Theory and Applications of Categories, Vol. 30, No. 23, 2015, pp. 808–835.

REFLEXIVITY AND DUALIZABILITY IN CATEGORIFIED LINEAR ALGEBRA

MARTIN BRANDENBURG, ALEXANDRU CHIRVASITU, AND THEO JOHNSON-FREYD

Abstract. The “linear dual” of a cocomplete linear category C is the category of all cocontinuous linear functors C → Vect. We study the questions of when a cocomplete linear category is reflexive (equivalent to its double dual) or dualizable (the pairing with its dual comes with a corresponding copairing). Our main results are that the category of comodules for a countable-dimensional coassociative coalgebra is always reflexive, but (without any dimension hypothesis) dualizable if and only if it has enough projectives, which rarely happens. Along the way, we prove that the category Qcoh(X) of quasi-coherent sheaves on a stack X is not dualizable if X is the classifying stack of a semisimple algebraic group in positive characteristic or if X is a scheme containing a closed projective subscheme of positive dimension, but is dualizable if X is the quotient of an affine scheme by a virtually linearly reductive group. Finally we prove tensoriality (a type of Tannakian duality) for affine ind-schemes with countable indexing poset.

1. Introduction

Fix a field 𝕂. For cocomplete 𝕂-linear categories C and D, let Hom(C, D) = Hom_{c,𝕂}(C, D) denote the cocomplete 𝕂-linear category of cocontinuous 𝕂-linear functors and natural transformations from C to D, and let C ⊠ D = C ⊠_{c,𝕂} D denote the universal cocomplete 𝕂-linear category receiving a functor C × D → C ⊠ D which is cocontinuous and 𝕂-linear in each variable (while holding the other variable fixed). We will denote the image of (C, D) ∈ C×D under this functor by C⊠D ∈ C⊠D. If C and D are both locally presentable (see Definition 2.1), then Hom(C, D) and C ⊠ D both exist as locally presentable 𝕂linear categories, and Hom and ⊠ satisfy a hom-tensor adjunction. The unit for ⊠ is Vect = Vect_{𝕂}; the image of V ⊠ C under the equivalence Vect ⊠ C ≃ C deserves to be called V ⊗ C.

Thus the bicategory of locally presentable 𝕂-linear categories provides one possible categorification of linear algebra. It includes as a full sub-bicategory the Morita bicategory Alg of associative algebras, bimodules, and intertwiners; the inclusion sends an algebra A to the category M_{A} of right A-modules. But locally presentable 𝕂-linear cat-

We would like to thank the anonymous referee for their detailed and valuable comments. TJF is supported by the NSF grant DMS-1304054.

Received by the editors 2014-09-28 and, in revised form, 2015-06-16. Transmitted by Steve Lack. Published on 2015-06-22. 2010 Mathematics Subject Classification: 18A30, 18A35, 18A40, 14A15, 14R20. Key words and phrases: locally presentable, dualizable, cocomplete, cocontinuous. \begin{matrix}c \\ ◯ \end{matrix} Martin Brandenburg, Alexandru Chirvasitu, and Theo Johnson-Freyd, 2015. Permission to copy

for private use granted.

REFLEXIVITY AND DUALIZABILITY IN CATEGORIFIED LINEAR ALGEBRA 809

egories are more general: among them are, for any scheme or any Artin stack X over 𝕂, the category Qcoh(X) of quasi-coherent sheaves of O_{X}-modules, and also for any coassociative coalgebra C over 𝕂, the category M^{C} of right C-comodules.

In any generalization of linear algebra, it is interesting to ask what the “finite-dimen-sional” objects are. Finite-dimensionality of vector spaces is closely related to good behavior under taking dual vector spaces. In the context of cocomplete 𝕂-linear categories, the “linear dual” of C is C^{∗} = Hom(C, Vect). There are canonical functors

C ⊠ C^{∗} → End(C) : C ⊠ F ↦ (D ↦ F (C) ⊗ D),
C → (C^{∗})^{∗} : C ↦ (F ↦ F (C)).

Each of these functors corresponds to a possible generalization of finite-dimensionality:

1.1. Definition. A locally presentable 𝕂-linear category C is called dualizable if the canonical functor C ⊠ C^{∗} → End(C) is an equivalence. A locally presentable 𝕂-linear category C is called reflexive if the canonical functor C → (C^{∗})^{∗} is an equivalence.

Dualizability implies reflexivity, but the converse is generally false. Dualizability is particularly important in light of [BD95, Lur09]. For example, a well-known corollary of the Eilenberg–Watts theorem, which asserts that Hom(M_{A}, M_{B}) ≃ M_{A^{op}⊗B} (see Example 2.14), answers the question of dualizability in the affirmative for objects of the Morita bicategory Alg:

1.2. Theorem. [Folklore] For any associative algebra A, the category M_{A} of right Amodules is dualizable. The dual is (M_{A})^{∗} ≃ M_{A^{op}}.

There is no Eilenberg–Watts theorem for coassociative coalgebras. Thus the questions of reflexivity and dualizability are more subtle. Our main results on dualizability are:

1.3. Theorem. Let C be a coassociative coalgebra. Then the category M^{C} of right Ccomodules is dualizable if and only if it has enough projectives.

Such a coalgebra is called right semiperfect in [Lin77] (generalizing the notion from [Bas60]).

1.4. Theorem. Let X be a 𝕂-scheme. If X has a closed projective subscheme of positive dimension, then Qcoh(X) is not dualizable.

In order to state the next result, recall ([Don96]) that a linear algebraic group is linearly reductive if its category of representations is semisimple. Examples include the classical groups GL(n), SL(n), etc., in characteristic 0. It is virtually linearly reductive if it has a linearly reductive normal algebraic subgroup such that the quotient is a finite group scheme. Finite groups in characteristic dividing the order of the group are examples of virtually linearly reductive groups that are not linearly reductive. The additive group 𝔾_{a} = Spec(𝕂[x]) is not virtually linearly reductive (see Example 3.7).

810 MARTIN BRANDENBURG, ALEXANDRU CHIRVASITU, AND THEO JOHNSON-FREYD

1.5. Theorem. Let X be an affine scheme over 𝕂 and G a virtually linearly reductive group over 𝕂 acting on X. Let [X/G] denote the corresponding quotient stack. Then Qcoh([X/G]) is dualizable.

Theorems 1.3, 1.4, and 1.5 are proved in Section 3. (Section 2 reviews some of the theory of locally presentable categories.) In Theorem 1.5, the condition on G is important. Indeed, according to [Don96], corrected in [Don98], Qcoh([Spec(𝕂)/G]) ≃ Rep(G) ≃ M^{O(G)} contains a non-zero projective if and only if G is virtually linearly reductive, and moreover if any non-zero injective fails to be projective, then there are no non-zero projectives. In particular, with Theorem 1.3 this implies that Qcoh([Spec(𝕂)/G]) is not dualizable for G a semisimple group in positive characteristic, nor is it dualizable for G = 𝔾_{a}. Such nondualizability results are in stark contrast with [BZFN10], where it is shown that for many stacks of geometric interest, the corresponding derived category of quasi-coherent sheaves is dualizable (for the derived version of ⊠).

