Moduli Problems in Derived Noncommutative Geometry

Pranav Pandit

A Dissertation

Mathematics

Presented to the Faculties of the University of Pennsylvania in Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy

2011

Tony Pantev Supervisor of Dissertation

Jonathan Block Graduate Group Chairperson Acknowledgments

ii ABSTRACT

Moduli Problems in Derived Noncommutative Geometry

Pranav Pandit

Tony Pantev, Advisor

We study moduli spaces of boundary conditions in 2D topological ﬁeld theories. To a compactly generated linear ∞-category X , we associate a moduli functor

MX parametrizing compact objects in X . Using the Barr-Beck-Lurie monadicity theorem, we show that MX is a ﬂat hypersheaf, and in particular an object in the

∞-topos of derived stacks. We find that the Artin-Lurie representability criterion makes manifest the relation between finiteness conditions on X , and the geometricity of MX . If X is fully dualizable (smooth and proper), then MX is geometric, recovering a result of Toën-Vacqiue from a new perspective. Properness of X does not imply geometricity in general: perfect complexes with support is a counterexample. However, if X is proper and perfect (symmetric monoidal, with “compact

= dualizable”), then MX is geometric.

The ﬁnal chapter studies the moduli of oriented 2D topological ﬁeld theories

(Noncommutative Calabi-Yau Spaces). The Cobordism Hypothesis, Deligne’s con- jecture, and formal En-geometry are used to outline an approach to proving the unobstructedness of this space and constructing a Frobenius structure on it.

iii Contents

1 Introduction 1

1.1 Brave New Mathematics ...... 2

1.2 Brane Theory ...... 2

1.3 Derived Noncommutative Geometry ...... 2

1.4 About this work ...... 3

1.5 Background and Notation ...... 3

2 Brane Moduli 8

2.1 Commutative spaces ...... 8

2.2 Noncommutative spaces ...... 8

2.3 Moduli of perfect branes ...... 8

3 Brane Descent 18

3.1 Perfect Branes Descend ...... 18

3.2 Gluing Compact Objects ...... 20

3.3 Bootstrapping: from Covers to Hypercovers ...... 26

iv 3.4 The Barr-Beck-Lurie Theorem ...... 26

3.5 Flat Hyperdescent: the proof ...... 28

4 Geometricity 38

4.1 Geometric Stacks ...... 38

4.2 The Artin-Lurie Criterion ...... 38

4.3 Inﬁnitesimal theory: Brane Jets ...... 38

4.4 Dualizability implies Geometricity ...... 38

4.5 A Proper Counterexample ...... 38

5 Moduli of 2D-TFTs 39

5.1 The Cobordism Hypothesis ...... 39

5.2 Moduli of Noncommutative Calabi-Yau Spaces ...... 39

5.3 Geometricity ...... 39

5.4 Unobstructedness ...... 39

5.5 Frobenius Structures: from TFTs to CohFTs ...... 39

v Chapter 1

Introduction

The subject of this thesis, derived noncommutative geometry, is the natural coming together of two fundamental paradigm shifts: one within mathematics, and the other in physics. The ﬁrst is a movement away from mathematics based on sets, to a mathematics where the primitive entities are shapes. The second, is the radical idea in physics that the notion of space-time is not intrinsic to a physical theory.

Sections §1.1 and §1.2 are devoted to a cursory overview of these two incipient revolutions in the way we perceive reality. In section §1.3, we sketch in quick, broad strokes the emerging contours of derived noncommutative geometry. Our purpose in including these sections is to place this thesis within the broader context into which it naturally ﬁts, and to lend some perspective to the results proven here.

Section §1.4 details, in a semi-informal tone, the main results of this article, and outlines the organizational structure of the document.

1 1.1 Brave New Mathematics

For several centuries, scientiﬁc thought has been conﬂated with reductionist paradigm.

The tremendous advances in human knowledge during the aforementioned era stand testimony, no doubt, to the power and practical utility of reductionism. Neverthe- less, as with any approach that has thought as its basis, it is limited. What follows is a discussion of a particular limitation.

One glaring manifestation of the reductionist approach is the fact that modern mathematics is based on an axiomatic framework where the primordial entities are sets.

1.2 Brane Theory

1.3 Derived Noncommutative Geometry

The author feels that identifying the act of doing DNG, with “replacing a scheme

X by its derived category of sheaves”, is decidedly undemocratic, and somewhat limiting, placing little emphasis, for instance, on the symplectic point of view on the Elephant. Furthermore, it obscures the physical origins of DNG.

2 1.4 About this work

1.5 Background and Notation

Throughout this work, we will assume familiarity with the language of homotopical mathematics as developed by Lurie in [Lur09, Lur11b, Lur04]. Speciﬁcally, we will assume that the reader has at least a ﬂeeting acquaintance with the rudiments of topology, algebra and algebraic geometry in the ∞-categorical context. Having said that, we would like to emphasize that an intimate knowledge of the inner workings of the theory in loc. cit. is not needed in order to read this paper.

An attempt has been made to keep the statements of the results and the proofs devoid of references to a particular model for ∞-categories. Any equivalent (in a suitable sense) model will suffice. In particular, the reader who is more comfortable with the parlance of Toën/Toën-Vezzosi [Toë07,TV05, TV08], should encounter little difficulty in translating most results of this paper into that language. There is one caveat: for statements that involve functor categories, monads and the Barr-

Beck-Lurie theorem, model categories must be replaced by a more ﬂexible notion such as Segal Categories, as is done, for instance, in [TV02].

To a large extent, the notation used in this paper is consistent with the notation in [Lur09, Lur11b, Lur04]. The following is a list of some frequently used notation.

Notation 1.5.1 (Bibliographical Convention). We will use the letter

• “T” to refer to the book Higher Topos Theory [Lur09].

3 • “A” to refer to the book, Higher Algebra [Lur11b].

• “G” to refer to the thesis Derived Algebraic Geometry [Lur04].

Thus, for example, T.3.2.5.1. refers to [Lur09, Remark 3.2.5.1.], while A.6.3.6.10. refers to [Lur11b, Theorem 6.3.6.10].

Notation 1.5.2 (Spaces). We will denote by S (resp. Sb) the ∞-category of small

(resp. large) spaces (T.1.2.1.6.), and by S∞ := Stab(S) the stable ∞-category of spectra. For an ∞-category C, Stab(C) is its stabilization (A.1.4.)

Notation 1.5.3 (∞-categories). Throughout, κ will denote an arbitrary regular cardinal, and ω is the smallest one. We will denote by

• Cat∞ (resp. Catd∞) the ∞-category of small (resp. large) ∞-categories.

