Moduli Problems in Derived Noncommutative Geometry

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Moduli Problems in Derived Noncommutative Geometry

Pranav Pandit

A Dissertation

in

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

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ABSTRACT

Moduli Problems in Derived Noncommutative Geometry

Pranav Pandit Tony Pantev, Advisor

We study moduli spaces of boundary conditions in 2D topological field theo-ries. To a compactly generated linear∞-categoryX, 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 geometric-ity of MX. If X is fully dualizable (smooth and proper), then MX is geometric,

recovering a result of To¨en-Vacqiue from a new perspective. Properness of X does not imply geometricity in general: perfect complexes with support is a counterex-ample. 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

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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.

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1.1 Brave New Mathematics

For several centuries,scientificthought has been conflated withreductionistparadigm. 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

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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.

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• “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. S) theb ∞-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. Cat∞) thed ∞-category of small (resp. large) ∞-categories.

• CatEx

∞ (resp. Cat∨∞) the subcategory of Cat∞ consisting of small stable (resp. idempotent complete stable) ∞-categories and exact functors.

• PrL (resp. PrR) the subcategory of Cat∞d consisting of presentable

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

• PrL

κ (resp. PrRκ) the subcategory ofPrL(resp. PrR) consisting ofκ-compactly

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

are κ-accessible). See T.5.5.7.

• PrL

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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. Fun}R_(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. Fun}RAd_(C_,_{D)) the full subcategory of Fun(C}_,_D)

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

• FunL

κ(C,D) (resp. FunRκ(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.), ifC andDare presentable, then we have natural equivalences FunL_(C_,_D)_'_FunLAd_(C_,_{D) and Fun}R_(C_,_D)_'_FunRAd_(C_,_D).

Notation 1.5.5 (Categorical hom and tensor). The categories PrL _and _P_rL κ are

symmetric monoidal, and the inclusion functor PrL

κ ⊆ PrL 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 Cat∞. The functorsd

FunL₍₋_,_{−) (resp. Fun}L

κ(−,−)) defines an internal hom on PrL (resp. PrLκ).

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Alg_O(C), we will write ModO_A(M) for the ∞-category of A-modules in M. When O is the commutative operad CAlg(C) := AlgO(C), and ModA(M) := Mod

O

A(M).

When O is the associative operad (A.4.1.1.), Alg(C) := Alg_O(C). We will use the abbreviations CAlg_A := 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 equiv-alences. 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 tohC 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 derived affine schemes. By definition Affk := CAlgop. We will denote by Spec :

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Notation 1.5.10(Diagrams and limits). ForK 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→ ∞).

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Chapter 2 Brane Moduli

V

2.1 Commutative spaces

2.2 Noncommutative spaces

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There is a functorM: 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.], Mis the composite:

CAlgk //Alg(Modk) //_Cat_dalg ∞

b

Θ _//

d

CatMod_∞ //Cat∞d

Recall that, roughly speaking, the∞-categoryCatd

alg

∞ consists of pairs (C⊗, A), where where C⊗ _{is a (not necessarily small) symmetric monoidal} _{∞-category and} _A _is an object in Alg(C). Similarly, the ∞-category Catd

Mod

∞ consists of pairs (C⊗,N), where C⊗ _{is a symmetric monoidal} _{∞-category, and}_N _{is a (not necessarily small)} ∞-category tensored overC⊗_{. 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(Mod}

k) →

d

Catalg_∞ is the inclusion of the subcategory consisting of pairs (C⊗_{, A}_{), where} _C⊗ _'

Mod⊗_k, and the morphisms are equivalent to the identity on C⊗_{. Roughly speaking,}

b

Θ associates to (C⊗_{, A}_{) the pair (C}⊗_,_RMod

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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

in-duced by the symmetric monoidal structure on S∞. Thus, M(A) can be viewed as

commutative algebra object in PrL. In particular, itM(A) is a module over itself; i.e., it can viewed as an object in ModM(A)(PrL) =: PrLA.

