& categories of disks

Peter J. Haine February 13, 2019

Abstract In this note we recall some facts cofinality and a few cofinality results that show that it is possible to compute factorization homology over three different categories of disks.

Contents

1 Cofinality & localiations 1

2 Categories of disks 2

1 Cofinality & localiations

1.1 Definition. Let 퐶 and 퐷 be ∞-categories. We say that a 퐹∶ 퐶 → 퐷 is: (1.1.1) colimit-cofinal if for every right fibration 푋 → 퐷, precomposition with 퐹 in- duces an equivalence

Map/퐷(퐷, 푋) → Map/퐷(퐶, 푋) .

(1.1.2) limit-cofinal if for every left fibration 푋 → 퐷, precomposition with 퐹 induces an equivalence Map/퐷(퐷, 푋) → Map/퐷(퐶, 푋) . 1.2 Lemma. Let 퐶 and 퐷 be ∞-categories. If a functor 퐹∶ 퐶 → 퐷 is colimit-cofinal or limit-cofinal, then 퐹 is a weak homotopy equivalence.

1.3 Theorem ([HTT, Proposition 4.1.1.8 & Theorem 4.1.3.1]). Let 퐶 and 퐷 be ∞-categories. The following are equivalent for a functor 퐹∶ 퐶 → 퐷: (1.3.1) The functor 퐹 is colimit-cofinal.

(1.3.2) Quillen’s Theorem A: For every object 푑 ∈ 퐷, the ∞-category 퐶 ×퐷 퐷푑/ is weakly contractible.

1 (1.3.3) For every ∞-category 퐸 and functor 퐺∶ 퐷 → 퐸, the colimit colim퐷 퐺 exists if and only if the colimit colim퐶 퐺퐹 exists, and when both colimits exist the natural colim퐷 퐺 → colim퐶 퐺퐹 is an equivalence in 퐸. 1.4 Definition. Let 퐶 be an ∞-category and 푊 ⊂ Mor(퐶) a class of in 퐶. The localization of 퐶 at 푊 is an ∞-category 퐶[푊−1] equipped with a functor 퐿∶ 퐶 → 퐶[푊−1] satisfying the following universal property: (1.4.1) The functor 퐿 sends all morphisms in 푊 to equivalences in 퐶[푊−1]. (1.4.2) For any ∞-category 퐷, precomposition with 퐿 defines a fully faithful functor

퐿⋆ ∶ Fun(퐶[푊−1], 퐷) ↪ Fun(퐶, 퐷)

with essential those 퐶 → 퐷 that send every morphism in 푊 to an equivalence in 퐷. 1.5 Remark. See [2, §7.1] for an excellent discussion of localizations of ∞-categories. In particular, [2, Proposition 7.1.3] shows that localizations always exist.

1.6 Lemma. Let 퐶 be an ∞-category and 푊 ⊂ Mor(퐶) a class of morphisms in 퐶. Then the localization functor 퐶 → 퐶[푊−1] is both limit-cofinal and colimit-cofinal. Proof. Recall that a functor 푝∶ 푋 → 푌 is conservative if a morphsim 푓 in 푋 is an equiv- alence if and only if 푝(푓) is an equivalence in 푌. Since both left and right fibrations are conservative functors [2, Proposition 3.4.8], it suffices to prove that for any conservative functor 푝∶ 푋 → 퐶[푊−1], the induced map

−1 Map/퐶[푊−1](퐶[푊 ], 푋) → Map/퐶[푊−1](퐶, 푊) is an equivalence. Since 푝∶ 푋 → 퐶[푊−1] is a conservative functor, any functor 퐶 → 푋 over 퐶[푊−1] necessarily carries all morphisms in 푊 to equivalences in 푋. The unvier- sal property of the localization 퐶[푊−1] implies that any functor 퐶 → 푋 over 퐶[푊−1] factors essentially uniquely through 퐶[푊−1], that is to say,

−1 Map/퐶[푊−1](퐶[푊 ], 푋) ⥲ Map/퐶[푊−1](퐶, 푊) . 1.7 Remark. See [2, Proposition 7.1.10] for an alternative proof of Lemma 1.6.

2 Categories of disks

2.1 Notation. For a nonnegative 푛, write Mfld푛 for the 1-category of 푛-manifolds and embeddings. Write Disk푛 ⊂ Mfld푛 for the full subcategory spanned by those 푛- manifolds isomorphic to a finite disjoint union of Euclidean spaces. Both Disk푛 and Mfld푛 have symmetric monoidal structures given by disjoint union.

2 1 We write Mfld푛 for the ∞-category associated to the topological category with ob- jects 푛-manifolds and morphism spaces the spaces Emb(푀, 푁) of embeddings 푀 ↪ 푁 with the compact-open topology. We write Disk푛 ⊂ Mfld푛 for the full subcategory spanned by those 푛-manfiolds isomorphic toa finite disjoint union of Euclidean spaces.