Ex ∨ • Cat∞ (resp. Cat∞) the subcategory of Cat∞ consisting of small stable (resp.

idempotent complete stable) ∞-categories and exact functors.

L R •P r (resp. Pr ) the subcategory of Catd∞ consisting of presentable ∞-

categories and left adjoints (resp. accessible right adjoints). See T.5.5.

L R L R •P rκ (resp. Prκ ) the subcategory of Pr (resp. Pr ) consisting of κ-compactly

generated ∞-categories and functors that preserve κ-compact objects (resp.

are κ-accessible). See T.5.5.7.

L L •P rst the subcategory of Pr consisting of stable ∞-categories.

4 Notation 1.5.4 (Functor categories). For C, D in Cat∞, Fun(C, D) denotes the

∞-category of functors C → D. We will denote by

• FunL(C, D) (resp. FunR(C, D)) the full subcategory of Fun(C, D) consisting of

functors that preserve all small colimits (resp. are accessible, and preserve all

small limits).

• FunLAd(C, D) (resp. FunRAd(C, D)) the full subcategory of Fun(C, D) consist-

ing of functors that have right adjoints (resp. have left adjoints).

L R • Funκ(C, D) (resp. Funκ (C, D)) the full subcategory of Fun(C, D) consisting

of functors that preserve all small colimits and κ-compact objects (resp. are

κ-accessible, and preserve all small limits).

By the adjoint functor theorem (T.5.5.2.9.), if C and D are presentable, then we have natural equivalences FunL(C, D) ' FunLAd(C, D) and FunR(C, D) ' FunRAd(C, D).

L L Notation 1.5.5 (Categorical hom and tensor). The categories Pr and Prκ are

L L symmetric monoidal, and the inclusion functor Prκ ⊆ Pr is symmetric monoidal

(A.6.3.) with unit S. We will denote by ⊗ the tensor product on PrL. This is not to be confused with the Cartesian monoidal structure “×”on Catd∞. The functors

L L L L Fun (−, −) (resp. Funκ(−, −)) deﬁnes an internal hom on Pr (resp. Prκ ).

Notation 1.5.6 (Algebras and modules). Let O⊗ be an ∞-operad, C⊗ → O⊗ be an O-monoidal category and let M be an ∞-category tensored over C (A.4.2.1.9.).

We will denote by AlgO(C) the ∞-category of O-algebra objects in C. For A in

5 O AlgO(C), we will write ModA (M) for the ∞-category of A-modules in M. When

O O is the commutative operad CAlg(C) := AlgO(C), and ModA(M) := ModA (M).

When O is the associative operad (A.4.1.1.), Alg(C) := AlgO(C). We will use the abbreviations CAlgA := CAlg(ModA(S∞)) for any A in CAlg(S∞) and ModA :=

ModA(S∞).

Notation 1.5.7 (1-Categories). We will denote by Cat the ∞-category of 1- categories. In the quasicategorical model, this is the simplicial nerve of the Dwyer-

Kan localization of the 1-category of categories along the subcategory of weak equivalences. We will denote by

• N(−) the natural inclusion Cat → Cat∞. In the quasicategorical model, this

is the nerve functor.

• h : Cat∞ → Cat the left adjoint to N(−). We will refer to hC as the homotopy

category of C.

Notation 1.5.8 (Ground ring). Throughout, we will ﬁx an E∞-ring k. We will assume that k is a Derived G-ring in Chapter 4.

Notation 1.5.9 (Algebraic geometry). . We will denote by Aﬀk the category of

op derived affine schemes. By definition Affk := CAlg . We will denote by Spec :

op CAlg → Aﬀk and O : Aﬀk → CAlg the tautological equivalences. We will denote by Stk the ∞-topos of derived stacks over k.

6 / . Notation 1.5.10 (Diagrams and limits). For K in Cat∞, K (resp. K ) will denote the category obtained from K by adjoining an initial (resp. ﬁnal) object {∞}. For x in K we will usually denote by ψx the unique morphism ∞ → x (resp. x → ∞).

Our terminology regarding limits follows T.4.

7 Chapter 2

Brane Moduli

2.1 Commutative spaces

2.2 Noncommutative spaces

2.3 Moduli of perfect branes

8 There is a functor M : CAlgk → Catd∞ whose action on objects and 1-morphisms can be described as follows. To an object A in CAlgk, M assigns the ∞-category of A-modules, ModA. The action of M on 1-morphisms f : A → B in CAlgk is given by base change (left Kan extension). In symbols:

M(A) := ModA

M(f) := B ⊗A (−)

The existence of the ∞-functor M can be established in several ways. In the language of [Lur11b, §6.6.3., §6.3.5.9.], M is the composite:

alg Θ Mod CAlg / Alg(Mod ) / b / / k k Catd∞ Catd∞ Catd∞

alg ⊗ Recall that, roughly speaking, the ∞-category Catd∞ consists of pairs (C ,A), where where C⊗ is a (not necessarily small) symmetric monoidal ∞-category and A is

Mod ⊗ an object in Alg(C). Similarly, the ∞-category Catd∞ consists of pairs (C , N ), where C⊗ is a symmetric monoidal ∞-category, and N is a (not necessarily small)

∞-category tensored over C⊗. In the diagram above, the ﬁrst arrow is the forgetful functor from E∞-algebras to E1-algebras, and the last arrow is the forgetful functor

⊗ that sends (C , N ) to the underlying ∞-category N . The functor Alg(Modk) →

alg ⊗ ⊗ Catd∞ is the inclusion of the subcategory consisting of pairs (C ,A), where C '

⊗ ⊗ Modk , and the morphisms are equivalent to the identity on C . Roughly speaking,

⊗ ⊗ Θb associates to (C ,A) the pair (C , RModA(C)).

9 Remark 2.3.1. The ∞-category M(A) = ModA is presentable. This follows, for instance, from A.4.2.3.7., and the fact the ∞-category of spectra is presentable.

Furthermore, for any f : A → B in CAlgk, the functor M(f) has a right adjoint, namely, the forgetful functor ModB → ModA.

Remark 2.3.2. More is true: the right adjoint M(B) → M(A) preserves all colimits. In particular, it is ω-accessible. It follows that M(f): M(A) → M(B) preserves ω-compact objects.

Remark 2.3.3. The categories ModA have a symmetric monoidal structure induced by the symmetric monoidal structure on S∞. Thus, M(A) can be viewed as commutative algebra object in PrL. In particular, it M(A) is a module over itself;

L L i.e., it can viewed as an object in ModM(A)(Pr ) =: PrA.

For A in CAlgk, functor M(θA): M(k) → M(A) induced by the structure morphism θA : k → A is symmetric monoidal (A.4.4.3.1), i.e., it is a morphism in

L CAlg(Pr ). Restricting the action of M(A) along M(θA), we get an induced

M(k)-module structure on M(A). Morphisms in CAlgk commute with the structure maps θ(−) by deﬁnition; this immediately implies that the functors M(f) are

M(k)-linear.