We address reflexivity in Section 4. Our main result there is:

1.6. Theorem. Suppose that A = (⋯ ↠ A_{2} ↠ A_{1} ↠ A_{0}) is an ℕ-indexed projective system of associative 𝕂-algebras. Let C denote the category of injective systems (M_{0} ↪ M_{1} ↪ M_{2} ↪ …) where each M_{i} is a right A_{i}-module and each inclusion M_{i} ↪ M_{i+1} identifies M_{i} as the maximal A_{i+1}-submodule on which the A_{i+1}-action factors through A_{i}. Let B be any associative 𝕂-algebra. Then the “double dual” functor

C ⊠ M_{B} → Hom(C^{∗}, M_{B}),

which maps X ⊠ M to (F ↦ F (X) ⊗ M), is an equivalence.

An immediate corollary establishes reflexivity for some categories that are not dualizable (as usual, the dimension of a coalgebra refers to the underlying vector space):

1.7. Corollary. Let C be a countable-dimensional coassociative coalgebra. Then M^{C} is reflexive.

Proof. By [Swe69, Theorem 2.2.1], there exists an increasing sequence of finite-dimen-sional subcoalgebras C_{0} ⊆ C_{1} ⊆ … with C = ⋃ C_{i}. Let A_{i} = C_{i}^{∗} and C as in Theorem 1.6. Then C ≃ M^{C} by Example 2.15. Theorem 1.6 with B = 𝕂 completes the proof.

We do not know if countable dimensionality in Corollary 1.7 can be dropped. Note that the coalgebra C = 𝕂[x] with Δ(x^{n}) = ∑_{i=0}^{n} x^{i} ⊗ x^{n−i} provides an example of a countable-dimensional coalgebra for which, by Theorem 1.3, M^{C} is not dualizable. (See Example 3.7 for an elaboration.) In particular, Theorem 1.3 and Corollary 1.7 together illustrate that dualizability and reflexivity are very different categorifications of finitedimensionality.

The main step in the proof of Theorem 1.6 is a description given in Proposition 4.8 of Hom(C^{∗}, M_{B}) in terms of certain modules over certain algebras. We end by applying this description to prove the following result, which can be thought of as a symmetric monoidal version of Theorem 1.6:

1.8. Theorem. Affine ind-schemes indexed by countable posets are tensorial.

Tensoriality for schemes was introduced in [Bra11] and studied for quasi-compact quasi-separated schemes and Artin stacks in [BC14, HR14]. We review the definition, along with its natural extension to ind-schemes, in Definition 4.10. Roughly speaking, a geometric object is tensorial if it can be recovered functorially in a “Tannakian” way from its symmetric monoidal category of quasi-coherent sheaves.

2. Recollections on locally presentable categories

We begin by recalling a few basic facts about locally presentable 𝕂-linear categories. The primary reference on locally presentable categories is [AR94]. Many of the results in this section appear as early as [GU71].

2.1. Definition. For an infinite cardinal λ, a partially ordered set I is λ-directed if any subset of I of cardinality strictly less than λ has an upper bound. A λ-directed colimit is a colimit of a diagram indexed by a λ-directed partially ordered set. An object X in a category C is λ-presentable (also called “λ-compact”) if hom(X, −) commutes with λdirected colimits. A colimit is λ-small if its indexing diagram has strictly fewer than λ arrows (including identity arrows).

A set of objects Γ in C is strongly generating if the closure of Γ in C under small colimits is all of C.

A 𝕂-linear locally small category C is locally presentable if it is cocomplete (i.e. has all small colimits) and admits a strongly generating set consisting entirely of objects that are λ-presentable for some λ. We may imagine C as a “categorified vector space over 𝕂”, since coproducts are categorified sums. Notice, however, that we don’t have any additive inverses, and that sums in a vector space have no universal property and therefore are additional structure, in contrast to colimits in a category.

Locally presentable 𝕂-linear categories are the objects of a bicategory Pres_{𝕂} whose 1morphisms are cocontinuous 𝕂-linear functors and whose 2-morphisms are natural transformations. The special adjoint functor theorem implies that every cocontinuous functor C → D with C locally presentable has a right adjoint (in the bicategory of all functors).

2.2. Remark. A set of objects Γ in C is generating if ∏_{X∈Γ} hom(X, −) is faithful. Strongly generating sets are generating, and the converse holds if the category in question is abelian, since then all epimorphisms are coequalizers. But the converse generally fails: the category of two-term filtered vector spaces (V_{0} ↪ V_{1}) is locally presentable, but ({0} ↪ 𝕂) is generating but not strongly generating.

For general categories, our definition of “strongly generating” is stronger than the usual one, but the two definitions are the same for locally presentable categories [Kel05, Propostion 3.40]. Our definition of local presentability is equivalent to the usual one by [AR94, Theorem 1.20].

Note that the forgetful functor Vect → Set preserves λ-directed colimits, so there is no difference between Set- and Vect-valued homs for the purposes of defining notions like λ-presentability. A λ-presentable object X ∈ C has “size less than λ”; even “smaller” than this are the objects X ∈ C such that hom(X, −) : C → Vect preserves all colimits, and not just the λ-directed ones. If C is abelian, such objects are precisely the compact projective ones: in a general category, an object is compact if it is ℵ_{0}-presentable; in an abelian category, an object X is projective if hom(X, −) preserves coequalizers; all colimits are compositions of finite direct sums, ℵ_{0}-filtered colimits and coequalizers. As such, for C an arbitrary locally presentable 𝕂-linear category, we will call an object X ∈ C compact projective if hom(X, −) : C → Vect is cocontinuous. Compact projectivity is called “small projectivity” in [Kel05].

The fundamental theorem of locally presentable categories is:

2.3. Proposition. Let C be a locally presentable 𝕂-linear category with a strongly generating set consisting of λ-presentable objects, for λ a regular infinite cardinal. Then the full subcategory C_{<λ} ⊆ C of λ-presentable objects is essentially small and closed in C under λ-small colimits. Moreover,

C 𝕂-linear functors C_{<λ}^{op} → Vect preserving λsmall limits “Yoneda” functor

Y ↦ restriction of hom(−, Y) to C_{<λ}

is an equivalence, as is

cocontinuous 𝕂-linear functors C → D 𝕂-linear functors C_{<λ} → D preserving λ-small colimits

restriction to C_{<λ}

for any cocomplete 𝕂-linear category D.

Recall that a cardinal λ is regular if it is not the union of strictly fewer than λ sets, each of which has cardinality strictly less than λ. See [AR94, Exercise 1.b] for why one may assume λ to be regular in Definition 2.1.