Ayala and Francis show that the ∞-category Disk푛,/푀 can be obtained from the 1- category Disk푛,/푀 by inverting those morphisms that are isotopic to an isomorphism. This is a somewhat technical result.

2.2 Theorem ([1, Proposition 2.19]). Let 푀 be a manifold and write 퐼푀 for the collection of morphisms in Disk푛,/푀 that are isotopic to an isomorphism. Then the functor

Disk푛,/푀 → Disk푛,/푀

−1 exhibits Disk푛,/푀 as the localization Disk푛,/푀[퐼푀 ] of Disk푛,/푀 at 퐼푀. In particular, the functor Disk푛,/푀 → Disk푛,/푀 is colimit-cofinal. 2.3 Notation. Let 푀 be an 푛-manifold. Write Disj(푀) ⊂ Open(푀) for the full subposet spanned by those open sets 푈 ⊂ 푀 that are homeomorphic to a finite disjoint union of Euclidean spaces 퐑푛. 2.4 Observation. Let 푀 be an 푛-manifold. Choose a parametrization of each open disk in 푀. This choice of parameterization gives rise to a functor 훾∶ Disj(푀) → Disk푛,/푀. We also write 훾 for the composite 훾∶ Disj(푀) → Disk푛,/푀. Essentially the same argument that Minta presented last talk using Lurie’s Seifert– van Kampen Theorem [HA, Theorem A.3.1] proves the following: 2.5 Proposition ([HA, Proposition 5.5.2.13]). Let 푀 be an 푛-manoifold and choose a parameterization of each open disk in 푀. Then the functor 훾∶ Disj(푀) → Disk푛,/푀 of Observation 2.4 is colimit-cofinal. 2.6. Let 푀 be an 푛-manifold. Theorem 2.2 and Proposition 2.5 show that we can com- pute the factorization homology of 푀 valued in an 푛-disk algebra 퐴∶ Disk푛 → 푉 with values in a symmetric monoidal ∞-category 푉 with colimits in three ways: (2.6.1) By definition, as the colimit of the diagram

퐴 Disk푛,/푀 Disk푛 푉 .

(2.6.2) As the colimit of the diagram

퐴 Disk푛,/푀 Disk푛,/푀 Disk푛 푉

indexed by the 1-category Disk푛,/푀.

1As a quasicategory, is the simplicial nerve of the fibrant simplicial category obtained from the topological category by applying the singular simplicial set functor Sing∶ Top → 푠Set.

3 (2.6.3) After choosing a parametrization of every disk in 푀, as the colimit of the dia- gram 훾 퐴 Disj(푀) Disk푛,/푀 Disk푛 푉 indexed by the poset Disj(푀).

2.7 Question. So what is the advantage of the ∞-category Disk푛,/푀 over the 1-category Disk푛,/푀, or the poset Disj(푀) if we can compute factorization homology in any of these three ways?

2.8. First of all, the functor 훾∶ Disj(푀) → Disk푛,/푀 depends on a choice of parametriza- tion of every disk in 푀, which is unnatural and cannot be made functorially. Second, the poset Disj(푀) generally doesn’t have any nice properties; Disj(푀) generally is not 2 filtered . The 1-category Disk푛,/푀 generally isn’t filtered either. The important feature of Disk푛,/푀 is that it is sifted (whereas Disk푛,/푀 generally is not). 2.9 Definition. An ∞-category 퐶 is sifted if 퐶 is nonempty and the 퐶 → 퐶 × 퐶 is colimit-cofinal. 2.10 Examples.

(2.10.1) Filtered ∞-categories are sifted [HTT, Example 5.5.8.3]. (2.10.2) The 1-category 횫op is sifted [HTT, Lemma 5.5.8.4]. (2.10.3) If 퐶 is a sifted ∞-category, then 퐶 is weakly contractible [HTT, Proposition 5.5.8.7].

2.11 Proposition ([HA, Proposition 5.5.2.15]). For every 푛-manifold 푀, the ∞-category Disk푛,/푀 is sifted.

References

HTT J. Lurie, Higher theory, Annals of Studies. Princeton, NJ: Princeton University Press, 2009, vol. 170, pp. xviii+925, isbn: 978-0-691-14049-0; 0-691-14049- 9. HA , Higher algebra, Preprint available at math.harvard.edu/~lurie/papers/HA.pdf, Sep. 2017.

1. D. Ayala and J. Francis, Factorization homology of topological manifolds, J. Topol., vol. 8, no. 4, pp. 1045–1084, 2015. doi: 10.1112/jtopol/jtv028. 2. D.-C. Cisinski, Higher categories and homotopical algebra, Preprint available at www. mathematik.uni-regensburg.de/cisinski/CatLR.pdf, May 2018.

2Although Disj(푀)op is filtered, but that’s not helpful for this purpose since we want to take a colimit over Disj(푀).

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