L L L L Recall that Prω (resp. Prω,k) denotes the subcategory of Pr (resp. Prk ) consisting of all compactly generated ∞-categories (resp. all Modk-linear compactly generated

∞-categories), and morphisms that preserve small colimits and ω-compact objects.

The preceding three remarks are summarized by the following lemma:

10 L Lemma 2.3.4. There exists a functor M1 : CAlgk → Prω,k such that the diagram below is (homotopy) commutative:

L L Prω,k / Prk D

M L L 1 Prω / Pr

CAlg / k M Catd∞

Notation 2.3.5. Let QCaﬀ denote the composite

op O M1 L L Aﬀk / CAlgk / Prω,k / Prk

aff For a derived affine scheme X in Affk, QC (X) is the ∞-category of quasicoherent sheaves on X. We would like to extend this functor to arbitrary derived stacks.

Let j : Affk → P(Affk) denote the Yoneda embedding. By the universal property of categories of presheaves, left Kan extension defines an equivelance

L Fun(Aﬀk, C) ' Fun (P(Aﬀk), C), for any C that admits all small colimits. Take

L op aﬀ C = (Prk ) , and let QCg denote that image of QC under the induced equivalence

op L R op L Fun(Aﬀk , Prk ) ' Fun (P(Aﬀk) , Prk ).

Notation 2.3.6. Let a : P(Aﬀk) → Stk be the localization functor, with right adjoint i, and let QC denote the composite QCg ◦ iop. We will often implicitly identify Stk with the essential image of the fully faithful functor i.

11 Deﬁnition 2.3.7. For a derived stack X over k, the ∞-category QC(X) is called the ∞-category of quasicoherent sheaves on X.

Remark 2.3.8. The ∞-category QC(X), is stable. This follows, for instance, from the fact that Modk-linear ∞-categories are stable [].

Remark 2.3.9. Let X be a discrete scheme. Then the relationship between the

∞-category QC and the abelian category Qcoh(X) of quasicoherent sheaves on X is as follows: there is a t-structure on QC such that QC♥ ' Qcoh(X), and we have an equivalence hQC ' D(Qcoh(X)).

Remark 2.3.10. The `etaletopology is subcanonical, so for A in CAlgk, Spec(A) is a derived stack. Furthermore, we have QCaﬀ (Spec(A)) = QC(Spec(A)).

Remark 2.3.11. Let F ∈ P(Aﬀk), and Φ : (j/F) → P(Aﬀk) be the functor that carries Spec(A) → F to Spec(A). Then we have a natural equivalence F' colim Φ.

Take F = i(X) for some derived stack X. Using the preceding remark and the fact that QCg preserves limits, we have

QC(X) ' lim(QCg ◦ Φop) ' lim QC(Spec(A)) Spec(A)→X

The diagram Φ : (j/F) → P(Aﬀk) is large, and consequently the description of

QC in 2.3.11 is not very useful in practice. In the category Stk, one often has small diagrams taking values in (the essential image of) Affk whose colimit is a given derived stack X. For example, if U• → X is an ètale(or flat) hypercover then we

op have colimn Un ' X. However, since i does not preserve limits, one has to work

12 much harder to show that QC(X) ' limnQC(Un). The following proposition is the homotopical/derived analogue of ﬂat descent for quasicoherent sheaves on ordinary schemes:

op L Proposition 2.3.12. The functor QC : Stk → Prk is a sheaf for the ﬂat hypertopology.

Proof. This is known to the experts; see e.g. [Lur04, Example 4.2.5] and [TV08,

Theorem 1.3.7.2]. It also follows from Theorem 3.1.1; indeed, it is the special case of that theorem when X' kˆ.

L L Remark 2.3.13. The inclusion Prω,k ⊆ Prk does not reﬂect limits in general:

L the limit in Prk (or equivalently in Catd∞) of a diagram of compactly generated categories need not be compactly generated. Following [BZFN10], we make the following deﬁnition:

Deﬁnition 2.3.14. A derived stack X is perfect if QC(X) is an ω-compactly gener-

perf ated ∞-category. Let Stk denote the full subcategory of Stk consisting of perfect stacks.

Proposition 2.3.15 ([To¨e07],[BZFN10]). The cartesian symmetric monoidal struc-

perf ture on Stk restricts to a symmetric monoidal structure on Stk . Furthermore, the

perf restriction of QC to Stk is symmetric monoidal. In other words, if X and Y are perfect stacks over k, then X ×k Y is perfect and we have a natural equivalence:

QC(X) ⊗Modk QC(Y ) ' QC(X ×k Y )

13 Furthermore, we have a natural equivalence

L QC(X ×k Y ) ' Funk(QC(X), QC(Y ))

One can associate with a derived stack X a functor:

QC op L MX : Aﬀk → Prk

S 7→ QC(X ×k S)

QC op QC Definition 2.3.16. The induced functor MX : Affk → Sb defined by MX (S) :=

QC ' (MX ) is the moduli of quasicoherent sheaves on X.

QC The drawback of MX is that it takes values in the category Sb of large spaces.

Since the category of derived stacks is as a localization of the category of presheaves on Aﬀk taking values in small spaces, it is more convenient to have a moduli functor that a priori takes values in small spaces. Fortunately, if X is a perfect stack, i.e., if

QC(X) is ω-compactly generated, then the large ∞-category QC(X) is determined

ω by the small ∞-category Perf(X) := QC(X) : we have QC(X) = Indω(Perf(X)).

This suggests that one should restrict attention to perfect stacks and replace the

QC pe functor MX by the functor MX :

pe MX

op ( Aﬀ / PrL / PrL / Cat k QC ω,k ω (−)ω ∞ MX

Here (−)ω is the functor that associates to compactly generated ∞-category the full

pe ω subcategory of ω-compact objects. Explicitly, we have MX (S) := M(X ×k S) =:

Perf(X ×k S).

14 pe Deﬁnition 2.3.17. The moduli of perfect complexes on X is the functor MX :

op pe pe ' ' Aﬀk → S deﬁned by MX (S) := (MX (S)) ' Perf(X ×k S) .

