Deformations of Categories

Let kbe a field and let C be ak-linear ∞-category. In §5.2, we studied the problem of deforming a fixed object M ∈C. In this section, we will study the deformation theory of the∞-category C itself. For every small E2-algebra R over k, we will define a classifying space CatDef_C(R) for R-linear ∞-categoriers CR

equipped with an equivalenceC'Modk⊗Mod_RCR. We will show that, modulo size issues, the construction R 7→ CatDef_C(R) is a 2-proximate formal moduli problem (Corollary 5.3.8; in good cases, we can say even more: see Proposition 5.3.21 and Theorem 5.3.33). Using Theorem 5.1.9, we deduce that there is a 0-truncated natural transformation CatDef_C → CatDef^∧_C, where CatDef^∧_C is a formal E2 moduli problem (which is uniquely determined up to equivalence: see Remark 5.1.11). According to Theorem 4.0.8, the formal moduli problem CatDef^∧_Cis given byR7→Map_Alg(2),aug

k

(D⁽²⁾(R), A) for an essentially unique augmentedE2 -algebra A over k. The main result of this section identifies the augmentation ideal mA (as a nonunital E2-algebra) with thek-linear center of the∞-category C(Theorem 5.3.16): in other words, with the chain complex of Hochschild cochains onC.

We begin with a more precise description of the deformation functor CatDef_C.

Notation 5.3.1. Letkbe anE∞-ring. We let LinCat_k'Mod_Modk(Pr^L) denote the∞-category ofk-linear

∞-categories, which we regard as as a symmetric monoidal∞-category. According to Theorem A.6.3.5.14, the constructionA 7→LModA(Modk) determines a symmetric monoidal functor Alg_k →LinCatk. Passing to algebra objects, we obtain a functor Alg⁽²⁾_k 'Alg(Alg_k)→Alg(LinCatk). We set

LCat(k) = Alg⁽²⁾_k ×Alg(LinCat_k)LMod(LinCatk) RCat(k) = Alg⁽²⁾_k ×_Alg(LinCatk)RMod(LinCatk).

The objects of LCat(k) are pairs (A,C) whereA is anE2-algebra over k andC is anA-linear∞-category (that is, an∞-category left-tensored over LModA). Similarly, the objects of RCat(k) are pairs (B,C) where B is anE2-algebra overkandCis an∞-category right-tensored over LModB.

Construction 5.3.2. Letk be a field, and let q: LCat(k)→Alg⁽²⁾_k be the evident coCartesian fibration.

We let LCat(k)^coCart denote the subcategory of LCat(k) spanned by the q-coCartesian morphisms, so that qrestricts to a left fibration LCat(k)^coCart→Alg⁽²⁾_k .

Let C be ak-linear ∞-category and regard (k,C) as an object of LCat(k). We let Defor[C] denote the

∞-category LCat(k)^coCart_/(k,C). We will refer to Defor[C] as the ∞-category of deformations of C. There is an evident forgetful functor θ : Defor[C] → Alg^(2),aug_k . The fiber of θ over an augmented k-algebra A can be identified with the ∞-category of pairs (CA, µ), where CA is an A-linear ∞-category and µ is a k-linear equivalence

LModk(CA)'Modk⊗LMod_ACA→C.

The mapθ is a left fibration, classified by a functorχ : Alg^(2),aug_k →bS; here bS denotes the∞-category of spaces which are not necessarily small. We let CatDef_C denote the composition of the functor χ with the fully faithful embedding Alg^(2),sm_k →Alg^(2),aug_k .

Let us now fix a k-linear ∞-category C and study the properties of the functor χ : Alg^(2),aug_k → bS introduced in Construction 5.3.2. We begin with a simple observation. Let A be an E2-ring and let CA

be an A-linear∞-category. For every map of E2-rings A→ B, letCB = LMod_B⊗LModAC 'LMod_B(C).

Proposition IX.7.4 implies that if we are given a pullback diagramσ:E2-rings A //

A⁰

B //B⁰, ofE2-rings then the induced map

CA→CA⁰×_CB0CB

is fully faithful. LetDA be anotherA-linear∞-category. For every map ofE2-ringsA→B, we letDB ' LMod_B(D) be defined as above, and Fun_B(DB,CB) denote the∞-category of LMod_B-linear functors from CBtoDB which preserve small colimits, so there is a canonical equivalence Fun_B(DB,CB)'Fun_A(DA,CB).

