Tannaka duality over ring spectra
Dualité de Tannaka sur les spectres d’anneauxJames Wallbridge
Kavli IPMU (WPI), UTIAS, University of Tokyo
5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan
james.wallbridge@ipmu.jp
March 7, 2016

Abstract
We prove a Tannaka duality theorem for (∞, 1)-categories. This is a duality between certain
derived group stacks, or more generally certain derived gerbes, and symmetric monoidal (∞, 1)categories endowed with particular structure. This duality theorem is defined over commutative
ring spectra and subsumes the classical statement. We show how the classical theory arises asa special case of our more general theory. The application to perfect complexes is explored.
Résumé
Nous prouvons un théorème de dualité de Tannaka pour les (∞, 1)-catégories. Cette dualité
est une dualité entre certains champs en groupes dérivés, ou plus généralement les gerbes dérivés,et (∞, 1)-catégories monoı̈dales munies d’une structure particulière. Cette dualité supérieure estdéfinie sur les spectres d’anneaux commutatifs et englobe la théorie de dualité classique. Nous
montrons que la théorie classique apparat comme un cas spécial de notre théorie plus générale.
L’application à des complexes parfaits est explorée.

1

Contents
1 (∞, 1)-categories

92 Symmetric monoidal structures

14

3 Spectral algebra

22

4 Stacks, gerbes and Hopf algebras

31

5 The positive, flat and finite topologies

41

6 Rigid (∞, 1)-categories

44

7 Neutralized Tannaka duality for (∞, 1)-categories

49

8 Proof of the neutralized theorem

54

9 Neutral Tannaka duality for (∞, 1)-categories

59

10 Comparison with the classical theory

61

11 Perfect complexes and schematization

63

Introduction
Classical Tannaka duality is a duality between certain groups and certain monoidal categories
endowed with particular structure. Tannaka duality for (∞, 1)-categories refers to a duality
between certain derived group stacks and certain monoidal (∞, 1)-categories endowed with particular structure. This duality theorem subsumes the classical statement. Our starting pointis the philosophy developed by Grothendieck which is to consider fundamental groupoids (ie.
1-truncated homotopy types) as automorphism groupoids of certain “fiber” functors.
This philosophy began with Grothendieck’s study of Galois theory axiomatically using purely
categorical methods [SGA]. He introduced the notion of a Galois category, that is, a category C
endowed with a fiber functor satisfying conditions such that C is equivalent to the category ofrepresentations of a profinite group. More precisely, let (C, ω) be a Galois category and define
the fundamental group of C at the base point ω to be the collection
π1 (C, ω) := Aut(ω)
of automorphisms of the fiber functor. Then π1 (C, ω) is a profinite group and the functor
C → π1 (C, ω)-FSetfrom C to the category FSet of finite sets endowed with an action of π1 (C, ω), is an equivalenceof categories. This is the Galois duality statement. By looking at the problem categoricallyGrothendieck was able to transfer the study of 1-truncated homotopy types to contexts where2such a notion was previously difficult to define. In this way he defined a new topological invariant- the étale fundamental group.
An analogous notion in the case of compact topological groups was initiated much earlier
by Tannaka [Ta] who showed that a compact group can be reconstructed from its categoryof representations. The group arises as the group of tensor preserving automorphisms of theforgetful fiber functor from the category of representations to its underlying category of vector
spaces. In [Kr], Krein characterised those categories of the form Rep(G) which arise in this way.
The passage from Galois theory to Tannaka theory is the linearization process of replacing
sets by vector spaces. Following the Galois philosophy of Grothendieck above, Saavedra developed a Tannaka duality theory for affine group schemes where the abstract category dual
is termed a neutralized Tannakian category [Sa]. The neutralized Tannaka duality statement isthen that the automorphism group of fiber functors is an affine group scheme and the Tannakian
category is equivalent to the category of representations of this affine group scheme.
More precisely, let k be a field. Then
Rep∗ : AffGpopk → (Tank )∗is an equivalence of categories where (Tank )∗ is the category of pairs (T, ω) where T is a kTannakian category and ω is a fiber functor. The category AffGpk on the left hand side is thecategory of affine k-group schemes.
Let (T, ω) be a neutralized Tannakian category and define the algebraic fundamental groupof T at the base point ω to be the collection
π1 (T, ω)alg := Aut⊗ (ω)
of tensor-perserving automorphisms of the fiber functor. Then π1 (T, ω)alg is an affine groupscheme and the functor
T → Rep(π(T, ω)alg )is an equivalence of categories. These affine group schemes are considered as algebraic versionsof 1-truncated homotopy types and lead to Deligne’s definition of the algebraic fundamentalgroup [D1].
More generally, Saavedra wrote a non-neutral Tannaka statement which characterises thosecategories T which are equivalent to the category of representations of the stack Fib(T ) of fiber
functors on T . In [D2], Deligne completed the proof that this stack is an affine gerbe (in thef f qc-topology). More precisely they showed that the functor
Rep : Ger(k, f f qc)op → Tank
is an equivalence of categories where Tank is the category of k-Tannakian categories and Ger(k, f f qc)
is the category of affine gerbes over Spec(k) in the f f qc-topology. When Fib(T ) is the neutral
gerbe of G-torsors, for G an affine group scheme, we recover the neutral Tannaka statement.
In order to study higher homotopy types it is necessary to move to a higher categoricalgeneralisation of the above ideas [Gr]. Work in this direction began in [T1] by Toën. It involves
the use of (∞, 1)-categorical techniques developed in work by Joyal [Jo] and Lurie [L1] and in
the theory of derived algebraic geometry by Lurie [L2] and Toën and Vezzosi [TV1, TV2].
3

Informally, the passage from categories to (∞, 1)-categories involves replacing the category ofsets by the (∞, 1)-category of spaces (topological spaces, Kan complexes or one such equivalentmodel). Indeed, one possible (although not so convenient) model for an (∞, 1)-category isa simplical category. This forces generalisations of other familiar categorical concepts. The
category of abelian groups is replaced by the (∞, 1)-category of spectra, an abelian category
is replaced by a stable (∞, 1)-category, commutative rings are replaced by commutative ring
spectra (called E∞ -rings) and a rigid category is replaced by a rigid (∞, 1)-category. More
examples are discussed througout the text.
In the spirit of the above, we prove the following pointed or neutralized Tannaka duality
statement for (∞, 1)-categories.
Theorem 1 (Neutralized (∞, 1)-Tannaka duality : see Theorem 7.14). Let R be a (connective,
bounded connective) E∞ -ring and τ a subcanonical topology. Then the map
Perf∗ : TGp(R, τ )op → (Tensrig
R )∗is fully faithful. Moreover, the adjunction Fib∗ a Perf∗ induces the following :1. Let R be a E∞ -ring. Then (T, ω) is a pointed finite R-Tannakian ∞-category if and only
if it is of the form Perf∗ (G) for G a finite R-Tannakian group stack.
2. Let R be a connective E∞ -ring. Then (T, ω) is a pointed flat R-Tannakian ∞-category if
it is of the form Perf∗ (G) for G a flat R-Tannakian group stack.
3. Let R be a bounded connective E∞ -ring. Then (T, ω) is a pointed positive R-Tannakian∞-category if it is of the form Perf∗ (G) for G a positive R-Tannakian group stack.
We will briefly explain the objects in this statement. The category TGp(R, τ ) is the (∞, 1)category of R-Tannakian group stacks. These are affine group stacks which are weakly rigid in
an appropriate sense. The category (Tensrig
R )∗ is the (∞, 1)-category of pointed rigid R-tensor(∞, 1)-categories. The objects in this category are rigid stable R-linear symmetric monoidal(∞, 1)-categories together with an R-linear symmetric monoidal functor to the (∞, 1)-categoryof rigid R-modules.
We will introduce three topologies on the (∞, 1)-category of R-algebras for an E∞ -ring R
called the finite, flat and positive topologies (denoted by f in, f l and ≥ 0 respectively). A rigid
R-tensor (∞, 1)-category will be called pointed Tannakian with respect to one of these topologies
if it is equipped with a fiber functor satisfying certain properties that reflect this topology (seeDefinition 7.10). A τ -R-Tannakian group stack for τ ∈ {f in, f l, ≥ 0} is an R-Tannakian group
stack such that its associated Hopf R-algebra reflects the topology τ (see Definition 7.13). Notethat we prove an equivalence of (∞, 1)-categories in the finite case but we rest on showing thatthe functor Perf∗ on Tannakian objects is fully faithful in the flat and positive cases.
The upshot of the above theorem is that we can define full homotopy types categorically as
follows. Let (T, ω) be a pointed τ -R-Tannakian ∞-category and define the algebraic homotopy
type of T at the base point ω to be the collection
π(T, ω)alg := Aut⊗ (ω)
4of τ -fiber functors. Then π(T, ω)alg is a τ -R-Tannakian group stack and the functor
T → Rep(π(T, ω)alg )is an equivalence of (∞, 1)-categories.
We also have the more general neutral Tannaka duality statement for (∞, 1)-categories.
Theorem 2 (Neutral (∞, 1)-Tannaka duality : see Theorem 9.3). Let R be a (connective,
bounded connective) E∞ -ring and τ a subcanonical topology. Then the map
Perf : TGer(R, τ )op → Tensrig
Ris fully faithful. Moreover, the adjunction Fib a Perf induces the following :1. Let R be a E∞ -ring. Then T is a finite R-Tannakian (∞, 1)-category if and only if it is of
the form Perf(G) for G a neutral finite R-Tannakian gerbe.
2. Let R be a connective E∞ -ring. Then T is a flat R-Tannakian (∞, 1)-category if it is of
the form Perf(G) for G a neutral flat R-Tannakian gerbe.
3. Let R be a bounded connective E∞ -ring. If (T, ω1 ) and (T, ω2 ) are two pointed positive
R-Tannakian (∞, 1)-categories then there exists a positive cover R → Q such that
ω1 ⊗R Q → ω2 ⊗R Q
is an equivalence.
An (∞, 1)-category is R-Tannakian with respect to a topology τ ∈ {f in, f l, ≥ 0} if it is a rigid
R-tensor (∞, 1)-category such there exists a fiber functor with respect to τ (see Definition 9.1).
A τ -Tannakian gerbe is a stack on the site of R-algebras with respect to τ which is locallyequivalent to the classifying stack of a τ -R-Tannakian group stack. It is said to be a neutral if
there exists a global point (see Definition 9.2).
Here TGer(R, τ ) is the (∞, 1)-category of neutral τ -R-Tannakian gerbes. Since the positive
topology is not subcanonical we rest with the weaker statement of (3). There may also exista reasonable notion of non-neutral (∞, 1)-Tannaka duality where the duality holds over an
extension of the base E∞ -ring but we will not consider this more general case here.
One application of our main theorem is to the study of homotopy types of complex algebraic
varieties. Let C be the field of complex numbers. To a complex algebraic variety X we can∗ (X) which are finite dimensional complex vector
associate its algebraic de Rham cohomology HdR
spaces. An important property of these vector spaces is that they are endowed with a pure Hodge
structure
M
p,q
n
HdR
(X, Z) ⊗Z C =
HdR
(X)
p+q=n

reflecting the arithmetic properties of the base field C. This pure Hodge structure is encoded in∗ (X)the Tannakian formalism by the action of a pro-algebraic group π1 (X, x)alg on the space HdR
which is the dual of the Tannakian category of pure Hodge structures. This idea extends tothe case of mixed Hodge structures and reflects the general principle in algebraic geometry that5

cohomology theories take their values in the category of linear representations of a pro-algebraic
group.
If we extend our scope beyond cohomology and consider possible full homotopy types of ourvariety X we arrive at the following question. Can one produce a functorial geometric object
H(X) which can be thought of as the homotopy type of X, together with a cohomology functor
H 7→ H ∗ (H(X))
in such a way that the action of the pro-algebraic group π1 (X, x)alg on H ∗ (H(X)) recovers∗ (X). In fact this has been done in [KPT1] using Toën’s theory
the linear representation HdR
of schematic homotopy types [T4]. A special example of a schematic homotopy type is theschematization of a space. The fundamental group of this schematization is isomorphic to the
pro-algebraic completion of the fundamental group of the underlying space. Thus one way tointerpret the schematization of a space is as a generalization of the pro-algebraic completionfunctor.
The final task in this paper is then to supply the missing Tannakian interpretation of thehomotopy type of a complex variety, already conjectured in [T4] and [T3]. More precisely weprove the following in Theorem 11.7.
Theorem 3. Let k be a commutative ring and X a finite CW complex. Then (Perf(X, k), ωx )is a pointed k-Tannakian ∞-category with respect to the positive topology. When k is a fieldof characteristic zero, the dual of this pointed positive k-Tannakian ∞-category is the schematization of X.
This theorem, apart from providing an interpretation of the dual of the Tannakian ∞category of perfect complexes of a space, extends the schematization functor over any base ringk. These results have an application to non-abelian Hodge theory and in particular, to thequestion of derived non-abelian mixed Hodge structures.
OutlineWe include here a brief overview of the contents in this paper. We begin in Section 1
by recalling the basic theory of (∞, 1)-categories [L1][S1]. All the objects in our Tannakian
statements are either (∞, 1)-categories (with extra structure) or form (∞, 1)-categories in anatural way. Due to the foundational work of Simpson and Hirschowitz in [HS], there exists a
Quillen model structure on the category of (∞, 1)-precategories which is cartesian closed. The
existence of this model structure facilitates the study of all the standard categorical notions
as applied to (∞, 1)-categories such as the theory of limits, colimits and Kan extensions. A
particularly important role will be played by the category of ind-objects of a (∞, 1)-category.
One side of our Tannaka duality statement describes certain symmetric monoidal (∞, 1)categories. In Section 2 we introduce the general theory of symmetric monoidal (∞, 1)-categories.
A key theorem (see [L2][TV3]) is an equivalence between a symmetric monoidal (∞, 1)-category
considered either as a functor from the category of pointed finite ordinals Γ to the (∞, 1)category of (∞, 1)-categories or as a cofibered (∞, 1)-category over Γ. We have corresponding
notions of commutative monoid objects, module objects and algebra objects in a symmetricmonoidal (∞, 1)-category. An important example is the symmetric monoidal (∞, 1)-category of6

presentable (∞, 1)-categories. This will allow us to define the notion of linearity using module
objects and the (∞, 1)-category of linear symmetric monoidal (∞, 1)-categories with respect toan arbitrary commutative monoid object.
In Section 3 we recall the natural extensions of abelian categories and commutative rings tothe (∞, 1)-categorical realm in the form of stable (∞, 1)-categories and E∞ -rings. The categoryof abelian groups in the familiar setting is replaced by the (∞, 1)-category of spectra and every
stable (∞, 1)-category is naturally enriched over spectra. A E∞ -ring is then simply a commuative
monoid object in the (∞, 1)-category of spectra. We then consider t-structures on a stable (∞, 1)category and recall how our t-structured spectral algebraic objects collapse to their classicalcounterparts by taking the heart construction.
The other side of our Tannaka duality statement describes derived group stacks, or moregenerally, derived gerbes. In Section 4 we introduce and study these objects with a particularinterest in those on the sites of R-algebras, for R an E∞ -ring. These are the homotopicalanalogues of (affine) group schemes and (affine) gerbes. They arise from the (∞, 1)-category ofderived stacks which are simply sheaves of spaces on the (∞, 1)-category of “derived rings”, theE∞ -rings of Section 3, endowed with a (Grothendieck) topology. We also include the theory of
Hopf R-algebras and Hopf R-algebras with respect to a site. We conclude the section with acouple of technical results for cosimplicial objects needed in the proof of our main theorem.
In Section 5 we include the study of the topologies of interest to us : the positive, flat andfinite topologies on the (∞, 1)-category of E∞ -rings. The latter two are shown to be subcanonicalmeaning that every prestack on their site is automatically a stack. This means that we will becomposing three seperate Tannaka statements depending on the topology chosen. In Section 6
we introduce a necessary finiteness condition on a symmetric monoidal (∞, 1)-category. This(∞, 1)-category of dualizable objects is the (∞, 1)-category analogue of a rigid category in thestandard theory. Also included is the study of ind-rigidity and some of its consequences in thetheory at hand.
We embark on the study of the duality theorem proper in Section 7 by introducing R-tensor(∞, 1)-categories for an E∞ -ring R. These are (∞, 1)-categories which are stable, presentable
and R-linear. The full subcategory of R-tensor (∞, 1)-categories spanned by dualizable objects
is the (∞, 1)-category of rigid R-tensor (∞, 1)-categories. Those that are pointed admit a Rtensor functor to the (∞, 1)-category of rigid R-modules. The aim of the neutralized Tannaka
duality theorem is to understand under what conditions the adjunction, whose left adjoint sends
a pointed rigid R-tensor (∞, 1)-category to its derived group stack of tensor endomorphisms of
its R-tensor functor and whose right adjoint sends a derived group stack to its pointed rigid
R-tensor (∞, 1)-category of representations, is an equivalence of (∞, 1)-categories. For thisreason, the notion of a pointed R-Tannakian (∞, 1)-category and R-Tannakian group stack
is introduced. Both notions depend on the topology chosen, in particular the fiber functor
is topology dependent, which gives rise to the three seperate cases. The neutralized (∞, 1)Tannaka duality theorem then describes how this equivalence is acheived, or how close one canget, in these cases.
The proof of the neutralized (∞, 1)-Tannaka duality theorem is deferred to Section 8 wherecare is taken to consider each case, one for each of the topologies in Section 5, in turn. In
Section 9 the neutral version of the theory is stated and proven. In this case only the existenceof a fiber functor is assumed. A comparison of our work with the classical results of Saavedra
7and Deligne is covered in Section 10 using the flat topology. The classical theory embeds fully
faithfully into our theory. In particular, the analogue of a k-Tannakian category T for the ffqc
topology in our theory is the Hk-Tannakian (∞, 1)-category of bounded complexes in T in theflat topology for the Eilenberg-Mac Lane ring spectrum Hk.
Finally, Section 11 addresses some applications of the theory when k is a field. We provethat the (∞, 1)-category of representations of a pointed schematic homotopy type of finite cohomological dimension is a pointed Hk-Tannakian (∞, 1)-category for the positive topology.
Moreover, the (∞, 1)-category of perfect complexes over a finite CW complex has the structureof a pointed positive Hk-Tannakian (∞, 1)-category and when k is of characteristic zero. Itsdual is precisely the schematization of the CW complex.
Relations to other work
The first article which discusses a theory of higher Tannaka duality is the paper [T1] byToën. Here, the theory is motivated and many of the key ingredients are introduced. Several
conjectures are then made. These ideas are then refined and the conjectures stated clearly inthe authors influential habilitation memoir [T3]. Tannakian (∞, 1)-categories over fields arealso discussed in loc. cit. and the present paper can be seen as one approach to answering
the conjectures posed in that paper. In order for our proofs to be realised, we rely much on
the foundational work on (∞, 1)-categories developed in [L2] by Lurie. We also mention thereferences [FI] and [L3] for other approaches to derived Tannaka duality.

Acknowledgements
This work is part of the authors doctorate thesis [W1] completed in Toulouse and Montpellier(connected by the now infamous TEOZ 4765). Therefore I would like to give special thanks to
Bertrand Toën. I would also like to thank Mathai Varghese, Jacob Lurie and Carlos Simpson.
This research was supported by the University of Adelaide, Université Paul Sabatier, World
Premier International Research Center Initiative (WPI) MEXT Japan and the Agence Nationale
de la Recherche grant ANR-09-BLAN-0151 (HODAG). I thank the Institut des Hautes Études
Scientifiques for hospitality during the completion of this work.