According to Proposition 2.3.15, we have the following alternate description of

QC the functor MX , when X is perfect:

QC MX (S) ' QC(X) ⊗QC(Spec(k)) QC(S)

QC pe Thus, MX and MX are manifestly invariants of the noncommutative shadow

QC(X) of the commutative space X. Furthermore, it suggests that for an arbitrary

perf op perf noncommutative space X , the functor MX : Aﬀk → S deﬁned by MX (S) :=

ω ' ((X ⊗k QC(S)) ) is a commutative reﬂection of X , which one might think of as the moduli of perfect branes on X . As above, it will be useful to keep track of somewhat more reﬁned information. To this end, let M˜ denote the composite functor:

M˜

L op id×O L id×M1 L L ⊗k * L Prω,k × Aﬀk / Prω,k × CAlgk / Prω,k × Prω,k / Prω,k

ω op L Notation 2.3.18. Composition with the functor (−) deﬁnes a functor Fun(Aﬀk , Prω,k) →

op perf Fun(Aﬀk , Cat∞). Deﬁne M to be the composite of M with this functor:

Mperf

L M op L * op Prω,k / Fun(Aﬀk , Prω,k) / Fun(Aﬀk , Cat∞)

' L ' Likewise, the functors (−) : Prω,k → Sb and (−) : Cat∞ → S which carry

op L an ∞-category to the underlying ∞-groupoid induce functors Fun(Aﬀk , Prω,k) →

ˆ op perf P(Affk) and Fun(Affk , Cat∞) → P(Affk). Let M and M be the functors which

15 make the following diagrams commutative:

M * L M op L ˆ Prω,k / Fun(Aﬀk , Prω,k) / P(Aﬀk)

Mperf

L Mperf op ) Prω,k / Fun(Aﬀk , Cat∞) / P(Aﬀk)

L We will write MX (A) for M(X )(A). For S in Aﬀk and X in Prω,k, one has the following explicit formulae summarizing the above deﬁnitions:

MX (S) = X ⊗k QC(S)

' MX (S) = (X ⊗k QC(S))

perf ω MX (S) = (X ⊗k QC(S))

perf ω ' MX (S) = ((X ⊗k QC(S)) )

Deﬁnition 2.3.19. Let X be a derived noncommutative space, i.e., an object of

L perf Prω,k. The moduli of perfect branes on X is the object MX in P(Aﬀk).

perf Remark 2.3.20. According to Theorem 3.1.1, the presheaf MX is in fact a derived stack.

Remark 2.3.21. It immediately from the deﬁnitions that for a commutative space

X we have:

QC MX ' MQC(X)

pe perf MX ' MQC(X)

16 pe Of course, there are similar statements for MX and MX .

Remark 2.3.22. Using A.6.3.4.1 and A.6.3.4.6, and the fact that for a commutative algebra A, left modules are canonically identiﬁed with right modules (upto a contracticble ambiguity), one sees that we have natural equivalences:

M (Spec(A)) ' X ⊗ Mod ' Mod (X ) ' FunL (Mod , X ) X Modk A A Modk A

17 Chapter 3

Brane Descent

3.1 Perfect Branes Descend

L op L Theorem 3.1.1. For any X in Prω,k, the functor MX : Aﬀk → Prω,k is a sheaf for the ﬂat hypertopology.

perf Corollary 3.1.2. The presheaf MX is a sheaf for the ﬂat hypertopology on Aﬀk.

∧ In particular, it deﬁnes an object in Stk, the hypercompletion of the ∞-topos Stk.

Before we proceed to outline the proof of the theorem, we make a simple observation that will play a central role in the way we organize our eﬀorts:

Lemma 3.1.3. Let (C, τ) be an ∞-site and let D, E be ∞-categories. Let F be a

D valued presheaf on C, and let D → E be a functor.

1. Assume that F is a sheaf for the τ-hypertopology, and that f preserves products

18 and limits of cosimplicial objects. Then f ◦ F is an E-valued sheaf for the τ-

hypertopology.

2. Assume that f ◦ F is a sheaf for the τ-hypertopology, and that f reﬂects

products and limits of cosimplicial objects. Then F is a sheaf for the τ-

hypertopology.

Proof. Follows immediately from the deﬁnitions.

The proof of Theorem 3.1.1 will occupy the rest of this chapter. We will proceed in several steps:

L L 1. We will see that the forgetful functor Prω,k → Prk reﬂects products and

limits of cosimplicial objects (Lemmas 3.2.6 and 3.2.7), while the forgetful

L functor Prk → Catd∞ preserves and reﬂects all limits (Lemma 3.2.2).

2. Step 1. and Lemma 3.1.3 reduce the problem to showing that the composite

] functor MX :

op MX L Aﬀk / Prω,k / Catd∞

is a sheaf for the ﬂat hypertopology. Standard techniques from the theory of

cohomological descent (Theorem 3.3.3) allow us to further reduce the problem

] to showing that MX is a sheaf for the ﬂat topology; i.e., it will suﬃce to

consider only 1-coskeletal hypercovers.

3. We will appeal to a corollary (Proposition 3.4.2) of the Barr-Beck-Lurie theo-

\ rem (Theorem 3.4.1) to show that MX is a sheaf for the ﬂat topology (Propo-

19 sition 3.5.2). To apply this corollary, we must show that certain “Beck-

Chevalley” conditions (see loc. cit) are satisﬁed: this follows from a base

change lemma for branes (Proposition 3.5.3).

3.2 Gluing Compact Objects

The fact that compact objects do not glue in general, or equivalently, the fact that

L the functor Prω ⊆ Catd∞ does not preserve and reﬂect limits, will be a recurring theme through this paper. Our purpose in this section is to note this fact, and to point out conditions on a diagram category K that ensure that the inclusion

L L Prω ⊆ Pr does preserve and reﬂect limits of shape K. In particular we will see

L L that the forgetful functor Prω,k → Prk reﬂects products and limits of cosimplicial objects (Lemmas 3.2.6 and 3.2.7)

We begin with the some observations about the relationship between linear structures and limits, which will be used in the sequel, and which, together with the lemmas mentioned above, complete Step 1 in the proof of Theorem 3.1.1 as outline in §3.1.

L L L Lemma 3.2.1. The forgetful functor Prω,k := ModModk (Prω ) → Prω preserves and reﬂects all small limits.

Proof. This follows from the general statement that the forgetful functor from a module category to the underlying category preserves and reﬂects all small limits

20 A.4.2.3.3.

L L Lemma 3.2.2. The forgetful functor ik : Prk → Catd∞ preserves and reﬂects all limits.

L L L L L Proof. We have ik = i ◦ πk , where i : Pr → Catd∞ is the natural inclusion,

L L L L L and πk : Prk := ModModk (Pr ) → Pr is the forgetful functor. The functor πk preserves and reﬂects all limits by A.4.2.3.1. According to T.5.5.3.13., the categories

L L Pr and Catd∞ admits all small limits and i preserves all small limits. The fact that iL is conservative, together with the following lemma, implies that iL reﬂects all small limits.