It follows that ifσis a pullback diagram as above, then it induces a fully faithful functor FunA(DA,CA)→FunA⁰(DA⁰,CA⁰)×Fun_B0(DB0,CB0)FunB(DB,CB).

This immediately implies the following result:

Proposition 5.3.3. Let k be a field, let C be a k-linear ∞-category, and let χ : Alg^(2),aug_k → bS be as in Construction 5.3.2. Then for every pullback diagram

A //

A⁰

B //B⁰

in Alg^(2),aug_k , the induced mapθ :χ(A)→χ(A⁰)×_χ(B⁰₎χ(B)is 0-truncated (in other words, the homotopy fibers of θare discrete, up to homotopy).

Variant 5.3.4. Letkbe a field,Cak-linear∞-category, andκa regular cardinal such thatCisκ-compactly generated. For eachA∈Alg^(2),aug_k , we letχ_κ(A) denote the summand ofχ(A) spanned by those pairs (CA, µ) where CA isκ-compactly generated. We can regardχκ as a functor Alg^(2),aug_k →bS. It follows immediately from Proposition 5.3.3 that for each pullback diagram

A //

A⁰

B //B⁰ in Alg^(2),aug_k , the induced map

χ_κ(A)→χ_κ(A⁰)×χ_κ(B⁰)χ_κ(B)

has discrete homotopy fibers. We claim that each of these homotopy fibers is essentially small. Unwinding the definitions, we must show that for every compatible triple ofκ-compactly generated∞-categoriesCA⁰ ∈ LinCat_A0,CB⁰ ∈LinCat_B0, andCB∈LinCat_B, there is only a bounded number of equivalence classes of full A-linear subcategories CA⊆CA⁰×_CB0CB which areκ-compactly generated and induce equivalences

CA⁰ 'LModA⁰(CA) CB'LModB(CA).

For in this case, CA must be generated generated (under κ-filtered colimits) by some subcategory of the essentially small ∞-category C^κA⁰×_C^κ

B0C^κB, where C^κA⁰ denotes the full subcategory of CA⁰ spanned by the κ-compact objects, andC^κB⁰ andC^κB are defined similarly.

Corollary 5.3.5. Let k be a field, let C be a k-linear ∞-category, and let χ : Alg^(2),aug_k → bS be as in Construction 5.3.2. Then:

(1) The space χ(k)is contractible.

(2) Let V ∈Mod_k. Then χ(k⊕V) is locally small, when regarded an∞-category. In other words, each path component of χ(k⊕V)is essentially small.

(3) Let A∈Alg^(2),aug_k be small. Then the spaceχ(A)is locally small.

Proof. Assertion (1) is immediate from the definitions. To prove (2), we note that for eachA ∈Alg^(2),aug_k and every pointη ∈χ(A) corresponding to a pair (CA, µ), the space Ω²(χ(A), η) can be identified with the homotopy fiber of the restriction map

Map_FunA(CA,CA)(id,id)→Map_Funk(C,C)(id,id) and is therefore essentially small. We have pullback diagrams

k⊕V //

k

k⊕V[1] //

k

k //k⊕V[1] k //k⊕V[2]

so that Proposition 5.3.3 guarantees that the mapχ(k⊕V)→Ω²χ(k⊕V[2]) has discrete homotopy fibers.

It follows that each path component ofχ(k⊕V) is a connected covering space of the essentially small space Ω²χ(k⊕V[2]), and is therefore essentially small.

We now prove (3). Assume thatAis small, so that there exists a finite sequence of maps A'A0→A1→ · · · →An'k

and pullback diagrams

Ai //

k

A_i+1 //k⊕k[m_i].

We prove that χ(Ai) is locally small using descending induction on i. Using (1), (2), and the inductive hypothesis, we deduce that X =χ(k)×_{χ(k⊕k[mi])}χ(Ai+1) is locally small. Proposition 5.3.3 implies that the map χ(Ai+1)→X has discrete homotopy fibers. It follows that every path component of χ(Ai+1) is a connected covering space of a path component ofX, and therefore essentially small.