8

1(∞, 1)-categories

In this section we will give a brief review of the theory of (∞, 1)-categories. All the results inthis article are stated and proved in this setting so it will be worthwhile recalling the relevant
definitions. We will henceforth refer to a (∞, 1)-category as simply an ∞-category.
There are several different equivalent approaches to defining ∞-categories. Knowing thatcertain models are equivalent enables us to move between one model or another depending on
the given context or calculation. The nature of the equivalence is a Quillen equivalence between
certain model categories of ∞-categories : we take the point of view that we are ultimatelyinterested in the objects of the homotopy category.
We will concentrate on two models which are known to be Quillen equivalent [B1] :PC(K )  Cat(K ).
The category K is the model category of simplicial sets with the Kan model structure. We
define an (∞, 0)-category to be a Kan complex. Thus K will be the model category of (∞, 0)categories. The category on the left-hand side is the category of K -enriched precategories (see[S1] for a self contained account of the theory) with the injective or Reedy model structure. This
will play the principal role for our model category of ∞-categories.
We recall that a K -enriched precategory is a functor
A : ∆op → K
such that A0 is discrete. An ∞-category is then a K -enriched precategory, which we will call an∞-precategory, satisfying the Segal condition [HS]. The category on the right-hand side is themodel category of simplicial categories, that is, the model category of categories enriched overK , see [B2], and is often useful when one would like to choose a strict model. For this paper thereader can substitute freely any choice of model category of ∞-categories which is equivalent to
the model category of simplicial categories (see for example those reviewed in [B3]).
Every K -enriched category is an ∞-precategory with the same set of objects and wherecomposition of maps is strictly defined. This induces a fully faithful functor
G : Cat(K ) → PC(K ).
We will very often consider a K -enriched category C as an ∞-precategory by identifying C with
G(C). The advantage of empolying the model category PC(K ) is that it is a cartesian closed
model category. Therefore, for any two objects A and B in PC(K ), there exists an internal Homobject Hom(A, B) in PC(K ) and thus an internal Hom object RHom(A, B) in the homotopy
category of PC(K ). This is in contrast to the model category of simplicial categories which does
not form an internal theory (analogous to the the theory of model categories itself). Secondly, incontrast to some other approaches, the definition of an ∞-precategory is based on an inductive
procedure which allows one to construct a cartesian closed model category of (∞, n)-categories
paving the way for further extensions of the theory presented here.
We will denote by hK the homotopy category of K obtained from K by formally adjoining
inverses to all weak equivalences. If C is an ∞-category, the homotopy category of C is thehK -enriched category hC with the same set of objects and such that for any x, y ∈ C,
MaphC (x, y) = [C(x, y)]9where [•] : K → hK . For objects x0 , . . . , xn ∈ C, composition MaphC (x0 , x1 ) × . . . ×
MaphC (xn−1 , xn ) → MaphC (x0 , xn ) is given by composing the inverse of the weak equivalenceC(x0 , . . . , xn ) → C(x0 , x1 ) × . . . × C(xn−1 , xn ) with the map C(x0 , . . . , xn ) → C(x0 , xn ) andapplying the functor [•]. We obtain in this way a functor h : PC(K ) → Cat(hK ).
A map F : C → D between ∞-categories is said to be an equivalence if the induced functor
hF : hC → hD is an equivalence of hK -enriched categories, ie.
• For every x, y ∈ C, the map C(x, y) → D(F (x), F (y)) is a weak equivalence in K .
• Every y ∈ D is equivalent to F (x) in the homotopy category hD for some x ∈ C.
A functor between ∞-categories satisfying these two conditions is said to be fully faithful andessentially surjective respectively.
Model categories provide a very powerful tool for proving results in the theory of ∞categories. Apart from being the natural setting to undertake comparison results as mentioned
above, model categories themselves can be used to model ∞-categories. The construction taking a model category to an ∞-category is called localisation. In fact any ∞-category which ispresentable in an appropriately defined sense is equivalent to the localisation of a combinatorial
simplicial model category. Moreover, any ∞-category can be fully embedded into the localisationof a model category.
Definition 1.1. Let C be an ∞-category and S a set of arrows in C. A localisation of C along
S is a pair (LS C, l) where LS C is an ∞-category and l : C → LS C is a functor such that thefollowing universal property is satisfied: for any ∞-category D, the induced map
RHom(LS C, D) → RHom(C, D)
is fully faithful and its essential image consists of those functors F : C → D which send eacharrow in S to an equivalence in D.
We will often refer to a localisation (LS C, l) of C along S as simply LS C. An explicit modelfor the localisation (LS C, l) is given by composing the homotopy pushout diagramf(S × [1])

h
a

C

S×[1]

f is the groupoid generated by one isoin PC(K ) with a fibrant replacement functor where [1]∼morphism {0 −→ 1} (see Section 8.2 of [T2] for this existence result). It follows thath(LS C) → S −1 (hC)is an equivalence of categories where S −1 (hC) is the category obtained by formally inverting theelements of S.
Every category C can be regarded as an ∞-category by considering the set C(x, y) for two
objects x, y ∈ C as a discrete simplicial set (a simplicial category and thus an ∞-category).
More generally we may consider a pair (C, S) consisting of a category C together with a set of10

morphisms S of C and construct the localised category S −1 C. This procedure can be refinedusing the simplicial localisation construction of Dwyer and Kan [DK]. The simplicial localisation
DK
LDK
S (C) of the pair (C, S) has the property that there exists a natural isomorphism π0 LS (C) '−1
DKS C showing that in general, LS (C) contains higher homotopical information not encoded in
S −1 C. If C is a category, then LS C is an ∞-category and LS C → LDK
S (C) is an equivalence of∞-categories [DK]. When M is a model category we will let L(M ) := LW M be the localisationof M along the set of weak equivalences W of M . Thus hL(M ) → hM is an equivalence of
categories.
Let M be an excellent model category (Definition A.3.2.16 of [L1]) and A a combinatorial
M -enriched model category. Let C be a M -enriched category and A C be endowed with the
projective model structure. Then there exists an equivalence
RHom(C, L(A )) → L(A C )of M -enriched categories. This is called the strictification theorem.
Example 1.2. Let A be an ∞-precategory, D a simplicial category and F(A) → D an equivalence where F is the left adjoint to the fully faithful functor G. Then the induced map
RHom(A, L(PC(K ))) → L(PC(K )D )is an equivalence of ∞-categories.
Definition 1.3. We denote by Cat∞ := L(PC(K )) the ∞-category of ∞-categories and S :=L(K ) the ∞-category of (∞, 0)-categories.
The ∞-category S of spaces will play an important role in the remainder of the text fulfilling
an analogous role as the category of sets does in ordinary category theory. Let X be an ∞category and consider the endofunctor
PrX : A 7→ RHom(Aop , X)
between the ∞-category of ∞-categories. The ∞-category PrX (A) will be called the ∞-categoryof X-valued prestacks on A. When A is an ∞-precategory and X is the ∞-category S of (∞, 0)categories, we write Pr(A) for PrS (A) and refer to Pr(A) as the ∞-category of prestacks on A.
This ∞-category will also be denoted A∧ .
Let A be an ∞-precategory. Then we can replace A by a simplicial category C := F(A). Letop
C op × C → K be the natural K -enriched bifunctor. By adjunction this gives a map C → K C
where the right hand side is equivalent to A∧ by the strictification theorem. We will refer to thecompositionop
A ' C → K C ' A∧ ,which is well defined in hPC(K ), as the Yoneda embedding. The ∞-categorical Yoneda Lemmathen states that the Yoneda embedding A → Pr(A) is fully faithful. By Proposition 5.1.3.2of [L1], the Yoneda embedding preserves small limits. If C be an ∞-category, a prestack F
in Pr(C) is said to be representable if it lies in the essential image of the Yoneda embedding
C → Pr(C).
11Proposition 1.4. Let C be an ∞-category. Then there exists a simplicial model category A
and a fully faithful map C → LA of ∞-categories.
Proof. Let D := F(C) be a strict model for C. Then the proposition follows from the composition
y

∼

∼

op

C−
→ RHom(C op , L(K )) −→ RHom(Dop , L(K )) −→ L(K D )using the fully faithful ∞-Yoneda lemma and the strictification theorem. We conclude by settingop
A := K D .
We will use this property to characterise ∞-categories having special properties by placing
natural conditions on the model category A and asking that the fully faithful map C → L(A )be an equivalence. Our first example is the following.
Definition 1.5. An ∞-category C is said to be presentable if it is equivalent to the localisationof a combinatorial simplicial model category.
Let Catp∞ denote the full subcategory of Cat∞ spanned by the presentable ∞-categories andcolimit preserving functors. If C is a presentable ∞-category and A is an ∞-precategory then
the ∞-category RHom(A, C) is presentable. In particular, the ∞-category of prestacks Pr(A)is presentable.
Let C be an ∞-category and S a set of arrows of S. In the setting of presentable ∞categories, the theory of localisations has a simple characterisation : LS C can be identified witha full subcategory of C. An object x in C is said to be S-local if for every arrow f : y → z in S,
the induced map C(z, x) → C(y, z) is an equivalence in S. An arrow f : x → y in C is said tobe an S-equivalence if for every S-local object z in C, the induced map C(y, z) → C(x, z) is anequivalence in S.
Let C be a presentable ∞-category and (LS C, l) a localisation of C in the ∞-category Catp∞ .
Then an object x in C is S-local if and only if it belongs to LS C. Furthermore, every elementof S is an S-equivalence in C. A localisation in the setting of presentable ∞-categories will becalled a Bousfield localisation to distinguish from the more general localisation of Definition 1.1.
An ∞-category admits all finite limits if and only if it admits pullbacks and a final object. A
functor between ∞-categories preserves finite limits if and only if it preserves pullbacks and final
objects. An analogous statement holds for finite colimits by passing to the opposite ∞-category.
Let F : C → D be a functor between ∞-categories. If C admits finite limits then F is said to beleft exact if it preserves finite limits. If C admits finite colimits then F is said to be right exact
if it preserves finite colimits. It is said to be exact if it is both left and right exact. We denoteby Homlex (C, D) (resp. Homrex (C, D)) the full subcategory of Hom(C, D) spanned by the left
exact (resp. right exact) functors.
When A is a combinatorial simplicial model category, then the existence of homotopy limits
and colimits in A ensures the existence of limits and colimits in the ∞-category L(A ). An∞-category is said to be complete if it admits all (small) limits and cocomplete if it admits all(small) colimits. It is said to be bicomplete if it is both complete and cocomplete. Let F : C → D
be a functor between ∞-categories. If C admits small limits then F is said to be continuous
if it preserves small limits. If C admits small colimits then F is said to be cocontinuous if it
preserves small colimits. It is said to be bicontinuous if it is both continuous and cocontinuous.
12

We denote by Homct (C, D) (resp. Homcoct (C, D)) the full subcategory of Hom(C, D) spannedby the continuous (resp. cocontinuous) functors.
The following general theory of limits and colimits in an ∞-category is contained in [L1].
Let A be a collection of ∞-precategories. An ∞-category C is said to admit A-indexed colimits
if it admits colimits along all diagrams indexed by elements in A. A functor F : C → D is said
to preserve A-indexed colimits if it preserves colimits along all diagrams indexed by elements in
A. We denote by RHomA (C, D) the full subcategory of RHom(C, D) spanned by those functors
which preserve A-indexed colimits. If C is an ∞-category and D an ∞-category which admits
A-indexed colimits then any functor functor F : C → D can be extended, essentially uniquely,
to an A-colimit preserving functor G : PrA (C) → D where PrA (C) is an ∞-category admitting
A-indexed colimits.
More precisely, by Proposition 5.3.6.2 of [L1], there exists an ∞-category PrA (C) admitting
A-indexed colimits and a fully faithful functor y : C → PrA (C) satisfying the following universal
property : for any ∞-category D admitting A-indexed colimits, composition with y induces anequivalence
RHomA (PrA (C), D) → RHom(C, D)
of ∞-categories. When A is the collection of all ∞-precategories, the ∞-category PrA (C) is
identified with the ∞-category of prestacks. Thus composition with the Yoneda embedding
A → Pr(A) induces an equivalence
RHomcoct (Pr(A), C) → RHom(A, C)
of ∞-categories.
Given an ∞-category C, we may also define other ∞-categories, thought of informally as
the ∞-categories associated to C by formally adding colimits of type A, using this universal
property. If C is an ∞-category then the ∞-category of prestacks on C is freely generated undersmall colimits by the image of the Yoneda embedding. The ∞-category of ind-objects of C isthen the smallest full subcategory of this ∞-category of prestacks which contains the image of
the Yoneda embedding and is stable under filtered colimits. Thus it is freely generated underfiltered colimits by C.
Definition 1.6. Let C be an ∞-category and A the class of all small filtered ∞-precategories.
Then the ∞-category of ind-objects of C is given by Ind(C) := PrA (C).
Let A be the class of all small κ-filtered ∞-precategories. The ∞-category of ind-objects ofC admits the following characterisation : the objects of Ind(C) are functors I → C where I ∈ A
and given two objects F : I → C and G : J → C in Ind(C), the mapping space is given by
Map(F, G) = lim colim C(F (i), G(j)).
i∈I

j∈J

By Proposition 5.3.5.14 of [L1], the Yoneda embedding y : C → Ind(C) taking x to the functor
x : ∗ → C preserves all small colimits which exist in C. The essential image of y consists ofobjects x in C such that the corepresentable functor
C(x, •) : C → S
13preserves filtered colimits. Such objects are said to be compact in C. Let C cpt denote the fullsubcategory of C spanned by the compact objects.
An ∞-category C is said to be closed if every diagram in C indexed by a small ∞-precategory
admits a colimit in C. Clearly, an ∞-category C is equivalent to Ind(D) for some small ∞category D if and only if the ∞-category C is closed and has a small subcategory D consisting ofcompact objects such that every object of C is a filtered colimit of objects of D. This motivates
the following.
Definition 1.7. An ∞-category C is said to be accessible if there exists a small ∞-category D
such thatC → Ind(D)is an equivalence of ∞-categories.
Let C be a presentable ∞-category. By Proposition 5.5.2.2 of [L1], a functor F : C op → S
is representable if and only if it preserves small limits. Similarly, a functor F : C → S is
corepresentable if and only if it is accessible and preserves small limits (Proposition 5.5.2.7 of[L1]). An important ramification of this is the adjoint functor theorem which states that if C
and D are presentable ∞-categories then a functor F : C → D admits a right adjoint if and only
if it preserves small colimits and admits a left adjoint if and only if it is accessible and preserves
small limits (see Corollary 5.5.2.9 of [L1]).

2

Symmetric monoidal structures

In this section we define symmetric monoidal structures in the setting of ∞-categories. References include [L2] and [TV3]. In particular we define the ∞-category of presentable symmetricmonoidal ∞-categories and the symmetric monoidal ∞-category of modules over a commutative
monoid object in a symmetric monoidal ∞-category. These notions are used to define linear∞-categories which are necessary for the definition of a Tannakian ∞-category in Section 7.
We define a symmetric monoidal ∞-category using the language of cofibered ∞-categories(see Section 3.2 of [L1] and Section 1.3 of [TV3]). Let Γ denote the category of pointed finite
ordinals and point preserving maps. This is equivalent to the category of `all linearly orderedfinite sets with a distinguished point ∗. We denote the pointed ordinal n {∗} by hni. The
category Γ is a monoidal category with monoidal structure (Γ, ∨, h0i).
An arrow f : hni → hmi in Γ is said to be inert (resp. semi-inert) if f −1 {j} = {i} (resp.
−1
f {j} ∈ {∅, {i}}) for all j ∈ hmi−∗. It is said to be null if f (i) = ∗ for all i ∈ hni. It issaid to be active if f −1 {∗} = {∗}. Every arrow f in Γ admits a factorisation f = f 00 ◦ f 0 by
an inert arrow f 0 followed by an active arrow f 00 . This factorisation is unique up to (unique)isomorphism.
Consider an object p : A → Γ in the category PC(K )/Γ . An arrow f in A(a, b) is said to be
p-cocartesian if for all c ∈ A, the induced morphism
A(b, c) → A(a, c) ×I(p(a),p(c)) I(p(b), p(c))
is a weak equivalence in K . An object p : A → Γ in the category PC(K )/Γ is said to be acofibered ∞-category if for every arrow u : hni → hmi in Γ and every object a in A with p(a) =14

hni, there exists a p-cocartesian arrow f such that p(f ) is isomorphic to u in the undercategory
Γhni/ . A morphism in the homotopy category of PC(K )/Γ is said to be cocartesian if it preservescocartesian arrows. The (non-full) subcategory of h(PC(K )/I ) consisting of cofibered objects
and cocartesian morphisms will be denoted by h(PC(K )/I )cc . An important observation is thatthe condition to be cofibered is stable by equivalences in PC(K ).
Let A be an ∞-precategory and Ahni the fiber of the map p : A → Γ at hni ∈ Γ. For eachn ≥ 1 and 0 < i ≤ n, consider the n pointed maps pi : hni → h1i in Γ given by pi (j) = {j} if
i = j and pi (j) = ∗ otherwise. These induce natural maps (pi )! : Ahni → Ah1i .
Definition 2.1. Let A be a ∞-precategory. A symmetric monoidal ∞-category is a cofiberedobject p : A → Γ in PC(K )/Γ such thatQ(pi )!

Ahni −−i−−→ (Ah1i )n
is an equivalence for each n ≥ 0.
We let Ah1i be the underlying ∞-category of A. We will often abuse notation by referring
to a symmetric monoidal ∞-category p : A → Γ as simply A. By Proposition 1.4 of [TV3] there
exists an equivalenceh(PC(K )I ) → h(PC(K )/I )cc
of categories. Thus a symmetric monoidal ∞-category coincides with the definition as a functor
into PC(K ).
Let p : A → Γ be a symmetric monoidal ∞-category. Then an arrow f in A is said to be
p-inert if f is a cocartesian arrow in A such that p(f ) is inert in Γ.
Definition 2.2. Let p : A → Γ and q : B → Γ be two symmetric monoidal ∞-categories. A
functor F : A → B is said to be symmetric monoidal if the diagram
F

A

- B

q

p



-

Γ
commutes and F carries p-cocartesian arrows to q-cocartesian arrows. It is said to be lax
symmetric monoidal if F carries p-inert arrows to q-cocartesian arrows.
Let RHom⊗Γ (A, B) denote the full subcategory of RHomΓ (A, B) spanned by the symmetric monoidal functors. Let Cat⊗∞ denote the ∞-category consisting of symmetric monoidal∞-categories and symmetric monoidal functors. Likewise, let RHomlax
Γ (A, B) denote the fullsubcategory of RHomΓ (A, B) spanned by the lax symmetric monoidal functors.
Definition 2.3. Let p : C → Γ be a symmetric monoidal ∞-category. A commutative monoid
object in C is a lax symmetric monoidal section of p (where the identity map idΓ endows thetrivial category h0i with a symmetric monoidal structure).
15

The ∞-category of commutative monoid objects in C will be denoted
CMon(C) := RHomlax
Γ (Γ, C).
A commutative comonoid object in C is a commutative monoid object in C op . The ∞-category
CMon(C) has an initial object A such that the unit map 1Ch1i → A(h1i) is an equivalence in
Ch1i (Corollary 3.2.1.9 of [L2]). For a commutative (co)monoid object A, we will sometimes use
the notation An := A(hni).
Example 2.4. Let p : C → Γ be a symmetric monoidal ∞-category and A an ∞-precategory.
Then the ∞-category RHom(A, Ch1i ) inherits the structure of a symmetric monoidal ∞-category
RHom(A, C) → Γ called the pointwise symmetric monoidal structure where we define
RHom(A, C)hni := RHom(A, C) ×RHom(A,Γ) {hni}giving RHom(A, C)hni ' RHom(A, Chni ). As a result, there exists an equivalence
CMon(RHom(A, C)) → RHom(A, CMon(C))
of ∞-categories.
Example 2.5. Let C be a symmetric monoidal ∞-category and Dh1i a full subcategory of
Ch1i . Assume that for every equivalence x → y in Ch1i , if y ∈ Dh1i then x ∈ Dh1i . Define asubcategory
D of C by letting an object x ∈ Chni belong to D if and only if the image under
Q
n
n
(p
)
:
C
hni → (Ch1i ) belongs to (Dh1i ) . Then it is clear that the restriction map D → Γi i ∗is a symmetric monoidal ∞-category if D is closed under tensor products and contains the unit
object of C.
Example 2.6. Let p : C → Γ be a symmetric monoidal ∞-category. Then by Example 3.2.2.4 of[L2], there exists a symmetric monoidal structure on the ∞-category CMon(C) of commutative
monoid objects in C induced by that on C. Moreover, by Proposition 3.2.4.7 of loc. cit., thetensor product of commutative monoid objects corresponds to the coproduct.
Example 2.7. If C admits finite colimits and the tensor product bifunctor preserves finite
colimits seperately in each variable then the ∞-category Ind(C) of ind-objects of C admits asymmetric monoidal structure which is characterised, up to symmetric monoidal equivalence,by the properties that the tensor product bifunctor ⊗ : Ind(C) × Ind(C) → Ind(C) preserves
small colimits seperately in each variable and that the Yoneda embedding C → Ind(C) canbe extended to a symmetric monoidal functor. This statement follows from a more generalstatement for a symmetric monoidal ∞-category admitting colimits indexed by an arbitrarycollection of ∞-precategories. See Proposition 6.3.1.10 of [L2] for a precise statement.
Let C be an ∞-category. There exists a natural analogue of the localisation of C as definedin Definition 1.1 when C is endowed with a symmetric monoidal structure whose underlying∞-category is LS C. We will abuse notation by denoting the resulting symmetric monoidal ∞category by LS C. The existence result follows from Proposition 4.1.3.4 of [L2]. When M is asymmetric monoidal model category we will define L(M ) := LW (M c ) (the monoidal producton M c preserves the weak equivalences W in M c ).
16

Example 2.8. The model category PC(K ) of ∞-categories is a symmetric monoidal simplicial
model category for the cartesian product. Thus Cat∞ := L(PC(K )) is a symmetric monoidal∞-category. Explicitly, the cofibered ∞-category Cat∞ is given as follows :• The objects are pairs (hni, (C0 , . . . , Cn )) where hni is an object of Γ and each Ci is a fibrant∞-precategory.
• A map between two objects (hni,
Q C• ) and (hmi, D• ) is a map u : hni → hmi in Γ together
with a collection of functors u(i)=j Ci → Dj .
By Proposition 4.4.4.6 and Theorem 4.4.4.7 and of [L2], if M is a combinatorial symmetricmonoidal model category with the conditions that M is left proper, the class of cofibrations in
M is generated by cofibrations between cofibrant objects, M satisfies the monoid axiom andevery cofibration in M is a power cofibration (see Definition 4.4.4.2 of loc. cit.) then CMon(M )admits a combinatorial model structure such that the map
L(CMon(M )) → CMon(L(M ))is an equivalence of ∞-categories. When the ∞-category Cat∞ is equipped with the cartesianmonoidal structure thenCMon(Cat∞ ) → Cat⊗∞is an equivalence of ∞-categories. This follows from the more general equivalence of Proposition
2.4.2.5 of [L2].
Let C be a symmetric monoidal ∞-category. A symmetric monoidal structure on C issaid to be compatible with countable colimits if for any simplicial set A with only countably
many simplices, the ∞-category C admits A-indexed colimits and for any x in C, the functor• ⊗ x : C → C preserves these colimits. If C is compatible with countable colimits then theforgetful functor CMon(C) → C admits a left adjoint
Fr : C → CMon(C)which we refer to as the free functor. A precise statement can be found in Corollary 3.1.3.5of [L2]. If C is equivalent to L(M ) for M a symmetric monoidal model category then Fr isequivalent to a functor Fr : L(M ) → L(CMon(M )) so
Fr(x) = LSym(x)where Sym(x) :=

`

n≥0 x

⊗n /Σ

n.

