The Bar Construction

LetCbe a monoidal∞-category, letA∈Alg(C) be an algebra object ofC, and let:A→1be an augmen-tation on the algebra A. IfC admits geometric realizations of simplicial objects, then thebar construction onAis defined to be the relative tensor product 1⊗_A1(here we regard1as both a right and left module overA, by means of the augmentation): that is, the geometric realization of the simplicial object

· · · ////////A⊗A //////A ////1.

We will denote this geometric realization by BarA. Our goal in this section is to study some of the properties of the constructionA7→BarA.

(a) Consider the composition

BarA = 1⊗A1 ' 1⊗AA⊗A1

→ 1⊗A1⊗A1 ' 1⊗A1⊗11⊗A1

→α (1⊗_A1)⊗(1⊗_A1)

= BarA⊗BarA.

(here the map α is an equivalence if we assume that the tensor product functor ⊗ : C×C → C preserves geometric realizations of simplicial objects). We can view this composite map as giving a comultiplication ∆ : BarA → BarA⊗BarA. We will show that this comultiplication is coherently associative: that is, it exhibits BarAas an associative algebra object in the monoidal∞-categoryC^op. Moreover, this algebra object is equipped with a canonical augmentation, given by the morphism

1'1⊗1→1⊗A1= BarA

in C. We may therefore identify the constructionA7→BarAwith a functor Bar : Alg^aug(C)^op→Alg^aug(C^op).

(b) Assume thatCadmits totalizations of cosimplicial objects. ThenC^op admits geometric realizations of simplicial objects, so that we can apply the bar construction to augmented associative algebra objects ofC^op. We therefore obtain a functor

CoBar : Alg^aug(C^op)→Alg^aug(C)^op,

which we will refer to as the cobar construction. If C is an augmented algebra object of C^op (which we can think of as a augmented coalgebra object ofC), then CoBarC is given by the totalization of a cosimplicial diagram

1 ////C //////C⊗C ////////· · ·

We will show that the functor CoBar : Alg^aug(C^op)→Alg^aug(C)^op is adjoint to the bar construction Bar : Alg^aug(C)^op→Alg^aug(C^op).

To simplify the discussion, we note that there is no harm in assuming that the unit object1∈Cis both initial and final. This can always be achieved by replacingCbyC1/ /1(which inherits a monoidal structure:

see§A.2.2.2). We have canonical equivalences

Alg(C1/ /1)'Alg^aug(C) Alg(C^op_{1/ /1})'Alg^aug(C^op), which allows us to ignore augmentations in the discussion below.

Let us now describe the adjunction appearing in assertion (b) more explicitly. Suppose we are given an algebra objectA∈Alg^aug(C) and a coalgebra objectC ∈Alg^aug(C^op). According to (b), we should have a canonical homotopy equivalence

Map_Alg(C)(A,CoBarC)'Map_Alg(Cop)(C,BarA).

We will prove this by identifying both sides with a classifying space for liftings of the pair (A, C) ∈ Alg(C×C^op) to an algebra object of the twisted arrow∞-category TwArr(C).

Theorem 4.3.1. Let C be a monoidal ∞-category, so that C^op and TwArr(C) inherit the structure of monoidal ∞-categories (see Example 4.3.6). Assume that the unit object 1 ∈ C is both initial and final.

Then:

(1) The induced map λ: Alg(TwArr(C))→Alg(C)×Alg(C^op) is a pairing of∞-categories.

(2) Assume that the unit object1∈Cis final (so that every algebra object ofCis equipped with a canonical augmentation) and thatCadmits geometric realizations of simplicial objects. Then the pairingλis left representable, and therefore determines a functor Dλ: Alg(C)^op→Alg(C^op). The composite functor

Alg(C)^opD→^λAlg(C^op)→C^op is given by A7→BarA.