Variant 5.3.6. Letkbe a field,Cak-linear∞-category, andκa regular cardinal such thatCisκ-compactly generated. Letχ_κ: Alg^(2),aug_k →bSbe as in Variation 5.3.4. The proof of Corollary 5.3.5 yields the following results:

(1⁰) The spaceχκ(k) is contractible.

(2⁰) LetV ∈Mod_k. Thenχ_κ(k⊕V) is essentially small.

(3⁰) LetA∈Alg^(2),aug_k be small. Then the spaceχ(A) is essentially small.

Notation 5.3.7. Letkbe a field andCak-linear∞-category. We letχ: Alg^(2),aug_k →bSbe as in Construction 5.3.2, and let CatDef_C : Alg^(2),sm_k → bS denote the composition of χ with the fully faithful embedding ν: Alg^(2),sm_k ,→Alg^(2),aug_k . Ifκis a regular cardinal such thatCisκ-compactly generated, we let CatDef_C,κ

denote the compositionχκ◦ν, whereχκ is as in Variation 5.3.4. It follows from Variation 5.3.6 that we can identify CatDef_C,κwith a functor Alg^(2),sm_k →S(and the functor CatDef_Cis given by the filtered colimit of the transfinite sequence of functors{CatDef_C,κ}, whereκranges over all small regular cardinals).

Corollary 5.3.8. Let k be a field and Cak-linear ∞-category. Then there exists a formal moduli problem CatDef^∧_C : Alg^(2),sm_k → S and a natural transformation α: CatDef_C → CatDef^∧_C which is 0-truncated. In particular, we can regardCatDef_C: Alg^(2),sm_k →bSas a 2-proximate formal moduli problem after a change of universe (see Theorem 5.1.9).

Proof. Combining Corollary 5.3.5, Proposition 5.3.3, and Theorem 5.1.9, we deduce the existence of a formal moduli problem CatDef^∧_C: Alg^(2),sm_k →bSand a 0-truncated natural transformationα: CatDef_C→CatDef^∧_C. For eachm≥0, we see that the space

CatDef^∧_C(k⊕k[m])'Ω²CatDef^∧_C(k⊕k[m+ 2])'Ω²CatDef_C(k⊕k[m+ 2])

is essentially small (see the proof of Corollary 5.3.5). For an arbitrary objectA∈Alg^(2),sm_k , we can choose a finite sequence of mapsA=A0→A1→ · · · →An'kand pullback diagrams

Ai //

k

A_i+1 //k⊕k[m_i].

Using the fact that CatDef^∧_C is a formal moduli problem, we deduce that each CatDef^∧_C(A_i) is essentially small by descending induction oni, so that CatDef^∧_C(A) is essentially small.

Remark 5.3.9. In the situation of Corollary 5.3.8, letκbe a regular cardinal such that Cis κ-compactly generated. Then the composite map

CatDef_C,κ→CatDef_C→CatDef^∧_C

is 0-truncated, so that CatDef_C,κis a 2-proximate formal moduli problem by Theorem 5.1.9.

Our next goal is to describe the formal E2-moduli problem CatDef^∧_C more explicitly. Using Theorem 4.0.8 (and its proof), we see that the functor CatDef^∧_Cis given by

CatDef^∧_C(R) = Map_Alg(2),aug k

(D⁽²⁾(R), k⊕m),

for some nonunitalE2-algebram overk. We would like to make the dependence ofmonCmore explicit.

Definition 5.3.10. Letkbe an E∞-ring and letCbe ak-linear∞-category. We let RCat(k)_C denote the fiber product RCat(k)×LinCat_k{C}. We will say that an object (B,C)∈RCat(k)_C of RCat(k)_C exhibitsB as thek-linear center ofCif (B,C) is a final object of RCat(k)_C.