Definition 2.9. Let p : D → Γ be a symmetric monoidal ∞-category. An ∞-category C is said
to be tensored over D if there exists a map F : C → D in PC(K ) such that:
1. The composition (p ◦ F ) : C → Γ is cofibered in PC(K )/Γ .
2. The map F carries (p ◦ F )-cocartesian arrows of C to p-cocartesian arrows of D.
3. For each n ≥ 0, the inclusion {n} ⊆ hni induces an equivalence Chni → Dhni × C{n} of∞-categories.
17

Let C be an ∞-category tensored over D. We will refer to the fiber Ch0i as the underlying∞-category of C and by abuse, also denote it by C. We obtain a natural diagram∼

C{0} ← Ch1i −→ Dh1i × C{1}which induces a symmetric monoidal bifunctor ⊗ : Dh1i × Ch0i → Ch0i , together with its higher
symmetric monoidal structure, which is well defined up to homotopy.
Definition 2.10. Let A be a symmetric monoidal ∞-category and C a ∞-category tensoredover A. Let a ∈ A and x, y ∈ C. Then C is said to be enriched over A if the functor Aop → S
given bya 7→ C(a ⊗ x, y)
is representable for all x, y ∈ C. The representing object will be denoted Mor(x, y) and called
the morphism object of x and y.
It follows directly from Definition 2.10 that morphism objects are characterised by the following universal property: there exists a map ev : Mor(x, y) ⊗ x → y such that composition withev yields an equivalenceA(a, Mor(x, y)) → C(a ⊗ x, y)evevof (∞, 0)-categories. The composition Mor(y, z) ⊗ Mor(x, y) ⊗ x −→ Mor(y, z) ⊗ y −→ z yields acomposition map
Mor(y, z) ⊗ Mor(x, y) → Mor(x, z)
and the chain of equivalencesA(b, Mor(a, Mor(x, y))) ' A(b ⊗ a, Mor(x, y)) ' C(b ⊗ a ⊗ x, y) ' A(b, Mor(a ⊗ x, y))yields an equivalence
Mor(a, Mor(x, y)) → Mor(a ⊗ x, y)of morphism objects.
Example 2.11. Let C be a symmetric monoidal ∞-category. Then the symmetric monoidal
product ⊗ : C × C → C endows C with the structure of an ∞-category tensored over itself. If
it is furthermore enriched, then the morphism object Mor(c, d) is just the internal Hom object
Hom(c, d) in C.
Let A be a monoidal ∞-category and C a ∞-category tensored over A. Suppose further
that C and A are presentable ∞-categories. Let a = colimi ai be an object in A. A prestack isrepresentable if and only if it preserves small limits. Therefore, the ∞-category C is enriched over
A if C(colimi ai ⊗x, y) ' limi C(ai ⊗x, y). By assumption, C(colimi ai ⊗x, y) ' C(colimi (ai ⊗x), y)which is naturally equivalent to limi C(ai ⊗ x, y). Thus the ∞-category C is enriched over A if
the functor•⊗x:A→C
preserves small colimits for all x ∈ C.
We will now define the ∞-category of module objects in a symmetric monoidal ∞-category.
A more detailed and general exposition can be found in Section 3.3.3 of [L2].
18

Notation 2.12. Let K si denote the full subcategory of RHom(h1i, Γ) spanned by the semi-inert
arrows. Let K null denote the full subcategory of K si spanned by the null arrows. There are twonatural maps evi : K • → Γ given by evaluation on i ∈ {0, 1}. A morphism in K • is said to beinert if its images under e0 and e1 are inert in Γ.
Let p : C → Γ be a symmetric monoidal ∞-category. We define an ∞-precategory M(C)through the following universal property: for every ∞-precategory A equipped with a map
A → Γ the map
RHomΓ (A, M(C)) → RHomRHom({1},Γ) (A ×RHom({0},Γ) K si , C)
is an equivalence. Thussi
M(C)hni → RHomRHom({1},Γ) (Khni>, C)
si
is an equivalence where Khni>
denotes the homotopy fiber of K si → RHom({0}, Γ) at hni. An
object of M(C)h1i is then given by a commutative diagram
F

si
Kh1i>

- C

p

ev1



-

Γ.
Let M(C) denote the full subcategory of M(C) spanned by those vertices for which the functor
F preserves inert morphisms.
Similarly we define an ∞-precategory A(C) through the following universal property : forevery ∞-precategory A equipped with a map A → Γ the map
RHomΓ (A, A(C)) → RHomRHom({1},Γ) (A ×RHom({0},Γ) K null , C)
is an equivalence. Thus
null
A(C)hni → RHomRHom({1},Γ) (Khni>, C)
null denotes the homotopy fiber of K null → RHom({0}, Γ) at hni. Let
is an equivalence where Khni>

A(C) denote the full subcategory of A(C) spanned by those vertices for which the functor F
preserves inert morphisms. We define
ModR (C) := M(C) ×A(C) {R}.
By Theorem 4.4.2.1 of [L2], if C admits colimits of simplicial objects such that for every object
x in C the functor • ⊗ x preserves these colimits, then the projection
p : ModR (C) → Γ19

is a symmetric monoidal ∞-category. The unit object of p is canonically equivalent to R. The
symmetric monoidal product is called the relative tensor product and for two R-modules M andN will be denoted M ⊗R N . See Theorem 4.4.2.1 [L2] for a detailed discussion of the relative
tensor product functor.
The above construction is functorial in p and hence we obtain a functor
Mod(C) : CMon(C) → Cat⊗∞sending a commutative monoid object R to ModR (C) and a map f : R → Q to the functor• ⊗R Q. Here • ⊗R Q is the symmetric monoidal base change functor left adjoint to the forgetful
functor ModQ (C) → ModR (C).
Let C be a symmetric monoidal ∞-category and R a commutative comonoid object in C.
By definition, the ∞-category of comodules in C over R is
ComodR (C) := ModR (C op )op .
Example 2.13. Many examples of ∞-categories of modules arise from the localisation of model
categories of modules. More precisely, let M be a combinatorial symmetric monoidal model
category and R a commutative monoid object of M . Assume that M satisfies the monoid axiom[SS2]. Then the category ModR (M ) of R-modules admits a combinatorial model structure wherea map is a fibration if and only if it is a fibration in M and a weak equivalence if and only if it isa weak equivalence in M . If M is endowed with a simplicial model structure, then ModR (M )is a simplicial model category. If we further assume that M satisfies the conditions furnishingan equivalence s : L CMon(M ) → CMon(LM ), then the natural map
L ModR (M ) → Mods(R) (LM )is an equivalence of ∞-categories. This follows from Theorem 4.3.3.17 of [L2].
Definition 2.14. Let C be a symmetric monoidal ∞-category and R a commutative monoid
object in C. A commutative R-algebra object in C is a commutative monoid object in thesymmetric monoidal ∞-category ModR (C) of R-modules.
Let CAlgR (C), or simply CAlgR , denote the ∞-category CMon(ModR (C)) of commutative
R-algebra objects in C. It follows from Corollary 3.4.1.7 of [L2] that if C be a symmetricmonoidal ∞-category and R is a commutative monoid object in C, then there exists an equivalenceθR : CAlgR (C) → CMon(C)R/of ∞-categories. Furthermore, by Corollary 6.3.5.16 of [L2], there exists a natural fully faithfulmapCMon(C) → CMon(ModC (Cat∞ ))and so the map
CMon(C) → CMon(Cat∞ )C/
given by composing this fully faithful map with θC is fully faithful.
20

The ∞-category CAlgR (C) inherits the structure of a symmetric monoidal ∞-category where
the tensor product is given by the tensor product in C (see Example 2.6). Furthermore, thistensor product coincides with the coproduct in the ∞-category of commutative R-algebras.
Let C be a symmetric monoidal ∞-category such that the symmetric monoidal product preserves (small) colimits separately in each variable and the fiber Chni is a presentable ∞-category
for all n > 0. In this case we will say that C is a presentable symmetric monoidal ∞-category. If
C is a presentable symmetric monoidal ∞-category then CMon(C) is a presentable ∞-category.
This follows from Corollary 3.2.3.5 of [L2]. Moreover, if C is a presentable symmetric monoidal∞-category then the ∞-category ModR (C) is a presentable symmetric monoidal ∞-category by
Theorem 3.4.4.2 of [L2]. Combining these two results, the ∞-category CAlgR (C) is a presentable∞-category.
There exists a symmetric monoidal structure on the ∞-category Catp∞ of presentable ∞categories by Proposition 6.3.1.14 of [L2]. This is a subcategory of the symmetric monoidal∞-category Cat∞ which can be explicitly described as follows :• The objects are pairs (hni, (C0 , . . . , Cn )) where hni is an object of Γ and each Ci is apresentable fibrant ∞-precategory.
• A map between two objects (hni,
Q C• ) and (hmi, D• ) is a map u : hni → hmi in Γ together
with a collection of functors u(i)=j Ci → Dj which preserve colimits seperately in eachvariable.
One can show (see loc. cit.) that the unit object of Catp∞ with this symmetric monoidal⊗
structure is the ∞-category S of spaces. Let Catp,⊗∞ denote the subcategory of Cat∞ spannedby presentable symmetric monoidal ∞-categories whose monoidal bifunctor preserves colimits
seperately in each variable and whose morphisms are colimit preserving symmetric monoidal
functors. Then we have an equivalence
CMon(Catp∞ ) → Catp,⊗∞of ∞-categories. Thus a symmetric monoidal ∞-category C belongs to CMon(Catp∞ ) if and only
if C is presentable and the tensor product bifunctor ⊗ : Ch1i × Ch1i → Ch1i preserves (small)colimits seperately in each variable.
Let C be a presentable symmetric monoidal ∞-category. Then the ∞-category ModR (C) of
R-modules in C is a presentable symmetric monoidal ∞-category with a bicontinuous monoidal
product. Thus ModR (D) belongs to CMon(Catp∞ ). We can then make the following definition.
Definition 2.15. Let D be a presentable symmetric monoidal ∞-category. A presentable ∞category is said to be R-linear if it is endowed with the structure of a ModR (D)-module object
in the symmetric monoidal ∞-category Catp∞ of presentable ∞-categories.
The ∞-category of R-linear ∞-categories is given by ModModR (D) (Catp∞ ). Note that an
R-linear ∞-category is a presentable ∞-category C which is tensored over the ∞-category
ModR (D). Then the functor • ⊗ x : ModR (D) → C preserves colimits for all x ∈ C owing to themonoidal structure on Catp∞ . Thus the presentable ∞-category C is enriched over ModR (D) as
expected. Therefore there exists an equivalence
CMon(ModModR (Catp∞ )) ' CMon(Catp∞ )ModR / .
21

of ∞-categories. The term on the left hand side is the ∞-category of R-linear presentable
symmetric monoidal ∞-categories and R-linear symmetric monoidal functors.
Example 2.16. Let D be a presentable symmetric monoidal ∞-category. Then ModR (D) isan R-linear ∞-category.
Example 2.17. Let k be a commutative ring and C a presentable k-linear symmetric monoidal
category (ie. a presentable symmetric monoidal category with a Modk (Ab)-module structure).
Then L(C) is a k-linear ∞-category, ie. if the map Modk → C of categories endows C witha k-linear structure then the map of ∞-categories L(Modk ) → L(C) endows L(C) with a klinear structure. More generally, if M is a k-linear symmetric monoidal model category, ie. a
Modk (Ab)-enriched symmetric monoidal model category, then L(M ) is a k-linear ∞-category.

3

Spectral algebra

In this section we address the algebraic structures appearing in our Tannaka duality theorem.
We begin by reviewing the basic theory of stable ∞-categories. A more detailed account canbe found in [L2]. Stable ∞-categories replace the role of abelian categories in the ∞-categoricalrealm. Indeed, after defining the ∞-category of spectra, together with its natural symmetricmonoidal structure, we will see that every stable ∞-category is naturally enriched over spectra.
Endowing the ∞-category of spectra with its natural t-structure we see that its heart is equivalent
to the category of abelian groups. Since a commutative ring is simply a commutative monoid
object in the category of abelian groups, it is natural to define a “commutative derived ring”,
or E∞ -ring, as a commutative monoid object in the ∞-category of spectra.
Let C be an ∞-category. We call an object which is both initial and terminal in C a zero
object and denote it by 0 ∈ C. An ∞-category is said to be pointed if it contains a zero object.
Definition 3.1. An ∞-category C is said to be stable if it is pointed, admits finite limits andcolimits and pullback and pushout squares coincide.
Note that if a functor between stable ∞-categories is left or right exact it is automaticallyexact. Let Cat⊥∞ denote the full subcategory of Cat∞ spanned by stable ∞-categories and exact
functors.
Let C be a pointed ∞-category and `f : x → y an arrow in C. A kernel of f is a pullback
x ×y 0 and a cokernel of f is a pushout y x 0. They are uniquely determined up to equivalencein C. A full subcategory of a stable ∞-category is said to be a stable subcategory if it contains
a zero object and is closed under the formation of kernels and cokernels.
The ∞-category Cat⊥∞ admits all (small) limits and all (small) filtered colimits (Theorem
1.1.4.4 and Proposition 1.1.4.6 of [L2]). The structure of a stable ∞-category induces a heavy
simplification of the nature of its limits and colimits : if κ is a regular cardinal, then a stable∞-category has all κ-small limits (resp. colimits) if and only if it has κ-small products (resp.
coproducts). Furthermore, an exact functor between stable ∞-categories preserves κ-small limits(resp. colimits) if and only if it preserves κ-small products (resp. coproducts).
Let C be a pointed ∞-category with finite limits. The loop functor Ω of C is the endomorphism of C given byΩ : x 7→ 0 × 0.
x22This functor admits a left adjoint
Σ : x 7→ 0

a

0

x

called the suspension functor. A pointed ∞-category is stable if and only if it admits finite
limits and the loop functor is an equivalence of ∞-categories. Likewise, it is stable if and only
if it admits finite colimits and the suspension functor is an equivalence of ∞-categories. This
follows from Proposition 1.1.3.4 and Corollary 1.4.2.20 of [L2]. If C is a stable ∞-category and
A is a ∞-precategory then the ∞-category RHom(A, C) is stable. In particular, Indκ (C) is astable ∞-category (see Proposition 1.1.3.6 of [L2]).
Example 3.2. A pointed, closed model category M is said to be stable if the adjunction Σ a Ω
is an equivalence in the homotopy category hM . Thus for any stable model category M , the∞-category L(M ) is stable. Moreover, if M is a cofibrantly generated, proper, stable, simplicial
model category with a set P of compact generators, the authors in [SS1] prove an equivalence
between M and the model category ModE(P ) of modules over a certain spectral endomorphism
category E(P ) (see Definition 3.7.5 of loc. cit.). We thus obtain an equivalence of stable ∞categories L(M ) → L(ModE(P ) ).
Example 3.3. Let A be an abelian category with enough projective objects and C(A) the
simplicial model category of chain complexes in A. Then the derived ∞-category L(C(A)) of
A is a stable ∞-category. The homotopy category hL(C(A)) can be identified with the derived
category D(A) of A. Likewise, one can define the bounded (resp. bounded above, bounded
below) derived ∞-category of A. See Section 1.3.1 of [L2] for more details.
Let C be an ∞-category with finite limits and Z the linearly ordered set of integers which
we consider as a filtered category. Let T be an endofunctor on C. We construct the following
endofunctor
φ : RHom(Z, C) → RHom(Z, C)
defined by φ(F )(n) := T (F (n + 1)).
Definition 3.4. Let C be an ∞-category with finite limits and T an endofunctor on C. A
T -spectrum object of C is a functor F : Z → C such that F → φ(F ) is an equivalence in
RHom(Z, C).
The ∞-category of T -spectrum objects in C, denoted SpT (C), is given by the homotopy
pullback
SpT (C)- RHom(Z, C)(φ,id)?

?

d
RHom(Z, C) - RHom(Z, C) × RHom(Z, C)

where d denotes the diagonal map. The equivalence d(G) ' (φ, id)(F ) induces the equivalences
∼

∼

F ←
−G−
→ φ(F )
23

whose composition gives the equivalence required in Definition 3.4. The ∞-category of T spectrum objects in C comes naturally equipped with an evaluation functor Evn : SpT (C) → C
for every n ∈ Z which acts on a spectrum F and picks out its n-th term F (n). If C is apresentable ∞-category then this evaluation functor admits a left adjoint Frn : C → SpT (C).
We will be particularly interested in the case where the endofunctor T is the loop functor.
In this case, if C is a pointed ∞-category with finite limits then the ∞-category SpΩ (C) is astable ∞-category (see Proposition 1.4.2.18 of [L2]). This defines a natural functor SpΩ from the∞-category of pointed ∞-categories with finite limits and left exact functors to the ∞-categoryCat⊥∞ of stable ∞-categories whose right adjoint is the forgetful functor.
Let C∗ denote the full subcategory of RHom([1], C) spanned by those morphisms x → y forwhich x is a terminal object of C. We call C∗ the ∞-category of pointed objects of C. If C ispointed, then the forgetful functor C∗ → C is an equivalence of ∞-categories.
Definition 3.5. A spectrum is a Ω-spectrum object of the ∞-category S∗ of pointed spaces.
Let Sp := SpΩ (S∗ ) denote the ∞-category of spectra. The ∞-category Sp is stable andpresentable. It follows from Definition 3.4 that the ∞-category Sp of spectra can be identified
with the homotopy limit of the towerΩΩ{. . . → S∗ −→ S∗ −→ S∗ }.
of ∞-categories.
Let S f in denote the smallest full subcategory of S which contains the final object and isstable under finite colimits. Then Ind(S∗f in ) → S∗ is an equivalence of ∞-categories and thusS∗ is compactly generated. Moreover, let
Σ

Σ

f in
S∞
:= colim {S∗f in −→ S∗f in −→ . . .}

in the ∞-category of ∞-categories and exact functors. Then the ∞-category of spectra iscompactly generated andf in
Ind(S∞) → Spis an equivalence of ∞-categories.
Let C be a presentable ∞-category. Then the ∞-category SpΩ (C∗ ) is presentable and thenatural functor Evn : SpΩ (C∗ ) → C admits a left adjoint Frn : C → SpΩ (C∗ ). Let C = S and∗ be the final object of S. The object Fr0 (∗) of Sp will be called the sphere spectrum and will
be denoted by S. Recall the homotopy group functor on spectra πn : Sp → Ab which takes aspectrum A to the abelian group HomhSp (S[n], A). A map f : A → B of spectra is an equivalence
if and only if it induces isomorphisms πn A → πn B for all n ∈ Z.
The ∞-category Sp admits a symmetric monoidal structure which is uniquely characterized
by the property that the unit object of Sp is the sphere spectrum S and the bifunctor ⊗ :Sp × Sp → Sp preserves colimits seperately in each variable (Corollary 6.3.2.16 of [L2]). If S p is
the category of symmetric spectra endowed with the S-model structure [Sh] and smash productsymmetric monoidal structure then S p can be lifted to a simplicial symmetric monoidal model
category and there exists an equivalence L(S p) → Sp of symmetric monoidal ∞-categories.
24

Notation 3.6. Let C be a stable ∞-category and x be any object of C. Let(Σn x
if n ≥ 0,x[n] :=−nΩ x if n ≤ 0,
given by taking the nth power of the suspension and loop functors. We use the same notation
for the corresponding object in hC.
The homotopy category of a stable ∞-category C is a triangulated category where thesuspension functor Σ : x 7→ x[1] denotes the translation functor. See Theorem 1.1.2.13 of [L2]for the proof. Furthermore, there exists a spectrum of maps between any two objects in C. For
all x, y ∈ C, since x ' 0 ×x[1] 0, the space MapC (x, y) (pointed by the zero map) is the zeroth
space of the spectrum
Ω

Ω

Ω

Ω

Ω

Ω

... −
→ MapC (x, y[2]) −→ MapC (x, y[1]) −→ MapC (x, y) −→ MapC (x[1], y) −→ MapC (x[2], y) −→ ....
pMore precisely, let Cat⊥,p∞ denote the full subcategory of Cat∞ spanned by stable, presentableobjects.