For A in CAlgk, functor M(θA) : M(k) → M(A) induced by the structure

mor-phism θA : k → A is symmetric monoidal (A.4.4.3.1), i.e., it is a morphism in

CAlg(PrL_). _{Restricting the action of} _M(_A_{) along} _M(_θ

A), we get an induced

M(k)-module structure on M(A). Morphisms in CAlgk commute with the

struc-ture maps θ(−) by definition; this immediately implies that the functors M(f) are M(k)-linear.

Recall thatPrL

ω(resp. Prω,Lk) denotes the subcategory ofPrL(resp. PrkL) consisting of all compactly generated∞-categories (resp. all Modk-linear compactly generated ∞-categories), and morphisms that preserve small colimits and ω-compact objects.

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Lemma 2.3.4. There exists a functor M1 : CAlgk → Prω,Lk such that the diagram

below is (homotopy) commutative:

Pr_ω,L_k //

PrL

k

PrL

ω //PrL

CAlgk M1

DD

M //Catd∞

Notation 2.3.5. Let QCaff denote the composite

Affop_k O //CAlgk M1 //PrL

ω,k //PrLk

For a derived affine scheme X in Affk, QCaff(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 Fun(Affk,C) ' FunL(P(Affk),C), for any C that admits all small colimits. Take C = (Pr_kL)op, and let gQC denote that image of QCaff under the induced equivalence

Fun(Affop_k,PrL

k)'FunR(P(Affk)op,PrLk).

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

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Definition 2.3.7. For a derived stackX over k, the ∞-category QC(X) is called

the ∞-category of quasicoherent sheaves onX.

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 onX

is as follows: there is at-structure on QC such that QC♥'Qcoh(X), and we have

an equivalence hQC'D(Qcoh(X)).

Remark 2.3.10. The `etale topology 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 equivalenceF 'colim Φ. TakeF =i(X) for some derived stack X. Using the preceding remark and the fact

that QC preserves limits, we haveg

QC(X)'lim(gQC◦Φop)' lim

Spec(A)→XQC(Spec(A))

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

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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:

Proposition 2.3.12. The functor QC : Stop_k → Pr_kL is a sheaf for the flat hyper-topology.

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.

Remark 2.3.13. The inclusion PrL

ω,k ⊆ PrLk does not reflect limits in general: the limit in PrL

k (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 stackXisperfect if QC(X) is anω-compactly gener-ated ∞-category. LetStperf_k denote the full subcategory ofStk consisting of perfect stacks.

Proposition 2.3.15([To¨e07],[BZFN10]). The cartesian symmetric monoidal struc-ture on Stk restricts to a symmetric monoidal structure onStperf_k . Furthermore, the

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

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Furthermore, we have a natural equivalence

QC(X×kY)'FunLk(QC(X),QC(Y))

One can associate with a derived stack X a functor:

MQC_X : Affop_k → Pr_kL

S 7→QC(X×kS)

Definition 2.3.16. The induced functorMQC_X : Affop_k →Sbdefined byMQC_X (S) :=

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

The drawback ofMQC_X is that it takes values in the category Sbof large spaces.

Since the category of derived stacks is as a localization of the category of presheaves on Affktaking values insmall spaces, it is more convenient to have a moduli functor thata priori takes values in small spaces. Fortunately, ifX is aperfect stack, i.e., if QC(X) isω-compactly generated, then the large ∞-category QC(X) is determined by the small ∞-category Perf(X) := QC(X)ω_{: we have QC(}_X_{) = Ind}

ω(Perf(X)).

This suggests that one should restrict attention to perfect stacks and replace the functor MQC_X by the functorMpe_X:

Affop_k

Mpe_X

((

MQC_X

//_P_rL

ω,k //PrLω ₍₋₎ω//Cat∞

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

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Definition 2.3.17. The moduli of perfect complexes on X is the functor Mpe_X : Affop_k → S defined by Mpe_X(S) := (M_Xpe(S))' 'Perf(X×kS)'.