Proof. The statement consists of Theorems 1.20 and 1.46 and Proposition 1.45 from [AR94] (an error in the proof was corrected in [AHR99]). To incorporate linearity is straightforward: one can systematically develop a 𝕂-linear theory exactly parallel to the usual theory with no surprises, making only a few changes as necessary; see [Kel82, Kel05]. (Indeed, one can systematically develop a version for categories enriched in any closed symmetric monoidal locally presentable category, although at that level of generality one must use weighted colimits and not just colimits.)

An important corollary of Proposition 2.3 is that Hom_{Pres_{𝕂}}(C, D) is locally small, being equivalent to a full subcategory of the category of functors from a small category to a locally small category.

We will use the following variant of Proposition 2.3:

2.4. Proposition. [Kel05, Theorem 5.26] Let C be a locally presentable 𝕂-linear category with a strongly generating set Γ consisting of compact projective objects. Then

C 𝕂-linear functors Γ^{op} → Vect “Yoneda” functor

Y ↦ restriction of hom(−, Y) to Γ

is an equivalence, as is

cocontinuous 𝕂-linear functors C → D 𝕂-linear functors Γ → D

restriction to Γ

for any cocomplete 𝕂-linear category D.

Conversely, note that for any small 𝕂-linear category Γ, in the category Fun_{𝕂}(Γ^{op}, Vect) of all 𝕂-linear functors, every representable functor hom(−, X) for X ∈ Γ is compact projective, and the Yoneda Lemma implies that the representable functors are a strongly generating set. Thus Proposition 2.4 identifies categories of the form Fun_{𝕂}(Γ^{op}, Vect) as precisely those 𝕂-linear categories strongly generated by a set of compact projectives.

2.5. Lemma. [Bir84],[AR94, Exercise 2.n],[CJF13, Proposition 2.1.11] The bicategory Pres_{𝕂} of locally presentable 𝕂-linear categories has all small 2-limits and all small 2colimits:

• To compute a 2-limit in Pres_{𝕂}, simply compute the same 2-limit in the bicategory Cat of categories, ignoring that the arrows happen to be left adjoints.

• To compute a 2-colimit in Pres_{𝕂}, replace every 1-morphism by its right adjoint — this is a contravariant bifunctor Pres_{𝕂} → Cat — and compute the corresponding 2-limit in Cat, ignoring that the arrows happen to be right adjoints.

What we call “2-limits” and “2-colimits” are also called “bilimits” and “bicolimits” in the literature. We will often drop the prefixes “2-,” calling them just “limits” and “colimits.”

2.6. Remark. We will be particularly interested in (2-)limits and (2-)colimits indexed by partially ordered sets I. Let C : I → Pres_{𝕂} be an I-indexed diagram. Thus it consists of categories C_{i} ∈ Pres_{𝕂} for each i ∈ I, 𝕂-linear cocontinuous functors F_{i<j} : C_{i} → C_{j} for each i < j, and natural isomorphisms F_{i<k} ≅ F_{j<k} ∘ F_{i<j} which are compatible for 4-tuples in I.

Consider first the limit \underleftarrow{lim}_{i∈I} C_{i} in Pres_{𝕂}. By Lemma 2.5, it is nothing but the corresponding limit in Cat. Thus an object of \underleftarrow{lim}_{i∈I} C_{i} consists of an object X_{i} ∈ C_{i} for each i ∈ I and an isomorphism X_{j} ≅ F_{i<j}(X_{i}) for each i < j such that these isomorphisms commute for triples i < j < k. (Such commutativity uses the isomorphisms F_{i<k} ≅ F_{j<k} ∘ F_{i<j}.) A morphism \{X_{i}\}_{i∈I} → \{Y_{i}\}_{i∈I} in \underleftarrow{lim}_{i∈I} C_{i} consists of a morphism X_{i} → Y_{i} in C_{i} for each i, with the ith morphism mapping under F_{i<j} to the jth morphism for i < j.

Let F_{i<j}^{R} : C_{j} → C_{i} denote the right adjoint to F_{i<j}. Under the adjunction, the isomorphism F_{i<j}(X_{i}) →^{∼} X_{j} in C_{j} corresponds to a map X_{i} → F_{i<j}^{R}(X_{j}) which is isomorphic to the unit-of-the-adjunction map X_{i} → F_{i<j}^{R}(F_{i<j}(X_{i})). It therefore realizes F_{i<j}^{R}(X_{j}) as the object in the image of F_{i<j}^{R} which is universal (among morphisms in the image of F_{i<j}^{R}) for receiving a morphism from X_{i}. In many cases of interest, the map X_{i} → F_{i<j}^{R}(X_{j}) is an epimorphism. In general, the object \{X_{i}\}_{i∈I} ∈ \underleftarrow{lim}_{i} C_{i} is a “formal limit” \underleftarrow{lim} X_{i} along units of adjunctions.

Consider second the colimit \underrightarrow{lim}_{i∈I} C_{i} in Pres_{𝕂}. By Lemma 2.5 this is the limit in Cat of the I^{op}-indexed diagram i ↦ C_{i} along the functors F_{i<j}^{R} : C_{j} → C_{i}. An object of \underrightarrow{lim}_{i∈I} C_{i} is therefore an object X_{i} ∈ C_{i} for each i along with compatible isomorphisms X_{i} ≅ F_{i<j}^{R}(X_{j}). Under the adjunction, these isomorphisms correspond to maps F_{i<j}(X_{i}) → X_{j} realizing F_{i<j}(X_{i}) as the universal object in the image of F_{i<j} that maps to X_{j}; in many cases of interest, these maps are monomorphisms. In general, the object \{X_{i}\}_{i∈I} ∈ \underrightarrow{lim}_{i} C_{i} can be thought of as a “formal colimit” \underrightarrow{lim} X_{i} along counits of adjunctions.

One corollary of Lemma 2.5 is that if I is just a set then products and coproducts in Pres_{𝕂} indexed by I agree (and agree with the product, but not the coproduct, of underlying categories). It thus makes sense to call this (co)product the direct sum ⨁_{i∈I} C_{i}.

Just like vector spaces, in addition to a direct sum, locally presentable 𝕂-linear categories also admit a tensor product:

2.7. Lemma. [Kel05, Section 6.5],[AR94, Exercise 1.l]

1. For C, D ∈ Pres_{𝕂}, the category Hom(C, D) is also locally presentable and 𝕂-linear.

2. For each C, the bifunctor Hom(C, −) : Pres_{𝕂} → Pres_{𝕂} has a left adjoint (−) ⊠ C, making Pres_{𝕂} into a closed symmetric monoidal bicategory.

3. The tensor product C ⊠D satisfies the following universal property: For any cocomplete 𝕂-linear category E, cocontinuous functors C ⊠ D → E correspond to functors C × D → E that are cocontinuous and 𝕂-linear in each variable, holding the other variable fixed.