Proposition 3.7. Let C be a stable, presentable ∞-category. Then C is tensored and enrichedover the ∞-category Sp of spectra.
Proof. By Proposition 6.3.2.15 of [L2] there exists an equivalence
ModSp (Catp∞ ) → Cat⊥,p∞of ∞-categories. A presentable ∞-category which is tensored over Sp where the tensored structure preserves (small) colimits is automatically stable and since the functor • ⊗ x : Sp → C
preserves (small) colimits for all x ∈ C, the result follows.
Definition 3.8. A spectrum A is said to be connective if πn A ' 0 for all n < 0. It is said to bediscrete if it is connective and 0-truncated.
We denote by Spc (resp. Spd ) the full subcategory of Sp spanned by the connective (resp.
discrete) spectra. The ∞-category of connective spectra is the smallest full subcategory of Sp
closed under colimits and extensions which contains the sphere spectrum S. It is projectively
generated with the sphere spectrum being a compact projective generator.
Definition 3.9. A commutative ring spectrum is a commutative monoid object in the ∞category Sp of spectra with respect to the smash product monoidal structure.
A commutative ring spectrum will be referred to as an E∞ -ring. The ∞-category of E∞ rings will be denoted E := CMon(Sp). The ∞-category E will play the role of our generalised
theory of rings. If R is an E∞ -ring then the ∞-category of commutative R-algebras
CAlgR := CMon(ModR )in Sp is equivalent to ER/ .
25

Example 3.10. Let S n be the simplicial n-sphere. For any commutative ring k, one canassociate an E∞ -ring spectrum Hk called the Eilenberg Mac Lane ring spectrum which is thesequence of simplicial abelian groups k ⊗ S n where (k ⊗ S n )m is the free abelian group on the
non-basepoint m-simplices of S n . This construction defines a fully faithful functor H : Rng → E
from the ∞-category of commutative rings to the ∞-category of E∞ -rings with π0 (Hk) = 0 andπi (Hk) 6= 0 for all i > 0.
For R ∈ E and n ∈ Z, let πn R denote the nth homotopy group of the underlying spectrumof R. We can identify πn R with the set π0 MapSp (S[n], R). In particular, π0 R is a discrete
commutative ring and πn R has the natural structure of a π0 (R)-module. An E∞ -ring R is said
to be connective if πn R ' 0 for all n < 0. The full subcategory of commutative ring spectra
spanned by the connective objects, denoted Ec , is equivalent to the ∞-category CMon(Spc ) ofcommutative monoid objects in the ∞-category of connective spectra.
We can think of connective E∞ -rings as simply spaces endowed with an addition and multiplication satisfying the axioms for a commutative ring up to coherent homotopy. More precisely,
let Ev0 : Ec → S denote the composition Ec → Spc → S. Then
θ : Ec → ModT (S)is an equivalence of ∞-categories for the monad T = Ev0 ◦ Fr0 on S (see Remark 7.1.1.8 of [L2]).
An E∞ -ring R is said to be bounded if πi (R) = 0 for i > n for some n. It is said to bediscrete if it is connective and 0-truncated. We let Ed denote the full subcategory of E spannedby the discrete objects. A connective E∞ -ring R is discrete if and only if for all n > 0 thehomotopy group πn R is trivial. The ∞-category Ed is equivalent to the ∞-category CMon(Spd )of commutative monoid objects in the ∞-category of discrete spectra.
A convenient way to manipulate algebra in the context of ring spectra is to utilise its model
categorical interpretation. Let S p be the model category of symmetric spectra. Then the
category CMon(S p) inherits the structure of a simplicial model category again by loc. cit. andby Section 2 there exists an equivalences : L(CMon(S p)) → E
of ∞-categories. Furthermore, by Example 2.13, there exists an equivalence
L(ModR (S p)) → Mods(R) (Sp)of ∞-categories.
Lemma 3.11. Let R be an E∞ -ring. The ∞-category of modules ModR is a stable ∞-category.
Proof. By the above discussion, the ∞-category ModR is equivalent to the localisation of astable model category. The localisation of a stable model category is a stable ∞-category(Example 3.2).
Example 3.12. When R ∈ E is discrete, ie. an ordinary commutative ring, the triangulated
category hModR corresponds to the classical derived category of ModR .
26

Example 3.13. Let k be a field of characteristic zero and cdgak the model category of commutative differential graded k-algebras where the weak equivalences are given by the quasiisomorphisms (see Section 5 of [SS2]), ie. the model category of commutative monoid objects inthe symmetric monoidal model category C(k) := C(Modk (Ab)) of chain complexes of k-modules.
By Theorem 5.1.6 of [SS1] there exists a Quillen equivalence ModHk (S p) → C(k) where themodel category S p of symmetric spectra is endowed with the S-model structure. Thus we havea diagram∼CMon(L(ModHk (S p))) - CMon(L(C(k)))?

?
- L(cdga )CAlgHk (Sp)kof ∞-categories. The left vertical arrow is an equivalence by Example 2.13 and the right verticalarrow is an equivalence by the discussion following Example 2.8. Therefore the ∞-category of
Hk-algebras in Sp can be identified with the localisation of the model category of commutative
differential k-algebras.
Let R be a connective E∞ -ring and CAlgcR := CMon(ModcR ) the ∞-category of connective
commutative R-algebras. The ∞-category CAlgcR is projectively generated with the compact
projective objects being identified with those connective commutative R-algebras that are retracts of a finitely generated free commutative R-algebra. If (CAlgcR )f p denotes the smallest fullsubcategory of CAlgcR which contains all finitely generated free R-algebras and is stable underfinite colimits thenInd((CAlgcR )f p ) → CAlgcRis an equivalence of ∞-categories. Moreover, the ∞-category CAlgcR is compactly generated
with the compact objects being the finitely presented R-algebras.
Consider the ∞-category ModR of R-modules for R an E∞ -ring. If M is an R-module wewill denote by πn M the homotopy group of its underlying spectrum. The ∞-category (ModR )≥0
is the smallest full subcategory of ModR which contains R and is stable under small colimits.
A module M in ModR is said to be connective if πn M = 0 for all n < 0 and we call (ModR )≥0
the ∞-category of connective R-modules. Likewise, a module M in ModR is said to be anticonnective if πn M = 0 for all n > 0. An R-module is said to be free if it is equivalent to a
coproduct of copies of R and finitely generated if it can be written as a finite coproduct of copies
of R.
Let R be a connective E∞ -ring. An R-module M is said to be projective if it is a projective
object of the ∞-category (ModR )≥0 of connective R-modules (note that the ∞-category ModR
has no nonzero projective objects). The R-module M is projective if and only if there exists
a free R-module N such that M is a retract of N . If N is moreover finitely generated, thenM is a compact projective object of (ModR )≥0 . This is equivalent to M being projective andπ0 M being finitely generated as a π0 R-module. The ∞-category of of connective modules overa connective E∞ -ring is projectively generated.
27

Let R be a connective E∞ -ring. Then the inclusion CAlgcR ,→ CAlgR commutes with colimits
so there exists an adjunction
i : CAlgcR  CAlgR : (•)cwhere the right adjoint is called the connective cover. Explicitly, a connective cover of an Ralgebra A is a map f : A → Ac such that Ac is connective and for any connective R-algebra B,
there exists an equivalence
MapCAlgR (B, A) → MapCAlgcR (B, Ac )of (∞, 0)-categories.
Let C be an ∞-category and x, y ∈ C. Then y is said to be a retract of x if it is a retract ofpi
x in hC, ie. there exists a diagram y →− x→− y which coincides with the identity idy in hC.
Definition 3.14. Let C be a stable ∞-category. Let C perf denote the smallest stable subcategory of C which contains the unit object and is closed under retracts. An object x ∈ C is said
to be perfect if it is an object of C perf .
Let R be an E∞ -ring. Then there exists an equivalence Modperf→ Modcpt
R
R between the∞-category of perfect R-modules and the full subcategory of ModR spanned by the compact
objects. Furthermore if R is a connective E∞ -ring and Modfgp
R denotes the smallest stablesubcategory of ModR which contains all finitely generated projective modules then there exists
perfof ∞-categories.
an equivalence Modfgp
R → ModR
Definition 3.15. Let C be a stable ∞-category. Then C is said to admit a t-structure if there
exists a t-structure on the homotopy category hC.
Let C be a stable ∞-category. The full subcategory of C spanned by the objects of (hC)≤nand (hC)≥n will be denoted by C≤n and C≥n respectively.
Example 3.16. When C is a presentable stable ∞-category, any small collection of objects{xα } determines a t-structure on C. The construction is as follows (see Proposition 1.4.5.11 of[L2]). One builds a subcategory C 0 of C as the smallest full subcategory of C containing {xα }which is closed under small colimits and such that for every distinguished triangle
x → y → z → x[1]for which x and z are in C 0 , then y is in C 0 . In this case, there exists a t-structure on C suchthat C 0 = C≥0 and C≥0 is presentable.
We call a t-structure on a presentable stable ∞-category C accessible if the subcategory C≥0
is presentable. It follows that if C admits an accessible t-structure then C≤0 is also presentable.
The t-structures that we will be concerned with in this paper are the following accessible tstructures on the ∞-category of spectra and the ∞-category of R-modules for R a connective
E∞ -ring.
Example 3.17. Let R be a connective E∞ -ring. Then there exists an accessible t-structure on
the ∞-category ModR of R-modules where28• (ModR )≤0 is the full subcategory of ModR spanned by the objects {M ∈ ModR |∀n >0, πn M ' 0}.
• (ModR )≥0 is the full subcategory of ModR spanned by the objects {M ∈ ModR |∀n <0, πn M ' 0}.
This t-structure is left and right complete. With this t-structure, the heart of ModR is equivalent
to the abelian category of discrete modules over the ring π0 (R).
Let C be a stable ∞-category. A t-structure (C≤0 , C≥0 ) on C is non-degenerate if and only
if ∪i C≤i = ∪i C≥i = C and ∩i C≤i = ∩i C≥i = 0. The t-structure of Example 3.17 are nondegenerate. The usefulness of this non-degeneracy is that it enables us to check equivalences inthese ∞-categories on their corresponding truncations in the following sense. The ∞-categoryC≤n is stable under limits in C and the ∞-category C≥n is stable under colimits in C. Hence
there exists a left adjointτ≤n : C → C≤nto the inclusion map C≤n ,→ C and a right adjointτ≥n : C → C≥nto the inclusion map C≥n ,→ C. By Proposition 1.2.1.10 of [L2] there exists an equivalenceτ≤m ◦τ≥n ' τ≥n ◦τ≤m of functors from C to C≤m ∩C≥n which we will denote by τn,m : C → C[n,m] .
Definition 3.18. Let C be a stable ∞-category. A t-structure on C is said to be non-degenerate
if for all objects x in C, if τn,n x = 0 for all n then x = 0.
Definition 3.19. Let C and D be stable ∞-categories admitting t-structures. A functor F :C → D is said to be left (resp. right) t-exact if it is exact and sends C≤0 into D≤0 (resp. sends
C≥0 into D≥0 ). It is said to be t-exact if it is both left and right t-exact.
Lemma 3.20. Let C and D be stable ∞-categories admitting t-structures and let F a G be an
adjunction between them. Then F is right t-exact if and only if G is left t-exact.
Proof. Let F : C → D be right t-exact. Then for any x ∈ C≥0 we have F (x) ∈ D≥0 . Thus forany y ∈ D≤−1 , HomhD (F (x), y) ' HomhC (x, G(y)) = 0. Therefore G(y) ∈ C≤−1 .
Another consequence of having a non-degenerate t-structure on a stable ∞-category is thefollowing. We say that a stable ∞-category C endowed with a t-structure is left t-complete if
the natural map
τ≤1

τ≤0

τ≤−1

C → holimn C≤n := holimn {. . . → C≤2 −−→ C≤1 −−→ C≤0 C −−−→ . . .}is an equivalence of ∞-categories. By Proposition 1.2.1.19 of [L2], if a stable ∞-category witha t-structure admits countable products such that C≥0 is stable under countable products, thenC is left t-complete if and only if ∩i C≥i = 0.
Definition 3.21. Let C be a stable ∞-category and D a stable ∞-category admitting a tstructure (D≤0 , D≥0 ). A functor F : C → D is said to create a t-structure on C if C≤0 := {x ∈C|F (x) ∈ D≤0 } and C≥0 := {x ∈ C|F (x) ∈ D≥0 } define a t-structure on C.
29

Definition 3.22. Let C be a stable ∞-category with a t-structure (C≤0 , C≥0 ). The heart of C
is the full subcategory C ♥ := C≤0 ∩ C≥0 .
Let C be a stable ∞-category admitting a t-structure. Then for any object x ∈ C andn ≥ −1, the object x belongs to C≤n if and only if the space C(y, x) is n-truncated for ally ∈ C≥0 . Thus for x and y in C ♥ , the group πn C(x, y) vanishes for all n > 0 and so there exists
an equivalence C ♥ → hC ♥ of ∞-categories.
Let C be a stable symmetric monoidal ∞-category. A t-structure on C is said to be compatible
with the symmetric monoidal structure if for all x ∈ C the functor x⊗• is exact and C≥0 is closed
under tensor products and contains the unit object. The t-structure on the ∞-category Sp ofspectra is compatible with the symmetric monoidal structure. As a result, by Proposition 2.5,
the ∞-category Spc of connective spectra inherits the structure of a symmetric monoidal ∞category.
Let R be a connective E∞ -ring. By extension, the ∞-category ModcR of connective R-modules
and connective R-algebras CAlgcR := CMon(ModcR ) inherit symmetric monoidal structures. Let
C be a stable symmetric monoidal ∞-category which admits a t-structure that is compatible
with the symmetric monoidal structure. Then by Example 2.2.1.10 of [L2], the heart C ♥ of C
inherits the structure of a symmetric monoidal category.
Definition 3.23. Let C be a stable ∞-category admitting a t-structure and let n ∈ Z. We
define a functor
πnt : C → C ♥sending x to τ0,0 (x[−n]).
Let C be a stable ∞-category. It follows from Theorem 1.3.6 of [BBD] that the category C ♥is abelian. One can show that the heart of a presentable stable ∞-category equipped with anadmissible t-struture is a presentable abelian category. Let Ab denote the category of abeliangroups. When C is the ∞-category Sp of spectra endowed with its natural t-structure then thefunctorSp♥ → Abis an equivalence of ∞-categories. This follows from Proposition 1.4.3.5 of [L2]. Furthermore,
the functor
π0t : Ed → Rng
is an equivalence of ∞-categories (Proposition 7.1.3.18 of [L2]) showing that the ∞-category ofcommutative ring spectra contains, as a fully subcategory, the ordinary theory of commutative
rings Rng.
To end this section, we prove an important lemma which provides conditions under whicha functor between stable ∞-categories admitting non-degenerate t-structures commutes withlimits of cosimplicial objects needed in the proof of our main duality theorem.
Lemma 3.24. Let C and D be stable ∞-categories admitting non-degenerate t-structures. Let
f : C → D be a t-exact functor. If X is a cosimplicial object in C such that there exists k ≥ 0with πit (Xn ) = 0 for all i > k and all n thenf ( lim Xn ) → lim f (Xn )
n∈∆

n∈∆

30

is an equivalence in D.
Proof. We can choose k = 0. Since the t-structure on D is non-degenerate, we can check theequivalence on the truncation
τ≥−N (f (lim Xn )) → τ≥−N (lim f (Xn )).
nn

The functor f is t-exact and the truncation commutes with limits so we are reduced to proving
f (lim(τ≥−N Xn )) → lim f (τ≥−N Xn )).
nn

The limits in C[−N,0] are considered as limits in C≤0 . Note that C[−N,0] is a subcategory of C
where the mapping spaces are N -truncated. Any limit along ∆ in an ∞-category whose mapping
spaces are N -truncated is a finite limit. Since f is t-exact it commutes with finite limits andtruncations so the induced functor f : C[−N,0] → D[−N,0] preserves finite limits and the result
follows.

4

Stacks, gerbes and Hopf algebras

This section is devoted to the group side of the correspondence in our Tannaka duality theorem.
The first main result gives conditions under which the ∞-category of sheaves valued in somepresentable symmetric monoidal ∞-category is tensored and enriched, both over itself and the∞-category which it is valued in. If these sheaves are merely presheaves then we give conditions
for the presheaf of maps between two of these objects to be a sheaf. The main example of interest
is when the presentable symmetric monoidal ∞-category is the ∞-category of ∞-categories. In
this case, we provide sufficient conditions for a presheaf valued in ∞-categories to be a sheaf of∞-categories.
After defining group stacks and gerbes we show that gerbes are exactly those stacks which
are locally equivalent to the classifying stack of a group stack. Furthermore, affine group stacks
correspond to Hopf algebras in a symmetric monoidal ∞-category. We finish the section withsome technical results concerning Hopf algebras needed in the proof of our duality theorem.
Let (C, τ ) be a site, ie. an ∞-category endowed with a topology. A topology on an ∞category C is equivalent to a Grothendieck topology on the homotopy category h(C) (see Remark
6.2.2.3 of [L1]). If C is an ∞-category with pullbacks, to prove the existence of a topology τ on
C it suffices to prove the existence of a pretopology on C. That is, a function covτ which assigns
to each object x in C a collection covτ (x) of subsets of objects in C/x called covering familiesof x satisfying the three axioms :• Stability : If f : y → x is an equivalence in C then the singleton {f : y → x} is in covτ (x).
• Composition : If {fi : yi → x}i∈I is in covτ (x) and if for each i ∈ I one has a family{gij : zij → yi }j∈Ji in covτ (yi ) then the family {fi ◦ gij : zij → x}i∈I,j∈Ji is in covτ (x).
• Base change : If {fi : yi → x}i∈I is in covτ (x) then for any morphism g : z → x, thepullbacks z ×x yi exist and the family {z ×x yi → z}i∈I is in covτ (z).
31A pretopology on C determines a topology on C : a sieve R on an object x in C is in τ (x) if
and only if there exists a covering family J in covτ (x) such that J is a subset`of R.
Let (C, τ ) be a site and {ui → x}i∈I a covering family of x ∈ C. Let u = i ui . We will saythat the map u → x is a covering of x. A cover of x (associated to u) is the simplicial prestacku∗ ∈ sPr(C)/x given byu∗ : [n] 7→ u × . . . × u.
xxDefinition 4.1. Let C be a site and X an ∞-category with limits. An X-valued prestack
F : C op → X on C is said to be an X-valued stack if for all x ∈ C and all coverings u∗ insPr(C)/x the map
F (x) → lim F (u∗ )∆is an equivalence in X.
The full subcategory of PrX (C) spanned by the X-valued stacks on the site (C, τ ) will bedenoted ShX (C, τ ). A S-valued stack will simply be called a stack and we will denote the ∞category of stacks ShS (C, τ ) by Sh(C, τ ). A topology τ on C is said to be subcanonical if every
representable functor on C is a stack with respect to τ .
QRemark 4.2. Note that the adjunction: K  PC(K ) : K between model categories (see[HS]) induces an adjunction S a Cat∞ between ∞-categories and thus an adjunction
ShS (C, τ )  ShCat∞ (C, τ )where (K∗ F )(x) = KF (x) for a stack F in ShCat∞ (C, τ ).
Example 4.3. Let (C, τ ) be a site for a subcanonical topology τ and hx the representable
prestack on an object x in C. Then there exists an equivalence Sh(C, τ )/hx → Sh(C/x , τ ) of∞-categories.
Let C be a symmetric monoidal ∞-category. We give the opposite ∞-category of commutative monoid objects in C the following special notation :AffC := CMon(C)op .
When C is the symmetric monoidal ∞-category ModR (D) of R-modules in a symmetric monoidal∞-category D, we will write AffR := AffModR (D) . In other words, the ∞-category AffR is the
opposite of the ∞-category of commutative R-algebras in D. The Yoneda embedding AffC →(AffC )∧ will be denoted by Spec.
Example 4.4. Let C be a symmetric monoidal ∞-category and R a commutative monoid objectof C. Then we will denote by Sh(R, τ ) := Sh(AffR , τ ) the ∞-category of stacks with respect
to the site (AffR , τ ) of R-algebras in C. By Example 4.3 we have an equivalence Sh(R, τ ) 'Sh(AffC , τ )/ Spec R of ∞-categories.
The ∞-category of stacks on a site can be obtained by the localisation (in the sense of
Definition 1.1) of the ∞-category of prestacks. The following proposition gives two possible
choices for the set of maps from which to localise.
32