(3) Assume that the unit object1is initial (so that every coalgebra object ofCis equipped with a canonical augmentation) and that C admits totalizations of cosimplicial objects. Then the pairing λ is right representable, and therefore determined a functorD⁰_λ: Alg(C^op)^op→Alg(C). The composite functor

Alg(C^op)^opD

0

→λAlg(C)→C is given by A7→CoBarA.

Remark 4.3.2. Assertion (3) of Theorem 4.3.1 follows from assertion (2), applied to the opposite∞-category C^op.

Remark 4.3.3. In the situation of Theorem 4.3.1, suppose that the unit object 1is both initial and final, and thatCadmits both geometric realizations of simplicial objects and totalizations of cosimplicial objects.

Then the pairingλ: Alg(TwArr(C))→Alg(C)×Alg(C^op) is both left and right representable. We therefore obtain adjoint functors

Alg(C)

D^op_λ

//cAlg(C),

D⁰_λ

oo

given by the bar and cobar constructions, where cAlg(C) = Alg(C^op)^opis the∞-category of coalgebra objects ofC.

More generally, ifCis an arbitrary monoidal∞-category which admits geometric realizations of simplicial objects and totalizations of cosimplicial objects, then by applying Theorem 4.3.1 to the ∞-categoryC1/ /1

we obtain an adjunction

Alg^aug(C) ^Bar //cAlg^aug(C).

CoBar

oo

The proof of Theorem 4.3.1 will require some general remarks about pairings between monoidal ∞-categories.

Definition 4.3.4. LetO^⊗ be an∞-operad. A pairing ofO-monoidal ∞-categoriesis a triple (p:C^⊗→O^⊗, q:D^⊗ →O^⊗, λ^⊗:M^⊗→C^⊗×_O⊗D^⊗)

wherepandqexhibitC^⊗ andD^⊗asO-monoidal∞-categories andλ^⊗:M^⊗ →C^⊗×_O^⊗D^⊗ is aO-monoidal functor which is a categorical fibration and which induces a right fibrationλ_X:MX →CX×DX after taking the fiber over any objectX∈O.

Remark 4.3.5. In the situation of Definition 4.3.4, we will generally abuse terminology by simply referring to the O-monoidal functor λ^⊗ : M^⊗ → C^⊗×_O⊗D^⊗ as a pairing of monoidal ∞-categories. In the special case where O^⊗ = Ass^⊗ is the associative ∞-operad, we will refer to λ^⊗ simply as a pairing of monoidal

∞-categories. IfO^⊗= Comm^⊗ is the commutative∞-operad, we will refer toλ^⊗ as apairing of symmetric monoidal∞-categories.

Example 4.3.6. Recall that the forgetful functor

(λ:M→C×D)7→C

induces an equivalence CPair^perf → Cat_∞, whose homotopy inverse is given on objects by C 7→ (λ : TwArr(C)→C×C^op) (see Remark 4.2.12). LetO^⊗be an∞-operad and letCbe aO-monoidal∞-category, which we can identify with aO-monoid object in the∞-categoryCat_∞. It follows that TwArr(C) admits the structure of aO-monoid object of CPair^perf, which we can identify with a pairing ofO-monoidal∞-categories

TwArr(C)^⊗→C^⊗×_O^⊗(C^op)^⊗.

Remark 4.3.7. Letλ^⊗ :M^⊗ →C^⊗×_O^⊗D^⊗ be a pairing ofO-monoidal∞-categories. Then the induced map Alg_/O(M) → Alg_/O(C)×Alg_/O(D) is a pairing of ∞-categories. This follows immediately from Corollary A.3.2.2.3.

In particular, ifλ^⊗ is a pairing of monoidal∞-categories, then it induces a pairing Alg(λ) : Alg(M)→Alg(C)×Alg(D).

The key step in the proof of Theorem 4.3.1 is to establish a criterion for verifying that Alg(λ) is left (or right) representable.

Proposition 4.3.8. Let λ^⊗:M^⊗→C^⊗×_Ass^⊗D^⊗ be a pairing of monoidal∞-categories. Assume that:

(1) If 1denotes the unit object ofD, then the right fibrationM×_D{1} →C is a categorical equivalence.