According to Proposition 2.3.15, we have the following alternate description of the functor MQC_X , when X is perfect:

MQC_X (S)'QC(X)⊗QC(Spec(k))QC(S)

Thus, MQC_X and Mpe_X are manifestly invariants of the noncommutative shadow

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

noncommutative space X, the functor Mperf_X : Aff_kop → S defined by Mperf_X (S) := ((X ⊗kQC(S))ω)'is acommutative reflection ofX, which one might think of as the

moduli of perfect branes onX. As above, it will be useful to keep track of somewhat more refined information. To this end, let ˜M denote the composite functor:

PrL

ω,k×Aff

op

k

˜ M

**

id×O_// PrL

ω,k×CAlgk

id×M1//

PrL

ω,k× PrLω,k ⊗k //

PrL ω,k

Notation 2.3.18. Composition with the functor (−)ω_{defines a functor Fun(Aff}op

k ,Pr

L ω,k)→ Fun(Affop_k,Cat∞). Define Mperf _{to be the composite of} _M _{with this functor:}

PrL ω,k

Mperf

**

M _//

Fun(Affop_k,PrL

ω,k) //Fun(Aff

op

k ,Cat∞)

Likewise, the functors (−)' _: _P_rL

ω,k → Sb and (−)' : Cat∞ → S which carry

an ∞-category to the underlying∞-groupoid induce functors Fun(Affop_k ,Pr_ω,L_k)→ ˆ

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make the following diagrams commutative:

PrL ω,k

M

**

M _//

Fun(Affop_k,PrL

ω,k) //Pˆ(Affk)

Pr_ω,L_k

Mperf

))

Mperf _//

Fun(Affop_k,Cat∞) //P(Affk)

We will write MX(A) for M(X)(A). For S in Affk and X in Prω,Lk, one has the

following explicit formulae summarizing the above definitions:

MX(S) =X ⊗kQC(S)

MX(S) = (X ⊗kQC(S))'

Mperf_X (S) = (X ⊗kQC(S))ω

Mperf_X (S) = ((X ⊗kQC(S))ω)'

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

ω,k. Themoduli of perfect branes on X is the object M perf

X inP(Affk).

Remark 2.3.20. According to Theorem 3.1.1, the presheaf Mperf_X is in fact a

de-rived stack.

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

X we have:

MQC_X 'MQC(X)

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Of course, there are similar statements for MX and MpeX.

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

MX(Spec(A))' X ⊗ModkModA'ModA(X)'Fun

L

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Chapter 3 Brane Descent

3.1 Perfect Branes Descend

Theorem 3.1.1. For any X in PrL

ω,k, the functor MX : Aff

op

k → PrLω,k is a sheaf

for the flat hypertopology.

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

In particular, it defines an object in St∧_k, the hypercompletion of the ∞-topos Stk. Before we proceed to outline the proof of the theorem, we make a simple

obser-vation 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.

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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:

1. We will see that the forgetful functor PrL

ω,k → PrkL reflects products and limits of cosimplicial objects (Lemmas 3.2.6 and 3.2.7), while the forgetful functor PrL

k →Cat∞d 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 M]_X:

Affop_k MX //PrL

ω,k //Cat∞d

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 M]_X is a sheaf for the flat topology; i.e., it will suffice to consider only 1-coskeletal hypercovers.

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(Propo-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 the functor PrL

ω ⊆ Cat∞d 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 PrL

ω ⊆ PrL does preserve and reflect limits of shape K. In particular we will see

that the forgetful functor PrL_ω,_k → Pr_kL 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.

Lemma 3.2.1. The forgetful functor Pr_ω,L_k := ModMod_k(Pr_ωL) → PrL_ω preserves and reflects all small limits.

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A.4.2.3.3.

Lemma 3.2.2. The forgetful functor iL_k : PrL_k → Cat∞d preserves and reflects all

limits.