2.8. Remark. One can present C ⊠ D from part (3) of Lemma 2.7 in a number of ways. For example, it is (up to canonical equivalence) the category of continuous functors C^{op} → D. Indeed, choose any regular cardinal λ for which C admits a strongly generating set consisting of λ-presentable objects. The first part of Proposition 2.3 identifies C^{op} with the category of functors C_{<λ} → Vect^{op} preserving λ-small limits, and the second part identifies this with Hom(C, Vect^{op}), so that together two parts of Proposition 2.3 provide a canonical equivalence C ≃ (Hom(C, Vect^{op}))^{op} identifying C ∈ C with hom(−, C). (As in the introduction, we write Hom(C, Vect^{op}) for the category of cocontinuous 𝕂-linear functors C → Vect^{op}, even though Vect^{op} is not locally presentable.) Then we have equivalences

C ⊠D ≃ (Hom(C ⊠D, Vect^{op}))^{op} ≃ (Hom(C, Hom(D, Vect^{op})))^{op} ≃ (Hom(C, D^{op}))^{op},

from which the claim follows.

2.9. Definition. A symmetric monoidal locally presentable 𝕂-linear category is by definition a symmetric monoidal category S = (C, ⊗) whose underlying category C is locally presentable 𝕂-linear such that the tensor product ⊗ : C × C → C is cocontinuous and 𝕂-linear in each variable, so that it extends to a 1-morphism C ⊠ C → C. In fact, S is a symmetric pseudomonoid in the symmetric monoidal bicategory (Pres_{𝕂}, ⊠). Therefore, we may imagine S as a “categorified commutative algebra over 𝕂.”

We denote by Pres_{⊗,𝕂} the bicategory of symmetric monoidal locally presentable 𝕂linear categories, where for two such objects C, D the category Hom_{⊗,c,𝕂}(C, D) consists of cocontinuous strong symmetric monoidal 𝕂-linear functors and symmetric monoidal natural transformations. We will often simply write Hom_{⊗} in place of Hom_{⊗,c,𝕂}, as 𝕂 will often be implicit and we will never use non-cocontinuous symmetric monoidal functors.

Let S = (C, ⊗) ∈ Pres_{⊗,𝕂}. An S-module in Pres_{𝕂} is a locally presentable 𝕂-linear category D together with an action C ⊠ D → D, denoted by X ⊠ M ↦ X ▷ M, as well as unit and associativity data making the appropriate triangles and pentagons commute.

2.10. Remark. Module categories are in particular enriched: For any object M ∈ D, the functor (−) ▷ M : C → D is cocontinuous, hence has a right adjoint hom(M, −)_{S} : D → C. It is straightforward to check that the associativity and unit data make hom(−, −)_{S} into an S-enrichment of D. See for example [GP97, Theorem 3.7].

2.11. Remark. For a field extension 𝕂 ↪ 𝕃, a locally presentable 𝕃-linear category is precisely a Vect_{𝕃}-module in Pres_{𝕂}. It follows that we have a 2-functor

• ⊠ Vect_{𝕃} : Pres_{𝕂} → Pres_{𝕃}

(the tensor product is in Pres_{𝕂}) which preserves ⊠ in the sense that it lifts to a symmetric monoidal 2-functor.

The following enriched version of compact projectivity will be useful in the course of the proof of Theorem 1.5.

2.12. Definition. Let S = (C, ⊗) ∈ Pres_{⊗,𝕂} and (D, ▷) be an S-module in the sense of Definition 2.9. An object M ∈ D is compact projective over S if the enriched hom functor hom(M, −)_{S} : D → C is cocontinuous. A set Γ of objects in D is strongly generating over S if D is the closure of Γ under S-weighted colimits.

2.13. Proposition. With the notations above, assume that C admits a strongly generating set of compact projective (over Vect) objects, and that D admits a strongly generating set over S of objects which are all compact projective over S. Then D is strongly generated by compact projective (over Vect) objects.

Proof. Let {c_{i}} be a strongly generating set of compact projective objects in C and {d_{j}} a strongly generating over S set of compact projective over S objects in D. We have

hom_{D}(c_{i} ▷ d_{j}, −) ≅ hom_{C}(c_{i}, hom_{D}(d_{j}, −)_{S}) : D → Vect.

The right hand side is cocontinuous by assumption, so the left hand side is as well. Hence, c_{i} ▷ d_{j} are compact projective. The fact that they strongly generate follows from the fact that c_{i} and d_{j} strongly generate and an unpacking of the notion of weighted colimit.

We may now introduce our main examples:

2.14. Example. Let A be an associative algebra over 𝕂. Then the category M_{A} of all right A-modules is a locally presentable 𝕂-linear category. Since every module has a presentation, the strongly generating set Γ may be taken to consist of the rank-one free module A, which is compact projective. Proposition 2.4 implies that for any cocomplete 𝕂-linear category D, cocontinuous 𝕂-linear functors M_{A} → D are equivalent to 𝕂-linear functors Γ → D. It then follows from Lemma 2.7 that there is a canonical equivalence of categories Hom(M_{A}, D) ≃ _{A}M ⊠ D for any D ∈ Pres_{𝕂}, where _{A}M ≅ M_{A^{op}} is the category of left A-modules. The Eilenberg–Watts theorem and its corollary Theorem 1.2 follow.

2.15. Example. Let C be a coassociative coalgebra over 𝕂. We will describe in some detail the category M^{C} of right C-comodules, showing in particular that it is locally presentable.

By the so-called fundamental theorem of coalgebras ([Swe69, Theorem 2.2.1]), C ≃ \underrightarrow{lim}_{i∈I} C_{i}, where \{C_{i}\}_{i∈I} is the partially ordered set of finite-dimensional sub-coalgebras of C_{i}. We claim first that M^{C_{i}}, the category of right C_{i}-comodules, is locally presentable for each i, and second that

M^{C} ≃ \underrightarrow{lim}_{i∈I} M^{C_{i}},

where the colimit is computed in Pres_{𝕂} along the scalar corestriction functors M^{C_{i}} → M^{C_{j}} for the inclusions C_{i} ≤ C_{j}.

To see the first claim, give the linear dual C_{i}^{∗} the algebra structure (α · β)(c) = ∑ α(c_{(2)})⊗β(c_{(1)}), where the comultiplication on C_{i} in Sweedler’s notation is c ↦ c_{(1)}⊗c_{(2)}. Then the category M^{C_{i}} of right C_{i}-comodules is canonically equivalent to the category M_{C_{i}^{∗}} of right C_{i}^{∗}-modules, hence locally presentable by Example 2.14.

To see the second claim, recall that the fundamental theorem of coalgebras moreover asserts that every right C-comodule X is canonically a colimit X = \underrightarrow{lim}_{i∈I} X_{i} where X_{i} is the largest subcomodule for which the coaction X_{i} → X_{i} ⊗ C factors through X_{i} ⊗ C_{i}, and conversely the union of any filtered system of C_{i}-comodules like this is a C-comodule. Moreover, every C-comodule morphism f : X → Y restricts to a system of compatible C_{i}-comodule morphisms f_{i} : X_{i} → Y_{i}, and any such compatible system {f_{i} : X_{i} → Y_{i}}_{i∈I} defines a C-comodule morphism f.