(2) The underlying pairing λ:M→C×Dis left representable.

(3) The ∞-categoryD admits totalizations of cosimplicial objects.

Then the induced pairingAlg(λ) : Alg(M)→Alg(C)×Alg(D)is left representable.

The proof of Proposition 4.3.8 will occupy our attention for most of this section. We begin by treating an easy special case.

Proposition 4.3.9. Let λ^⊗:M^⊗ →C^⊗×_Ass^⊗D^⊗ be a pairing between monodial∞-categories, and assume that the underlying pairing of∞-categoriesλ:M→C×D is left representable. LetA∈Alg(C)be a trivial algebra object of C(see§A.3.2.1). Then:

(1) There exists a left universal object of Alg(M)lying over A∈Alg(C).

(2) An object M ∈Alg(M)lying over A∈Alg(C)is left universal if and only if the image of M in Mis left universal (with respect to the pairing λ:M→C×D).

Proof. We can identify Alg(M)×_Alg(C) {A} with Alg(N), where N^⊗ denotes the monoidal ∞-category M^⊗×_C^⊗Ass^⊗. An object of Alg(M)×_Alg(C){A}is left universal if and only if it is a final object of Alg(N).

Sinceλis left representable,Nhas a final object. Assertions (1) and (2) are therefore immediate consequences of Corollary A.3.2.2.5.

To prove Proposition 4.3.8 in general, we must show that an arbitrary algebra object A ∈ Alg(C) can be lifted to a left universal object of Alg(M). This object is not as easy to find: for example, its image in M is generally not left universal for the underlying pairing λ : M → C×D. In order to construct it, we would like to reduce to the situation where A is a trivial algebra object of C. We will accomplish this by replacingCby another monoidal ∞-category havingA as the unit object: namely, the∞-category Mod^A_A^ss(C)'ABModA(C) (see Theorem A.4.3.4.28).

Lemma 4.3.10. Let λ^⊗:M^⊗→C^⊗×_Ass^⊗D^⊗ be a pairing of monoidal∞-categories, and letM ∈Alg(M) have image(A, B)∈Alg(C)×Alg(D). Then the induced map

MBMod_M(M)^⊗→_ABMod_A(C)^⊗×_Ass⊗BBMod_B(D)^⊗ is also a pairing of monoidal ∞-categories.

Proof. It will suffice to show that the mapMBModM(M)→ABModA(C)×BBModB(D) is a right fibration.

This map is a pullback of the categorical fibration

θ: BMod(M)→(BMod(C)×BMod(D))×_{Alg(C)×Alg(D)}Alg(M).

It will therefore suffice to show thatθis a right fibration. Let

θ⁰ : (BMod(C)×BMod(D))×_Alg(C)2×Alg(D)²Alg(M)²→BMod(C)×BMod(D)

be the projection map. Since θ is a categorical fibration, it will suffice to show thatθ⁰ andθ⁰◦θ are right fibrations. The map θ⁰ is a pullback of the forgetful functor Alg(λ) : Alg(M)→ Alg(C)×Alg(D). It will therefore suffice to show that Alg(λ) andθ⁰◦θare right fibrations, which follows immediately from Corollary A.3.2.2.3.

Proposition 4.3.11. Letλ^⊗:M^⊗→C^⊗×_Ass^⊗D^⊗be a pairing of monoidal∞-categories. LetM ∈Alg(M) have image (A, B) ∈ Alg(C)×Alg(D), and identify A and B with their images in C and D, respec-tively. Assume that B is a trivial algebra object of D and that, for every object C ∈ C, the Kan complex λ⁻¹{(C, B)} ⊆ M is contractible. Then the forgetful functor Alg(MBModM(M)) → Alg(M) carries left universal objects ofAlg(_MBMod_M(M))to left universal objects of Alg(M).