Proof. We have iL

k = iL ◦πLk, where iL : PrL → Cat∞d is the natural inclusion,

and πL_k : Pr_kL := ModMod_k(PrL) → PrL is the forgetful functor. The functor π_kL

preserves and reflects all limits by A.4.2.3.1. According to T.5.5.3.13., the categories PrL _and

d

Cat∞ admits all small limits and iL _{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, thatf 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.

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

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{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 thatβ◦f(α) is an equivalence. By the two out of three property,

f(α) is an equivalence.

Lemma 3.2.4. Let K be a simplicial set, and let ν : K/ → PrL_ω be a diagram classifying a coCartesian fibration ν[ _: _{V →} _K/_{. Let} _ψ

x : V∞ → Vk be the natural

functor (see....). Let i:PrL

ω →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.

(2) The induced diagram (−)ω_◦_ν_:_K/ _→_Cat∞ _{is a limit diagram.}

Proof. Assume that the hypotheses (i) and (ii) are satisfied. We will now prove (1). The limitV∞ofν|K inPrLk is charaterized upto equivalence by FunLk(W,V∞)'

limx∈KFunLk(W,Vx), for anyW inPrkL. Our hypothesis thatν takes values inPr

L ω,k

implies that each of the functorsψx preservesω-compact objects, and therefore this

equivalence restricts to a fully faithful functor FunL

ω,k(W,V∞)→limx∈KFunLω,k(W,Vx),

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Now letW ∈ PrL

ω,k, and letw[ :W]→K/ be a cocartesian fibration classified by the constant functorK/ → Pr_ω,L_kthat sends every object toW, and letσx:W∞] → W]

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

X ∈limx∈KFunLω,k(W,Vx), and letχ:W_|]_K → V|K be the corresponding cocartesian

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

equivalence FunL

k(W,V∞) ' limx∈KFunLk(W,Vx) implies that χ extends to a map

χ : W] _{→ V} _{defined by a cocartesian section such that} _χ

∞ ∈FunLk(W∞,V∞). We have natural equivalences χx◦σx 'ψx◦χx, sinceχ is cocartesian.

Let X ∈ W]

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

ψ0(χ∞(X)) in V0. Sinceσ0 is an equivalence and σ0 preserves compact objects, we

conclude thatψ0(χ∞(X)) is compact.Condition (ii) now implies thatχ∞(X) is com-pact. Thus χ∞ ∈ FunL_ω,_k(W∞] ,V∞), so χ defines an element X0 in FunLω,k(W,V∞) that maps to X. This proves essential surjectivity of the natural functor mapping

FunL_ω,_k(W,V∞) to limx∈KFunLω,k(W,Vx).

So we have FunL

ω,k(W,V∞) ' limx∈KFunLω,k(W,Vx). This equivalence

charac-terizes V∞ as a limit of ν|K in Prω,Lk, so φ : K/ → PrLω.k is a limit diagram. This

proves (1).

The following lemma is the only simple observation that one needs to add to the results of “T” to prove thatPrL

ω,k → PrkL reflects limits of cosimplicial objects.

Lemma 3.2.5. Let K be an ∞-category and let ν : K/ _{→ P}_rL

ω,k be a diagram,

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(i) The induced diagram ν :K/ _{→ P}_rL

k 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.

Proof. (1) ⇒(2)⇒(3) is trivial: indeed, the hypothesis thatνtakes values inPrL ω,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

let A = `

λ∈ΛAλ. Since ν is a limit diagram in PrLk (or equivalently, in Cat∞d by

Lemma 3.2.2), we may identify V∞ with cocartesian sections ofν[_{. Let} _χ_, _α

λ, α in

FunK/(K/,V) be cocartesian sections with χ_∞ = X, α_λ,_∞ =A_λ 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∈Λαx,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

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colimits (recall thatν[_{is classified by a diagram}_K/ _{→ P}_rL

k), we have a commutative diagrams:

φx(χ0) o

φx(θ0) _//

φx(α0) o

`

λ∈Λφx(αx,0) ∼

oo

o

χx θx //αx oo ∼ `_λ_∈Λαx,λ

Since χ0 is a compact object of V0 by hypothesis, there exists an ω-small set Λω _⊂ _{Λ such that} _θ

0 : χ0 → α0 '

`

λ∈Λα0,λ factors through

`

λ∈Λωα0,λ. From

the diagram above, we see that this implies that θx factors through `_λ∈Λωαx,λ for

all x. Thus, θ factors through `

λ∈Λωαλ, or equivalently, t: X →`_λ∈ΛAλ factors

through `

λ∈ΛωAλ. 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 3.2.5. Then forgetful functor π : PrL

ω,k → PrkL 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.

Lemma 3.2.7. The functor π :PrL_ω,_k→ PrL_k reflects ω-small products.

Proof. This follows from Lemma 3.2.1, Lemma 3.2.4 and the following fact: If{Cα}

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3.3 Bootstrapping: from Covers to Hypercovers

Notation 3.3.1. LetX be an object inPrL

ω,k. The functor M

]

X : Aff

op

k →Catd∞ is defined by the commutativity of the following diagram:

Affop_k MX //

M]_X FFF## F F F F F

F PrLω,k

d

Cat∞

Definition 3.3.2. Let U• : N(∆op₊) → Affk be an augmented simplicial derived

affine scheme. Put U := U−1. We will say that U• is of cohomological descent if the natural map M]_X(U)→limM]₍_U•_{) is an equivalence for every}_X _in_P_rL

ω,k. We will say thatU• isuniversally of cohomological descent, if any base change ofU• is of cohomological descent.

For a map f : U0 _→ _U _{in Aff}

k, the ˇCech nerve of f, denoted ˇC(f), is the 0-coskeleton of f computed in (Affk)/U. Let P denote the class of morphisms in

Affk whose ˇCech 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

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comon-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]). Letf :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(∆+)} _→

d

Cat∞ be a coaugmented cosimplicial ∞ -category, and set C :=C−1_{. Let} _f _:_{C → C}0 _{be the evident functor. Assume that:}

1. The∞-categoryC admits totalizations off-split cosimplicial objects, and those totalizations are preserved by f.

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

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for every α, the diagram

Cm

α

d0

//_Cm+1

˜

α

Cn d0 //_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 thatMX defines aCat∞-valuedd

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).

Notation 3.5.1. LetX be an object inPrL

ω,k. The functor M

]

X : Aff

op

k →Cat∞d is

defined by the commutativity of the following diagram:

Affop_k MX //

M]_X FFF## F F F F F

F PrLω,k

d

Cat∞

The functorM\_X : CAlgk→Cat∞d is defined to be the compositeM

\

X =M

]

X◦O,

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

d

Cat∞ valued presheaf M]_X 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

of Theorem 3.3.3, the essential point is to verify that M\_X carries the ˇCech nerve of a faithfully flat morphism f : A→A0 _{to a limit diagram in}

d

Cat∞. 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 con-ditions” of Proposition 3.4.2:

Lemma 3.5.3. (Base change). For any X in Pr_ω,L_k, the functor M\_X : CAlgk →

d

Cat∞ 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

A p //

f A0 f0 B p0

//_B0

be a cocartesian square in CAlgk, and let

ModA(X)

p∗ _//

f∗

ModA0(X)

f0∗

p∗

xx

ModB(X)

p0∗

//_Mod_B0(X)

p0∗

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be the diagram in Cat∞d by induced by M

\

X. Here, for a morphism p :A → A0 in CAlgk, p∗ := MX\ (p) = MX(p) = A⊗A0 (−), and p_∗ : Mod_A0(X) → Mod_A(X) is

the forgetful functor, which is right adjoint to p∗ (see Remark ??).