Proposition 4.5. Let (C, τ ) be a site. The following classes of maps give the same localisationof Pr(C) :1. The set of all covering sieves on x.
2. The set of maps u∗ → x.
Proof. We note that Pr(C) is a presentable ∞-category. Thus the localisation is a Bousfield
localisation in the sense of Section 1. The result now follows from Proposition A1 of [DHI].
Let S denote the equivalent set of maps of Proposition 4.5. Then Sh(C, τ ) ' LS (Pr(C)).
Proposition 4.6. Let (C, τ ) be a site and X a presentable symmetric monoidal ∞-category
whose tensor product preserves colimits seperately in each variable. Then the ∞-category ShτX (C)is tensored and enriched over itself. Moreover, it is tensored and enriched over X.
Proof. We first observe that the ∞-category ShX (C, τ ) of X-valued stacks on C is naturally asymmetric monoidal ∞-category with respect to the pointwise symmetric monoidal structure of
Example 2.4. Explicitly, ShX (C, τ )hni ' ShXhni (C, τ ). To show that it is enriched over itself it
remains to show that • ⊗ F : ShX (C, τ ) → ShX (C, τ ) preserves colimits (see Section 2). This
follows by the assumption that the tensor product on X preserves colimits seperately in eachvariable (colimits are calculated pointwise in functor categories).
To show that ShX (C, τ ) is tensored and enriched over X it suffices to show that there exists a
colimit preserving symmetric monoidal functor X → ShX (C, τ ). First consider the ∞-category
PrX (C) of X-valued prestacks endowed with its pointwise symmetric monoidal structure. The
constant prestack functor induces a symmetric monoidal functor X ' PrX (∗) → PrX (C). Thisfunctor preserves colimits seperately in each variable owing to the assumption that they do in X.
Finally, the stackification functor LS preserves colimits (it is a left adjoint) so the composition
X → PrX (C) → LS PrX (C) ' ShX (C, τ ) preserves colimits seperately in each variable.
The internal Hom provided by Proposition 4.6, and more generally for X-valued prestacks,
will be denoted by Hom(F, G). We will now demonstrate that Hom(F, G) is an X-valued stack
under the weaker condition that F is only an X-valued prestack when the conditions of Proposition 4.6 are satisfied.
Proposition 4.7. Let (C, τ ) be a site, X a presentable symmetric monoidal ∞-category whose
tensor product preserves colimits separately in each variable and F and G two X-valued prestacks
on C. If G is an X-valued stack then Hom(F, G) is an X-valued stack on C.
Proof. Let F be an X-valued prestack and G an X-valued stack. Then Hom(F, G) is an Xvalued stack if and only if the map
Mor(x, Hom(F, G)) → Mor(colim u∗ , Hom(F, G))n

is an equivalence in X. This is equivalent to the condition that Mor(x⊗F, G) → Mor(colimn (u∗ ⊗F ), G) is an equivalence in X and subsequently to colimn (u∗ ⊗ F ) → x ⊗ F being an equivalence of X-valued stacks. Let B be an object of X. Since G is an X-valued stack, the map
33

Map(B, G(x)) → limn Map(B, G(u∗ )) is an equivalence and so Map(x⊗B, G) → Map(colimn (u∗ ⊗B), G) is an equivalence for any B ∈ X.
Now recall that any X-valued prestack can be written as a colimit given by colimα (vα ⊗ Bα )for vα a set of prestacks and Bα a set of generators for the presentable ∞-category X. Therefore,x ⊗ F ' x ⊗ colimα (vα ⊗ Bα ) ' colimα (x ⊗ vα ) ⊗ Bα ' colimn (colimα (u∗ ⊗ vα ) ⊗ Bα ) 'colimn (u∗ ⊗ colimα (vα ⊗ Bα )) ' colimn (u∗ ⊗ F ) and the result follows.
Let n ≥ 0 be an integer. Recall that a (∞, 0)-category A is said to be n-truncated (resp.
n-connective) if for every i > n (resp. i < n)πi (A, a) ' ∗for all objects a ∈ A. An (∞, 0)-category which is 1-connective will be called connected. A
map of (∞, 0)-categories f : A → B is said to be n-truncated (resp. n-connective) if thehomotopy fibers of f taken over any base point of B is n-truncated (resp. n-connective). A
prestack F : C op → S is said to be n-truncated (resp. n-connective) if F (x) is n-truncated(resp. n-connective) for all x ∈ C. A map of prestacks F → G is said to be n-truncated (resp.
n-connective) if F (x) → G(x) is n-truncated (resp. n-connective) for all x ∈ C.
Let C be an ∞-category and n ≥ 0 an integer. An object x in C is said to be n-truncated
if the representable prestack C(•, x) is n-truncated. An arrow f : x → y in C is said to be ntruncated if the map of prestacks C(•, f ) is n-truncated. Let τ≤n C denote the full subcategory of
C spanned by the n-truncated objects. Then there exists an equivalence τ≤0 C → h(τ≤0 C) of ∞categories. Note that objects in the ∞-category τ≤0 C are objects x ∈ C such that for all y ∈ C,
the (∞, 0)-category C(y, x) is homotopy equivalent to a discrete space, ie. C(y, x) → π0 C(y, x)
is a homotopy equivalence for all y ∈ C. We can construct a left adjoint to the inclusion functor
i : τ≤n C → C given bytn : C → τ≤n C.
Proposition 4.8. Let (C, τ ) be a site andF : C op → Cat∞a prestack of ∞-categories satisfying the following conditions :1. For each object x in C, the ∞-category F (x) admits limits.
2. For each object x in C and for any covering {ui → x}i∈I in cov τ (x), the functor F (x) →F (ui ) preserves limits.
3. Q
For each object x in C and for any covering {ui → x}i∈I in cov τ (x), the functor F (x) →i F (ui ) is conservative.
4. For any map f : y → x in C, the functor f ∗ := F (f ) : F (x) → F (y) admits a right adjointf∗ : F (y) → F (x).
34

5. For each pullback square
f

y

- x
6

6
p

g
q

z

- y0

in C, the natural morphism g ∗ f∗ ⇒ q∗ p∗ is an equivalence in the ∞-category RHom(F (y), F (y 0 )).
Then F is a stack of ∞-categories.
Proof. We need to show that for any covering u → x in cov τ (x) the map
F (x) → lim F (un )∆is an equivalence of ∞-categories. Consider the pullback diagram
f

u

- x

6

6

p

f
q

v

- u

in C where f ∈ cov τ (x). Construct the section δ : u → u ×x u of the map q where q ◦ δ = idu .
Taking the nerve of the maps f and q and using (1) we obtain a homotopy commutative diagram
F (x)

?

F (u)

A

- lim F (un )∆A0?
- lim F (vn )∆in Cat∞ . Using (2), (4) and (5) we obtain an adjoint homotopy commutative diagram
F (x) ?

F (u) B

lim F (un )
∆

B0

?

lim F (vn )∆in Cat∞ . To complete the proof it will suffice to show that the unit and counit of the adjunction
A a B are equivalences. By (3) and the fact that both squares commute, we are able to checkthe corresponding statement for the adjunction A0 a B 0 . This adjunction is an equivalence owing
to the fact that for any covering admitting a section, the prestack F satisfies descent.
35

We now define the notion of a group object in an ∞-category. We start with the moregeneral notion of a groupoid object. Let C be an ∞-category with pullbacks. A category object
in C is a functor F : ∆op → C such that for all n ≥ 0, the canonical map
F ([n]) → F ([1]) ×F ([0]) × . . . ×F ([0]) F ([1])
is an equivalence in C. Let Ct(C) denote the full subcategory of RHom(∆op , C) spanned by thecategory objects of C. A category object F in C is said to be a groupoid object in C if it takes
every partition [2] = {S ∪ S 0 |S ∩ S 0 = {s}, s ∈ S} to a pullback squareF ([2])- F (S 0 )?

?
- F ({s})F (S)in C. Let Gpd(C) denote the full subcategory of Ct(C) spanned by the groupoid objects of C.
We have an adjoint pairi : Gpd(C)  Ct(C) : j
where j(F ) is the groupoid object of isomorphisms of a category object F in C.
Definition 4.9. Let C be an ∞-category and G a groupoid object in C. Then G is said to bea group object in C if G([0]) is a terminal object in C.
Let Gp(C) denote the full subcategory of Gpd(C) spanned by the group objects of C. The∞-category Gp(S) will play the analogue of the category of groups in the ∞-categorical context.
If C is an ∞-category then there exists an equivalence
Gp(Pr(C)) → RHom(C op , Gp(S))
of ∞-categories (since limits in functor categories are computed pointwise). If (C, τ ) is a sitethen a Gp(S)-valued stack will be called a group stack on C. The ∞-category Gp(Sh(C, τ )) ofgroup stacks on C will be denoted Gp(C, τ ).
Example 4.10. If C is the site (AffR , τ ) of commutative R-algebras, we will denote the ∞category Gp(AffR , τ ) by Gp(R, τ ).
Let (C, τ ) be a site. A stack F in Sh(C, τ ) is said to be locally non-empty if for all x ∈ C
there exists a τ -covering u → x such that F (u) is non-empty. It is said to be locally connected
if t0 (F ) → ∗ is an isomorphism (of sheaves of sets). A morphism of prestacks φ : F → G is said
to be a local equivalence if it is fully faithful, ie. φx : F (x) → G(x) is fully faithful for all x ∈ C,
and locally essentially surjective, ie. for all x ∈ C and a ∈ G(x) there exists a covering α : u → xsuch that α∗ (a) is equivalent to α∗ (φx (b)) (ie. an isomorphism in hG(u)) for some b ∈ F (x). If
F and G are stacks then φ is a local equivalence if and only if it is an equivalence of stacks.
Definition 4.11. Let (C, τ ) be a site and F a stack in Sh(C, τ ). Then F is said to be a gerbe
in Sh(C, τ ) if it is locally non-empty and locally connected.
36

The full subcategory of Sh(C, τ ) spanned by gerbes will be denoted by Ger(C, τ ). A gerbe
G in Ger(C, τ ) is said to be neutral if there exists a morphism ∗ → G in Ger(C, τ ).
Example 4.12. Let C be a symmetric monoidal ∞-category and R a commutative monoid
object of C. Then we will denote by Ger(R, τ ) := Ger(AffR , τ ) the ∞-category of stacks withrespect to the site (AffR , τ ) of R-algebras in C.
We now provide a requisite characterisation of a gerbe. First we will need a small lemma.
Lemma 4.13. Let f : C → D be a map of (∞, 0)-categories. Then the following are equivalent.
1. The map f is fully faithful, ie. for all x, y ∈ C,∼

x ×hC y = MapC (x, y) −→ MapD (f x, f y) = f x ×hD f y.
2. The map δ : C → C ×hD C is an equivalence.
3. The map π0 (C) → π0 (D) is a monomorphism and for all x ∈ C and i > 0, the map
πi (C, x) → πi (D, f (x))
is an equivalence.
Proof. (1) ⇒ (2). Since f is fully faithful, the map∼

C(x, y) −→ C(x, y) ×hD(f x,f y) C(x, y)
is an equivalence so δ : C → C ×hD C is fully faithful. For essential surjectivity, we need to show∼that any object (x, y, α : f x −→ f y) in C → C ×hD C is equivalent to an object δ(z) = (z, z, idz )∼for some z ∈ C. Let z = x. Since f is fully faithful, we set β : x −→ y and α = f (β). (2) ⇒ (3).
Let d ∈ D and (x, y) ∈ C ×hD C such that d ' f (x) ' f (y). Then δ −1 (x, y) is equivalent
to the path space between x and y in f −1 (d). Thus f −1 (d) is empty or contractible. Thus
π0 (C) → π0 (D) is a monomorphism. (3) ⇒ (1). This statement is clear.
Recall that the classifying space functor B : Gp(S) → S is given by
G 7→ BG : [n] 7→ Gn,n .
nIt admits a right adjoint Ω which sends an (∞, 0)-category A to Ω(A) : [n] 7→ A∆∗ . Let (C, τ )be a site. We construct the following classifying prestack functor B : Gp(Pr(C)) → Pr(C) by
sending
G 7→ BG : x 7→ B(G(x))
together with its right adjoint Ω, where Ω(F ) : x 7→ Ω(F (x)). Finally, the classifying stack
functore : Gp(C, τ ) → Sh(C, τ )Be : Sh(C, τ ) → Gp(C, τ ).
is the stackification of B and admits the right adjoint ΩProposition 4.14. Let (C, τ ) be a site and F a stack on C. The following are equivalent :37

1. The stack F is a gerbe.
e for G a group stack in Gp(C, τ ).
2. The stack F is locally equivalent to BG
Proof. Let Kan0 be the category of Kan complexes with a single 0-simplex. We have a natural string of equivalences Gp(S) ' L(Gp(Kan)) ' L(Kan0 ) ' L(sGp) of ∞-categories (seeCorollary 6.4 of [GJ]). By Section 4 and 5 of Chapter 5 of loc. cit there exists an adjunction
W : s Gp  Kan0 : Ω
where the construction W G is a model for BG where G ∈ sGp. Thus for a pointed stack F anda Gp(S)-valued prestack G, we have an equivalence
MapPr(C)∗ (BG, F ) → MapGp(Pr(C)) (G, ΩF )of (∞, 0)-categories where ΩF := Aut(s) for s ∈ F (∗).
We claim that if F is a stack which is locally non-empty and locally connected with F (∗) 6= ∅e for G = Aut(s), s ∈ F (∗). By the equivalence above, thethen F is locally equivalent to BG
identity map G → Aut(s) corresponds to a map of prestacks
φ : BG → F
sending ∗ to s. But since F is a stack, the universal property of stackification implies that φ ise → F.
actually a map of stacks φ : BG
It remains to show that φ is fully faithful and locally essentially surjective. By Lemma 4.13,
e is an equivalence ofe → BGe ×F BG
fully faithfulness is equivalent to the condition that BG
stacks. By the universal property if suffices to check it for a map of prestacks. By Lemma 4.13
again, it suffices to check the two conditions of Lemma 4.13 part (3). The first condition of (3)is clear. The second condition follows from the fact that for all x ∈ C we have πi (BG(x), ∗) 'πi−1 (G(x), ∗) := πi−1 (ΩF (x), s) ' πi (F (x), s).
Finally, since F is locally non-empty and locally connected there always exists a τ -covering
α : u → x such that for a ∈ F (x) the map α∗ (a) → α∗ (φx (∗)) is an equivalence.
Definition 4.15. Let C be a symmetric monoidal ∞-category and R a commutative monoid
object in C. A Hopf R-algebra in C is a cogroup object B in the symmetric monoidal ∞-category
CAlgR (C) of commutative R-algebras in C.
Let HopfR (C) denote the full subcategory of Comon(CAlgR (C)) spanned by the Hopf Ralgebras in C. We will call B1 := B([1]) the underlying R-algebra of B. We have a well defined
functor
Hopf : CMon(C) → Cat∞sending R to HopfR (C). When the ∞-category C is clear from the context we will simply write
Hopf(R) in place of HopfR (C). By Example 2.6 the relative tensor product monoidal structure
on CAlgR (C) coincides with the coproduct.
Definition 4.16. Let C be a symmetric monoidal ∞-category, R a commutative monoid object
in C and (AffR , τ ) a site. A Hopf R-algebra in C is said to be a τ -Hopf R-algebra if its underlying
R-algebra is an element of τ (R).
38

We denote by Hopf(R, τ ) the ∞-category of τ -Hopf R-algebras in C. When R is an E∞ -ring,
we will be primarily interested in the ∞-category of positive (resp. flat, finite) Hopf R-algebras
in the ∞-category Sp of spectra. See Section 5 for a definition of these topologies.
Proposition 4.17. Let C be a symmetric monoidal ∞-category, R a commutative monoid objectof C and (AffR , τ ) the site of R-algebras with respect to a subcanonical topology τ . Then thefunctorSpec : Hopf(R) → Gp(R, τ )is fully faithful.
Proof. Since τ is subcanonical, the Yoneda embedding AffR → Pr(AffR ) factors through thesubcategory of stacks and hence Spec : AffR → Sh(R, τ ) is fully faithful. The tensor product in
CAlgR corresponds to the coproduct by Example 2.6 and the Yoneda lemma preserves limits so
we have an induced fully faithful functor Spec : Comon(CAlgR ) → Mon(Sh(R, τ )) on monoid
objects. Restricting to group-like objects we find that Spec : Hopf(R) → Gp(R, τ ) is fully
faithful.
Informally, we have a diagram
AffR- Hopf(R)

Spec

Spec

?

?

Sh(R, τ ) - Gp(R, τ )of ∞-categories where the vertical arrows take an algebraic object to its corresponding affine
geometric object and the horizontal arrows pass to the corresponding group objects.
Let C be an ∞-category with finite colimits and X a cosimplicial object in C. For any
cosimplicial set A, we define a cosimplicial object in C given byY
X ⊗ A : [n] 7→
Xn .
AnLet h0 , h1 : X ⇒ Y be two arrows in cC := RHom(∆, C). A homotopy between h0 and h1 is a
map h : X ⊗ ∆1 → Y such thath ◦ (idX × i0 ) = h ◦ (i0 × idX ) = h0h ◦ (idX × i1 ) = h ◦ (i1 × idX ) = h1
where i0 , i1 : ∆0 → ∆1 denote the inclusion maps. A diagram
f :XY :gin cC is said to be a homotopy equivalence if there exists a map k : X ⊗ ∆1 → X satisfyingk ◦ (idX × i0 ) = k ◦ (i0 × idX ) = k0 := id39k ◦ (idX × i1 ) = k ◦ (i1 × idX ) = k1 := g ◦ ftogether with a map l : Y ⊗ ∆1 → Y satisfyingl ◦ (idY × i0 ) = l ◦ (i0 × idY ) = l0 := idl ◦ (idY × i1 ) = l ◦ (i1 × idY ) = l1 := f ◦ g.
Lemma 4.18. Let C be an ∞-category with finite limits. Then the functor holimn : cC → C
takes homotopy equivalences in cC to equivalences in C.
Proof. Let X be a cosimplicial object in C and A a simplicial set. It will suffice to show thatholimn (X ⊗ A) ' holimn (X) ⊗ A. Let X be a constant cosimplicial object in C. We have thatholimn (A ⊗ X) ' holimn (A) ⊗ X ' A ⊗ X ' A ⊗ holimn (X).
Now let X = X ⊗ B for B a simplicial set. We have thatholimn ((X ⊗ B) ⊗ A) ' holimn (X ⊗ (B ⊗ A)) ' holimn (X) ⊗ (B ⊗ A) ' holimn (X ⊗ B) ⊗ A.
Finally, X ⊗ ∆n generates the ∞-category cC by homotopy limits. Therefore, setting X 'holimα (X α ⊗ ∆nα ) we have
holimn (X ⊗ A) ' holimn (holimα (X α ⊗ ∆nα ) ⊗ A)
' holimn (holimα ((X α ⊗ ∆nα ) ⊗ A))' holimα (holimn (X α ⊗ ∆nα ) ⊗ A)
' holimα (holimn (X α ⊗ ∆nα )) ⊗ A ' holimn (X) ⊗ A.

Notation 4.19. Let ∆+ be the category of augmented simplicial sets. We define a categrory∆−∞ given as follows :• The set of objects Ob(∆−∞ ) is given by Ob(∆+ ).
• The set of maps Hom∆−∞ ([n], [m]) is the set of order preserving maps f : {−∞} ∪ [n] →{−∞}∪[m] which preserve the base point {−∞} thought of as a least element of {−∞}∪[p]for any [p] ∈ ∆−∞ .
We have the natural sequence of inclusions ∆ ⊆ ∆+ ⊆ ∆−∞ where ∆+ is identified with thefull subcategory of ∆−∞ with the same set of objects and a where a map f in ∆−∞ belongs to∆+ if and only if f −1 (−∞) = {−∞}. The forgetful functor ∆−∞ → ∆ admits a left adjoint.
Let C be an ∞-category and let c−∞ C denote the ∞-category RHom(∆−∞ , C). We havean induced adjunction
c−∞ C  cCbetween ∞-categories. Let X be a cosimplicial object in C. We let Dec+ denote the comonad+ ◦ Dec(X) on the ∞-category cC of cosimplicial objects in C.
40

Proposition 4.20. Let C be an ∞-category and X a cosimplicial object in C. Then there exists
a homotopy equivalence X0 → Dec+ (X) between cosimplicial objects in C.
Proof. See Proposition 1.4 of [Il].
Proposition 4.21. Let C be a symmetric monoidal ∞-category and B a Hopf R-algebra in C.
Then there exists an equivalence
R → holim B ⊗R (n+1)
n∈∆in the ∞-category CAlgR .
Proof. This follows from Lemma 4.18, Proposition 4.20 and the equivalence Dec+ (B)n = Bn+1 'B ⊗R (n+1) .
Definition 4.22. Let C be an ∞-category. An augmented cosimplicial object X : ∆+ → C issaid to be split if there exists a map X → Dec(X). A cosimplicial object is said to be split if it
extends to a split augmented cosimplicial object. Let F : C → D be a functor. A (augmented)
cosimplicial object X of C is said to be F -split if F ◦ X is split as a (augmented) cosimplicial
object of D.
Proposition 4.23. Let C be an ∞-category and X : ∆+ → C a split augmented cosimplicial
object in C. Then X is a limit diagram.
Proof. This is essentially (the dual of) Lemma 6.1.3.16 of [L1].