Proof. It will suffice to show that for everyA⁰ ∈Alg(_ABMod_A(C))'Alg(C)_A/ having imageA⁰₀∈Alg(C), the left fibration Alg(_MBMod_M(M))×Alg(_ABMod_A(C)){A⁰} →Alg(M)×Alg(C){A⁰₀} is an equivalence of ∞-categories (and therefore carries final objects to final objects). SinceB is a trivial algebra object ofD, the forgetful functor Alg(_BBMod_B(D))→Alg(D) is an equivalence of∞-categories. It will therefore suffice to show that for eachB⁰∈Alg(BBModB(D)) having imageB₀⁰ in Alg(BBModB(D)), the induced map

Alg(MBModM(M))×Alg(_ABMod_A(C))×Alg(BBMod_B(D)){(A⁰, B⁰)} →Alg(M)×Alg(C)×Alg(D){(A⁰₀, B₀⁰)}

is a homotopy equivalence of Kan complexes. For this, it suffices to show that M ∈ Alg(M) is a p-initial object, wherep: Alg(M)→Alg(C)×Alg(D) denotes the projection. Sincepis a right fibration, it suffices to verify that M is an initial object of Alg(M)×_{Alg(C)×Alg(D)}{(A, B)}. This∞-category is is given by a homotopy fiber of the mapφ : Alg(N)→Alg(C), whereN^⊗ =M^⊗×^⊗_DAss^⊗. We now observe that φ is a categorical equivalence, since the monoidal functorN→Cis a categorical equivalence (by virtue of the fact that it is a right fibration whose fibers are contractible Kan complexes).

Notation 4.3.12. Letλ^⊗ :M^⊗ → C^⊗×_Ass^⊗D^⊗ be a pairing of monoidal ∞-categories. If M ∈ Alg(M) has image (A, B)∈Alg(C)×Alg(D), we letλM denote the induced pairingMBModM(M)→ABModA(C)×

BBModB(D).

Lemma 4.3.13. Let λ^⊗:M^⊗→C^⊗×_Ass^⊗D^⊗ be a pairing of monoidal∞-categories, and letM ∈Alg(M) be an object having image(A, B)∈Alg(C)×Alg(D), whereB is a trivial algebra object ofD. LetF :M→

MBMod_M(M) be a left adjoint to the forgetful functor, given by V 7→M ⊗V ⊗M (Corollary A.4.3.3.14).

Then F carries left universal objects of M (with respect to the pairing λ : M → C×D) to left universal objects of_MBMod_M(M)(with respect to the pairingλ_M :_MBMod_M(M)→ABMod_A(C)×BBMod_B(D)).

Proof. Let F⁰ : C → _ABMod_A(C) and F⁰⁰ : D → _BBMod_B(D) be left adjoints to the forgetful functors G⁰ : _ABMod_A(C) → C and G⁰⁰ : _BBMod_B(D) → D. We may assume without loss of generality that the diagram

M ^F //

MBMod_M(M)

C×D^F0×F00//_ABModA(C)×BBModB(D)

is commutative. For eachC∈C,F induces a functor

f :M×_C{C} →_MBMod_M(M)×_ABModA(C){F⁰(C)}.

We note that f has a right adjoint g, given by composing the forgetful functor MBModM(M)×_ABModA(C) {F⁰(C)} →M×_C{(G⁰◦F⁰)(C)}with the pullback functorM×_C{(G⁰◦F⁰(C)} →M×_C{C}associated to the unit mapC→(G⁰◦F⁰)(C). To show thatf preserves final objects, it will suffice to show thatgis a homotopy inverse tof. Letu: id→g◦f be the unit map. For every objectV ∈M×_C{C} having imageD∈D, the unit mapuV :V →(g◦f)(V) has image inDequivalent to the unit mapD→(G⁰⁰◦F⁰⁰)(D)'B⊗D⊗B in D. Since B is a trivial algebra, we conclude that the image ofuV inD is an equivalence. Because the map M×_C{C} →D is a right fibration, we conclude thatuV is an equivalence. A similar argument shows that the counit mapv:f◦g→id is an equivalence of functors, so thatgis homotopy inverse tof as desired.

