ScalingStacks

Proof. The functor $\phi_{n}^{0}$ is simply the identity. We now define $\phi_{n}^{k}$ recursively. Suppose that $k>0$, and assume that the equivalence $\phi_{n-1}^{k-1}$ has been defined.

Let us define the functor $\phi_{n}^{k}\colon(\gaunt_{n}/C_{k})\to\corr_{n}^{k}$. For any object $p\colon X\to C_{k}$ of $(\gaunt_{n}/C_{k})$, let $\phi_{n}^{k}(p)$ be the triple $(X_{0},X_{1},F)$, where $X_{0}$ and $X_{1}$ are the fibers of $p$ over $0$ and $1$, respectively, and the functor $F\colon X_{0}^{\mathrm{op}}\times X_{1}\to\corr_{n-1}^{k-1}$ is the composite $\phi_{n-1}^{k-1}\circ F^{\prime}$, where the functor

$$F^{\prime}\colon X_{0}^{\mathrm{op}}\times X_{1}\to(\gaunt_{n-1}/C_{k-1})$$

is defined as follows:

• •

  For any objects $(x_{0},x_{1})\in X_{0}^{\mathrm{op}}\times X_{1}$, let $F(x_{0},x_{1})$ be the $(n-1)$-category $X(x_{0},x_{1})$, equipped with the functor $X(x_{0},x_{1})\to C_{k}(0,1)=C_{k-1}$ induced by $p$.

• •

  For any objects $(x_{0},x_{1})$ and $(y_{0},y_{1})$ of $X_{0}^{\mathrm{op}}\times X_{1}$, the functor

  $$X_{0}(y_{0},x_{0})\times X_{1}(x_{1},y_{1})\to\Fun_{/C_{k-1}}(X(x_{0},x_{1}),X(y_{0},y_{1}))$$

  is simply composition.

For any commutative triangle

of gaunt $n$-categories, we define

$$\phi_{n}^{k}(f)\mathrel{\mathop{:}}=(f_{0},f_{1},\alpha)\colon\phi_{n}^{k}(p)=(X_{0},X_{1},F^{\prime}\circ\phi_{n}^{k})\to(Y_{0},Y_{1},G^{\prime}\circ\phi_{n}^{k})=\phi_{n}^{k}(q),$$

where $f_{0}$ and $f_{1}$ are the restrictions of $f$ to the fibers, and $\alpha$ is the composite $\phi_{n-1}^{k-1}\ast\alpha^{\prime}$, in which the natural transformation $\alpha^{\prime}$ is the one whose components are given by the functor $X(x_{0},x_{1})\to Y(f(x_{0}),f(x_{1}))$ induced by $f$.

We now construct a quasi-inverse $\psi_{n}^{k}\colon\corr_{n}^{k}\to(\gaunt_{n}/C_{k})$ to $\phi_{n}^{k}$. Again, when $k=0$, we let $\psi_{n}^{0}$ be the identity, and we proceed recursively. We assume $k>0$ and that the quasi-inverse $\psi_{n-1}^{k-1}$ to $\phi_{n-1}^{k-1}$ has been defined.

For any object $(X_{0},X_{1},F)\in\corr_{n}^{k}$, define a gaunt $n$-category $U(X_{0},X_{1},F)$ with object set $\mathrm{ob}X_{0}\sqcup\mathrm{ob}X_{1}$ and

$$U(X_{0},X_{1},F)(a,b)\mathrel{\mathop{:}}=\begin{cases}X_{0}(a,b)&\textrm{ if }a,b\in X_{0}\\ X_{1}(a,b)&\textrm{ if }a,b\in X_{1}\\ \psi_{n-1}^{k-1}(F(a,b))&\textrm{ if }a\in X_{0},b\in X_{1}\\ \varnothing&\textrm{else}\end{cases}$$

The composition in $U(X_{0},X_{1},F)$ is the obvious one, and it is clear that this defines a functor $U\colon\corr_{n}^{k}\to\gaunt_{n}$. We now apply this functor to the terminal object of $\corr_{n}^{k}$, namely the triple $(C_{0},C_{0},\phi_{n-1}^{k-1}(C_{k-1}))$. Since $U(C_{0},C_{0},\phi_{n-1}^{k-1}(C_{k-1}))=C_{k}$, it follows that $U$ factors through a functor $\psi_{n}^{k}\colon\corr_{n}^{k}\to(\gaunt_{n}/C_{k})$.

It is now a simple matter to observe that $\psi_{n}^{k}$ is indeed quasi-inverse to $\phi_{n}^{k}$. ∎