5

The positive, flat and finite topologies

In the introduction we stated three duality theorems in both the neutralized and neutral contexts
depending on the topology chosen on the underlying site of our group stack or gerbe. We
introduce and study these topologies in this section which go under the names positive, flat andfinite. The two latter topologies are shown to be subcanonical, the proof of which uses the resultthat the presheaf of ∞-categories sending an algebra in spectra to its ∞-category of modules isa sheaf of ∞-categories.
Recall that a module M over an ordinary ring R is said to be flat if the functor • ⊗R M :ModR (Ab) → ModR (Ab) is exact (ie. preserves finite limits and colimits).
Definition 5.1. Let R be an E∞ -ring and A an R-algebra. An A-module M is said to be
1. Positive if the functor • ⊗A M : ModA → ModA preserves anti-connective objects.
2. Flat if the abelian group π0 M is flat as a module over the ordinary commutative algebra
π0 A and for each n ∈ Z, the map πn A ⊗π0 A π0 M → πn M is an isomorphism of abeliangroups.
3. Finite if the functor • ⊗A M : ModA → ModA preserves all (small) limits.
41

A map A → B of R-algebras is said to be positive (resp. flat, finite) if B is positive (resp.
flat, finite) when considered as an A-module. If R is a connective E∞ -ring then every flat Rmodule is also connective. If R is a discrete E∞ -ring then every R-module M is flat if and only
if M is discrete and π0 (M ) is flat over π0 (R) in the classical sense. A module M is finite overan E∞ -ring if and only if it is perfect (see Proposition 6.7).
Let k be a commutative ring and M and N be k-modules. Recall the construction of theabelian groups Torkn (M, N ) (see for example [We]). Recall also that a k-module M is flat if andonly if for any k-module N , the group Tork1 (M, N ) = 0. Let R be a discrete E∞ -ring and M
and N be two discrete R-modules. Then the canonical map
πn (M ⊗R N ) → Torπn0 R (π0 M, π0 N )is an isomorphism.
flf in
Let Mod≥0
R (resp. ModR , ModR ) denote the full subcategory of ModR spanned by the
positive (resp. flat, finite) R-modules. These full subcategories are closed under taking tensor
products and contain the unit object R of ModR . Hence by Example 2.5, these ∞-categories
inherit a symmetric monoidal structure. We deduce that the functor CMon(Mod≥0R ) → ER/ isfully faithful and its essential image consists of positive maps R → R0 . Similarly statements
hold for the flat and finite examples.
Lemma 5.2. Let R be an E∞ -ring.
1. Maps of positive, flat and finite R-algebras are stable under composition.
2. Let
A

?

C

f

g

- B

?
- D

be a pushout in ER/ of R-algebras. If f is positive (resp. flat, finite) then g is positive(resp. flat, finite).
Proof. For part (1), let A → B → C be two maps of R-algebras. For any A-module M , there
exists a natural equivalenceC ⊗B (B ⊗A M ) ' C ⊗A M
showing that the functor C ⊗A • is equivalent to the composition C ⊗B (B ⊗A •) of functors.
Since the composition of two functors preserving connective objects is connective this proves the
positive part. Since the composition of two exact functors is exact and the above equivalence is
an isomorphism on π0 objects, the flat case is satisfied. Finally, the composition of two functors
preserving limits preserves limits which proves the finite case. To prove (2), observe that there
exists a natural equivalence D ' B ⊗A C. Thus for any C-module M , there exists an equivalence
D ⊗C M ' (B ⊗A C) ⊗C M ' B ⊗A M
42

of B-modules. Following the argument above, this shows that if f is positive (flat, finite) theng is also.
Let R be an E∞ -ring and A → B a map of R-algebras. Consider the base change functor
B ⊗A • : ModA → ModB
M 7→ B ⊗A M.
A map of R-algebras A → B is said to be conservative if the base change functor B ⊗A • isconservative, ie. B ⊗A M ' 0 if and only if M ' 0.
Definition 5.3. Let R be an E∞ -ring. A finite family of maps {A → Bi }i∈I of R-algebras issaid to be a positive (resp. flat, finite) covering if A → Bi is positive (resp. flat, finite) andconservative for each i ∈ I.
Proposition 5.4. Let R be an E∞ -ring. The positive, flat and finite coverings define a topology
on the ∞-category AffR .
Proof. The conservative property is clearly stable under composition and pushouts. Thus thethree cases can be deduced from Lemma 5.2.
The positive, flat and finite topologies will be denoted by “≥ 0”, “fl” and “fin” respectively.
The most important example of a sheaf with respect to these topologies in our context is thesheaf of ∞-categories sending an R-algebra to the ∞-category of modules over R. We begin
with the preasheaf
Mod : Affop
R → Cat∞of ∞-categories sending A to ModA and a map f : A → B to B ⊗A •.
Proposition 5.5. Let R be an E∞ -ring. The functor Mod is a stack of ∞-categories over thesite AffR with respect to the flat and finite topologies.
Proof. We begin with the finite topology. We will show that Mod : Affop
R → Cat∞ satisfies eachof the conditions of Proposition 4.8. For any A ∈ CAlgR , the ∞-category ModA has limits since
ModA is presentable (and presentable ∞-categories admit all limits). Given any u : B → A in
AffR the base change functor u∗ : ModA → ModB commutes with limits along ∆ by virtue ofthe flat and finite topologies. Its right adjoint u∗ is given by the conservative forgetful functor.
For any pushout square
A

u

- B
u0

v

?

C

v0

?
- D

in CAlgR we have, for M ∈ ModB ,(v 0 )∗ u0∗ (M ) = (v 0 )∗ (M ⊗B D) ' (v 0 )∗ (M ⊗A C) ' v∗ (M ⊗B A) = v∗ u∗ (M )43`where the first equivalence follows from the natural equivalence B C ' B ⊗A C in CAlgR of
Example 2.6. The proof of the flat case can be extracted from Lemma 2.2.2.13 of [TV2] (thesame arguments hold here).
Proposition 5.6. Let R be an E∞ -ring. The flat and finite topologies on AffR are subcanonical.
Proof. This follows from the fact that if Mod is a stack on AffR with respect to a topology
τ , then τ is subcanonical. This can be seen as follows. Assume Mod is a stack. Then bydefinition we have an equivalence ModA → lim∆ ModB∗ for any covering B → A of A. Thus
for all M ∈ ModA , the unit map M → lim∆ (M ⊗A B∗ ) is an equivalence. Take M = A. Then
A → lim∆ B∗ is an equivalence and for all C ∈ CAlgR the composition
Map(C, A) → Map(C, lim∆ B∗ ) → lim∆ Map(C, B∗ )is an equivalence. Thus the representable prestack hC is a stack. The result now follows from
Proposition 5.5.
Let τ ∈ {fl, fin}. Since τ is subcanonical, we have a fully faithful morphism
AffR → Sh(R, τ )of ∞-categories given by the Yoneda embedding. We denote a stack in the essential image of thisfunctor by Spec A for an R-algebra A. A stack F in Sh(R, τ ) is said to be affine if F → Spec A
is an equivalence of stacks for some R-algebra A. An affine stack is called an affine group stack
if the affine stack is a group stack.

6

Rigid (∞, 1)-categories

Here we describe what it means for an ∞-category to be rigid. This amounts to every object
being dualizable and is a strong condition which gives much of the Tannakian theory its flavour.
The notion of a dual object in an ordinary category has its origins in the example of the categoryof vector spaces : a vector space admits a dual if and only if the vector space is finite dimensional.
Thus the rigidification of an ∞-category, that is, discarding all objects that do not admit duals,
can be thought of as the implementation of a “finiteness condition” on its objects.
Many of the ∞-categories we work with are ind-rigid. That is, they are equivalent to the∞-category of ind-objects of its full subcategory of dualizable objects. For example the ∞category of modules over an E∞ -ring is ind-rigid : any module over an E∞ -ring is a given bya filtered colimit of rigid objects. We prove the very useful projection formula for ∞-categories
and later in this section show that under certain conditions, the ∞-category of endomorphismsof a functor valued in R-modules is an affine group stack.
Recall that an object y in a symmetric monoidal category C is said to be a dual of an object
x in C if there exists maps evx : x ⊗ y → 1 and coevx : 1 → y ⊗ x such that the compositions
id ⊗coev

ev ⊗idcoevx ⊗idy

idy ⊗evxxxx
x −−x−−−−→x ⊗ y ⊗ x −−−−−→
xy −−−−−−→ y ⊗ x ⊗ y −−−−−→ y44

coincide with the identity maps of x and y. Let C now be a symmetric monoidal ∞-category.
An object x in C is said to be dualizable if it admits a dual when considered as an object of thesymmetric monoidal category hC. We will denote the dual of an object x by x∨ .
Definition 6.1. A symmetric monoidal ∞-category is said to be rigid if all objects are dualizable.
See also Proposition 2.6 of [TV3] for more equivalent characterisations of rigidity. If C is asymmetric monoidal ∞-category then the unit object 1C is dualizable. Moreover, the dualizable
objects are stable by isomorphism in hC and stable by the tensor product. We denote the fullsubcategory of C consisting of dualizable objects by C rig . Let Catrig∞ denote the full subcategoryof Cat⊗spanned
by
the
rigid
∞-categories.
By
Theorem
2.10
andLemma 2.11 of [TV3] we∞deduce that there exist adjunctions
rig

Fr
i
Cat⊗
Catrig
Cat⊗
∞ 
∞ 
∞
(•)rigiof ∞-categories where Frrig (C) is the free rigid ∞-category generated by the symmetric monoidal∞-category C. The right adjoint (•)rig will be called the rigidification functor and C rig therigidification of C.
Proposition 6.2. Let C be a symmetric monoidal ∞-category. Then C is rigid if and only if
it satisfies the following conditions :1. The ∞-category C is enriched over itself.
2. The map Hom(x, 1) ⊗ y → Hom(x, y) is an equivalence in C for all x, y ∈ C.
Proof. It is enough to prove this statement in hC. The classical statement can then be foundfor example in Section 2 of [D2].
Let C be a rigid ∞-category. To any map f : x → y in C there corresponds a transpose map
given by the compositiont1⊗coev

1⊗f ⊗1evy ⊗1xy ∨ ⊗ x ⊗ x∨ −−−−→ y ∨ ⊗ y ⊗ x∨ −−−−→ x∨ .
f : x −−−−−→

Similarly, to any f : y ∨ → x∨ we associate a composition map
coevy ⊗1

1⊗f ⊗1y⊗evx

f t : x −−−−−→ y ⊗ y ∨ ⊗ x −−−−→ y ⊗ x∨ ⊗ x −−−−→ y.
This induces an equivalenceC(x, y) → C(y ∨ , x∨ ).
of (∞, 0)-categories. The following Lemma follows straightforwardly from Proposition 6.2.
Lemma 6.3. Let C be a rigid ∞-category and x, y, x0 , y 0 be objects in C. Then the following
hold.
1. The map Hom(x, y) ⊗ Hom(x0 , y 0 ) → Hom(x ⊗ x0 , y ⊗ y 0 ) is an equivalence in C.
45

2. The map x ⊗ y → Hom(x∨ , y) is an equivalence in C.
3. The map Hom(x, y)∨ → Hom(y, x) is an equivalence in C.
Proof. Set x∨ = Hom(x, 1). For (1), we have a chain of equivalences
Hom(x, y) ⊗ Hom(x0 , y 0 ) ' x∨ ⊗ (x0 )∨ ⊗ y ⊗ y 0 ' Hom(x ⊗ x0 , y ⊗ y 0 )since x∨ ⊗ y ∨ is dual to x ⊗ y. For (2), we have a chain of equivalencesC(z, x ⊗ y) ' C(x∨ ⊗ y ∨ , z ∨ ) ' C(x∨ , Hom(y ∨ , z ∨ )) ' C(x∨ , Hom(z, y)) ' C(z, Hom(x∨ , y))which is functorial in z. Finally, for (3), we have a chain of equivalences
Hom(x, y)∨ ' Hom(Hom(x, y), 1) ' Hom(x∨ ⊗ y, 1) ' y ∨ ⊗ x
due to (2). Thus Hom(x, y)∨ ' Hom(y, x).
Proposition 6.4. Let C and D be rigid ∞-categories and F, G : C → D be two symmetricmonoidal functors. Then any map α : F → G is an equivalence.
Proof. As shown in [Sa], an explicit inverse to α is given by the map β : G → F making thefollowing diagram
F (x∨ )

?

F (x)

∨

αx∨

- G(x∨ )tβ?
- G(x)∨

x

commute for all x ∈ C.
Definition 6.5. Let C be a presentable symmetric monoidal ∞-category. Then C is said to be
ind-rigid if Ind(C rig ) → C is an equivalence of ∞-categories.
Let R be an E∞ -ring. It follows from the proof of Proposition 7.2.5.2 of [L2] that
Ind(Modperf
R ) → ModR
rig
is an equivalence of ∞-categories. Combining this with the equivalence Modperf
R → ModR we findthat the ∞-category ModR is ind-rigid. Let G = F ◦ y : Modrig
R → ModR denote the composition
rigof F : Ind(ModR ) → ModR with the Yoneda embedding. It follows from Proposition 5.3.5.11
of [L1] that the set of objects {G(M )}M ∈Modrig generate ModR under filtered colimits.
R

Proposition 6.6. Let C and A be symmetric monoidal ∞-categories, f : C → A a symmetricmonoidal functor and g its right adjoint. Assume C is ind-rigid. Then for any object x in C
and a in A, the map
g(a) ⊗ x → g(a ⊗ f (x))
is an equivalence.
46

Proof. Let x be a dualizable object of C and y be an arbitrary object of C. Then C(y, ga ⊗ x) ∼=C(y ⊗ x∨ , ga) ∼= A(f (y ⊗ x∨ ), a) ∼= A(f y ⊗ f x∨ , a) ∼= A(f y ⊗ (f x)∨ , a) ∼= A(f y, a ⊗ f x) ∼=C(y, g(a ⊗ f x)). Since any object in C is given by a colimit of dualizable objects by assumption
and the above demonstration is functorial in x, the result follows.
The equivalence in Proposition 6.6 is often called the projection formula. Setting a = 1,
f (x) = b and applying f to the projection formula gives the equivalence f g(1) ⊗ b ' f g(b).
Proposition 6.7. Let R be an E∞ -ring. If the functor • ⊗R M : ModR → ModR commuteswith limits then M is dualizable.
Proof. Let X and Y be R-modules. We can write any R-module as a colimit of perfect Rmodules so we set X = colimα Xα . Assume that the functor • ⊗R M commutes with limits andrecall that the ∞-category Modperf
is equivalent to Modrig
R
R . We have that
Hom(X, Y ) ⊗ M ' lim Hom(Xα , Y ) ⊗ M
Rα' lim (Hom(Xα , Y ) ⊗ M )α' lim (Xα∨ ⊗ Y ⊗ M )α' lim Hom(Xα , Y ⊗ M ) ' Hom(X, Y ⊗ M ).
α

Setting X = M and Y = R we see that M ∨ ⊗R M ' Hom(M, M ) so M is dualizable by
Proposition 6.2.
Lemma 6.8. Let C and D be presentable symmetric monoidal ∞-categories. Assume that C is
ind-rigid. Then there exists an equivalence
Hom⊗ (C, D) → Hom⊗ (C rig , Drig )of ∞-categories.
Proof. By the universal property of ind-objects, the map Hom⊗ (C, D) → Hom⊗ (C rig , D) is anequivalence. The result now follows from the fact that symmetric monoidal functors preserve
rigid objects.
Let F : C → D be a functor between ∞-categories. Then we denote by End(F ) the mapping
space Map(F, F ) taken in the ∞-category RHom(C, D). If C and D are symmetric monoidal∞-categories we let End⊗ (F ) denote the mapping space Map(F, F ) in RHom⊗Γ (C, D).
Let R be an E∞ -ring. First recall that given two rigid R-modules M and N , the mapping
space MapModR (M, N ) as a functor on the ∞-category CAlgR of R-algebras is given byMap(M, N )(A) := Map(M ⊗R A, N ⊗R A).
This functor is representable by the chain of equivalences
MapModR (M ⊗R A, N ⊗R A) ' MapModR (M ⊗R N ∨ , A) ' MapCAlgR (Fr(M ⊗R N ∨ ), A)
47

where the second equivalence follows from Proposition 6.2 and the third follows from the equivalence
CAlgR (Fr(M ), •) ' ModR (M, •)arising from the adjunction CAlgR a ModR (see Section 2).
Lemma 6.9. Let R be an E∞ -ring, C be a symmetric monoidal ∞-category and F : C → ModrigRa symmetric monoidal functor. Then End⊗ (F ) is representable.
Proof. Any symmetric monoidal ∞-category is of the form hocolimα Cα where Cα is the freesymmetric monoidal ∞-category over an ∞-graph Gα defined through the following universal
property : for any symmetric monoidal ∞-category D, there exists an equivalence
RHom⊗ (Cα , D) ' RHomgrph (Gα , D)
where RHomgrph denotes the ∞-category of functors between ∞-graphs. Thus we have an
equivalence
grph
RHom⊗ (C, Modrig(Gα , Modrig
R ) → holim RHom
R )α

F 7→ Fαwhere Fα sends an object x in Gα to a rigid module Mx and the mapping space Gα (x, y) to themapping space Modrig
R (Mx , My ).
When Gα consists of a single object x, then End(Fα ) ' End(Mx ) = Spec Fr(Mx ⊗R (Mx )∨ ).
When Gα consists of two objects x and y and the simplicial set A of arrows between x and ythen
End(Fα ) = End(Mx ) ×hMap(Mx ,My )A End(My ).
Since representable objects are stable under homotopy limits and any ∞-graph is generated
under homotopy colimits by the above two simple graphs, the functor holimα End⊗ (Fα ) 'End⊗ (F ) is representable.
Proposition 6.10. Let R be an E∞ -ring, C be a presentable ind-rigid symmetric monoidal ∞category and F : C → ModR a symmetric monoidal functor. Then End⊗ (F ) is a representable
Gp(S)-valued prestack. Hence it is an affine group stack with respect to any subcanonical topology.
Proof. Since the ∞-category C is ind-rigid, the map RHom⊗ (C, ModR ) → RHom⊗ (C rig , Modrig
R )is an equivalence by Lemma 6.8. Thus we have an equivalence End⊗ (F ) ' End⊗ (F rig ). By
Lemma 6.9, End(F rig ) is representable. Finally, End⊗ (F ) ' RAut⊗ (F ) by Proposition 6.4 soEnd⊗ (F ) is in fact a representable Gp(S)-valued prestack and hence an affine group stack forany subcanonical topology.
48

7

Neutralized Tannaka duality for (∞, 1)-categories

We begin by constructing the ∞-category of R-linear, stable, presentable, symmetric monoidal∞-categories which we will call R-tensor ∞-categories. In Example 2.8 we saw an explicit
construction of the cartesian monoidal structure on the ∞-category Cat∞ of ∞-categories. LetpCatp,⊥∞ denote the full subcategory of Cat∞ spanned by presentable ∞-categories which aremoreover stable ∞-categories. The projection Cat⊥,p∞ → Γ determines a symmetric monoidalp,⊥structure on the ∞-category Cat∞ of stable, presentable ∞-categories (see Proposition 6.3.2.15
of [L2]). The unit object of Catp,⊥∞ with this symmetric monoidal structure is the ∞-category
Sp of spectra.
Let Catp,⊥,⊗denote the subcategory of Cat⊗∞∞ spanned by stable, presentable symmetricmonoidal ∞-categories whose symmetric monoidal bifunctor preserves colimits seperately ineach variable and whose morphisms are colimit preserving symmetric monoidal functors. Then
we have an equivalencep,⊥,⊗CMon(Catp,⊥∞ ) → Cat∞of ∞-categories. Thus a symmetric monoidal ∞-category C belongs to CMon(Catp,⊥∞ ) if andonly if C is stable, presentable and the bifunctor ⊗ : Ch1i × Ch1i → Ch1i preserves colimits
separately in each variable.
Let R be an E∞ -ring. Then the stable, presentable symmetric monoidal ∞-category ModR
of R-modules belongs to CMon(Catp,⊥∞ ). Applying this observation to the results of Section 2
gives the equivalencep,⊥CMon(ModModR (Catp,⊥∞ )) ' CMon(Cat∞ )ModR / .

of ∞-categories. The term on the left hand side is the ∞-category of R-linear, stable, presentable
symmetric monoidal ∞-categories and R-linear symmetric monoidal functors.
Definition 7.1. Let R be an E∞ -ring. A symmetric monoidal ∞-category is said to be atensor ∞-category if it is stable and presentable. It is said to be an R-tensor ∞-category if it
is R-linear, stable and presentable.
We will denote the ∞-category of tensor ∞-categories by Tens⊗ := Catp,⊥,⊗and the ∞∞category of R-tensor ∞-categories and R-linear symmetric monoidal functors byp,⊥,⊗)ModR / .
Tens⊗
R := (Cat∞We now introduce the stack of fiber functors and state our duality theorems for neutralized
higher Tannaka duality. In the next section, we will describe the proofs. Let R be an E∞ -ringand τ ∈ {f l, f in}. The stack Mod of modules of Proposition 5.5 can naturally be extended toact on the ∞-category of stacks Sh(R, τ ) as follows. The objects of Sh(R, τ ) can be considered
as stacks associated to the functors
F : CAlgR → Cat∞taking values in the ∞-category of ∞-categories using the inclusion of Remark 4.2. The actionof Mod on the ∞-category of stacks Sh(R, τ ) is then given by
Mod : Sh(R, τ )op → Cat∞F → Mor(F, Mod)49

which by abuse we also denote Mod where Mor is the morphism object of ShCat∞ (R, τ ) given
by Proposition 4.6.
The ∞-category Mor(F, Mod) is naturally endowed with the structure of an R-tensor ∞category. The tensor ∞-structure on Mor(F, Mod) is induced from that on Mod : it is presentable, stable and is given the pointwise symmetric monoidal structure
Mor(F, Mod)hni := Mor(F, Mod) ×Mor(F,Γ) {hni}of Example 2.4 where Modhni (A) := (ModA )hni for an R-algebra A and Γ is the constantprestack. The R-linear structure on Mor(F, Mod) is also induced from that on Mod through thecomposition
ψ

ModR −→ Mor(F, ModR ) → Mor(F, Mod)
where ModR is the constant prestack and ψ is the natural constant map. Thus we obtain afunctorMod : Sh(R, τ )op → Tens⊗
R.
Definition 7.2. Let R be an E∞ -ring. A dualizable object of Tens⊗
R will be called a rigid
R-tensor ∞-category.
⊗ rig
Let Tensrig
denote the ∞-category of rigid R-tensor ∞-categories. Restricting
R := (TensR )the functor Mod to rigid objects we obtain a natural functor

Perf : Sh(R, τ )op → Tensrig
R
F 7→ Mor(F, Perf)
where the stack Perf : CAlgR → Tensrig
R on the right hand side sends a commutative R-algebrarig
A to the ∞-category ModA of rigid A-modules.
Lemma 7.3. The functor Perf admits a left adjoint.
Proof. Let C be a rigid R-tensor ∞-category. We have the following chain of equivalences
MapTensrig (C, Perf(F )) ' MapSh(R,τ ) (C × F, Perf) ' MapSh(R,τ ) (F, Hom(C, Perf)).
R

Here Hom(C, Perf) is a stack by Proposition 4.7, since Perf is a stack, where we regard C as aconstant prestack.
The left adjoint to Perf of Lemma 7.3 will be denoted
op
Fib : Tensrig
R → Sh(R, τ )C 7→ Hom(C, Perf)
where Fib(C)(A) := MapTensrig (C, Modrig
A ) for a commutative R-algebra A.
R
We would now like to consider conditions on rigid R-tensor ∞-categories and stacks oncertain sites of R-algebras for which the adjunction Fib a Perf is an equivalence. We begin withsome preliminary definitions and results.
50

Definition 7.4. Let R be an E∞ -ring. The ∞-category of Segal comodules over a Hopf Ralgebra B is given by the following limit
SeComodB := lim ModBnn∈∆of ∞-categories.
We will often abuse terminology by calling a Segal comodule over a Hopf R-algebra simply a
comodule over a Hopf R-algebra. This is justified since by Theorem 6.2.4.2 of [L2], there exists
an equivalence
SeComodB → ComodB
rigof ∞-categories. Restricting to rigid objects we have an identification SeComodrig
B := limn ModBn .
The forgetful functor SeComod → ModR is given by the evaluation map ev0 : limn ModBn →ModB0 = ModR .