Thus M^{C} is equivalent to the category of I-indexed systems of vector spaces {X_{i}}_{i∈I} such that for each i, X_{i} is a C_{i}-comodule, and for i ≤ j, there maps X_{i} ↪ X_{j} (compatible for i ≤ j ≤ k) realizing X_{i} as the largest C_{i}-sub-comodule of X_{j}. Said another way, X_{i}, scalar corestricted to become a C_{j}-comodule, is the universal object in the image of scalar corestriction M^{C_{i}} → M^{C_{j}} that maps to X_{j}. But Remark 2.6 identifies such systems \{X_{i}\}_{i∈I} as the objects of \underrightarrow{lim} M^{C_{i}}, and systems \{f_{i} : X_{i} → Y_{i}\}_{i∈I} as the morphisms of \underrightarrow{lim} M^{C_{i}}.

2.16. Example. Let X be a scheme. We can present X as a colimit of open affine subschemes: X = \underrightarrow{lim}_{i∈I} Spec(A_{i}). Let Qcoh(X) denote the category of quasi-coherent sheaves of O_{X}-modules. We have

Qcoh(X) ≃ \underleftarrow{lim}_{i∈I^{op}} Qcoh(Spec(A_{i})) ≃ \underleftarrow{lim}_{i∈I^{op}} _{A_{i}}M,

since a quasicoherent sheaf is a module on each open in an affine open cover along with compatibility data on overlaps.

Even though a priori this limit is computed in Cat, the pullback functors_{A_{i}}M → _{A_{j}}M involved are cocontinuous, and so by Lemma 2.5 it is also the limit in Pres_{𝕂}. In particular, Qcoh(X) is locally presentable. More generally, any Artin stack X can be presented as a 2-colimit of affine schemes, and a similar argument applies to show that Qcoh(X) is locally presentable, using faithfully flat descent of quasi-coherent sheaves.

3. (Non)dualizability

In this section we will prove Theorems 1.3, 1.4, and 1.5. First, let us briefly unpack the notion of dualizability from Definition 1.1.

We will make use of the notion of adjunction between bicategories, referring the reader to [Fio06, Chapter 9] for background. The term used in that reference is “biadjunction,” but we will simply say “adjunction.” The following Lemma is implicit in [DS97, Proposition 3] and reproduces at the 2-categorical level facts that are essentially standard within symmetric monoidal 1-categories:

3.1. Lemma. For a locally presentable 𝕂-linear category C the following conditions are equivalent:

1. C is dualizable in the sense of Definition 1.1.

2. For any D ∈ Pres_{𝕂} the canonical functor can_{D} : D ⊠ C^{∗} → Hom(C, D) is an equivalence.

3. If ev : C^{∗} ⊠ C → Vect is the standard evaluation pairing, then

\begin{matrix}(• ⊠ C^{∗}) ⊠ C ≃ • ⊠ (C^{∗} ⊠ C) \\ id \end{matrix}_{−⊠ev}  (∗)

is the counit of an adjunction between the 2-endofunctors •⊠C and •⊠C^{∗} of Pres_{𝕂}.

4. There is a cocontinuous linear functor coev : Vect → C ⊠ C^{∗} (the coevaluation) such that the two compositions

C ≃ Vect C C C^{∗} C C Vect ≃ C ⊠ ⊠ ⊠ ⊠ coev ev id id  (†)

and

C^{∗} ≃C^{∗} Vect C^{∗} C C^{∗} Vect C^{∗} ≃ C^{∗} ⊠ ⊠ ⊠ ⊠ coev ev id id  (‡)

are naturally isomorphic to the identity.

5. The identity functor id_{C} is in the essential image of the canonical functor C ⊠ C^{∗} → End(C).

6. The 2-endofunctor ⊠C of Pres_{𝕂} has a right adjoint of the form ⊠C′ for some C′ ∈ Pres_{𝕂}.

Proof. (1) ⇒ (5) This is immediate.

(5) ⇒ (4) An object x ∈ C ⊠C^{∗} that maps onto id_{C} through C ⊠C^{∗} → End(C) induces a left adjoint coev : Vect → C ⊠ C^{∗}, 𝕂^{⊕α} ↦ x^{⊕α}. We claim that as the name suggests, coev is a coevaluation in the sense of (4). In order to verify this, we have to show that the functors (†) and (‡) are (naturally isomorphic to) identities.

For (†) this is simply an unpacking of the fact that the image of x in End(C) is the identity. Indeed, the right-hand half of (†) is simply the C-valued pairing of C ⊠ C^{∗} with C obtained by first mapping the former into End(C) and then evaluating C-endofunctors at a given object in C.

The verification is almost as simple for (‡). The desired isomorphism can be tested against C by pairing via ev; in other words, it is enough to show that the composition ev ∘ ((‡) ⊠ id_{C}) is naturally isomorphic to ev : C^{∗} ⊠ C → Vect. A diagram chase shows that this composition is isomorphic to ev ∘ (id_{C^{∗}} ⊠ (†)), which in turn is isomorphic to ev because (†) ≅ id_{C}.

(4) ⇒ (3) For every D, E ∈ Pres_{𝕂}, the functors ev and coev induce functors

Hom(D ⊠ C, E) Hom(D, E ⊠ C^{∗}), R_{D,E} L_{D,E}

natural in D and E in the obvious sense. The functor R_{D,E}, for instance, sends F : D⊠C → E to

D ≃ D Vect D C C^{∗} EC^{∗} ⊠ ⊠ ⊠ ⊠ coev F id id ,

while L_{D,E} is defined similarly using ev.

Now, the conditions in (4) imply that R_{D,E} and L_{D,E} are mutually inverse, so in particular each R_{D,E} is an equivalence. Collectively, the R_{D,E} implement an adjunction between • ⊠ C and • ⊠ C^{∗} (where the former is the left adjoint) as in [Fio06, Definition 9.8]. The identification of the counit with ev as in (∗) is now easy.

(3) ⇒ (2) By definition Hom(C, •) is a right adjoint to • ⊠ C. By the uniqueness of adjoints between bicategories (e.g. [Fio06, Theorem 9.20]), we can find an equivalence η_{D} : D ⊠ C^{∗} ≃ Hom(C, D), natural in D, that intertwines the counits of the two adjunctions: the diagram

D ⊠ C^{∗} ⊠ C Hom(C, D) ⊠ C D ⇓ η_{D}⊠id_{C} id_{D}⊠ev

commutes up to natural isomorphism for every D ∈ Pres_{𝕂}. Moreover, we get a similar commutative diagram if we substitute can_{D} for η_{D}. But then, by the universality of the counit of an adjunction (the so-called biuniversality of [Fio06, Definition 9.4], or rather its dual), the two functors η_{D} and can_{D} are naturally isomorphic; see e.g. [Fio06, Lemma 9.7].