Lemma 3.2.3. Let f : C → D be a functor between ∞-categories, and let K be a simplicial set. Assume that C admits limits of diagrams of shape K, that f preserves these limits, and that f is conservative. Then f reﬂects limits of shape K.

Proof. Let φ : K/ → C be a diagram, and suppose that f ◦ φ : K/ → D is a limit diagram. Since C admits limits of diagrams of shape K, there exists a limit

/ diagram ψ : K → C with ψ|K ' φ|K . By the deﬁnition of limits, there is a

/ morphism α : φ → ψ in FunK (K , C). We will complete the proof by showing that

α is an equivalence. Since f is conservative, it will suﬃce to show that f(α) is an equivalence.

Since f preserves K-limit diagrams, f ◦ ψ is also a limit diagram. Furthermore,

/ we have (f ◦ ψ)|K = (f ◦ φ)|K . Since the restriction FunK (K , D) → FunK (K, D) '

21 {f ◦ φ} is a trivial Kan ﬁbration, we have a natural equivalence β : f ◦ ψ → f ◦ φ in

/ / FunK (K , D). Using the fact that FunK (K , D) → FunK (K, D) is a trivial ﬁbration again, we conclude that β ◦f(α) is an equivalence. By the two out of three property, f(α) is an equivalence.

/ L Lemma 3.2.4. Let K be a simplicial set, and let ν : K → Prω be a diagram

[ / classifying a coCartesian ﬁbration ν : V → K . Let ψx : V∞ → Vk be the natural

L functor (see....). Let i : Prω → Catd∞ be the natural inclusion. Assume that :

(i) i ◦ ν is a limit diagram.

(ii) An object X in V∞ is compact if and only if ψx(X) is compact for all x in K.

Then the following hold:

(1) ν is a limit diagram.

ω / (2) The induced diagram (−) ◦ ν : K → Cat∞ is a limit diagram.

Proof. Assume that the hypotheses (i) and (ii) are satisﬁed. We will now prove

L L (1). The limit V∞ of ν|K in Prk is charaterized upto equivalence by Funk(W, V∞) '

L L L limx∈K Funk(W, Vx), for any W in Prk . Our hypothesis that ν takes values in Prω,k implies that each of the functors ψx preserves ω-compact objects, and therefore this

L L equivalence restricts to a fully faithful functor Funω,k(W, V∞) → limx∈K Funω,k(W, Vx),

L L where Funω,k(−, −) ⊆ Funk(−, −) denotes the full subcategory of functors that preserve ω-compact objects. We will show tha this functor is essentially surjective.

22 L [ ] / Now let W ∈ Prω,k, and let w : W → K be a cocartesian ﬁbration classiﬁed by

/ L ] the constant functor K → Prω,k that sends every object to W, and let σx : W∞ →

] Wx denote the functor (equivalence) induced by the unique morphism ∞ → x. Let

L ] X ∈ limx∈K Funω,k(W, Vx), and let χ : W|K → V|K be the corresponding cocartesian

] section. Note that χx : Wx → Vx preserves ω-compact objects for all x in K. The

L L equivalence Funk(W, V∞) ' limx∈K Funk(W, Vx) implies that χ extends to a map

] L χ : W → V deﬁned by a cocartesian section such that χ∞ ∈ Funk(W∞, V∞). We have natural equivalences χx ◦ σx ' ψx ◦ χx, since χ is cocartesian.

] Let X ∈ W∞ be a compact object. Then we have an equivalence χ0(σ0(X)) '

ψ0(χ∞(X)) in V0. Since σ0 is an equivalence and σ0 preserves compact objects, we conclude that ψ0(χ∞(X)) is compact.Condition (ii) now implies that χ∞(X) is com-

L ] 0 L pact. Thus χ∞ ∈ Funω,k(W∞, V∞), so χ deﬁnes an element X in Funω,k(W, V∞) that maps to X . This proves essential surjectivity of the natural functor mapping

L L Funω,k(W, V∞) to limx∈K Funω,k(W, Vx).

L L So we have Funω,k(W, V∞) ' limx∈K Funω,k(W, Vx). This equivalence charac-

L / L terizes V∞ as a limit of ν|K in Prω,k, so φ : K → Prω.k is a limit diagram. This proves (1).

The following lemma is the only simple observation that one needs to add to

L L the results of “T” to prove that Prω,k → Prk reﬂects limits of cosimplicial objects.

/ L Lemma 3.2.5. Let K be an ∞-category and let ν : K → Prω,k be a diagram, classifying a coCartesian ﬁbration ν[ : V → K/. Assume that:

23 / L (i) The induced diagram ν : K → Prk is a limit diagram.

(ii) There is an object 0 in K such that for every object x in K there is an edge

fx : 0 → x.

Then the following are equivalent for an object X in V∞:

(1) X is a compact object of V−∞.

(2) ψx(X) is a compact object of Vx for all objects x in K.

(3) ψ0(X) is a compact object of V0.

L Proof. (1) ⇒ (2) ⇒ (3) is trivial: indeed, the hypothesis that ν takes values in Prω,k implies that for any x, the functor ψx preserves colimits and ω-compact objects.

We will complete the proof by showing that (3) ⇒ (1).

Suppose that ψ0(X) is compact. Let {Aλ}λ∈Λ be a set of objects in V∞, and

` L let A = λ∈Λ Aλ. Since ν is a limit diagram in Prk (or equivalently, in Catd∞ by

[ Lemma 3.2.2), we may identify V∞ with cocartesian sections of ν . Let χ, αλ, α in

/ FunK/ (K , V) be cocartesian sections with χ∞ = X, αλ,∞ = Aλ and α∞ = A. By deﬁnition of the ψx’s, we have ψx(X) = χx, ψx(Aλ) = αλ,∞ and ψx(A) = αx. Note ` that since the functors ψx preserve all colimits, we have αx ' λ∈Λ αx,λ, where the coproduct is computed in Vx.

Let t : X → A be a morphism in V∞, and let θ : χ → α be the corresponding morphism of cocartesian sections. Let φx : V0 → Vx be the functor induced by fx : 0 → x. Since χ, α and αλ, λ ∈ Λ are cocartesian, and since φx preserves

24 [ / L colimits (recall that ν is classiﬁed by a diagram K → Prk ), we have a commutative diagrams:

φx(θ0) ∼ ` φx(χ0) / φx(α0) o λ∈Λ φx(αx,0)

o o o θx ∼ ` χx / αx o λ∈Λ αx,λ

Since χ0 is a compact object of V0 by hypothesis, there exists an ω-small set

ω ` ` Λ ⊂ Λ such that θ0 : χ0 → α0 ' λ∈Λ α0,λ factors through λ∈Λω α0,λ. From ` the diagram above, we see that this implies that θx factors through λ∈Λω αx,λ for ` ` all x. Thus, θ factors through λ∈Λω αλ, or equivalently, t : X → λ∈Λ Aλ factors ` through λ∈Λω Aλ. The category V∞ is stable (because it is admits a k-linear structure, for instance). Therefore (A.1.4.5.1), this proves that X is compact.