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

νM :f∗p∗M →p0∗f0∗M is an equivalence. This follows from the following peculiarity of commutative algebras: pushouts coincide with tensor products in CAlgk, i.e., we have B0 'A0`

AB 'A

0_⊗

AB. Consequently, we have a chain of equivalences:

M ⊗A0 B0−→˜ M ⊗_A0(A0⊗_AB) ˜−→M⊗_AB

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

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

f : A → A0, and let X ∈ Pr_ω,L_k. In order to deduce from the base change lemma that the cosimplicial ∞-category M\_X(A•) satisfies the Beck-Chevalley conditions, we need following simple observation:

Lemma 3.5.4. Let f : A → A0 be a morphism in CAlgk, and let A• : N(∆+) → CAlgk be the ˇCech nerve Cˇ(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:

Am A(α)

d0

//_Am+1

A( ˜α)

An d0 //_An+1

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Proof. SinceA•is the 0-coskeleton off, we haveAn _{' ⊗}n+1

A A0 '

`n+1

A A

0_{, and}_d0 _is

the inclusion of the summand `n+1

A A

0 _→_A0`

A(

`n+1

A A

0_{). It follows immediately}

that the square (†) is cocartesian for anyα : [m]→[n].

Lemma 3.5.6 almost says that ifA• : N(∆+)→CAlgk is the ˇCech nerve of a flat morphism f :A→A0 _{in CAlg}

k, then M\X(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 ⊗AM. Assume that f is flat. Then

f∗ preserves totalizations of f∗-split cosimplicial objects.

Proof. LetM• ∈Fun(N(∆),ModA) be anf∗-split cosimplicial module. We wish to

show that the natural map

B⊗A|M•| −→ |B⊗AM•| (∗)

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 onπn

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.

There is a Bousfield-Kan spectral sequence with E₂p,q = π−p_π

qM•, and E∞p+q =

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

we have an induced spectral sequence with E₂p,q =π0B⊗π0Aπ

−p_π

qM• and E∞p+q =

π0B ⊗π0Aπp+q|M

•_{|. Finally, since} _B _{is flat over} _A_{, we have} _π

p+q(B ⊗A|M•|) '

π0B ⊗π0Aπp+q|M

•_{|. So, in summary, we have a spectral sequence}

E₂p,q=π0B⊗π0Aπ

−p

πqM• =⇒πp+q(B⊗A|M•|)

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

with E₂p,q =π−p_π

q(B⊗AM•) and E∞p+q =πp+q|B ⊗AM•|. Using the flatness of B

overAagain, we haveπ−p_π

q(B⊗AM•)'π−p(π0B⊗π0AπqM

•₎_'_π

0B⊗πoAπ

−p_π qM•.

So the spectral sequence becomes

E₂p,q=π0B⊗π0Aπ

−p_π

qM• =⇒πp+q|B⊗AM•|

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

haveπ−pπqN• = 0 forp6= 0, andπ0πqN• =πqN. Applying this toN• :=B⊗AM•,

and N =|B ⊗AM•|, 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

object of PrL

ω,k. Then the category M

\

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objects.

Proof. The first statement is clear: the ∞-category M\_X(A) ' X ⊗Mod_k 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

that the restricted Yoneda embedding gives an equivalence X ' Funk(Xω_,_Modk).

Furthermore, for any A in CAlgk, we have M\X(A) ' ModA(Funk(Xω,Modk)) ' Funk(Xω,ModA).

LetX ∈ Xω_{be an object classified by a morphism of small}_k_-linear_{∞-categories}

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

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

ModA.

Let f : A → B be a morphism in CAlgk. Under the identification M\X(A) ' Funk(Xω,ModA), the functor M\X(f) corresponds to the functor f

∗ _◦_{(−), where}

f∗ :=M\₁(f) = B ⊗A(−). Furthermore, for every X in Xω, we have a homotopy

commutative diagram in Cat∞d

Funk(Xω_,_Mod A)

f∗◦(−) _//

ψ∗_X,A

Funk(Xω_,_Mod B)

ψ∗_X,B

ModA _f∗ //ModB

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object in Funk(Xω,ModA), for which the induced cosimplicial objectf∗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 {ψ_X,B∗ }X∈Xω is jointly conservative, it is enough

to show that for each X in Xω_{, the morphism} _ψ∗

X,B(νM•) is an equivalence. The

commutativity of the diagram above, together with the fact that the functors ψ∗_X,₋

commute with all limits, implies that this equivalent to showing that the natural

morphism νψ∗_X,A(M•₎ : f∗(limψ_X,A∗ )→ limf∗(ψ_X,A∗ (M•)) is an equivalence for every

object X in Xω_.