Remark 5.3.11. In the situation of Definition 5.3.10 Corollary A.6.1.2.42 implies that the forgetful functor RMod(LinCatk)×LinCat_k{C} →Alg(LinCatk) is a right fibration. It follows that the map q: RCat(k)_C→ Alg⁽²⁾_k is a right fibration, so that an object (B,C)∈RCat(k)_C is final if and only if the right fibrationqis represented by the objectB ∈Alg⁽²⁾_k . In other words, the k-linear center B of C can be characterized by the following universal property: for everyA∈Alg⁽²⁾_k , the space Map_Alg(2)

k

(A, B) can be identified with the space RMod_LModA(Pr^L)×LinCat_k{C} ofk-linear right actions of LMod_AonC.

Proposition 5.3.12. Let k be anE∞-ring and letCbe a k-linear ∞-category. Then there exists an object (B,C)∈RCat(k)_C which exhibitsB as ak-linear center ofC.

Proof. LetEbe an endomorphism object ofCin LinCat_k: that is,Eis the ∞-category ofk-linear functors from C to itself. We regard E as a monoidal ∞-category via order-reversed composition, so that C is

a right E-module object of LinCatk. According to Theorem A.6.3.5.10, the symmetric monoidal functor Alg_k →(LinCatk)_Modk/admits a right adjointG. It follows thatGinduces a right adjointG⁰to the functor

Alg⁽²⁾_k 'Alg(Alg_k)→Alg((LinCat_k)_Modk/)'Alg(LinCat_k), and we can defineB=G⁰(E).

Remark 5.3.13. Letk be anE∞-ring and Ca k-linear∞-category The proofs of Proposition 5.3.12 and Theorem A.6.3.5.10 furnish a somewhat explicit description of the k-linear center B of C, at least as an E1-algebra overk: it can be described as the endomorphism ring of the identity functor id_E∈E, whereEis the∞-category ofk-linear functors fromCto itself.

Example 5.3.14. Let k be anE∞-ring, let R ∈Alg_k be anE1-algebra over k, and let Z(R) =Z_E1(R)∈ Alg⁽²⁾_k be a center of R (see Definition A.6.1.4.10). Then Z(R) is a k-linear center of the ∞-category RModR(Modk).

Remark 5.3.15. Let k be an E∞-ring, C a k-linear ∞-category, and A ∈ Alg⁽²⁾_k a k-linear center of C. The homotopy groups πnA are often called the Hochschild cohomology groups of C. In the special case where C = LModR(Modk) for someR ∈ Alg_k, Example 5.3.14 allows us to identify πnA with the group Ext⁻ⁿ

RBModR(Modk)(R, R).

We are now ready to formulate the main result of this section.

Theorem 5.3.16. Let k be a field and let C be a k-linear ∞-category. Then the functor ObjDef^∧_C : Alg^(2),sm_k →S is given by

ObjDef^∧_C(R) = Map_Alg(2),aug k

(D⁽²⁾(R), k⊕Z(C))'Map_Alg(2) k

(D⁽²⁾(R),Z(C)).

whereZ(C)denotes a k-linear center ofC.

Using Remark 5.1.11, we see that Theorem 5.3.16 is equivalent to the following:

Proposition 5.3.17. Letkbe a field, letCbe ak-linear∞-category, and letZ(C)∈Alg⁽²⁾_k denote ak-linear center ofC. LetX: Alg^(2),sm_k →Sbe the functor given by the formulaX(R) = Map_Alg(2)

k

(D⁽²⁾(R), B). Then there exists a0-truncated natural transformationβ: CatDef_C→X.

The first step in our proof of Proposition 5.3.17 is to construct the natural transformation CatDef_C→X. Construction 5.3.18. Let k be a field and let C be a k-linear ∞-category. We let λ⁽²⁾ : M⁽²⁾ → Alg^(2),aug_k ×Alg^(2),aug_k be the pairing of Construction 4.4.6, so that we can identify objects ofM⁽²⁾with triples (A, B, ) whereA, B∈Alg⁽²⁾_k and:A⊗kB→kis an augmentation. Given an object (A, B, )∈M⁽²⁾ and an object (A,CA, µ)∈Defor[C], we regardCA⊗LMod_B as an object of

LModA⊗LModBBMod_LModB(LinCat_k), so that

Mod_k⊗LModA⊗LModB(CA⊗LMod_B)

can be identified with an object of RModLMod_B(Modk) whose image in LinCatk is given by Modk⊗LMod_ACA'C.