Lemma 3.2.6. Let K be an ∞-category satisfying the hypothesis (ii) of Lemma

L L 3.2.5. Then forgetful functor π : Prω,k → Prk reﬂects limits of diagrams of shape

K. In particular, π reﬂects limits of cosimplicial objects.

Proof. This follows immediately from Lemma 3.2.1 and Lemma 3.2.5, since N(∆) satisﬁes the hypotheses of Lemma 3.2.5.

L L Lemma 3.2.7. The functor π : Prω,k → Prk reﬂects ω-small products.

Proof. This follows from Lemma 3.2.1, Lemma 3.2.4 and the following fact: If {Cα} is a ﬁnite family of ∞-categories with product C, then an object X in C is ω-compact as soon as its image in each Cα is ω-compact (T.5.3.4.10).

25 3.3 Bootstrapping: from Covers to Hypercovers

L ] op Notation 3.3.1. Let X be an object in Prω,k. The functor MX : Aﬀk → Catd∞ is deﬁned by the commutativity of the following diagram:

op MX L Aﬀk / Prω,k FF FF FF M] FF X F# Catd∞

• op Definition 3.3.2. Let U : N(∆+ ) → Affk be an augmented simplicial derived affine scheme. Put U := U −1. We will say that U • is of cohomological descent if

] ] • L the natural map MX (U) → lim M (U ) is an equivalence for every X in Prω,k. We will say that U • is universally of cohomological descent, if any base change of U • is of cohomological descent.

0 ˇ ˇ For a map f : U → U in Aﬀk, the Cech nerve of f, denoted C(f), is the

0-coskeleton of f computed in (Aﬀk)/U . Let P denote the class of morphisms in ˇ Aﬀk whose Cech nerve is universally of cohomological descent.

Theorem 3.3.3. P-hypercovers are universally of cohomological descent.

3.4 The Barr-Beck-Lurie Theorem

The involution on the (∞, 2)-category of ∞-categories that takes an ∞-category to its opposite, interchanges left adjoints with right adjoints, and monads with comon-

26 ads. Consequently, every theorem about monads has a dual comonadic analogue.

In particular, we have the following comonadic analogue of the Barr-Beck-Lurie theorem.

Theorem 3.4.1 (Lurie [Lur11b, Theorem 6.2.2.5]). Let f : C → D be an ∞-functor that admits a right adjoint g : D → C. Then the following are equivalent:

1. f exhibits C as comonadic over D.

2. f satisﬁes the following two conditions:

(a) f is conservative, i.e., it reﬂects equivalences.

(b) Let U be a cosimplicial object in C, which is f-split. Then U has a limit

in C, and this limit is preserved by f.

In practice, we will use the following consequence of the comonadic Barr-Beck-

Lurie theorem, which is the dual version of A.6.2.4.3:

• Proposition 3.4.2. Let C : N(∆+) → Catd∞ be a coaugmented cosimplicial ∞- category, and set C := C−1. Let f : C → C0 be the evident functor. Assume that:

1. The ∞-category C admits totalizations of f-split cosimplicial objects, and those

totalizations are preserved by f.

2. (Beck-Chevalley conditions) For a morphism α :[m] → [n] in ∆+, let α˜ be

the morphism deﬁned by α˜(0) = 0 and α˜(i) = α(i − 1) for 1 ≤ i ≤ m. Then

27 for every α, the diagram

d0 Cm / Cm+1

α α˜ d0 Cn / Cn+1 is right adjointable.

• Then the canonical map θ : C → lim∆ C admits a fully faithful right adjoint. If f is conservative, then θ is an equivalence.

3.5 Flat Hyperdescent: the proof

In this section, we will complete Step 4. by showing that MX deﬁnes a Catd∞-valued hypersheaf (Proposition 3.5.2). Finally, we will combine this with the results of the previous sections to prove the main theorem of this chapter (Theorem 3.1.1).

L ] op Notation 3.5.1. Let X be an object in Prω,k. The functor MX : Aﬀk → Catd∞ is deﬁned by the commutativity of the following diagram:

op MX L Aﬀk / Prω,k FF FF FF M] FF X F# Catd∞

\ \ ] The functor MX : CAlgk → Catd∞ is deﬁned to be the composite MX = MX ◦O,

op where O : Aﬀk → CAlgk is the tautological equivalence. Explicitly, we have

\ MX (A) ' X ⊗Modk ModA ' ModA(X ) (see Remark 2.3.22).

28 Proposition 3.5.2. Let X be a compactly generated k-linear ∞-category. The

] Catd∞ valued presheaf MX on Affk defined in Notation 3.5.1 is a sheaf for the flat hypertopology.

The proof of the proposition will be given at the end of this section. In light

\ ˇ of Theorem 3.3.3, the essential point is to verify that MX carries the Cech nerve

0 of a faithfully ﬂat morphism f : A → A to a limit diagram in Catd∞. For this, we will appeal to Proposition 3.4.2. We begin by collecting together some preliminary results that will allow us the verify the hypotheses of that Proposition.

The lemma that follows facilitates the veriﬁcation of the “Beck-Chevalley conditions” of Proposition 3.4.2:

L \ Lemma 3.5.3. (Base change). For any X in Prω,k, the functor MX : CAlgk →

Catd∞ of Notation 3.5.1 carries cocartesian squares to right adjointable squares.

Proof. The proof is essentially the same as [TV08, Prop 1.1.0.8]. Let

p A / A0

f f 0 B / 0 p0 B be a cocartesian square in CAlgk, and let

p∗

x p∗ ModA(X ) / ModA0 (X )

f ∗ f 0∗ 0 p∗ x ModB(X ) / ModB0 (X ) p0∗

29 \ 0 be the diagram in Catd∞ by induced by MX . Here, for a morphism p : A → A in

∗ \ CAlgk, p := MX (p) = MX (p) = A ⊗A0 (−), and p∗ : ModA0 (X ) → ModA(X ) is the forgetful functor, which is right adjoint to p∗ (see Remark ??).

Let M ∈ ModA0 (X ). To prove the lemma, we must show that the natural morphism

∗ 0 0∗ νM : f p∗M → p∗f M is an equivalence. This follows from the following peculiarity of commutative algebras: pushouts coincide with tensor products in CAlgk, i.e., we

0 0 ` 0 have B ' A A B ' A ⊗A B. Consequently, we have a chain of equivalences:

0 0 M ⊗A0 B −→˜ M ⊗A0 (A ⊗A B)−→ ˜ M ⊗A B

which, as the reader will readily check, is a homotopy inverse of νM .

