$-$\infty$-action on $F_U$ since $\mathbf{H}(U,-)$ preserves $\infty$-limits and since by assumption on the site $\mathcal{S}$ we have $\mathbf{H}(U,B O(n)) \simeq B O(n)$, so that the homotopy fiber sequence

$$ F \to F/\!/O(n) \to B O(n) $$

which exhibits the $\infty$-action of $O(n)$ on $F$ (here I am using arXiv:1207.0248) naturally induces a system of homotopy fiber sequences

$$ F_U \to F_U/\!/O(n) \to B O(n) $$

exhibiting $O(n)$-$\infty$-actions on each $F_U$.

It follows that a diagram

$$ \array{ F/\!/O(n) && \stackrel{L/\!/O(n)}{\longrightarrow} && \mathcal{C}/\!/O(n) \\ & \searrow && \swarrow \\ && B O(n) } $$

in $\mathbf{H}$ is equivalent to an $\infty$-sheaf of diagrams of the form

$$ U \;\;\; \mapsto \;\;\; \left\{ \array{ F_U/\!/O(n) && \longrightarrow && \mathcal{C}/\!/O(n) \\ & \searrow && \swarrow \\ && B O(n) } \right\} $$

in $\infty \mathrm{Grpd}$. Now the statement of the proposition for $\mathbf{H} = \infty \mathrm{Grpd}$ applies objectwise for each $U$, and since it is natural in $F_U$ (with its action) it is also natural in $U$, and so the above is equivalent to the $\infty$-sheaf ($(\infty,n)$-sheaf) of local unoriented-topological field theories in $\infty \mathrm{Grpd}$:

$$ U \;\;\; \mapsto \;\;\; \left\{ \array{ && (Fam_n(\mathcal{C}))^{\otimes} \\ & {}^{}\nearrow & \downarrow \\ Bord_n^\sqcup &\underset{F_U/\!/O(n)}{\longrightarrow}& (Fam_n)^\otimes } \right\} \,. $$

But this is equivalently a field theory

$$ \array{ && (Fam_n^{\mathbf{H}}(\mathcal{C}))^{\otimes} \\ & {}^{}\nearrow & \downarrow \\ Bord_n^\sqcup &\underset{F/\!/O(n)}{\longrightarrow}& (Fam_n^{\mathbf{H}})^\otimes } $$

with coefficients in $\mathbf{H}$.