ScalingStacks

Now we will apply the preceding formalism to calculate the center of the monoidal $\infty$-category $\qc(X\times_{Y}X)$. Recall that our aim is to show that there is a canonical equivalence

$$\mathcal{Z}(\qc(X\times_{Y}X))\simeq\qc(\mathcal{L}Y).$$

where $\mathcal{L}Y$ is the loop space of $Y$.

Consider the symmetric monoidal $\infty$-category $\qc(X)$, together with the monoidal functor

$$\Delta_{*}:\qc(X)\to\qc(X\times_{Y}X)$$

obtained via pushforward along the relative diagonal $\Delta:X\to X\times_{Y}X$.

Set $A=\qc(X\times_{Y}X)$, and $B=\qc(X)$. By the preceding discussion, the center $\mathcal{Z}(A)$ is the limit of the relative Hochschild cochain complex $\mathbf{N}^{*}_{B}(A)$. Furthermore, its terms can be calculated

$$\mathbf{N}_{B}^{n}(A)\simeq{\mathcal{H}om}_{B\otimes B^{\rm op}}(A_{B}^{\otimes n+1},B).$$

Applying the results of Section 4, we can rewrite each term in the form

$$\mathbf{N}_{B}^{n}(A)\simeq\qc(\mathcal{L}Y\times_{Y}X\times_{Y}\cdots\times_{Y}X),\quad\mbox{ with $n+1$ copies of $X$.}$$

Here we have used the elementary identification

$$\mathcal{L}Y\times_{Y}X\times_{Y}\cdots\times_{Y}X\simeq X\times_{X\times X}((X\times_{Y}X)\times_{X}\cdots\times_{X}(X\times_{Y}X))$$

where the left hand side has $n+1$ copies of $X$, and the right hand side has $n+1$ copies of $X\times_{Y}X$.

We conclude that the terms of $\mathbf{N}_{B}^{*}(A)$ are nothing more than the terms of the cosimplicial $\infty$-category obtained by applying $\qc(-)$ to the C̆ech simplicial stack induced by the map

$$\tilde{p}:\mathcal{L}Y\times_{Y}X\to\mathcal{L}Y$$

obtained by base change from the original map $p:X\to Y$. It is straightforward to check that under this identification the coboundary maps are given by the usual C̆ech pullbacks. By assumption, $p$ satisfies descent, hence $\tilde{p}$ satisfies descent, and thus the totalization $\lim\mathbf{N}_{B}^{*}(A)$ also calculates the $\infty$-category $\qc(\mathcal{L}Y)$.

Finally, note that under this identification, the composition $\delta_{*}\pi^{*}:\qc(\mathcal{L}Y)\to\qc(X\times_{Y}X)$ corresponds to the composition

$$\qc(\mathcal{L}Y)\simeq\lim\mathbf{N}_{B}^{*}(A)\to\mathbf{N}_{B}^{0}(A)\to\mathbf{N}_{cyc}^{0}(A)\simeq\qc(X\times_{Y}X)$$

which is precisely the central functor $\mathfrak{z}$. This concludes the proof of Theorem 5.3.