ScalingStacks

Notation 8.3. We consider the $\infty$-category $\pre(\Upsilon_{n})$ of presheaves on the category $\Upsilon_{n}$ of Definition 6.2 and the Yoneda embedding

$$f\colon\Upsilon_{n}\hookrightarrow\tau_{\leq 0}\pre(\Upsilon_{n})\hookrightarrow\pre(\Upsilon_{n}).$$

Let $S_{00}$ denote the image of the finite set of morphisms of the same name as defined in Notation 6.5, which also represent the morphisms that appeared in (C.3). Let $S_{0}$ be the smallest class of morphisms of $\pre(\Upsilon_{n})$ that is stable under equivalence, contains $S_{00}$, and is stable under the operation $X\times_{C_{i}}(-)$ for $X\in\Upsilon_{n}$. One may check that $S_{0}$ has countably many isomorphism classes of maps and agrees with the essential image of the class $S_{0}$ introduced in Notation 6.5. Let $S$ be the strongly saturated class of morphisms of $\pre(\Upsilon_{n})$ generated by the class $S_{0}$. Let us study the localization $S^{-1}\pre(\Upsilon_{n})$.

The inclusion $j\colon\Upsilon_{n}\hookrightarrow\gaunt_{n}$ induces a functor $j^{\ast}\colon\pre(\gaunt_{n}^{\omega})\to\pre(\Upsilon_{n})$, which admits a left adjoint $j_{!}$ (given by left Kan extension) and a right adjoint $j_{\ast}$ (given by right Kan extension). Since $j_{!}$ and $j^{\ast}$ each preserve those presheaves represented by objects of $\Upsilon_{n}$ as well as all colimits, it follows that

$$j_{!}(S)\subseteq T\quad\text{and}\quad j^{\ast}(T)\subseteq S.$$

Consequently,

$$j^{\ast}(\precat_{(\infty,n)})\subseteq S^{-1}\pre(\Upsilon_{n})\quad\text{and}\quad j_{\ast}(S^{-1}\pre(\Upsilon_{n}))\subseteq\precat_{(\infty,n)}.$$

And so $j^{\ast}\colon\precat_{(\infty,n)}\to S^{-1}\pre(\Upsilon_{n})$ admits a left adjoint $L_{T}j_{!}$ (where $L_{T}$ is the localization $\pre(\gaunt_{n}^{\omega})\to T^{-1}\pre(\gaunt_{n}^{\omega})=\precat_{(\infty,n)}$) and a right adjoint $j_{\ast}$.