This construction determines a functor Defor[C]×_Alg(2),aug

k M⁽²⁾→RMod(LinCatk)×LinCat_k{C}.

The induced map

Defor[C]×_Alg(2),aug

k M⁽²⁾ →Defor[C]×(Alg^(2),aug_k ×Alg(LinCat_k)RMod(LinCatk)×LinCat_k{C}) factors as a composition

Defor[C]×_Alg^aug

k M⁽²⁾→ⁱ Me⁽²⁾→^λ0 Defor[C]×(Alg^(2),aug_k ×Alg(LinCat_k)RMod(LinCatk)×LinCat_k{C}) whereiis an equivalence of∞-categories andλ⁰ is a categorical fibration. It is not difficult to see thatλ⁰ is a left representable pairing of∞-categories, which induces a duality functor

D⁽²⁾_C : Defor[C]^op→Alg^(2),aug_k ×_Alg(LinCatk)RMod(LinCat_k)×_LinCatk{C}.

Concretely, the functorD⁽²⁾_C assigns to each object (A,CA, α)∈Defor[C]^opthe object (D⁽²⁾(A),C), where we regardCas right-tensored over LMod_D(2)(A)via thek-linear equivalence

C'Modk⊗LMod_A⊗LMod_B(CA⊗LModB).

Letkbe a field and letCbe ak-linear∞-category. We letZ(C) denote ak-linear center ofC(Definition 5.3.10), so that we have a canonical equivalence of∞-categories

η: Alg⁽²⁾_k ×_Alg(LinCatk)RMod(LinCatk)×LinCat_kC'(Alg⁽²⁾_k )_/Z(C). Composingη with the functorD⁽²⁾_C , we obtain a diagram of∞-categories

Defor[C]^op //

(Alg^(2),aug_k )_/(k⊕Z(C))

(Alg^(2),aug_k )^{op D}

(2) //Alg^(2),aug_k

which commutes up to canonical homotopy, where the vertical maps are right fibrations. This diagram determines a natural transformationβ: ObjDef_C→X, whereX : Alg^(2),sm_k →Sis the functor given by the formulaX(R) = Map_Alg(2)

k

(D⁽²⁾(A),Z(C)).

We will prove Proposition 5.3.17 (and therefore also Theorem 5.3.16) by showing that the natural trans-formationβ of Construction 5.3.18 is 0-truncated. Since the functor X is a formal E2-moduli problem, β induces a natural transformationβ: CatDef^∧_C→X. We wish to prove thatβis an equivalence (which implies Proposition 5.3.17, by virtue of Theorem 5.1.9). According to Proposition 1.2.10, it suffices to show that β induces an equivalence of tangent complexes. Using the description of the tangent complex of CatDef^∧_C supplied by Lemma 5.1.12, we are reduced to proving the following special case of Proposition 5.3.17:

Proposition 5.3.19. Let k be a field and C ak-linear ∞-category. For each m≥0, the natural transfor-mation β: ObjDef_C→X of Construction 5.3.18 induces a 0-truncated map

ObjDef_C(k⊕k[m])→Map_Alg(2) k

(D⁽²⁾(k⊕k[m]),Z(C)).

Proof. We have a commutative diagram

ObjDef_C(k⊕k[m]) //

Map_Alg(2) k

(D⁽²⁾(k⊕k[m]),Z(C))

Ω²ObjDef_C(k⊕k[m+ 2]) ^θ //Ω²Map_Alg(2) k

(D⁽²⁾(k⊕k[m+ 2]),Z(C)),

where the left vertical map is 0-truncated by Corollary 5.3.8 and the right vertical map is a homotopy equivalence. It will therefore suffice to show that θ is a homotopy equivalence. Let A = k⊕k[m+ 2], let CA = LModA⊗C 'RModA(C), letE be the ∞-category of k-linear functors from C to itself, and let EA be the∞-category of LModA-linear functors fromCA to itself, so that there is a canonical equivalence γ:EA 'LModA(E). Let id∈E denote the identity functor fromCto itself. Under the equivalence γ, the identity functor fromCAto itself can be identified with the free moduleA⊗id∈LModA(E). Unwinding the definitions, we see that the domain ofθcan be identified with the homotopy fiber of the map

ξ: Map_E

A(A⊗id, A⊗id)'Map_E(id, A⊗id)→Map_E)(id,id).

