Section 11. (0MLZ) · ScalingStacks

11. Presentations of $(\infty,n)$ -categories[0MLZ]

The best-known examples of theories of $(\infty,n)$ -categories are given by presentations in terms of generators and relations. In order to show that these examples also satisfy our axioms, we can compare them directly to our colossal model $\cat_{(\infty,n)}$ of $(\infty,n)$ -categories. The main point is that $\Upsilon_{n}$ is so large that any ‘reasonable’ set of generators can be compared to it.

[0MJR]

Notation 11.1. Suppose $\mathcal{R}$ an ordinary category, and suppose $i\colon\mathcal{R}\to\Upsilon_{n}$ a functor. Suppose $T_{0}$ a set of morphisms of $\pre(\mathcal{R})$ , and write $T$ for the strongly saturated class of morphisms of $\pre(\mathcal{R})$ it generates.

[0MJS]

Theorem 11.2. Suppose that the following conditions are satisfied.

(R.1)

$i^{*}(S_{0})\subseteq T$ .
(R.2)

$i_{!}(T_{0})\subseteq S$ .
(R.3)

Any counit $R\to i^{*}i_{!}(R)=i^{*}(i(R))$ is in $T$ for any $R\in\mathcal{R}$ .
(R.4)

For each $0\leq k\leq n$ , there exists an object $R_{k}\in\mathcal{R}$ such that $i(R_{k})\cong C_{k}\in\Upsilon_{n}$ .

Then $i^{*}\colon\cat_{(\infty,n)}\to T^{-1}\pre(\mathcal{R})$ is an equivalence, and $T^{-1}\pre(\mathcal{R})$ is a theory of $(\infty,n)$ -categories.

[0MJT]

Proof. Condition (R.1) implies both that $i_{*}$ carries $T$ -local objects to $S$ -local objects and that we obtain an adjunction:

L^{T}\circ i^{*}\colon S^{-1}\pre(\Upsilon_{n})\rightleftarrows T^{-1}\pre(\mathcal{R})\colon i_{*}.

Similarly, condition (R.2) implies that $i^{*}$ carries $S$ -local objects to $T$ -local objects and that we obtain a second adjunction:

L^{S}\circ i_{!}\colon T^{-1}\pre(\mathcal{R})\rightleftarrows S^{-1}\pre(\Upsilon_{n})\colon i^{*}.

Since $i^{*}$ sends $S$ -local objects to $T$ -local objects, $i^{*}\simeq L^{T}\circ i^{*}$ when restricted to the $S$ -local objects of $\pre(\Upsilon_{n})$ . Thus $i^{*}\colon\pre(\Upsilon_{n})\to\pre(\mathcal{R})$ restricts to a functor

i^{*}\colon\cat_{(\infty,n)}=S^{-1}\pre(\Upsilon_{n})\to T^{-1}\pre(\mathcal{R}),

that admits a left adjoint $L^{S}\circ i_{!}$ and a right adjoint $i_{*}$ .

Notice that $i_{!}(R)\cong i(R)$ in $\pre(\Upsilon_{n})$ , where we have identified $\mathcal{R}$ and $\Upsilon_{n}$ with their images under the Yoneda embedding in, respectively, $\pre(\mathcal{R})$ and $\pre(\Upsilon_{n})$ . Thus by (R.3) the counit map applied to $r\in\mathcal{R}$ ,

R\to i^{*}\circ L^{S}\circ i_{!}(R)\cong i^{*}\circ L^{S}i(R)\cong i^{*}i(R)

becomes an equivalence in $T^{-1}\pre(\mathcal{R})$ (the last equality follows from Lemma 8.6, as the image of $i$ consists of $S$ -local objects). The endofunctor $i^{*}\circ L^{S}\circ i_{!}\colon T^{-1}\pre(\mathcal{R})\to T^{-1}\pre(\mathcal{R})$ is a composite of left adjoints, hence commutes with colimits. Therefore, as $\mathcal{R}$ is dense in $T^{-1}\pre(\mathcal{R})$ , the functor $i^{*}\circ L^{S}\circ i_{!}$ is determined by its restriction to $\mathcal{R}$ . It is equivalent the left Kan extension of its restriction to $\mathcal{R}$ . Consequently $i^{*}\circ L^{S}\circ i_{!}$ is equivalent to the identity functor.