• ˇ Let A : N(∆+) → CAlgk be the Cech nerve of a faithfully ﬂat morphism

0 L f : A → A , and let X ∈ Prω,k. In order to deduce from the base change lemma

\ • that the cosimplicial ∞-category MX (A ) satisﬁes the Beck-Chevalley conditions, we need following simple observation:

0 • Lemma 3.5.4. Let f : A → A be a morphism in CAlgk, and let A : N(∆+) →

ˇ ˇ CAlgk be the Cech nerve C(f) := cosk0(f). For a morphism α in ∆+, let α˜ be the morphism deﬁned in Proposition 3.4.2. Then for each α :[m] → [n] in ∆+, the following diagram is right adjointable:

d0 Am / Am+1 (†)

A(α) A(˜α) d0 An / An+1

30 • n n+1 0 `n+1 0 0 Proof. Since A is the 0-coskeleton of f, we have A ' ⊗A A ' A A , and d is

`n+1 0 0 ` `n+1 0 the inclusion of the summand A A → A A( A A ). It follows immediately that the square (†) is cocartesian for any α :[m] → [n].

• ˇ Lemma 3.5.6 almost says that if A : N(∆+) → CAlgk is the Cech nerve of a ﬂat

0 \ • morphism f : A → A in CAlgk, then MX (A ) satisﬁes condition 1. of Proposition

3.4.2. In preparation for the proof of Lemma 3.5.6, we prove the following special case of that lemma:

∗ Lemma 3.5.5. Let f : A → B be a morphism in CAlgk, and let f : ModA →

∗ ModB be the functor deﬁned by f (M) := B ⊗A M. Assume that f is ﬂat. Then f ∗ preserves totalizations of f ∗-split cosimplicial objects.

• ∗ Proof. Let M ∈ Fun(N(∆), ModA) be an f -split cosimplicial module. We wish to show that the natural map

• • B ⊗A |M | −→ |B ⊗A M | (∗)

is an equivalence. Since the forgetful functor, ModB → S∞ is conservative, White- head’s theorem implies that it will suﬃce to show that the induced morphism on πn is an isomorphism for all n ∈ Z. We will use the Bousﬁeld-Kan spectral sequence to compute the homotopy groups, and show that we have an isomorphism on the

E2 page.

p,q −p • p+q There is a Bousﬁeld-Kan spectral sequence with E2 = π πqM , and E∞ =

• −p • πp+q|M |. Here π πqM is the (−p)th cohomotopy group of the cosimplicial

31 • abelian group πqM . Since B is a ﬂat A-module, π0B is a ﬂat π0A-module. So

p,q −p • p+q we have an induced spectral sequence with E2 = π0B ⊗π0A π πqM and E∞ =

• • π0B ⊗π0A πp+q|M |. Finally, since B is ﬂat over A, we have πp+q(B ⊗A |M |) '

• π0B ⊗π0A πp+q|M |. So, in summary, we have a spectral sequence

p,q −p • • E2 = π0B ⊗π0A π πqM =⇒ πp+q(B ⊗A |M |)

Similarly, we have a Bousfeld-Kan spectral sequence for the right hand side of (∗),

p,q −p • p+q • with E2 = π πq(B ⊗A M ) and E∞ = πp+q|B ⊗A M |. Using the ﬂatness of B

−p • −p • −p • over A again, we have π πq(B⊗AM ) ' π (π0B⊗π0AπqM ) ' π0B⊗πoAπ πqM .

So the spectral sequence becomes

p,q −p • • E2 = π0B ⊗π0A π πqM =⇒ πp+q|B ⊗A M |

Thus, the E2 pages of the spectral sequences for the left and right hand sides of (∗) coincide. To complete the proof, it will suﬃce to show that these spectral sequences degenerate. Let N → N • be a split coaugmented cosimplicial B-module. Then

• πqN → πqN is a split coaugmented cosimplical abelian group for all q, and so we

−p • 0 • • • have π πqN = 0 for p 6= 0, and π πqN = πqN. Applying this to N := B ⊗A M ,

• and N = |B ⊗A M |, we see that both the spectral sequences above degenerate at the E2 page.

Lemma 3.5.6. Let f : A → B be a ﬂat morphism in CAlgk, and let X be an

L \ object of Prω,k. Then the category MX (A) admits all small limits, and the functor

\ \ \ \ MX (f): MX (A) → MX (B) preserves totalizations of MX (f)-split cosimplicial

32 objects.

\ Proof. The ﬁrst statement is clear: the ∞-category MX (A) ' X ⊗Modk ModA is presentable (2.3.1), and in particular admits all small limits and colimits.

Since X is a compactly generated k-linear ∞-category, Proposition ?? says

ω that the restricted Yoneda embedding gives an equivalence X' Funk(X , Modk).

\ ω Furthermore, for any A in CAlgk, we have MX (A) ' ModA(Funk(X , Modk)) '

ω Funk(X , ModA).

Let X ∈ X ω be an object classiﬁed by a morphism of small k-linear ∞-categories

ω ψX : Bk → X . Using the natural identiﬁcations Funk(Bk, ModA) ' Modk⊗A '

∗ ω ModA, we see that pullback along ψX deﬁnes a functor ψX,A : Funk(X , ModA) →

ModA.

\ Let f : A → B be a morphism in CAlgk. Under the identiﬁcation MX (A) '

ω \ ∗ Funk(X , ModA), the functor MX (f) corresponds to the functor f ◦ (−), where

∗ \ ω f := M1(f) = B ⊗A (−). Furthermore, for every X in X , we have a homotopy commutative diagram in Catd∞

∗ ω f ◦(−) ω Funk(X , ModA) / Funk(X , ModB)

∗ ∗ ψX,A ψX,B

Mod / Mod A f ∗ B

Now suppose that f : A → B is a ﬂat morphism. Let M • be a cosimplicial

33 ω ∗ • object in Funk(X , ModA), for which the induced cosimplicial object f M is split.

To complete the proof of the lemma, it will suﬃce to show that the natural morphism

∗ • ∗ • νM • : f (limM ) → limf (M ) is an equivalence.

∗ Since the family of functors {ψX,B}X∈X ω is jointly conservative, it is enough

ω ∗ to show that for each X in X , the morphism ψX,B(νM • ) is an equivalence. The

∗ commutativity of the diagram above, together with the fact that the functors ψX,− commute with all limits, implies that this equivalent to showing that the natural

∗ ∗ ∗ ∗ • morphism ν ∗ • : f (limψ ) → limf (ψ (M )) is an equivalence for every ψX,A(M ) X,A X,A object X in X ω.