We have a canonical fiber sequence

id[m+ 2]→A⊗id→id inE, so that the homotopy fiber ofξ is given by

Map_E(id,id[m+ 2])'Map_Modk(k[−m−2],Z(C)).

The mapθis induced by a morphismν:k[−m−2]→D⁽²⁾(k⊕k[m]) in Modk. Let Free⁽²⁾: Modk →Alg⁽²⁾_k be a left adjoint to the forgetful functor, so thatνdetermines an augmentation (k⊕k[m])⊗kFree⁽²⁾(k[−m−2])→ k. The proof of Proposition 4.5.6 shows that this pairing exhibits Free⁽²⁾(k[−m−2]) as the Koszul dual of k⊕k[m], from which it immediately follows thatθis a homotopy equivalence.

Our goal, for the remainder of this section, is to describe the formal moduli problem CatDef^∧_C more explicitly in terms of the ∞-category C. Assume that C is compactly generated. Let ω denote the first infinite cardinal and let CatDef_C,ω be the deformation functor of Notation 5.3.7 (so that CatDef_C,ω classifies compactly generateddeformations ofC). Our main result (Theorem 5.3.33) asserts that, under some rather restrictive assumptions, the composite map

CatDef_C,ω→CatDef_C→CatDef^∧_C

is an equivalence of functors. Since the natural transformation CatDef_C,ω →CatDef^∧_C is 0-truncated, this is equivalent to the assertion that CatDef_C,ω is itself a formal moduli problem (see Remark 5.1.11). The functor CatDef_C,ω is automatically a 2-proximate formal moduli problem (Remark 5.3.9). Our first step is to obtain a criterion which guarantees that CatDef_C,ω is a 1-proximate formal moduli problem. First, we need to introduce a bit of terminology.

Definition 5.3.20. LetC be a presentable stable∞-category. We let C^c denote the full subcategory of C spanned by the compact objects of C. We will say that C is tamely compactly generated if it satisfies the following conditions:

(a) The∞-categoryCis compactly generated (that is, C'Ind(C^c)).

(b) For every pair of compact objectsC, D∈C, the groups Extⁿ_C(C, D) vanish forn0.

Proposition 5.3.21. Let kbe a field, let C be a k-linear ∞-category which is tamely compactly generated, and letCatDef_C,ω: Alg^(2),sm_k →Sbe as in Notation 5.3.7. ThenCatDef_C,ω is a1-proximate formal moduli problem.

The proof of Proposition 5.3.21 will require some preliminaries.

Notation 5.3.22. Let R be an E2-ring and let C be an R-linear ∞-category. For every pair of objects C, D∈C, we let Mor_C(C, D)∈LModR be a classifying object for morphisms fromC to D. This object is characterized (up to canonical equivalence) by the requirement that there exists a mape: Mor_C(C, D)⊗C→ D such that, for everyM ∈LMod_R, the composite map

Map_LMod

R(M,Mor_C(C, D))→Map_C(M ⊗C,Mor_C(C, D)⊗C)→^e◦ Map_C(M⊗C, D) is a homotopy equivalence.

Lemma 5.3.23. Let R be an E2-ring and let Cbe an R-linear ∞-category. If C ∈C is compact, then the constructionD7→Mor_C(C, D)determines a colimit-preserving functorC→LModR.

Proof. It is clear that the construction D 7→ Mor_C(C, D) commutes with limits and is therefore an exact functor. To prove that it preserves colimits, it suffices to show that it preserves filtered colimits. For this, it suffices to show that the constructionD 7→Ω^∞Mor_C(C, D) preserves filtered colimits (as a functor fromC toS), which is equivalent to the requirement that Cis compact.

LetR andCbe as in Notation 5.3.22. Given an objectN ∈LMod_R, the induced map N⊗Mor_C(C, D)⊗C^id−→^N⊗eN⊗D

is classified by a mapλ:N⊗Mor_C(C, D)→Mor_C(C, N⊗D).