(2) ⇒ (1) Indeed, Definition 1.1 is the particular instance of (2) obtained by taking D = C.

(3) ⇒ (6) Simply set C′ = C^{∗}.

(6) ⇒ (3) The equivalence

C′ ≃ Hom(Vect, C′) ≃ Hom(C, Vect)

resulting from the adjunction identifies C′ with C^{∗} in such a way that the counit C′ ⊠ C → Vect gets identified with the evaluation ev : C^{∗} ⊠ C → Vect.

3.2. Remark. Condition (6) in Lemma 3.1 makes it clear that the base change 2-functor • ⊠ Vect_{𝕃} : Pres_{𝕂} → Pres_{𝕃} from Remark 2.11 preserves dualizability. Indeed, • ⊠ (C′ ⊠ Vect_{𝕃}) is right adjoint to • ⊠ (C ⊠ Vect_{𝕃}) on Pres_{𝕃} whenever • ⊠ C′ is right adjoint to • ⊠ C on Pres_{𝕂}.

3.3. Remark. Let (C_{i})_{i∈I} be a family of dualizable locally presentable 𝕂-linear categories. Then its direct sum ⨁_{i∈I} C_{i} is dualizable. This follows from (1) ⇔ (2) in Lemma 3.1 and Remark 2.6.

For future use, we note the following consequence of Lemma 3.1.

3.4. Corollary. Suppose that ι : D → C is a cocontinuous 𝕂-linear functor between locally presentable 𝕂-linear categories, and π : C → D a cocontinuous 𝕂-linear functor with π ∘ ι ≅ id_{D}. If C is dualizable, then so is D.

Proof. Consider the commutative diagram

D ⊠ D^{∗} Hom(D, D), D ⊠ C^{∗} Hom(C, D) id_{D}⊠ι^{∗} ι^{∗} ≃

where the vertical arrows are both given by restriction along ι : D → C.

If C is dualizable, then by Lemma 3.1 the upper horizontal functor is an equivalence. The hypotheses imply that id_{D} is in the essential image of the right hand vertical arrow (e.g. it is the image of π ∈ Hom(C, D)), and hence also in the image of the lower horizontal functor. But then the equivalence between (1) and (5) of Lemma 3.1 applies to prove dualizability.

We may now begin establishing certain categories as either dualizable or not. First, Example 2.14 generalizes immediately to all categories strongly generated by compact projectives:

3.5. Lemma. Suppose that a locally presentable 𝕂-linear category C has a strongly generating set Γ consisting entirely of compact projective objects. Then C is dualizable.

Proof. By Proposition 2.4, the Yoneda functor establishes an equivalence C ≃ Fun_{𝕂}(Γ^{op}, Vect). Its dual is then C^{∗} = Hom(C, Vect) ≃ Fun_{𝕂}(Γ, Vect), where the pairing C ⊠ C^{∗} → Vect is computed as a coend over Γ, and

Hom(Fun_{𝕂}(Γ^{op}, Vect), D) ≃ Fun_{𝕂}(Γ, D) ≃ Fun_{𝕂}(Γ, Vect) ⊠ D.

Both equivalences use Proposition 2.4; the latter also uses Remark 2.8. The composition is nothing but the functor Hom(Fun_{𝕂}(Γ^{op}, Vect), D) ← Fun_{𝕂}(Γ^{op}, Vect)^{∗} ⊠D induced by the pairing. By taking D = C we see that C ≃ Fun_{𝕂}(Γ^{op}, Vect) is dualizable.

3.6. Remark. We know of no dualizable locally presentable category not of this type, and conjecture that dualizability of a locally presentable 𝕂-linear category implies that the category is strongly generated by compact projectives. Theorem 1.3 implies that there are no counterexamples to this conjecture among categories of the form M^{C} for C a coassociative coalgebra.

Theorem 1.5 is now a simple consequence of Proposition 2.13.

Proof of Theorem 1.5. Let O(X) be the 𝕂-algebra of regular functions on X. It carries an action of G that is compatible with the multiplication; i.e. O(X) makes sense as an algebra object in Rep(G) = M^{O(G)}, the symmetric monoidal category of comodules over the Hopf algebra O(G) of regular functions on G.

A quasi-coherent sheaf on [X/G] is nothing but a quasi-coherenet sheaf on X with a compatible G-action. Since X is affine, quasi-coherent sheaves on X are just O(X)- modules, and so Qcoh([X/G]) is the category of O(X)-modules with compatible Gaction. But an O(X)-module with compatible G-action is nothing but an O(X)-module-object in the symmetric monoidal category Rep(G). This puts us well within the enriched world of Definition 2.9 and Remark 2.10: in particular, Qcoh([X/G]) is a Rep(G)- module.

We now apply Proposition 2.13 to the present situation, where we let C = Qcoh([X/G]) and S = Rep(G), to conclude that C is strongly generated by compact projectives over Vect:

First, O(X) is a Rep(G)-compact-projective Rep(G)-strongly-generating object in Qcoh([X/G]) (this is an enriched version of Example 2.14, and follows from [Kel05, Theorem 5.26]).

Secondly, Rep(G) is strongly generated by compact projectives over Vect essentially by [Don96]. One result that is part of the main theorem of that paper is that being virtually linearly reductive is equivalent to the monoidal unit 𝕂 in Rep(G) having finitedimensional injective envelope E. It is easy to see then that E^{∗} is projective in Rep(G) (and hence compact projective because it is also finite-dimensional).

For every finite-dimensional object V ∈ Rep(G) the endofunctor − ⊗ V of Rep(G) is left adjoint to the exact functor − ⊗ V^{∗}, and hence preserves projectivity. Since 𝕂 is surjected upon by E^{∗}, V is surjected upon by the compact projective object E^{∗}⊗V. Since objects in the comodule category Rep(G) are unions of finite-dimensional subobjects, the conclusion that Rep(G) is strongly generated by compact projectives follows.

Finally, dualizability of Qcoh([X/G]) is a consequence of Lemma 3.5.

We turn now to the category M^{C} of right comodules for C a coassociative coalgebra. Since M^{C} is abelian and in abelian categories generating sets are strongly generating, Lemma 3.5 implies that M^{C} is dualizable if it is generated by its compact projectives. This, of course, fails in general, as the following well-known example illustrates:

3.7. Example. Consider the coalgebra 𝕂[x] with comultiplication x^{n} ↦ ∑_{i=0}^{n} x^{i} ⊗ x^{n−i}. We will show that in the category M^{𝕂[x]}, there are no non-zero projectives. Indeed, suppose that V ∈ M^{𝕂[x]} is not the zero object. To see that V is not projective, it suffices to witness a surjection W ↠ V that does not split.

