Section 3. (0MLR) · ScalingStacks

3. Gaunt $n$ -categories[0MLR]

Definition 3.1. A strict $n$ -category $X$ is gaunt if for any $1\leq k\leq n$ , the $n$ -category $X$ is local with respect to the natural functor

\sigma^{k-1}E\to\sigma^{k-1}(C_{0})=C_{k-1};

that is, the induced map

\cat_{n}(C_{k-1},X)\to\cat_{n}(\sigma^{k-1}E,X)

is a bijection. Equivalently, a strict $n$ -category $X$ in gaunt just in case, for any $1\leq k\leq n$ , any invertible $k$ -morphism is an identity.

We write $\gaunt_{n}\subset\cat_{n}$ for the full subcategory spanned by the gaunt $n$ -categories.

[0MHG]

Remark 3.2. Observe that the suspension of a gaunt $n$ -category is again gaunt. Note also that if a strict $n$ -category $X$ is gaunt, then for any $0\leq k\leq n$ , so is the strict $k$ -category $j_{k}X$ .

[0MHH]

Remark 3.3. Rezk observed [34, § 10] that the $1$ -category $E$ may be exhibited in $\cat_{1}$ as a pushout of more elementary $n$ -categories:

E\cong\Delta^{3}\cup^{(\Delta^{\{0,2\}}\sqcup\Delta^{\{1,3\}})}(\Delta^{0}\sqcup\Delta^{0}).

Consequently, a strict $n$ -category $X$ is gaunt if and only if for each $k\geq 0$ the following natural map is a bijection:

\Fun(C_{k},X)\to\Fun(\sigma^{k}(\Delta^{3}),X)\times_{\Fun(\sigma^{k}(\Delta^{\{0,2\}}\sqcup\Delta^{\{1,3\}}),X)}\Fun(\sigma^{k}(\Delta^{0}\sqcup\Delta^{0}),X).

The following is an easy consequence of the fact that $\cat_{n}$ is a presentable category.

[0MHI]

Proposition 3.4. The inclusion

\gaunt_{n}\hookrightarrow\cat_{n}

admits a left adjoint $L^{G}$ that exhibits $\gaunt_{n}$ as a localization of $\cat_{n}$ . ∎

In particular, $\gaunt_{n}$ is a presentable category. In fact, we can be more precise.

[0MHJ]

Lemma 3.5. The category $\gaunt_{n}$ is locally finitely presentable.

[0MHK]

Proof. Since $\cat_{n}$ is locally finitely presentable, it suffices to show that the inclusion $\gaunt_{n}\hookrightarrow\cat_{n}$ commutes with filtered colimits. To this end, suppose $\Lambda$ a filtered category, and suppose $D\colon\Lambda\to\cat_{n}$ a diagram such that for any object $\alpha\in\Lambda$ , the $n$ -category $D_{\alpha}$ is gaunt. We claim that the colimit $D=\colim_{\alpha\in\Lambda}D_{\alpha}$ (formed in $\cat_{n}$ ) is gaunt as well. This claim now follows readily from the fact that both $C_{k}$ and $\sigma^{k}(E)$ are compact objects in $\cat_{n}$ . ∎

[0MHL]

Remark 3.6. The previous lemma now implies that the category $\gaunt_{n}$ can be identified with the category of Ind-objects of the full subcategory $\gaunt_{n}^{\omega}\subset\gaunt_{n}$ spanned by the compact objects of $\gaunt_{n}$ . That is [31, Corollary 2.1.9^′], for any category $\mathcal{D}$ that admits all filtered colimits, if $\Fun^{\omega}(\gaunt_{n},\mathcal{D})$ denotes the full subcategory of $\Fun(\gaunt_{n},\mathcal{D})$ spanned by those functors that preserve filtered colimits, then the restriction functor

\Fun^{\omega}(\gaunt_{n},\mathcal{D})\to\Fun(\gaunt_{n}^{\omega},\mathcal{D})

is an equivalence.

[0MHM]

Corollary 3.7. Suppose $0\leq k\leq n$ . Then the smallest full subcategory of $\gaunt_{n}$ that is closed under colimits and contains the cells $C_{r}$ for $r\leq k$ is $\gaunt_{k}$ .

[0MHN]

Proof. The inclusion of gaunt $k$ -categories commutes with colimits (as it admits a right adjoint, see Rk. 3.2), whence it is enough to consider the case $k=n$ . This now follows readily from Corollary 3.4 and Proposition 2.7. ∎

3. Gaunt nn-categories[0MLR]

3. Gaunt $n$ -categories[0MLR]