Lemma 5.3.24. LetRbe anE2-ring and letCbe anR-linear∞-category. LetC, D∈Cand letN ∈LModR. If C is a compact object ofC, then the map λ:N⊗Mor_C(C, D)→Mor_C(C, N⊗D)is an equivalence.

Proof. Using Lemma 5.3.23, we deduce that the functorN 7→Mor_C(C, N⊗D) preserves small colimits. It follows that the collection of objects N ∈LModR such thatλis an equivalence is closed under colimits in LModR. We may therefore suppose thatN 'R[n] for some integern, in which case the result is obvious.

Lemma 5.3.25. Suppose we are given a map ofE2-ringsR→R⁰, letCbe an R-linear ∞-category, and let C⁰= LMod_R⁰⊗_LModRC'LMod_R⁰(C). LetF :C→C⁰ be a left adjoint to the forgetful functorG:C⁰→C, so thatF is given by given byC7→R⁰⊗C. For every pair of objectsC, D∈C, the mapMor_C(C, D)⊗C→D induces a map

(R⁰⊗RMor_C(C, D))⊗F(C)'F(Mor_C(C, D)⊗C)→F(D),

which is classified by a map α:R⁰⊗_RMor_C(C, D)→Mor_C⁰(F(C), F(D)). If C∈C is compact, then αis an equivalence.

Proof. The image ofαunder the forgetful functor LMod_R⁰ →LMod_R coincides with the equivalenceR⁰⊗_R Mor_C(C, D)→Mor_C(C, R⁰⊗D) of Lemma 5.3.24.

Lemma 5.3.26. Suppose given a pullback diagram A //

B

A⁰ //B⁰

ofE2-rings. LetCA be anA-linear∞-category, letCB = LMod_B⊗_LModACA'LMod_B(CA), and defineCA⁰

andCB⁰ similarly. An objectC∈CA is compact if and only if its images in CB andCA⁰ are compact.

Proof. The “only if” direction is obvious, since the forgetful functors CB → CA ← CA⁰ preserve filtered colimits. For the converse, suppose that C∈ CA has compact imagesCB ∈CB andCA⁰ ∈CA⁰. Then the image ofCinCB×_CB0CA⁰is compact. Since the natural mapCA→CB×_CB0CA⁰is fully faithful (Proposition IX.7.4) and preserves filtered colimits, we conclude thatC is compact.

Lemma 5.3.27. Let f : A → B be a map of connective E2-rings, and let CA be an A-linear ∞-category which is tamely compactly generated. ThenCB = LModB(C)is tamely compactly generated.

Proof. We note thatCBis compactly generated: in fact,CBis generated under small colimits by the essential image of the composite functor map C^cA ,→ CA

→F CB, which consists of compact objects (since F is left adjoint to a forgetful functor). It follows that the∞-categoryC^cB is the smallest stable full subcategory of CB which containsF(C^cA) and is closed under retracts. LetX⊆CB be the full subcategory spanned by those objectsCsuch that for everyD∈C^cB, we have Extⁿ_CB(C, D)'0 forn0. It is easy to see thatXis stable

and closed under retracts. Consequently, to show that C^cB ⊆X, it will suffice to show thatF(C0)∈Xfor eachC0 ∈C^cA. Let us regardC0 as fixed, and letYbe the full subcategory ofCB spanned by those objects D for which the groups Extⁿ_CB(F(C0), D) vanish forn0. Since Yis stable and closed under retracts, it will suffice to show that F(D0)∈Yfor eachD0∈C^cA. In other words, we are reduced to proving that the homotopy groups π−nMor_CB(F(C0), F(D0)) vanish for n 0. Using Lemma 5.3.25, we must show that π−n(B⊗AMor_CA(C0, D0)) vanishes forn0. SinceAandB are connective, this follows from the fact that π_−nMor_CA(C₀, D₀)'0 forn0 (sinceCAis tamely compactly generated).