Lemma 4.3.14. Let f : C →C be a right fibration of ∞-categories, classified by a map χ :C^op →S and suppose we are given a diagram p:K^.→C. The following conditions are equivalent:

(1) For every commutative diagram σ:

K

q //C

f

K^{. p} //

q

>>

C

there exists an extension q as indicated, which is anf-colimit diagram.

(2) The restriction χ|(K^.)^op is a limit diagram inS.

If pis a colimit diagram inC, then these conditions are equivalent to the following:

(3) For every diagramσas in(1), the diagramq:K→Ccan be extended to a colimit diagram inC, whose image inC is also a colimit diagram.

Proof. The equivalence of (1) and (2) follows from Lemma VII.5.17, and the equivalence of (1) and (3) from Proposition T.4.3.1.5.

We will need the following refinement of Corollary A.4.2.3.5:

Lemma 4.3.15. Let Mbe an∞-category which is left-tensored over a monoidal∞-category C, letA be an algebra object ofC, and let θ: LModA(M)→M denote the forgetful functor. Letp:K →LModA(M)be a diagram and let p0=θ◦p. Suppose that p0 can be extended to an operadic colimit diagram p₀ :K^/ →M (in other words,p₀ has the property that for every objectC∈C, the composite map

K^{. p}→⁰ M^C⊗→ M is also a colimit diagram in M). Then:

(1) The diagrampextends to a colimit diagram p:K^/→LMod_A(M).

(2) Let p:K^/ →LModA(M) be an arbitrary extension of p. Then p is a colimit diagram if and only if θ◦pis a colimit diagram.

Proof. Letq : M →C be the locally coCartesian fibration defined in Notation A.4.2.2.16. The algebra object A determines a map N(∆)^op → C. Let X denote the fiber product N(∆)^op ×_C M and let LMod_A(M)→^∆LMod_A(M)⊆Fun_N(∆)op(N(∆)^op,X) be the equivalence of Corollary A.4.2.2.15. It follows from Lemma A.3.2.2.9 that the composite map K → LMod_A(M) → ^∆LMod_A(M) can be extended to a colimit diagram in Fun_N(∆)op(N(∆)^op,X), that this diagram factors through ^∆LMod_A(M), and that its image in M is a colimit diagram. This proves (1) and the “only if” direction of (2). To prove the “if”

direction of (2), let us suppose we are given an arbitrary extensionp:K^/→LModA(M), carrying the cone point to a leftA-moduleM. Thenpdetermines a mapα: lim

−→(p)→M. If the image of pin Mis a colimit diagram, then the image of α in M is a colimit diagram. Since the forgetful functor LModA(M) →M is conservative (Corollary A.4.2.3.2), we deduce thatαis an equivalence. It follows thatpis a colimit diagram as desired.

Example 4.3.16. Let M be an ∞-category which is left tensored over a monoidal ∞-category C^⊗, let A ∈Alg(C), and θ : LModA(M) →M denote the forgetful functor. IfM• is a θ-split simplicial object of LMod_A(M), then the underlying map N(∆)^op → LMod_A(M) satisfies the hypothesis of Lemma 4.3.15. It follows thatM_• admits a geometric realization in LMod_A(M) which is preserved by the forgetful functorθ.

Example 4.3.17. LetMbe an∞-category which is bitensored over the a pair of monoidal∞-categoriesC^⊗ andD^⊗, and suppose we are given algebra objectsA∈Alg(C) andB ∈Alg(D). Let θ: RModB(M)→M denote the forgetful functor, and regard RModB(M) as an ∞-category left-tensored overC (see§A.4.3.2).