Example 7.5. The object Dec+ (B) is a B-comodule which is just B thought of as a comodule
over itself. More precisely, a B-comodule, by definition, consists of objects Mn ∈ ModBn for all∼[n] ∈ ∆ and for every arrow [n] → [m] in ∆, an equivalence Mn ⊗Bn Bm −→ Mm in ModBm . We
have that Dec+ (B)n = Bn+1 and Bn+1 ⊗Bn Bm ' B ⊗R n+1 ⊗B ⊗n B ⊗m ' B ⊗m+1 where the firstequivalence follows from Segal maps of the Hopf R-algebra structure on B. Thus Dec+ (B) is a
B-comodule.
Proposition 7.6. Let R be an E∞ -ring and B a Hopf R-algebra. Then there exists an equivalencee
SeComodB ' Mod(BG)of ∞-categories for an affine group stack G = Spec B.
e = |BGe • | where BGe • : ∆op → Sh(C, τ ) is the functor that sends
Proof. Firstly, we have that BG[n] to Gn . Therefore
e Spec B = colimn∈∆ Spec Bn .
B
We thus have an equivalenceee Spec B, Mod) = Mor(colimn∈∆ Spec Bn , Mod) ' limn∈∆ ModBn =: SeComodB
Mod(BG):= Mor(Bgiven by the Yoneda Lemma.
Let R be an E∞ -ring and B a rigid Hopf R-algebra. A corollary of Proposition 7.6 is that
there exists an equivalencee
SeComodrig
B ' Perf(BG)eof ∞-categories for an affine group stack G = Spec B. We call Perf(BG)
the ∞-category ofrepresentations of G and denote it byeRep(G) := Perf(BG).
for G = Spec B where B is a Hopf R-algebra in the ∞-category Sp of spectra.
51

Definition 7.7. The ∞-category of rigid R-tensor ∞-categories over Modrig
R will be denoted
rig(TensrigR )∗ := (TensR )/ Modrig .
R

The objects of (Tensrig
R )∗ will be described as pairs (T, ω) where T is a rigid R-tensor ∞category and ω : T → Modrig
R is an R-tensor functor. They will be called pointed rigid R-tensor∞-categories. Let
opFib∗ : (Tensrig
R )∗ → Gp(R, τ )be the functor defined by Fib∗ (T, ω) := End⊗ (ω). Let
Perf∗ : Gp(R, τ )op → (Tensrig
R )∗be the functor defined by Perf∗ (G) := (Rep(G), ν) where ν := f ∗ : Rep(G) → Modrig
R is theefunctor induced by the natural map f : ∗ → BG.
Lemma 7.8. The maps of Definition 7.7 induce an adjunction
opFib∗ : (Tensrig
R )∗  Gp(R, τ ) : Perf∗of ∞-categories.
Proof. Consider the homotopy pullback diagrame Fib(T )) - MapSh(R,τ ) (BG,e Fib(T ))Map∗ (BG,

?

∗

?

ω

- Map

Sh(R,τ ) (∗, Fib(T ))and its corresponding adjoint diagram
e
e
- Map
Map∗ (T, Perf(BG))(T, Perf(BG))
Tensrig
R

?

∗

?

ω

- Map

Tensrig
R(T, Modrig
R ).

using Lemma 7.3. Since the two diagrams are equivalent, the homotopy pullbacks are equivalent.
Thus we have a chain of equivalencese Fib(T )) ' Map∗ (T, Perf(BG))
e
Map(G, Fib∗ (T )) ' Map∗ (BG,eaΩe ∗.
where the first map arises from the adjunction B
52

We now state the main results of the paper. They will be proven in the next section. We will
begin with the pointed case, otherwise known as neutralized Tannaka duality for ∞-categories.
We would like to study conditions on pointed rigid R-tensor ∞-categories and group stacks forwhich the adjunction of Lemma 7.8 is an equivalence of ∞-categories. We make use of thepositive, flat and finite topologies introduced in Definition 5.1.
We begin by defining the appropriate subcategory of pointed rigid R-tensor ∞-categories
which we call Tannakian.
Definition 7.9. Let R be an E∞ -ring, C a rigid R-linear symmetric monoidal ∞-category andω : C → Modrig
b . Then ω is said
R an R-linear symmetric monoidal functor. Denote Ind(ω) by ω
to be :
1. A finite fiber functor if ω
b is conservative and preserves (small) limits.
Let R be a connective E∞ -ring. Then ω is said to be :2. A flat fiber functor if ω
b is conservative, creates a t-structure on C, is exact and whoseright adjoint is t-exact.
Let R be a connective bounded E∞ -ring. Then ω is said to be :3. A positive fiber functor if ω
b is conservative, creates a t-structure on C and is exact.
Note that since ωb is a presentable symmetric monoidal functor it commutes with colimits andhence by the adjoint functor theorem admits a right adjoint pb which is a lax symmetric monoidal
functor. Also, we remark that since positive and flat fiber functors are conservative, t-exact anddefined over a connective bounded base E∞ -ring, the t-structures created are non-degenerate.
Definition 7.10. Let R be an E∞ -ring. A pointed R-Tannakian ∞-category with respect to τ
is a pair (T, ω) where T is a rigid R-tensor ∞-category and ω : T → Modrig
R is a τ -fiber functor.
Let (TanτR )∗ denote the full subcategory of (Tensrig
R )∗ spanned by pointed τ -R-Tannakian ∞categories. We will often abuse terminology by referring to a pointed R-Tannakian ∞-category(T, ω) as simply T .
A notion of rigidity manifests itself on the opposite side of the duality in the following sense.
LeteωbG : Mod(BG)→ ModReand ωG : Perf(BG)→ Modrig
R be the forgetful functors.
Definition 7.11. Let R be an E∞ -ring. An affine group stack G = Spec B in Gp(R, τ ) is said
to be
1. Weakly rigid if End⊗ (bωG ) → End⊗ (ωG ) is an equivalence.
2. Rigid if the map SeComodB → Ind(SeComodrig
B ) is an equivalence of ∞-categories.
Lemma 7.12. Let R be an E∞ -ring and G = Spec B be a rigid affine group stack. Then G isweakly rigid.
53

Proof. Since G is rigid, the ∞-category SeComodB is ind-rigid and by Lemma 6.8 there exists
an equivalencerig
Hom⊗ (SeComodB , ModR ) ' Hom⊗ (SeComodrig
B , ModR )of ∞-categories. The result now follows from Proposition 7.6.
We remark that these rigidity conditions on an affine group stack (or on a Hopf R-algebra)
are a new feature of the higher categorical approach : in the classical case, an affine groupscheme over a field is automatically rigid. We will now define the objects on the group side of
the correspondence.
Definition 7.13. Let R be an E∞ -ring. A group stack G in Gp(Pr(AffR )) is said to be RTannakian if it is of the form Spec B for B a weakly rigid Hopf R-algebra. It is said to beτ -R-Tannakian for a topology τ if it is R-Tannakian where B is a τ -Hopf R-algebra.
Let TGp(R, τ ) denote the full subcategory of Gp(R, τ ) spanned by τ -R-Tannakian groupprestacks.
Our first main theorem is the following generalization of neutralized Tannaka duality to the∞-categorical setting.
Theorem 7.14 (Neutralized ∞-Tannaka duality). Let τ be a subcanonical topology. Then themap
Perf∗ : TGp(R, τ )op → (Tensrig
R )∗is fully faithful. Moreover, the adjunction Fib∗ a Perf∗ induces the following :1. Let R be an E∞ -ring. Then (T, ω) is a pointed finite R-Tannakian ∞-category if and only
if it is of the form Perf∗ (G) for G a finite R-Tannakian group stack.
2. Let R be a connective E∞ -ring. Then (T, ω) is a pointed flat R-Tannakian ∞-category if
it is of the form Perf∗ (G) for G a flat R-Tannakian group stack.
3. Let R be a bounded connective E∞ -ring. Then (T, ω) is a pointed positive R-Tannakian∞-category if it is of the form Perf∗ (G) for G a positive R-Tannakian group stack.

8

Proof of the neutralized theorem

We will now embark on the proof of the higher Tannaka duality statement described at the endof the last section. For an R-linear tensor functor f : C → ModR , we will denote by fA thecomposition
f

•⊗ A

C−
→ ModR −−−R−→ ModA
given by composing f with the base change functor.
Proposition 8.1. Let R be an E∞ -ring, B a R-bialgebra and ωb : SeComodB → ModR theforgetful functor. Then there exists an equivalenceSpec B → End⊗ (bω ).
54Proof. Let G = Spec B and consider the homotopy pullback diagram∗ = Spec R

u

e
- BG
6

6

u

v

G

v
∗ = Spec R.

The induced functors between stable, presentable, R-linear symmetric monoidal ∞-categories
are given bye
pb = u∗ : ModR → Mod(BG)eωb = u∗ : Mod(BG)→ ModRv ∗ : ModR → ModBv∗ : ModB → ModR .
By Proposition 2.5.14 of [L3], we have the base change formula u∗ u∗ ' v∗ v ∗ . Therefore thecomposition ωb ◦ pb is given by B ⊗R •.
e
Observe now that the object u∗ (R) in Mod(BG)
is endowed with the structure of a commutative monoid object inherited from R. From Corollary 6.3.5.18 of [L2] (see also Section 2) we
deduce the existence of a fully faithful functoreρ : CMon(Mod(BG))→ (Tens⊗ )Mod(BG)/eewhich takes u∗ R to ρ(u∗ R) = u∗ : Mod(BG)→ ModR . Thus we have the following chain ofequivalencesEnd⊗ (bω ) ' MapCMon(Mod(BG))(u∗ R, u∗ R) ' MapCAlgR (u∗ u∗ R, R) ' MapCAlgR (B, R)ewhich completes the proof.
Corollary 8.2. Let R be an E∞ -ring. Then the counit map
G → Fib∗ (Perf∗ (G))
is an equivalence in Gp(R, τ ) when G is of the form Spec B for B a Hopf R-algebra and is
weakly rigid.
e
Proof. We need to show that Spec B → End⊗ (ω) is an equivalence where ω : Perf(BG)→ ModrigR
for G weakly rigid. We have a diagram∼∼

G = Spec B −→ End⊗ (bω) −→ End⊗ (ω)
where the first equivalence follows from Proposition 8.1 and the second equivalence follows fromthe assumption that G is weakly rigid. Thus Spec B is equivalent to End⊗ (ω).
55

Corollary 8.2 states that we have a full embedding of the ∞-category of R-Tannakian group
stacks into the ∞-category of pointed rigid R-tensor ∞-categories given by the rule Perf∗ : G 7→e
(Perf(BG),
ω). This is the first statement of Theorem 7.14. We now prove the remainder of theneutralized ∞-Tannaka duality theorem.
Proof of Theorem 7.14. Let τ ∈ {≥ 0, f l, f in} be a topology. We will first show that if ω is a
τ -fiber functor then the R-algebra B := ωb pb(R) is a τ -R-algebra. For all three cases we consider
the projection formula ωb pb(M ) ' B ⊗R M of Proposition 6.6 for an R-module M . For the
positive case, the functor ωb is t-exact by definition so by Lemma 3.20, the right adjoint pb is left
t-exact and so B is positive. For the flat case, R is connective and the functors ωb and pb are
t-exact so B = ωb pb(R) is a connective R-module. Also, the functor ωb pb ' B ⊗R • is left t-exact
by Lemma 3.20. It then follows from Theorem 7.2.2.15 of [L2] that B is flat over the connective
E∞ -ring R. The finite case follows simply since ωb preserves limits by definition and pb preserves
limits since it is a right adjoint. Thus the composition ωb pb preserves limits and by the projection
formula we are done.
Let (T, ω) be a τ -R-Tannakian ∞-category. We will now show that the unit (T, ω) →Perf∗ (Fib∗ (T, ω)) of the adjunction Fib∗ a Perf∗ is an equivalence when restricted to the subcateeegory (TanτR )∗ , ie. (T, ω) is equivalent to (Perf(BG),
ωG ) for G = End⊗ (ω) and ωG : Perf(BG)
→
rig
b
b
e
ModR the forgetful functor. We consider the map φ : T → Mod(BG). We have the commutae
tivity ωb 'ω
bG ◦ φb where ωbG : Mod(BG)→ ModR is the forgetful functor and we consider thefollowing diagram
b
φ

Tb 

-

e
Mod(BG)

b
ψ

ω
b ab
p

-



ω
bG ab
pG

ModR
where ψb is the right adjoint to φb owing to the fact that φb commutes with colimits (it is abb
map between presentable ∞-categories in Tan⊗
R ). Now observe that φ is conservative since ω
is conservative (by definition of a τ -fiber functor), ωbG is the conservative forgetful functor andb Therefore, we have the following :ωb'ω
bG ◦ φ.
(*) The map φb is an equivalence if and only if φb ◦ ψb → id is an equivalence.
We treat the three different topology cases separately.
1. Finite case : In other words, we assume that ωb is conservative and preserves limits.
b
We begin by proving that φb ◦ ψ(E)
→ E is an equivalence when E is of the form pbG (M ) forM ∈ ModR . We have that pbG (M ) ' B ⊗R M . Furthermore, by the commutativity of the
diagram we havebωbG ◦ φb ◦ ψ(E)'ωb ◦ ψb ◦ pbG (M ) ' ωb ◦ pb(M ).
By the projection formula of Proposition 6.6, we have the equivalence ωb ◦ pb(M ) ' M ⊗R ωb pb(R).
b
Therefore ωbG ◦ φb ◦ ψ(E)'ωbG (B ⊗R M ) ' ωbG (E) and the result follows from the conservativity
of ω
bG .
56

b
We now prove that φb ◦ ψ(E)
→ E is an equivalence for a general B-comodule E. The functorωbG is conservative and preserves ωbG -split limits by Proposition 6.2.4.1 of [L2]. Thus by the dualof Proposition 6.2.2.11 of [L2] we have that for all E in SeComodB , there exists an augmented
cosimplicial object E• : ∆+ → ComodB given by En ' (bpG ◦ ω
bG )n+1 E which is ωbG -split. It
follows from Proposition 4.23 that there exists an equivalence
E → holim ((E0 ⊗R B ⊗R n ) ⊗R B)
n∈∆where E0 = ωbG (E). Hence we can consider E to be of the form holimn pbG (Mn ) for Mn =(E0 ⊗R B ⊗R n ) ∈ ModR .
We have a diagramb
φb ◦ ψ(E)

- E

?

?

b pG (Mn )) - holim pbG (Mn ).
holim φb ◦ ψ(bnn

The lower horizontal arrow is an equivalence from the above case of E = pbG (M ). Also, the left
vertical arrow is an equivalence from the fact that φb ◦ ψb commutes with limits : since ψb is a rightadjoint functor it preserves limits and φb preserves limits since ωb preserves limits (being a finiteefiber functor), ωbG preserves limits (because its the forgetful functor from Mod(BG)
where G isb Thus the upper horizontal arrow is an equivalence. Thus Tb → Mod(BG)efinite) and ωb'ω
bG ◦ φ.
is an equivalence.
ee
The equivalence Tb → Mod(BG)
induces an equivalence T → Perf(BG)
on rigid objects since
symmetric monoidal functors preserve rigid objects. Thus SeComodB is ind-rigid (using Proposition 7.6) and hence by Lemma 7.12 the affine group stack G is weakly rigid.
2. Positive case : In other words, we assume that ωb is conservative, creates a non-degenerate
t-structure and is t-exact. We also assume that the base E∞ -ring R is bounded and connective.
e ≤n , where Mod(BG)e ≤n denotes the full subcategory of Mod(BG)
eLet E be an object in Mod(BG)
spanned by objects which get mapped to (ModR )≤n (with the t-structure of Example 3.17) under∼e ≤n as aboveωbG . We have that E −→ holimn ((E0 ⊗R B ⊗R n ) ⊗R B) is an equivalence in Mod(BG)
with E0 = ω
bG E in (ModR )≤n .
Let Mn := (E0 ⊗R B ⊗R n ) ∈ (ModR )≤n and consider the diagrambωb ◦ ψ(E)- ωbG (E)?

?

b n ⊗R B) - holim ωholim ωb ◦ ψ(MbG (Mn ⊗R B).
nn

The bottom horizontal arrow is an equivalence sinceb n ⊗R B) ' ωωb ◦ ψ(Mb ◦ ψb ◦ pbG (Mn ) ' ωb ◦ pb(Mn ) ' ωG (Mn ⊗R B).
57

The right vertical arrow is an equivalence since ωG preserves ωG -split limits by Proposition
6.2.4.1 of [L2]. The left vertical arrow is also an equivalence by the following. Note firstly thatb n ⊗R B) ' ψb ◦ pbG (Mn ) ' pb(Mn ) and so ψ(Mb n ⊗R B) is in T≤n since pb is left t-exact.
ψ(MTherefore, the cosimplicial objectb k ⊗ B)[k] 7→ ψ(Mb k ⊗R B)) = 0 for all i > n and all k.
in T satisfies the property that πit (ψ(MWe can then apply Lemma 3.24 to the t-exact fiber functor ωb : Tb → ModR to deduce that∼b n ⊗R B) −b n ⊗R B)
holim ωb ◦ ψ(M→ωb ◦ holim ψ(Mnn

is an equivalence. Secondly, the functor ψb preserves limits (it is a right adjoint) so the leftbb
vertical arrow is an equivalence. We then have that the map ωbG ◦ φb ◦ ψ(E)'ωb ◦ ψ(E)→ ωG (E)is an equivalence. Thus by the conservativity of ωbG , we obtain the equivalencee ≤n .
Tb≤n → Mod(BG)e ≤n between the unions over all n is an
We deduce that the map ∪n Tb≤n → ∪n Mod(BG)
equivalence. Rigid modules over a bounded E∞ -ring R are bounded and so T is contained in∪n Tb≤n (in fact Tb = ∪n Tb≤n since Tb is left t-complete owing to its non-degenerate t-structure) andee ≤n . Since φb is a symmetric monoidal functor it preserves
Perf(BG)
is contained in ∪n Mod(BG)erigid objects and so we obtain an induced fully faithful functor φb : T → Perf(BG).
eHowever, for any E, F ∈ Perf(BG), we have a diagrambb )) ∼- ωωb (ψ(E)⊗ ψ(FbG (E) ⊗ ωbG (F )∼?

b ⊗ F)ωb ◦ ψ(E∼?
- ωbG (E ⊗ F )using the symmetric monoidal structures on ωb and ωbG . Therefore the left vertical arrow of thebediagram is an equivalence. Conseqently, the map ψ : Perf(BG)→ T is symmetric monoidal andthus preserves rigid objects leading to the equivalence∼e
T −→ Perf(BG)

of ∞-categories. Finally, G is weakly rigid by combining this equivalence with Proposition 8.1.
3. Flat case: In other words, we assume that ωb is conservative, creates a non-degenerate
t-structure, is t-exact and whose right adjoint is t-exact. The first part of the proof can bee ≤n is an equivalence.
deduced directly from the positive case : it follows that Tb≤n → Mod(BG)
However, here the base E∞ -ring R is merely connective. In the flat case though, we actually
have an equivalencee ≤n ' Mod(BG)e
Tb ' holim Tb≤n → holim Mod(BG)
n

n

58

of ∞-categories where the identification on the left hand side follows from the left t-exactnessof Tb and the identification on the right hand side follows from the fact that a non-degenerateet-structure is created on Mod(BG).
Since monoidal functors preserve rigid objects we have aneequivalence T → Perf(BG) of ∞-categories. Combining this equivalence with Proposition 8.1
we find that G is rigid and hence weakly rigid.