The finite-dimensional subcoalgebras of the 𝕂[x] are dual to the algebras 𝕂[t]/(t^{n}); thus M^{𝕂[x]} is the category whose objects are vector spaces V equipped with a locally nilpotent endomorphism t, i.e. an endomorphism t such that for each v ∈ V, there is some n ∈ ℕ with t^{n}(v) = 0. In particular, ker t is not zero if V is not zero. Split V as a vector space as V = V̄ ⊕ ker t. The action of t then has form:

t = (\begin{matrix}t̄ & 0 \\ τ & 0 \end{matrix})

for some t̄: V̄ → V̄ and τ : V̄ → ker t. We then define W = V ⊕ ker t = V̄ ⊕ ker t ⊕ ker t, and give it the locally nilpotent endomorphism

t = (\begin{matrix}t̄ & 0 & 0 \\ τ & 0 & 0 \\ 0 & id_{ker t} & 0 \end{matrix}) .

The map W ↠ V kills the second copy of ker t. Any splitting will take v ∈ ker t to some element w ∈ W for which t(w) = (0, 0, v) ∈ V̄ ⊕ ker t ⊕ ker t. Thus no splitting is t-linear, and V is not projective.

Note that 𝕂[x] is the Hopf algebra of functions on the (non-reductive) additive group 𝔾_{a}. Thus M^{𝕂[x]} = Qcoh((Spec𝕂)/𝔾_{a}), and so Theorem 1.5 fails without the virtual linear reducibility requirement.

3.8. Lemma. Let C be a coassociative coalgebra and {C_{i}}_{i∈I} the partially ordered set of finite-dimensional sub-coalgebras of C. Let D be a locally presentable linear category. The canonical functor M^{C} ⊠ D^{∗} → Hom(D, M^{C}) is fully faithful. Its essential image consists of those functors F occurring as colimits F = \underrightarrow{lim}_{i} F_{i} where F_{i} : D → M^{C} factors through the inclusion M^{C_{i}} ↪ M^{C}.

Proof. The hom-tensor adjunction of Lemma 2.7 implies ⊠ distributes over colimits. Thus there are canonical equivalences:

M^{C} ⊠ D^{∗} ≃ (\underrightarrow{lim}_{i} M^{C_{i}}) ⊠ D^{∗} ≃ \underrightarrow{lim}_{i}(M^{C_{i}} ⊠ D^{∗}) ≃ \underrightarrow{lim}_{i} Hom(D, M^{C_{i}})

In the last step we used dualizability of M^{C_{i}} ≅ M_{C_{i}^{∗}} to imply that the canonical functor from M^{C_{i}} ⊠D^{∗} to Hom(D, M^{C_{i}}) induced by the pairing is an equivalence: by Lemma 3.1 part (3) and the hom-tensor adjunction,

M^{C_{i}} ⊠ D^{∗} = M^{C_{i}} ⊠ Hom(D, Vect) ≃ Hom((M^{C_{i}})^{∗}, Hom(D, Vect)) ≃ Hom((M^{C_{i}})^{∗} ⊠ D, Vect) ≃ Hom(D, Hom((M^{C_{i}})^{∗}, Vect)) ≃ Hom(D, M^{C_{i}}).

Since the pairing-induced functor C ⊠ D^{∗} → Hom(D, C) is natural in C, we conclude that the inclusion

\underrightarrow{lim}_{i} Hom(D, M^{C_{i}}) ≃ M^{C} ⊠ D^{∗} → Hom(D, M^{C})

is the one induced by the inclusions M^{C_{i}} ↪ M^{C}. Lemma 2.5 and Remark 2.6 complete the proof.

We are now equipped to prove Theorem 1.3, which asserts that M^{C} is dualizable if and only if it has enough projectives.

Proof of Theorem 1.3. According to part (d) of the main theorem of [Lin77], the category M^{C} of right C-comodules has enough projectives if and only if every finitedimensional right C-comodule has a projective cover; inspection of the proof reveals the attested projective cover to be finite-dimensional. The finite-dimensional right Ccomodules are precisely the compact ones, and so we see that M^{C} has enough projectives if and only if it is generated by its compact projective objects. Lemma 3.5 then implies one direction of the claim.

Suppose now that M^{C} does not have enough projectives. Then, again by [Lin77] (this time part (c) of the main theorem), there exists a simple left C-comodule S such that the injective hull of S is infinite-dimensional. It follows that we can find essential extensions S ↪ T with dim T finite but arbitrarily large (as otherwise there would be a maximal such T, which would therefore be injective). Recall that an extension S ↪ T is essential if any non-zero subobject of T intersects S nontrivially; the dual notion (for abelian categories) is an essential projection Q ↠ P, which is a surjection for which every proper subobject of Q fails to surject onto P. Thus, by dualization, we have found a simple right C-comodule S^{∗} ∈ M^{C} with essential projections T^{∗} ↠ S^{∗} of arbitrarily large dimension. (The map S ↦ S^{∗} is a contravariant equivalence between the abelian category of finite-dimensional left C-comodules and the abelian category of finite dimensional right C-comodules.)

Let F ∈ Hom(M^{C}, M^{C}) be in the essential image of (M^{C})^{∗} ⊠ M^{C}. By Lemma 3.8, F = \underrightarrow{lim} F_{i}, where F_{i} is the largest subfunctor of F factoring through M^{C_{i}}. We will prove that F ≅ i̸d_{M^{C}}. To do so, consider an arbitrary natural transformation θ : F → id_{M^{C}}, or, what is equivalent, a system of natural transformations θ_{i} : F_{i} → id_{M^{C}}. Since \underrightarrow{lim}(F_{i}(X)) = (\underrightarrow{lim} F_{i})(X) for all X ∈ M^{C}, it suffices to prove that θ_{i}(S^{∗}) : F_{i}(S^{∗}) → S^{∗} vanishes for all sufficiently large i and for S^{∗} the simple with arbitrarily large essential surjections from the previous paragraph. Since dim(S^{∗}) < ∞, for all sufficiently large i we have S^{∗} ∈ M^{C_{i}}, and it suffices to consider just these.

Thus fix i ∈ I with S^{∗} ∈ M^{C_{i}} ≅ M_{C_{i}^{∗}}. Since dim(C_{i}^{∗}) < ∞, there are bounds on the dimensions of essential surjections onto S^{∗} in M^{C_{i}}: a projective cover is as large as you can get. We can therefore choose T^{∗} mapping essentially onto S^{∗} so large that T^{∗} ∉ M^{C_{i}}. Consider the following commutative diagram:

F_{i}(S^{∗}) F_{i}(T^{∗}) S^{∗} T^{∗} θ_{i}(T^{∗}) θ_{i}(S^{∗})

The left arrow is a surjection because F_{i} is cocontinuous. Since F_{i}(T^{∗}) ∈ M^{C_{i}} but T^{∗} ∉ M^{C_{i}}, the image of θ_{i}(T^{∗}) must be a proper sub-co-module of T^{∗}. (Indeed, it is within the largest C_{i}-subcomodule of T^{∗}.) Since the surjection T^{∗} → S^{∗} is essential, the composition F_{i}(T^{∗}) → S^{∗} cannot be a surjection; since S^{∗} is simple, the composition must vanish. But F_{i}(T^{∗}) → F_{i}(S^{∗}) is a surjection; hence θ_{i}(S^{∗}) = 0.