Let µ : ABModB(M) ' LModA(RModB(M)) → RModB(M) be the forgetful functor, and let M_• be a simplicial object of ABModB(M). Assume thatM_• isθ◦µ-split. Then µ(M_•) is aθ-split simplicial object of RModB(M). It follows from Example 4.3.16 that µ(M•) admits a geometric realization in RModB(M).

For every objectC∈C, the diagram

RModB(M) ^C⊗ //

θ

RModB(M)

θ

M ^C⊗ //M

commutes up to homotopy. It follows that the formation of the geometric realization ofµ(M_•) is preserved by operation of tensor product with C. Applying Lemma 4.3.15, we deduce that M_• admits a geometric realization inABModB(M), which is preserved by the forgetful functorµ. This proves the following:

(∗) Let M• be ν-split simplicial object of ABModB(M), where ν = θ◦µ : ABModB(M) → M is the forgetful functor. ThenM• admits a geometric realization inABModB(M), which is preserved by the functorν.

Remark 4.3.18. In the situation of Example 4.3.17, the forgetful functorν:ABModB(M)→Mis conser-vative (since the functorsθandµare conservative, by Corollary A.4.2.3.2). Applying Theorem A.6.2.2.5, we deduce that the functorν ismonadic: that is, it exhibitsABModB(M) as the∞-category of representations of a monadT on the ∞-categoryM. The underlying functor ofT is given on objects byM 7→A⊗M ⊗B.

Remark 4.3.19. LetG:C→Dbe any monadic functor between∞-categories, so thatC'ModT(D) for some monad T on D. Let F be a left adjoint to G. Every objectC ∈C can be (canonically) realized as the geometric realization of aG-split simplicial objectC• ofC, where eachCn 'F◦Tⁿ◦G(C) lies in the essential image ofF.

Lemma 4.3.20. Letf :C→Cbe a right fibration of∞-categories and letC• be a simplicial object ofC. If f(C•) is a split simplicial object ofC, thenC• is a split simplicial object of C•.

Proof. This is an immediate consequence of the characterization of split simplicial objects given in Corollary A.6.2.1.7.

Lemma 4.3.21. Let λ^⊗:M^⊗→C^⊗×_Ass^⊗D^⊗ be a pairing of monoidal∞-categories, and letM ∈Alg(M) be an object having image(A, B)∈Alg(C)×Alg(D). Assume that:

(1) The objectB ∈Alg(D)is a trivial algebra inD. (2) The pairing λ:M→C×D is left representable.

(3) The ∞-categoryD admits totalizations of cosimplicial objects.

Then the induced pairingλ_M :_MBMod_M(M)→ABMod_A(C)×BBMod_B(D)is left representable.

Proof. Fix an object C ∈ ABMod_A(C); we wish to show that C can be lifted to a left universal object of

MBMod_M(M). Letp⁰:_ABMod_A(C)→Cdenote the forgetful functor. Using Example 4.3.17, we deduce that there is a p⁰-split simplicial object C_• of _ABMod_A(C) having colimitC, where eachC_n lies in the essential image of the left adjoint top⁰.

Fix an object D ∈ D, let MD denote the fiber product M×_D{D}, and consider the induced right fibration θ : MD → C. Condition (1) implies that D lifts uniquely to an object D ∈ BBModB(D). Set N = MBModM(M)×_BBModB(D){D⁰}, so that the projection map N → ABModA(C) is a right fibration classified by a mapχD :ABModA(C)^op →S. We claim that the canonical map χD(C)→lim

−→χD(C_•) is a homotopy equivalence. To prove this, it will suffice (Lemma 4.3.14) to show that for every simplicial object N_• of N lifting C_•, there exists a geometric realization |N_•| which is preserved by the forgetful functor q:N→ABModA(C). Letp:N→MD denote the forgetful functor. Sinceq⁰ :MD→Cis a right fibration, it follows from Lemma 4.3.20 thatp(N•) is a split simplicial object of MD. SinceNcan be identified with an∞-category of bimodule objects ofMD, Example 4.3.17 implies thatN• has a colimitN inNsuch that p(N)' |p(N•)|. Sincep(N•) is split, we conclude that the colimit ofN•is preserved byq⁰◦p'p⁰◦q. Using Lemma 4.3.20 again, we conclude that the colimit ofN_• is preserved byq.