9

Neutral Tannaka duality for (∞, 1)-categories

We now consider the case where there simply exists a τ -fiber functor. This is the non-pointed
case, otherwise known as neutral Tannaka duality for ∞-categories.
Definition 9.1. Let R be an E∞ -ring and τ ∈ {f in, f l, ≥ 0} a topology. A rigid R-tensor∞-category T is said to be a R-Tannakian ∞-category with respect to τ if there exists a τ -fiber
functor w : T → Modrig
R .
We denote the ∞-category of R-Tannakian ∞-categories with respect to τ by TanτR . The
objects on the other side of the correspondence are described as follows.
Definition 9.2. Let R be an E∞ -ring. A stack F in Sh(R, τ ) is said to be a τ -R-Tannakian
e for G a τ -R-Tannakian group stack. It is said to be agerbe if it is locally equivalent to BG
neutral τ -R-Tannakian gerbe if there moreover exists a morphism of stacks ∗ → F .
Let TGer(R, τ ) denote the ∞-category of neutral τ -R-Tannakian gerbe’s. We have a natural
inclusion TGer(R, τ ) ⊆ Ger(R, τ ).
We now state the Tannaka duality theorem for ∞-categories in the neutral setting. Notethat we have a weaker statement in positive case owing to the fact that the positive topology is
not subcanonical.
Theorem 9.3 (Neutral ∞-Tannaka duality). Let τ be a subcanonical topology. Then the map
Perf : TGer(R, τ )op → Tensrig
Ris fully faithful. Moreover, the adjunction Fib a Perf induces the following :1. Let R be an E∞ -ring. Then T is a finite R-Tannakian ∞-category if and only if it is of
the form Perf(G) for G a neutral finite R-Tannakian gerbe.
2. Let R be a connective E∞ -ring. Then T is a flat R-Tannakian ∞-category if it is of theform Perf(G) for G a neutral flat R-Tannakian gerbe.
3. Let R be a bounded connective E∞ -ring. If (T, ω1 ) and (T, ω2 ) are two pointed positive
R-Tannakian ∞-categories then there exists a positive cover R → Q such that
ω1 ⊗R Q → ω2 ⊗R Q
is an equivalence.
59

To prove the neutral ∞-Tannaka duality statement of Theorem 9.3 it suffices to combine
the neutralized statement of Theorem 7.14 with the demonstration that two fiber functors areequivalent after base change.
Proposition 9.4. Let R be an E∞ -ring and τ ∈ {≥ 0, f l, f in}. Given two τ -fiber functors ωand ν over R, there exists a τ -cover R → Q in E such that ω and ν are equivalent over Q.
Proof. Let (T, ω) and (T, ν) be two pointed R-Tannakian ∞-categories with respect to τ .
By Proposition 6.4 this amounts to showing that there exists a τ -cover R → Q such that
Hom(ω, ν)(Q) = Map(ωQ , νQ ) 6= ∅. It suffices to prove that Hom(bω , νb)(Q) 6= ∅. Let pb and qb
denote the right adjoints to ωb and νb respectively. We have equivalencesHom(bω , νb) ' Hom(bωG , νbG0 ) ' Map(bpG R, qbG0 R) ' Map(bνG0 pbG R, R)
where the first equivalence follows from Theorem 7.14 and the second equivalence from Corollary
6.3.5.18 of [L2] (see the proof of Proposition 8.1). Therefore, Hom(bω , νb)(Q) ' MapCAlgR (bν (B), Q)
and we set Q := νb(B) to consider the identity map. It remains to show that there exists a τ cover R → νb(B). Since νb is R-linear, νb(B) ⊗R • ' νb(B ⊗R •) so R → νb(B) is a τ -R-algebra
since B is a τ -Hopf R-algebra and ν is a τ -fiber functor.
We will now show that R → νb(B) is conservative, ie. given M ∈ ModR such that νb(B) ⊗R
M ' 0 then M ' 0. Let B 0 = B ⊗R R0 . By the projection formula, we have the following
statement :(*) For all R → R0 , the map νb(B 0 ) → νb(B) ⊗R R0 is an equivalence.
By Proposition 4.21, the map R0 → holimn∈∆ ((B 0 )⊗R (n+1) ) is an equivalence and since ν is a
τ -fiber functor we have the following statement :(**) The map R0 → holimn∈∆ (bν (B 0 )⊗R (n+1) ) is an equivalence.
`
Set R0 = SymR (M ) := p≥0 M ⊗R p /Σp and assume that νb(B) ⊗R M ' 0. Therefore
R0 ' holim(bν (B)⊗R n+1 ⊗ SymR (M ))
n∈∆
R
a
⊗n+1
' holim(b
ν (B)⊗ M ⊗R p /Σp )
n∈∆

R

p≥0

' R.
The first line is an equivalence in CAlgR and follows from (∗∗) and (∗). The second line is anequivalence in ModR and follows from the fact that the tensor product commutes with coproducts
and the forgetful functor CAlgR → ModR is conservative and commutes with limits (it is a rightadjoint). The third line is a result of the only non-zero term being p = 0 by assumption followedby (∗∗) applied to R. Thus M ' 0.
Proof of Theorem 9.3. The first part of the proof follows from Corollary 8.2. The remainder
follows directly from Theorem 7.14 and Proposition 9.4.
60

10Comparison with the classical theory

The following series of comparison results shows that the Tannaka duality theorem for ∞categories of Section 9 naturally generalises the classical theory. Let k be a field. In [Sa],
Saavedra defined the notion of a neutral Tannakian category over k. That is, a rigid abeliank-linear symmetric monoidal category T for which there exists an exact k-linear symmetricmonoidal functor T → Vectk (the fiber functor) taking values in the category of finite rank
projective k-modules.
The collection of ordinary fiber functors form a stack Fib(T ) over k in the faithfully flat
quasi-compact topology, denoted f f qc.
Q Recall that a finite family {Ai → A}i∈I of arrows in
Affk is an f f qc cover if the morphism i∈I Ai → A is faithfully flat (ie. exact and conservative).
Let Sh(k, f f qc) denote the ∞-category of stacks on the classical site (Affk , f f qc).
Proposition 10.1. Let k be a commutative ring and Hk its corresponding Eilenberg-Mac Lanering spectrum. The inclusioni : Sh(k, f f qc) → Sh(Hk, f l)of ∞-categories is fully faithful.
Proof. We will show that there exists a composition of fully faithful maps
Sh(k, f f qc) ,→ Shc (Hk, f l) ,→ Sh(Hk, f l)
where Shc (Hk, f l) is the ∞-category of stacks on the site of connective Hk-algebras. Firstly,
there exist adjunctions
i

j

τ≤0

(•)

CAlgk  - CAlgcHk  CAlgHk
cof ∞-categories (see Section 3). We then left Kan extend these adjunctions to obtain the
adjunctions
i!

j!

τ≤0

((•) )

RHom(CAlgk , S) ∗- RHom(CAlgcHk , S)  c-∗ RHom(CAlgHk , S)
This chain of adjunctions induces adjunctions on the subcategories of stacks by a property ofBousfield localisations since i preserves f f qc-covers. It follows from Lemma 2.2.4.1 of [TV2] thati! : Sh(k, f f qc) → Shc (Hk, f l) is fully faithful. A similar argument holds for j! : Shc (Hk, f l) ,→Sh(Hk, f l).
The right adjoint to the functor i of Proposition 10.1
τ≤0 : Sh(Hk, f l) → Sh(k, f f qc)
is explicitly given by τ≤0 (F )(k 0 ) := F (Hk 0 ) for a stack F in Sh(Hk, f l). If F is a neutral affine
gerbe in the sense of [D2] then i(F ) in Sh(Hk, f l) is a neutral flat Hk-Tannakian gerbe. Thus
61

neutral affine gerbes as defined in the classical Tannakian theory form a full subcategory ofneutral flat Hk-Tannakian gerbes in our sense.
For the following comparison results, we will need to introduce a finiteness condition onsome of our categories in order for the statements to make sense. The notion is that of finite
cohomological dimension of an arbitrary abelian category.
Definition 10.2. Let C be an abelian category. Then C is said to be of finite cohomologicaldimension if there exists n such that for every object x in C, the group Exti (y, x) = 0 for alli > n and y ∈ C.
Let T be an abelian category and C(T ) (resp. C b (T )) be the category of unbounded (resp.
bounded) complexes in T . We denote by LC b (T ) the ∞-category given by localising C b (T ) at
the set of quasi-isomorphisms. If T is a R-Tannakian ∞-category (resp. k-Tannakian category)
we will denote Fibτ (T ) the neutral R-Tannakian gerbe (resp. neutral affine gerbe) of fiber
functors on C with respect to the topology τ .
Proposition 10.3. Let k be a field and T a k-Tannakian category. Assume that T is of finite
cohomological dimension. Then the ∞-category LC b (T ) is a flat Hk-Tannakian ∞-category andthere exists an equivalencei(Fibf f qc (T )) → Fibf l (LC b (T ))of stacks in Sh(Hk, f l).
Proof. Since T is a k-Tannakian category there exists a natural k-tensor functor Modrig
k → T
between categories defined by M 7→ M ⊗ 1T . This exists since for any x, y ∈ T , the k-linear
structure on T induces an equivalence
HomModrig (M, HomT (x, y)) ' HomT (M ⊗ x, y).
k

rigUsing the equivalence LC b (Modrigk (Ab)) ' ModHk (Sp) (see Example 3.13) we obtain a mapbModrigHk → LC (T )of symmetric monoidal ∞-categories. This map commutes with colimits and induces the Hklinear structure on LC b (T ). Furthermore, the ∞-category LC b (T ) is stable by Example 3.3
which makes LC b (T ) a rigid Hk-tensor ∞-category. The fiber functor ω 0 : LC b (T ) → Modrig
Hk
induced from the fiber functor ω : T → ModrigonTisclearlyflat:
theforgetful
functorkω
b 0 : LC(T ) → LC(k) is conservative and creates a t-structure for which ωb 0 and its right adjoint
are t-exact.
For the second part of the proposition, consider the classical Tannaka duality theorem whichstates that Fibf f qc (T ) is naturally equivalent to a neutral gerbe H in the ffqc topology whiche for an affine group scheme G. We also have that Fibf l (LC b (T ))is locally equivalent to BG
e 0 for G0 a flat
is equivalent to a neutral flat Hk-Tannakian gerbe H 0 locally equivalent to BG
0e
Hk-Tannakian group stack. From Proposition 8.1, the gerbe H is locally equivalent to B(iG).
eeFrom the equivalence i(BG)' B(iG),
where i is the fully faithful functor of Proposition 10.1,
we obtain the desired result.
62

Let Tanfk f qc,! denote the category of k-Tannakian categories of finite cohomological dimension
with respect to the f f qc topology.
Corollary 10.4. Let k be a field. Then the functorl
LC b : Tanfk f qc,! → TanfHk

is an equivalence of ∞-categories.
Proof. This follows directly from Proposition 10.3, Proposition 10.1 and the flat ∞-Tannaka
duality Theorem 7.14.

11

Perfect complexes and schematization

Tannakian ∞-categories arise in a number of mathematical applications. We have seen in
Section 10 that the Tannaka duality statement for ∞-categories with the flat topology subsumes
the classical statement. It furthermore extends the 1-categorical duality by allowing the theorem
to be defined over arbitrary commutative rings as opposed to just fields. In this section we will
discuss the example of the ∞-category of perfect complexes on a topological space X. The∞-category of perfect complexes on X is a k-Tannakian ∞-category with respect to the positive
topology for k a field. When k is a field of characteristic zero, then the Tannakian dual of the∞-category of perfect complexes on X is precisely the schematization of X introduced by Toënin [T4].
In Section 7 we introduced the notion of a Tannakian ∞-category over a commutative ring
spectrum R. In fact one can define a whole family of Tannaka duality theorems based on the
topology chosen on the site of R-algebras. In Section 5 we concerned ourselves with three
examples called the flat, finite and positive topologies. The flat topology is the topology whichlinks us to the classical theory of Tannaka duality over fields for the ffqc topology (see Section 10
of loc.cit.) and the use of the finite topology is rather limited. The positive topology is however
something new and in this section we would like to explore some uses of the positive topology
and hence to the pointed positive Tannaka duality theorem for ∞-categories.
We will fix a field k throughout this section. In [T4], Toën introduced the notion of aschematic homotopy type. Given two pointed stacks F and G, let Map∗ (F, G) denote the(∞, 0)-category of pointed maps between them.
Definition 11.1. Let F and G be two pointed stacks in Sh(k, f f qc)∗ . A map F → G is said tobe a P-equivalence if the induced map
Perf(F ) → Perf(G)is an equivalence of ∞-categories. A pointed stack F in Sh(k, f f qc)∗ is said to be P-local if for
any P -equivalence H1 → H2 , the induced map
Map∗ (H2 , F ) → Map∗ (H1 , F )is an equivalence.
63Lemma 11.2. Let τ ∈ {≥ 0, f l, f in} and T be a pointed k-Tannakian ∞-category with respecteto τ . Then BFib
∗ (T ) is P -local.
Proof. Let H1 → H2 be a P -equivalence. From the pointed adjunction of Lemma 7.8 we have achain of equivalenceseeMap∗ (H2 , BFib
∗ (T )) ' Map∗ (T, Perf(H2 )) ' Map∗ (T, Perf(H1 )) ' Map∗ (H1 , BFib∗ (T ))of (∞, 0)-categories.
Let A ∈ cCAlgk be a cosimplicial k-algebra in the category Ab of abelian groups. We define
the following prestackSpec A : CAlgk → S
B 7→ Map(A, B)
where Map(A, B)n := Hom(An , B). This prestack can be shown to be a stack for the ffqctopology and thus defines a natural functor
Spec : cCAlgk → Sh(k, f f qc)
between ∞-categories. We are interested in objects which lie in the essential image of the Specfunctor.
Definition 11.3. Let k be a commutative ring. An affine stack over k is a stack in Sh(k, f f qc)
which is equivalent to an object of the form Spec A for A a cosimplicial k-algebra.
Let s : ∗ → F be a pointed stack in Sh(k, f f qc). We define the prestack Ω∗ F of loops at sby the formulaΩ∗ F : CAlgk → S
x 7→ Ωs(∗) F (x)where Ωs(∗) F (x) is the subsimplicial set of Map(∆1 , F (x)) which sends the endpoints {0, 1} of∆1 to s(∗).
Definition 11.4. Let k be a field. A pointed stack F on the site (Affk , f f qc) is said to be apointed schematic homotopy type over k if it is P -local, connected and Ω∗ F is an affine stackover k.
Let SHTk denote the full subcategory of Sh(k, f f qc) spanned by the pointed schematic
homotopy types. Every pointed connected affine stack is naturally a pointed schematic homotopy
type. Furthermore, a stack F is a pointed schematic homotopy type if and only if F is a pointed
connected stack in Sh(k, f f qc) such that the sheaf π1 (F, x) is represented by an affine groupscheme and for any i > 1, the sheaf πi (F, x) is represented by a unipotent affine group scheme(see Section 3.2 of [T4]).
Let k be a field and F a pointed schematic homotopy type over k. We will say that F isof finite cohomological dimension if the abelian category Perf(i(F ))♥ is of finite cohomologicaldimension.
64

Proposition 11.5. Let k be a field and F a pointed schematic homotopy type over k. Assume
that F is of finite cohomological dimension. Then Perf∗ (i(F )) is a pointed positive Hk-Tannakian∞-category.
Proof. The stack iF is clearly a positive Hk-Tannakian group stack in TGpf l (Hk). The result
then follows from the positive ∞-Tannaka duality Theorem 7.14.
The schematization is an important schematic homotopy type which we define through thefollowing universal property.
Definition 11.6. Let k be a field and X a fibrant simplicial set considered as a constant prestack.
Then the schematization of X over k is a schematic homotopy type Xk∼ over k together with amap f : X → Xk∼ satisfying the following universal property : for any stack F in Sh(k, f f qc),
composition with f induces an equivalence
MapSh(k,f f qc) (Xk∼ , F ) → MapSh(k,f f qc) (X, F )of (∞, 0)-categories.
The main result in the theory of schematizations is Theorem 3.3.4 of [T4] which states that fora pointed, connected simplicial set (X, x), a schematization always exists. The schematizationof a pointed space admits a number of fundamental properties which follow from Definition 11.6
together with this existence result. Firstly, the affine group scheme π1 (Xk∼ , x) is naturally
isomorphic to the pro-algebraic completion of the discrete group π1 (X, k) (thought of as aconstant group scheme over k). When X is finite and simply connected, then for i > 1, thegroup scheme πi (Xk∼ , x) is naturally isomorphic to the pro-unipotent completion of the discretegroup πi (X, x). Finally, if V is a local system of finite dimensional k-vector spaces on X, thenV corresponds to a linear representation of the affine group scheme π1 (Xk∼ , x). This induces a
local system V on the schematization Xk∼ (a sheaf of abelian groups V on (Affk , f f qc) together
with an action of π1 (Xk∼ ) on V) such that the map X → Xk∼ furnishes an isomorphism
H ∗ (Xk∼ , V) → H ∗ (X, V )in cohomology with local coefficients. See [T4] for further discussion.
Let k be a commutative ring. The category C(k) of (unbounded) complexes of k-modules
admits a cofibrantly generated model structure where the fibrations are degree-wise surjective
morphisms and the equivalences are the quasi-isomorphisms (those maps inducing isomorphisms
on homology groups) (see Theorem 2.3.11 of [H1]). Then for a topological space X, the category
C(X, k) of complexes of presheaves of k-modules on X is a C(k)-enriched model category. HereC(k) is endowed with its usual monoidal structure and C(X, k) is naturally tensored over C(k)since the category of presheaves of k-modules is naturally tensored over the category of k-modules(if F is a presheaf of k-modules on X and M is a k-module then M ⊗ F is defined to be thepresheaf U 7→ M ⊗ F (U )). The model category structure on C(X, k) arises from a more general
result of a model structure on the category C(A) of complexes in any Grothendieck category A(see [H2]).
We define Perf(X, k) to be the subcategory of LC(X, k) consisting of complexes of presheavesof k-modules which have locally constant cohomology sheaves. The objects of Perf(X, k) are
65

complexes of presheaves of k-modules which are, locally on X, quasi-isomorphic to a constantcomplex of presheaves associated with a bounded complex of finite rank projective k-modules.
Let x : ∗ → X be a point of X and define Perf(k) := Perf(∗, k). There exists a Hk-linear structure on the symmetric monoidal ∞-category Perf(X, k) from the symmetric monoidal functor
Perf(k) → Perf(X, k) and a Hk-linear symmetric monoidal functor
ωx : Perf(X, k) → Perf(k)induced by the point x.
Theorem 11.7. Let k be a field and X a finite CW complex. Then (Perf(X, k), ωx ) is a pointed
Hk-Tannakian ∞-category with respect to the positive topology. When k is a field of characteristic zero, the dual of this pointed positive Hk-Tannakian ∞-category is the schematization ofX.
Proof. We need to show that Perf(X, k) is a rigid Hk-tensor ∞-category and ωx a positive fiber
functor. The first part is clear by the above discussion and by Example 3.3 : Perf(X, k) is stablesince it is a stable subcategory of the derived ∞-category of an additive category. Secondly,ωbx : Ind(Perf(X, k)) → Mod(k) is the conservative forgetful functor and clearly creates a tstructure using the canonical t-structure on Perf(k) whilst preserving finite limits and colimits.
For the second part of the proof, let k be of characteristic zero and consider the fully faithfulclassifying prestack functor
e : Gp(k, f f qc) → Sh(k, f f qc)∗ .
Be is P -local. Let LP (Gp(k, f f qc))
We will say that a group stack G in Gp(k, f f qc) is P -local if BG
be the localization of Gp(k, f f qc) with respect to the P -equivalences. We can identify this ∞category with the full subcategory of Gp(k, f f qc) spanned by the P -local objects.
e induces an equivalence
From Theorem 3.14 of [KPT2], the functor B
LP (Gp(k, f f qc)) → SHTkof ∞-categories. Thus by Corollary 3.19 of loc. cit. there exists a Hopf k-algebra B P ine Spec B P is an equivalence inthe category Ab of abelian groups such that the map Xk∼ → B
Sh(k, f f qc)∗ . Therefore, there exists an equivalencee Spec B P
iXk∼ → iB
in Sh(Hk, f l)∗ where i is the fully faithful inclusion of Proposition 10.1.
Now Fib∗ (Perf(X, k)) = End⊗ (ωx ) = Spec B 0 for the Hopf Hk-algebra B 0 ' ω
cx ◦ pbx (k). It
0Peremains to show that B is equivalent to iB Spec B . However, this is clear from the explicit
form of B P [T4].

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