Section 15. (0MM3) · ScalingStacks

15. Epilogue: Model categories of $(\infty,n)$ -categories[0MM3]

We conclude with a brief discussion of model categories of $(\infty,n)$ -categories, in which we describe some interactions between our results here and those of Bergner, Lurie, Rezk, and Simpson. We first note that a spate of further corollaries to our main results can be obtained by employing the following.

[0ML7]

Construction 15.1. Suppose $\mathcal{A}$ a category equipped with a subcategory $w\mathcal{A}$ that contains all the objects of $\mathcal{A}$ (i.e., a relative category in the terminology of [6]). We call the morphisms of $w\mathcal{A}$ weak equivalences. In this situation, one may form the hammock localization $\mathrm{L}^{\!\!\mathrm{H}}\mathcal{A}$ of Dwyer–Kan [16]; this is a simplicial category. One may apply to each mapping space a fibrant replacement $R$ that preserves products (e.g., $\mathrm{Ex}^{\infty}$ ) to obtain a category enriched in Kan complexes, which we shall denote $\mathrm{L}^{\!\!\mathrm{H}}_{f}\mathcal{A}$ . We may now apply the simplicial nerve construction [28, 1.1.5.5] to obtain a $\infty$ -category $\mathrm{N}\mathrm{L}^{\!\!\mathrm{H}}_{f}\mathcal{A}$ , which we shall denote simply by $\mathrm{N}^{\mathrm{H}}\mathcal{A}$ . We shall call $\mathrm{N}^{\mathrm{H}}\mathcal{A}$ the $\infty$ -category underlying the relative category $\mathcal{A}$ .

When $\mathcal{A}$ is a simplicial model category, the simplicial localization $\mathrm{L}^{\!\!\mathrm{H}}\mathcal{A}$ is equivalent [17] to the full sub-simplicial category $\mathcal{A}^{\circ}$ spanned by the cofirant-fibrant objects. In this case, our $\mathrm{N}^{\mathrm{H}}\mathcal{A}$ is equivalent to $\mathrm{N}\mathcal{A}^{\circ}$ , as used by Lurie [28, A.2].

[0ML8]

Remark 15.2. When $\mathcal{A}$ and $\mathcal{B}$ are model categories, and $F\colon\mathcal{A}\rightleftarrows\mathcal{B}\colon G$ is a Quillen equivalence between them, there is [17] an induced equivalence of hammock localizations $\mathrm{L}^{\!\!\mathrm{H}}\mathcal{A}\simeq\mathrm{L}^{\!\!\mathrm{H}}\mathcal{B}$ , and thus of underlying $\infty$ -categories $\mathrm{N}^{\mathrm{H}}\mathcal{A}\simeq\mathrm{N}^{\mathrm{H}}\mathcal{B}$ .

[0ML9]

Example 15.3. The $\infty$ -category underlying the relative category of $n$ -relative categories [5] is a theory of $(\infty,n)$ -categories.

[0MLA]

Definition 15.4. Let us call a model category $\mathcal{A}$ a model category of $(\infty,n)$ -categories if its underlying $\infty$ -category $\mathrm{N}^{\mathrm{H}}\mathcal{A}$ is a theory of $(\infty,n)$ -categories.

[0MLB]

Example 15.5. By [25], the Joyal model category of simplicial sets is a model category of $(\infty,1)$ -categories. More generally, all of the following model categories are model categories of $(\infty,1)$ -categories:

(a)

the Joyal model category of quasicategories $\mathrm{QCat}$ [15, 24, 28],
(b)

the Rezk model category of complete Segal spaces $\CSS$ [33],
(c)

the Bergner model category of simplicial categories [18, 11],
(d)

the Tamsamani–Hirschowitz–Simpson–Pellissier model categories of Segal Categories [19, 22, 32, 10, 37, 12],
(e)

the Barwick–Kan model category of relative categories [6].

[0MLC]

Example 15.6. By Th. 13.15 and Th. 14.6, both Rezk’s model category $\Theta_{n}Sp$ of complete $\Theta_{n}$ -spaces [34] and the model category of $n$ -fold complete Segal spaces [4, 29] are model categories of $(\infty,n)$ -categories.