The pairing λ_M is classified by a functor χ⁰ :_ABMod_A(C)^op →Fun(_BBMod_B(D)^op,S). It follows from the above arguments thatχ⁰(C)'lim

←−χ⁰(C_•). We wish to prove thatχ⁰(C) is representable. Using condition (3), we are reduced to proving that eachχ⁰(C_n) is a representable functor. This follows immediately from (2) and Lemma 4.3.13.

Proof of Proposition 4.3.8. LetA∈Alg(C); we wish to show that there is a left universal object of Alg(M) lying overA. Let B be a trivial algebra object ofD, so that condition (1) implies that the right fibration M×_D{B} →C is an equivalence of (monoidal)∞-categories. It follows that the pair (A, B) can be lifted to an objectM ∈Alg(M) in an essentially unique way. Using Proposition 4.3.11, we are reduced to proving that there exists a left universal object of Alg(MBModM(M)) lying overA∈Alg(ABModA(C)). Since Ais the unit object ofABModA(C), it suffices to lift Ato a left universal object of MBModM(M) (Proposition 4.3.9). The existence of such a lift now follows from Lemma 4.3.21.

Remark 4.3.22. Let λ^⊗ : M^⊗ → C^⊗×_Ass⊗D^⊗ be a pairing of monoidal ∞-categories satisfying the hypotheses of Proposition 4.3.8. Then Alg(λ) : Alg(M)→Alg(C)×Alg(D) is a left representable pairing, and therefore induces a duality functor D_Alg(λ) : Alg(C)^op → Alg(D). By unravelling the proof, we can obtain a more explicit description of this functor. Namely, letBbe a trivial algebra object ofDand lift the pair (A, B) to an algebra object M ∈Alg(M). Then, as an object of D, we can identifyD_Alg(λ)(A) with

Dλ_M(A). To compute the latter, we need to resolve A∈ABModA(C) by free bimodules. The equivalence A'A⊗AA gives a simplicial resolutionA_• ofAby free bimodulesAm'A⊗(A^⊗m)⊗A. Then

D_Alg(λ)(A)'D_λM(A)'lim

←−D_λM(A_•)' lim

[m]∈∆←−

D_λ(A^⊗m), where the last equivalence follows from Proposition 4.3.11.

Proof of Theorem 4.3.1. Let C be a monoidal ∞-category and let Alg(λ) : Alg(TwArr(C)) → Alg(C)× Alg(C^op) be the canonical map. Assertion (1) of the Theorem follows from Remark 4.3.7, which asserts that λis a pairing of ∞-categories. To complete the proof, it will suffice to verify assertion (2) (Remark 4.3.2).

Assume that the unit object 1∈ Cis final and that C admits geometric realizations of simplicial objects.

Then the pairing λ : TwArr(C) → C×C^op satisfies the hypotheses of Proposition 4.3.8, so that λ is left representable. In particular, we have a duality functorDAlg(λ): Alg(C)^op→Alg(C^op).

Note that the underlying pairing of ∞-categories λ : TwArr(C) → C×C^op induces the identity map D_λ:C^op→C^op(Remark 4.2.6). LetA∈Alg(C) be arbitrary and letM ∈Alg(TwArr(C)) be as in the proof of Proposition 4.3.8. Then the duality functor

Dλ_M :ABModA(C)^op→C^op

is right adjoint to the forgetful functor ABModA(C)^op → C^op, and therefore given by the two-sided bar constructionC7→1⊗AC⊗A1. In particular, we see thatD_Alg(λ) carriesAto the object

1⊗AA⊗A1'1⊗A1∈C.

Dans le document Derived Algebraic Geometry X: Formal Moduli Problems (Page 112-119)