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

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

A-moduleψ∗_X,A(M•), we see that the morphismνψ∗_X,A(M•₎is an equivalence for every

X is Xω_.

Lemma 3.5.7. Letf :A→B be a faithfully flat morphism inCAlgk, and letX be

an object in PrL

ω,k. Then the functor M

\

X(f) :M \

X(A)→ M

\

X(A) is conservative.

Proof. We will retain the notation from Lemma 3.5.6. Since the family{ψ∗_X,B}X∈Xω

is jointly conservative,M\_X(f) is conservative if and only if{ψ_X,B∗ ◦ M\_X(f)}X∈Xω =

{f∗◦ψ_X,A∗ }X∈Xω is a jointly conservative family. Using the fact that{ψ∗

X,A}X∈Xω is

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

is conservative. Since ModB is stable, this is equivalent to asking that f∗ reflects

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Lemma 3.5.8. Let X be an object in PrL

ω,k. The functor M

\

X preserves finite

products.

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

A1×A2. Consider the adjunction

M\_X(A)

&&

M\_X(A1)× M\X(A2)

oo

The left adjoint, which is the natural morphism M\_X(A) → limM\_X(Ai), carries

M ∈ ModA(X) to (M ⊗AA1, 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 ⊗AA1)×p2∗(M ⊗AA2)'M ⊗Ap1∗A1×M ⊗Ap2∗A2

'(M ⊗Ap1∗A1)⊕(M ⊗Ap2∗A2)

'M ⊗A(p1∗A1⊕p2∗A2)

'M ⊗AA

'M

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ForMi in ModAi(X), we have natural equivalences pi∗Mi⊗AAi 'Mi andpi∗Mi⊗A

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

(p1∗M1 ×p2∗M2)⊗AAi ' (p1∗M1⊗AAi)⊕(p2∗M2⊗AAi)→ 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.

Proof of Proposition 3.5.2. Let X be in PrL

ω,k. We must show that M

]

X 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, it will suffice to show that M]_X carries the ˇCech nerve of a flat morphism in Affk to a limit diagram in Cat∞.d

Let U• : N(∆op₊) →Affk be a flat hypercover, and let A• :=O(U•) : N(∆+)→ CAlgk. Put A := A−1. Lemma 3.5.6 says that the associated diagram M\X(A•) : N(∆op₊)→Cat∞, satisifies condition 1d .of the corollary of the Barr-Beck-Lurie

theo-rem, Proposition 3.4.2. The base change lemma for branes (Lemma 3.5.3), together with Lemma 3.5.4, implies that M\_X(A•) satisfies condition 2.of Proposition 3.4.2. Finally, Lemma 3.5.7 tells us that the natural map M\_X(A) → M\_X(A0_{) is}

conser-vative. Thus, by Proposition 3.4.2, the natural map M\_X(A)→limM\

X(An) is an equivalence.

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

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inclusion PrL

ω,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(−)'_:

d

Cat∞ →Sband (−)' : 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 ´etale hypertopol-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, the induced functor MX(A•) : N(∆+) → PrLω,k satisfies the hypotheses of Lemma 3.2.4. It follows that the induced diagram (MX(A•))ω is a limit diagram in Cat∞.

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Chapter 4 Geometricity

4.1 Geometric Stacks

4.2 The Artin-Lurie Criterion

4.3 Infinitesimal theory: Brane Jets

4.4 Dualizability implies Geometricity

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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

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Moduli Problems in Derived Noncommutative Geometry