We may now use the construction above to find more examples of theories of $(\infty,n)$ -categories.

[0MLD]

Example 15.7 ([29, Proposition 1.5.4]). Let $\mathcal{M}$ be a left proper simplicial combinatorial model category which is an absolute distributor ([29, Definition 1.5.1]). Then the category $\Fun(\Delta^{\mathrm{op}},\mathcal{M})$ , of simplicial objects in $\mathcal{M}$ , admits the $\mathcal{M}$ -enriched complete Segal model structure $\CSS_{\mathcal{M}}$ , which is again left proper, simplicial, combinatorial, and an absolute distributor. If $\mathcal{M}$ is a model category of of $(\infty,n-1)$ -categories, then $\CSS_{\mathcal{M}}$ is a model category of $(\infty,n)$ -categories.

The condition of being an absolute distributor is needed in order to formulate the correct notion of complete $\mathcal{M}$ -enriched Segal object. We refer the reader to [29] for details, but note that being an absolute distributor is a property of the underlying $\infty$ -category of the given model category. In particular it is preserved under any Quillen equivalence.

[0MLE]

Example 15.8. Suppose that $\mathcal{M}$ is a model category satisfying the following list of conditions.

(M.1)

The class of weak equivalences of $\mathcal{M}$ are closed under filtered colimits.
(M.2)

Every monomorphism of $\mathcal{M}$ is a cofibration.
(M.3)

For any object $Y$ of $\mathcal{M}$ , the functor $X\mapsto X\times Y$ preserves colimits.
(M.4)

For any cofibrations $f\colon X\to Y$ and $f^{\prime}\colon X^{\prime}\to Y^{\prime}$ , the pushout product

$f\Box f^{\prime}\colon(X\times Y^{\prime})\cup^{(X\times X^{\prime})}(Y\times X^{\prime})\to Y\times Y^{\prime}$

is a cofibration that is trivial if either $f$ or $f^{\prime}$ is.
(M.5)

The $\infty$ -category $\mathrm{N}^{\mathrm{H}}\mathcal{M}$ is a homotopy theory of $(\infty,n-1)$ -categories.

Work of Bergner [12] and Lurie [29], combined with 14.6 above, shows that each of the following is an example of a model category of $(\infty,n)$ -categories:

•

the projective or (equivalently) the injective model category [29, 2.2.16, 2.3.1, 2.3.9] $\Seg_{\mathcal{M}}$ of $\mathcal{M}$ -enriched preSegal categories, and
•

the model category [28, A.3.2] $\cat_{\mathcal{M}}$ of categories enriched in $\mathcal{M}$ .

Moreover, following Simpson [37] the injective (aka Reedy) model category of Segal $(n-1)$ -categories [22, 32, 37] satisfies conditions (M.1-4); indeed, the most difficult of these to verify is (M.4), which Simpson does in [37, Th. 19.3.2 (using Corollary 17.2.6)].

Thus, for example, the injective and projective model categories of $\Theta_{n}Sp$ -enriched Segal categories $\Seg_{\Theta_{n}Sp}$ as well as the model category $\cat_{\Theta_{n}Sp}$ of categories enriched in $\Theta_{n}Sp$ are seen to be model categories of $(\infty,n)$ -categories. Indeed very recent work of Bergner and Rezk [13] discusses these model categories in detail and links them by an explicit chain of Quillen equivalences.

Additionally, we see that the injective model category of Segal $n$ -categories is also a model category of $(\infty,n)$ -categories, as is the model category of categories enriched in Segal $(n-1)$ -categories.

A partial converse to 15.2 holds, which allows one to deduce Quillen equivalences between these various model categories.

[0MLF]

Lemma 15.9 ([28, A.3.7.7]). Two combinatorial model categories $\mathcal{A}$ and $\mathcal{B}$ are connected by a chain of Quillen equivalences if and only if $\mathrm{N}^{\mathrm{H}}\mathcal{A}$ and $\mathrm{N}^{\mathrm{H}}\mathcal{B}$ are equivalent $\infty$ -categories.