Lemma 5.3.28. LetA be a connectiveE2-ring and let CA be anA-linear ∞-category which is tamely com-pactly generated. For every map ofE2-ringsA→R, we letCR denote the∞-category LMod_R⊗_LModACA' LModR(CA). Suppose we are given a pullback diagram

A //

B

A⁰ //B⁰

of connectiveE2-rings which induces surjective mapsπ0B→π0B⁰ andπ0A⁰ →π0B⁰. Then the induced map θ^c :C^cA→C^cB×_C^c

B0C^cA⁰ is an equivalence of ∞-categories.

Proof. The functorθ^cis given by the restriction of a functorθ:CA→CB×_CB0CA⁰, which is fully faithful by Proposition IX.7.4; this proves thatθ^c is fully faithful. We will show thatθis essentially surjective. We can identify objects ofCB×_CB0CA⁰ with triples (CB, CA⁰, η) whereCB ∈C^cB,CA⁰ ∈C^cA⁰, andηis an equivalence B⁰⊗BCB 'B⁰⊗A⁰CA⁰. In this case, we will denoteB⁰⊗BCB 'B⁰⊗A⁰CA⁰ byCB⁰. Note thatθ admits a right adjointG, given by (CB, CA⁰, η)7→ CB×C_B0 CA⁰. In view of Lemma 5.3.26, it will suffice to show that the counit transformation v :θ◦G→id is an equivalence when restricted to objects of C^cB×_C^c

B0C^cA⁰. Choose such an object (C_B, C_A⁰, η) (so thatC_B andC_A⁰ are compact) and letC_A=C_B×_CB0C_A⁰; we wish to show that the canonical maps

φ:B⊗AC_A→C_B φ⁰:A⁰⊗AC_AC_A⁰

are equivalences. We will show that φis an equivalence; the argument that φ⁰ is an equivalence is similar.

LetX⊆CB be the full subcategory spanned by those objects DB ∈CB such thatφinduces an equivalence φ0 : Mor_CB(DB, B⊗ACA) → Mor_CB(DB, CB). We wish to show that X =CB. Since Xis closed under small colimits, it will suffice to show thatXcontainsB⊗ADA for every compact objectDA∈CA. LetDA⁰

and D_B⁰ be the images of D_A in CA⁰ andCB⁰, respectively. Using Lemma 5.3.24, we can identify φ₀ with the canonical mapB⊗AMor_CA(D_A, C_A)→Mor_CB(D_B, C_B). Note that we have a pullback diagram

Mor_CA(DA, CA) //

Mor_CB(DB, CB)

Mor_CA0(DA⁰, CA⁰) //Mor_CB0(DB⁰, CB⁰) and that Lemma 5.3.24 guarantees that the underlying maps

B⁰⊗A⁰Mor_CA0(DA⁰, CA⁰)→Mor_CB0(DB⁰, CB⁰)←B⁰⊗BMor_CB(DB, CB)

are equivalences. It will therefore suffice to show that there exists an integernsuch that Mor_CB(D_B, C_B) and Mor_CA0(D_A⁰, C_A⁰) belong to (LMod_B)_≥n and (LMod_A⁰)_≥n, respectively (Proposition IX.7.6). This follows immediately from Lemma 5.3.27.

Notation 5.3.29. Let LinCat^tcg be the subcategory of LinCat whose objects are pairs (A,C), whereAis a connective E2-ring and Cis a tamely compactly generated A-linear∞-category, and whose morphisms are

maps (A,C)→ (A⁰,C⁰) such that the underlying functor C →C⁰ carries compact objects of C to compact objects ofC⁰. It follows from Lemma 5.3.27 that the forgetful functor LinCat^tcg→Alg^(2),cnis a coCartesian fibration. This coCartesian fibration is classified by a functorχ^tcg: Alg^(2),cn→dCat_∞.

maps (A,C)→ (A⁰,C⁰) such that the underlying functor C →C⁰ carries compact objects of C to compact objects ofC⁰. It follows from Lemma 5.3.27 that the forgetful functor LinCat^tcg→Alg^(2),cnis a coCartesian fibration. This coCartesian fibration is classified by a functorχ^tcg: Alg^(2),cn→dCat_∞.

Dans le document Derived Algebraic Geometry X: Formal Moduli Problems (Page 150-166)