On the other hand, for each $X\in\cat_{(\infty,n)}$ , consider the other counit map $X\to i_{*}i^{*}X$ . For each $k$ , we have natural equivalences,

	$\displaystyle\map(C_{k},i_{}i^{}X)$	$\displaystyle\simeq\map(i(R_{k}),i_{}i^{}X)$
		$\displaystyle\simeq\map(i^{}i(R_{k}),i^{}X)$
		$\displaystyle\simeq\map(R_{k},i^{*}X)$
		$\displaystyle\simeq\map(i(R_{k}),X)\simeq\map(C_{k},X),$

which follow from (R.3), (R.4), the identity $i_{*}(R_{k})\cong i(R_{k})$ , and the fact that $i^{*}X$ is $T$ -local. By Remark 7.1 this implies that the counit $X\to i_{*}i^{*}X$ is an equivalence. Thus $i^{*}$ is a functor with both a left and right inverse, hence is itself an equivalence of $\infty$ -categories. ∎

[0MJU]

Remark 11.3. Note that if the functor $i\colon\mathcal{R}\to\Upsilon_{n}$ is fully-faithful, then condition (R.3) is automatic. Note also that (R.3) and (R.4) together imply that the presheaves $i^{*}(C_{k})$ on $\mathcal{R}$ are each $T$ -equivalent to representables $R_{k}$ .

Condition (R.1) appears to be the most difficult to verify in practice. Heuristically, it states that $T$ contains enough morphisms. To verify it, it will be convenient to subdivide it into two conditions.

[0MJV]

Lemma 11.4. Condition (R.1) is implied by the conjunction of the following.

(R.1-bis(a))

$i^{*}(S_{00})\subset T$ .
(R.1-bis(b))

For any morphism $U^{\prime}\to V^{\prime}$ of $T_{0}$ , and for any morphisms $V^{\prime}\to i^{*}(C_{i})$ and $H\to C_{i}$ with $H\in\Upsilon_{n}$ , the pullback

$U^{\prime}\times_{i^{*}(C_{i})}i^{*}H\to V^{\prime}\times_{i^{*}C_{i}}i^{*}H$

lies in $T$ .

[0MJW]

Proof. First, consider the subclass $T^{\prime}\subset T$ containing those morphisms $U^{\prime}\to V^{\prime}$ of $T$ such that for any nondegenerate morphisms $V^{\prime}\to C_{k}$ and $H\to C_{k}$ , the pullback

U^{\prime}\times_{C_{k}}H\to V^{\prime}\times_{C_{k}}H

lies in $T$ . Since colimits in $\pre(\mathcal{R})$ are universal, one deduces immediately that the class $T^{\prime}$ is strongly saturated. Hence (R.1-bis(b)) implies that $T^{\prime}=T$ . Thus $T$ is closed under pullbacks along morphisms $H\to C_{k}$ and contains $i^{*}S_{00}$ (by (R.1-bis(a))), hence contains all of $i^{*}S_{0}$ . ∎

There are two main examples to which we shall apply Th. 11.2: Rezk’s model of complete Segal $\Theta_{n}$ -spaces [§ 13] and the model of $n$ -fold complete Segal spaces [§ 14].

Clark Barwick, Christopher Schommer-Pries

Original source: arXiv:1112.0040v6

Original source · 1112.0040v6

	$\displaystyle\map(C_{k},i_{}i^{}X)$	$\displaystyle\simeq\map(i(R_{k}),i_{}i^{}X)$
		$\displaystyle\simeq\map(i^{}i(R_{k}),i^{}X)$
		$\displaystyle\simeq\map(R_{k},i^{*}X)$
		$\displaystyle\simeq\map(i(R_{k}),X)\simeq\map(C_{k},X),$

11. Presentations of (∞,n)(\infty,n)-categories[0MLZ]

11. Presentations of $(\infty,n)$ -categories[0MLZ]