From this it follows that if $\mathcal{A}$ and $\mathcal{B}$ are combinatorial model categories with the property that both $\mathrm{N}^{\mathrm{H}}\mathcal{A}$ and $\mathrm{N}^{\mathrm{H}}\mathcal{B}$ are theories of $(\infty,n)$ -categories, then $\mathcal{A}$ and $\mathcal{B}$ are connected by a chain of Quillen equivalences. This applies to all of the model categories of $(\infty,n)$ -categories mentioned above.

A zig-zag of Quillen equivalences can be a troublesome gadget to work with. It is usually far more informative to have a single direct and explicit Quillen equivalence between competing model categories of $(\infty,n)$ -categories. While our techniques do not generally provide such a direct Quillen equivalence, we do offer the following recognition principle.

[0MLG]

Proposition 15.10. Let $\mathcal{A}$ and $\mathcal{B}$ be two model categories of $(\infty,n)$ -categories and let $L:\mathcal{A}\leftrightarrows\mathcal{B}:R$ be a Quillen adjunction between them. Then $(L,R)$ is a Quillen equivalence if and only if the left derived functor $\mathrm{N}^{\mathrm{H}}L:\mathrm{N}^{\mathrm{H}}\mathcal{A}\to\mathrm{N}^{\mathrm{H}}\mathcal{B}$ preserves the cells up to weak equivalence.

[0MLH]

Proof. A Quillen equivalence induces an equivalence $\mathrm{N}^{\mathrm{H}}L:\mathrm{N}^{\mathrm{H}}\mathcal{A}\to\mathrm{N}^{\mathrm{H}}\mathcal{B}$ of $\infty$ -categories. By Lemma 10.2 and Lemma 4.8 any such equivalence necessarily preserves the cells up to equivalence. Conversely, as the left-derived functor $\mathrm{N}^{\mathrm{H}}L:\mathrm{N}^{\mathrm{H}}\mathcal{A}\to\mathrm{N}^{\mathrm{H}}\mathcal{B}$ preserves (homotopy) colimits and $\mathrm{N}^{\mathrm{H}}\mathcal{A}$ and $\mathrm{N}^{\mathrm{H}}\mathcal{B}$ are generated under (homotopy) colimits by the cells (Axiom C.2), it follows that $\mathrm{N}^{\mathrm{H}}L$ induces an equivalence of $\infty$ -categories. In particular it induces an equivalence of homotopy categories, and hence $(L,R)$ is a Quillen equivalence. ∎

In particular the above applies when the cells are fibrant-cofibrant objects of $\mathcal{A}$ and $\mathcal{B}$ which are preserved by $L$ itself.

[0MLI]

Example 15.11. The standard Quillen adjunction (cf. [29, Lemma 2.3.13]) from Segal $n$ -categories to $n$ -fold complete Segal spaces is a Quillen equivalence.

[0MLJ]

Example 15.12. The functor $\delta_{n}$ induces a Quillen equivalence between the model category of complete Segal $\Theta_{n}$ -spaces [34] and the model category of $n$ -fold complete Segal spaces [29, 1.5.4]. (See also Bergner–Rezk [14]).

A category with a specified subcategory of weak equivalences is a relative category, and hence gives rise to a homotopy theory. Thus any theory of $(\infty,n)$ -categories arising this way may, in principle, be compared using our axioms. We therefore end with the following.

[0MLK]

Conjecture 15.13. The $\infty$ -category underlying Verity’s $n$ -trivial weak complicial sets [39, 40] is a homotopy theory of $(\infty,n)$ -categories. The relative category consisting of Batanin’s $\omega$ -categories [7] such that every $k$ -cell is an equivalence for $k>n$ , together with the class of morphisms which are essentially $k$ -surjective for all $k$ is a theory of $(\infty,n)$ -categories.

Clark Barwick, Christopher Schommer-Pries

Original source: arXiv:1112.0040v6

previous section

Original source · 1112.0040v6

15. Epilogue: Model categories of (∞,n)(\infty,n)-categories[0MM3]

15. Epilogue: Model categories of $(\infty,n)$ -categories[0MM3]