∗ ∗ • Note that the cosimplicial B-module f (ψX,A(M )) is split, being the image

∗ ∗ • under ψX,B of the split cosimplicial object f (M ). Applying Lemma 3.5.5 to the

∗ • A-module ψ (M ), we see that the morphism ν ∗ • is an equivalence for every X,A ψX,A(M )

X is X ω.

Lemma 3.5.7. Let f : A → B be a faithfully ﬂat morphism in CAlgk, and let X be

L \ \ \ an object in Prω,k. Then the functor MX (f): MX (A) → MX (A) is conservative.

∗ Proof. We will retain the notation from Lemma 3.5.6. Since the family {ψX,B}X∈X ω

\ ∗ \ is jointly conservative, MX (f) is conservative if and only if {ψX,B ◦MX (f)}X∈X ω =

∗ ∗ ∗ {f ◦ ψX,A}X∈X ω is a jointly conservative family. Using the fact that {ψX,A}X∈X ω is

∗ jointly conservative, we see that this is equivalent to asking that f : ModA → ModB

∗ is conservative. Since ModB is stable, this is equivalent to asking that f reflects zero objects. But this is what is means for a flat morphism to be faithfully flat.

34 L \ Lemma 3.5.8. Let X be an object in Prω,k. The functor MX preserves ﬁnite products.

Proof. This is formal. Let Ai, i = 1, 2, be commutative k-algebras, and let A :=

A1 × A2. Consider the adjunction

& \ \ \ MX (A) o MX (A1) × MX (A2)

\ \ The left adjoint, which is the natural morphism MX (A) → lim MX (Ai), carries

M ∈ ModA(X ) to (M ⊗A A1,M ⊗A A2). The right adjoint carries (M1,M2) to

p1∗M1 × p2∗M2, where pi : A → Ai is the natural projection, and pi∗ : ModAi (X ) →

ModA(X ) is the forgetful functor. We will show that the unit and counit of this adjunction are equivalences.

The ∞-category ModA(X ) is stable, and therefore we have natural equivalences

M ⊕ N ' M × N for M, N in ModA(X ). Using this, together with the projection formula, we have, for M in ModA(X ):

p1∗(M ⊗A A1) × p2∗(M ⊗A A2) ' M ⊗A p1∗A1 × M ⊗A p2∗A2

' (M ⊗A p1∗A1) ⊕ (M ⊗A p2∗A2)

' M ⊗A (p1∗A1 ⊕ p2∗A2)

' M ⊗A A

' M

One checks that the composite morphism p1∗(M ⊗A A1) × p2∗(M ⊗A A2) → M is inverse to the unit of the adjunction, proving that the unit is an equivalence.

35 For Mi in ModAi (X ), we have natural equivalences pi∗Mi ⊗A Ai ' Mi and pi∗Mi ⊗A

Aj ' 0 for i 6= j. From this, one immediately deduces that the natural maps

(p1∗M1 × p2∗M2) ⊗A Ai ' (p1∗M1 ⊗A Ai) ⊕ (p2∗M2 ⊗A Ai) → Mi are equivalences.

This shows that the counit is an equivalence.

We are now in a position to prove the main proposition of this section.

L ] Proof of Proposition 3.5.2. Let X be in Prω,k. We must show that MX preserves

ﬁnite products and carries ﬂat hypercover to limit diagrams. By virtue of Lemma

3.5.8, only the second statement remains to be proved. In view of Theorem 3.3.3,

] ˇ it will suffice to show that MX carries the Cech nerve of a flat morphism in Affk to a limit diagram in Catd∞.

• op • • Let U : N(∆+ ) → Aﬀk be a ﬂat hypercover, and let A := O(U ) : N(∆+) →

−1 \ • CAlgk. Put A := A . Lemma 3.5.6 says that the associated diagram MX (A ):

op N(∆+ ) → Catd∞, satisiﬁes condition 1. of the corollary of the Barr-Beck-Lurie theorem, Proposition 3.4.2. The base change lemma for branes (Lemma 3.5.3), together

\ • with Lemma 3.5.4, implies that MX (A ) satisﬁes condition 2. of Proposition 3.4.2.

\ \ 0 Finally, Lemma 3.5.7 tells us that the natural map MX (A) → MX (A ) is conser-

\ \ n vative. Thus, by Proposition 3.4.2, the natural map MX (A) → limM X (A ) is an equivalence.

Finally, we can now prove the main theorem of this chapter:

Proof of Theorem 3.1.1. By virtue of Lemmas 3.2.2, 3.2.6 and 3.2.7, the natural

36 L inclusion Prω,k → Catd∞ reﬂects ﬁnite products and limits of cosimplicial objects.

The theorem now follows from Proposition 3.5.2 and Lemma 3.1.3.

We need to simple observation before we can write down the proof of Corollary

3.1.2:

' ' Lemma 3.5.9. The functors (−) : Catd∞ → Sb and (−) : Cat∞ → S, which carry an ∞-category to the maximal ∞-groupoid that it contains, preserve all limits.

Proof. The functor (−)' is a right adjoint, and therefore preserves all limits: the

' natural inclusion π≤∞ of spaces into ∞-categories is left adjoint to (−) .

Proof of corollary.3.1.2. The flat hypertopology is finer than the étalehypertopol- ogy. Thus, the second statement follows immediately from the first statement and the fact that the hypercomplete objects in an ∞-topos of sheaves on an ∞-site (C, τ) are precisely those presheaves that are sheaves for the τ-hypertopology [Lur11a,

Prop. 5.12].

• Let A : N(∆+) → CAlgk be a ﬂat hypercover. By virtue of Propostion 3.5.2,

• L the induced functor MX (A ) : N(∆+) → Prω,k satisﬁes the hypotheses of Lemma

• ω 3.2.4. It follows that the induced diagram (MX (A )) is a limit diagram in Cat∞.

perf • • ω ' Applying Lemma 3.5.9, we see that MX (A ) := ((MX (A )) ) is a limit diagram.

perf perf Thus, MX carries ﬂat hypercovers to limit diagrams in S. The proof that MX preserves ﬁnite products is similar.

37 Chapter 4

Geometricity

4.1 Geometric Stacks

4.2 The Artin-Lurie Criterion

4.3 Inﬁnitesimal theory: Brane Jets

4.4 Dualizability implies Geometricity

4.5 A Proper Counterexample

38 Chapter 5

Moduli of 2D-TFTs

5.1 The Cobordism Hypothesis

5.2 Moduli of Noncommutative Calabi-Yau Spaces

5.3 Geometricity

5.4 Unobstructedness

5.5 Frobenius Structures: from TFTs to CohFTs

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