A similar argument works for Qcoh(X) when X is a projective scheme over 𝕂, and provides the basis of the proof of Theorem 1.4. Recall that X is a projective scheme if it is embeddable as a closed subscheme into some projective space ℙ_{𝕂}^{N}.

First, we specialize Corollary 3.4 as follows.

3.9. Corollary. If a 𝕂-scheme X is such that Qcoh(X) is dualizable, then the same is true of all closed subschemes i : Y ⊆ X.

Proof. Setting C = Qcoh(X), D = Qcoh(Y), π = i^{∗} and ι = i_{∗} places us within the scope of Corollary 3.4.

Indeed, note first that i_{∗} : Qcoh(Y) → Qcoh(X) is a left adjoint whenever i : Y ⊆ X is a closed subscheme, with right adjoint Hom(O_{X}/I_{Y} , −) where I_{Y} is the sheaf of ideals defining the inclusion Y ⊆ X ([Har77, Proposition II.5.9]) and Hom denotes the sheaf of homomorphisms ([Har77, Definitions preceding II.5.1]).

Second, note that the counit i^{∗}i_{∗} → id is an isomorphism whenever i is a closed embedding: i_{∗} implements an equivalence of Qcoh(Y) with the category of quasi-coherent sheaves on X annihilated by the sheaf of ideals I_{Y}. In particular i_{∗} is fully faithful, hence the claim.

Proof of Theorem 1.4. We make the problem progressively simpler as follows.

Step 1: Reduction to algebraically closed ground fields. We claim that for any commutative 𝕂-algebra A we have

Qcoh(X × Spec(A)) ≃ Qcoh(X) ⊠_{A}M.

To see this, note first that the canonical functor

Qcoh(X) ⊠_{A}M → Qcoh(X × Spec(A))

is an equivalence when X = Spec(B) is affine by the Eilenberg–Watts theorem. Then, covering X by affine open subschemes U_{i} = Spec(A_{i}), we have canonical functors

Qcoh(X) ⊠_{A}M →^{∼} (\underleftarrow{lim}_{A_{i}}M) ⊠_{A}M →^{∼} \underleftarrow{lim}(_{A_{i}}M ⊠_{A}M),

where the first equivalence follows from Example 2.16 and the second one follows from the fact that _{A}M is dualizable and hence − ⊠_{A}M preserves limits in Pres_{𝕂}.

We then further have

\underleftarrow{lim}(_{A_{i}}M ⊠_{A}M) →^{∼} \underleftarrow{lim} Qcoh(U_{i} × Spec(A)) →^{∼} Qcoh(X × Spec(A))

by another application of Example 2.16 to the open affine cover U_{i} ×Spec(A) ≅ Spec(A_{i} ⊗ A) of X × Spec(A).

Now apply this observation to an algebraic closure A = 𝕃 of 𝕂 and note that if Qcoh(X) is dualizable over Vect_{𝕂}, then by Remark 3.2 Qcoh(X × Spec(𝕃)) ≃ Qcoh(X) ⊠ Vect_{𝕃} is dualizable over Vect_{𝕃}. Consequently, throughout the rest of the proof we will assume that 𝕂 was algebraically closed to begin with.

Step 2: Reduction to integral projective schemes. Corollary 3.9 allows us to do this by transporting dualizability first from X to a closed projective subscheme of positive dimension and then to the reduced induced subscheme structure on a positive-dimensional irreducible component of that (c.f. [Har77, Example II.3.2.6]).

Step 3: The case of integral projective schemes over algebraically closed fields. Any embedding X ↪ ℙ^{N} into some projective space provides a strongly generating set of line bundles O(n) = O(1)^{⊗n} ∈ Qcoh(X), with O(1) being the pullback of the twisting sheaf O_{ℙ^{N}} (1) (see [Har77, Definition preceding Proposition II.5.12]; the fact that they generate is a consequence of [Har77, Theorem II.5.17]).

Note that there are no non-zero maps O(m) → O(n) for m > n, or in other words that O(−n) has no non-zero global sections if n > 0. Indeed, suppose for contradiction that O(−n) has a non-zero global section s : O → O(−n). Since X is integral, each line bundle O(n) is a subsheaf of the constant sheaf K associated to the field of rational functions on X (see e.g. [Har77, Proposition II.6.15] and surrounding discussion), and so s can be regarded as a rational function, and the section s^{⊗k} of O(−kn) ≅ O(−n)^{⊗k} is simply its kth power and hence again non-zero. Moreover, ⊗s^{⊗k} : O(kn) → O(kn) ⊗ O(−kn) ≅ O is an embedding on spaces of global sections.

But the function

m ↦ dimension of the space of sections of O(m)

is a polynomial of degree dim(X) for large m [Har77, Theorem I.7.5 and Exercise II.7.6], whereas the only global regular functions on an integral projective scheme over an algebraically closed field are the constants [Har77, Theorem I.3.4]. Thus for k sufficiently large, no map O(kn) → O can be an embedding on global sections. Thus the non-zero section s : O → O(−n) cannot exist.

If Qcoh(X)_{≥n} is the full, cocomplete subcategory of Qcoh(X) strongly generated by \{O(m)\}_{m≥n}, then Qcoh(X) ≃ \underrightarrow{lim}_{n→−∞} Qcoh(X)_{≥n}. Without dualizability of Qcoh(X)_{≥n}, the essential image of Qcoh(X)^{∗}⊠Qcoh(X) inside Hom(Qcoh(X), Qcoh(X)) may fail to include all colimits of the form F = \underrightarrow{lim} F_{n} for F_{n} factoring through Qcoh(X)_{≥n}, but the arguments of Lemma 3.8 do imply that any F in the essential image is of this type. Thus, as in the proof of Theorem 1.3, it suffices to find a non-zero object M ∈ Qcoh(X) such that for all sufficiently negative n, any natural transformation θ_{n} : F_{n} → id_{Qcoh(X)} satisfies θ_{n}(M) = 0 : F_{n}(M) → M.

Fix k arbitrarily and set M = O(k). For any n ≤ k, we can find some direct sum of O(m)s with m < n that surjects onto O(k). We thus build a commutative square similar to the one from the proof of Theorem 1.3:

F_{n}(O(k)) F_{n}(⨁ O(m)) O(k) ⨁ O(m) θ_{n}(⨁ O(m)) θ_{n}(O(k))

As before, the left arrow is a surjection since F_{n} is cocontinuous. But F_{n}(⨁ O(m)) ∈ Qcoh(X)_{≥n}, and so θ_{n}(⨁ O(m)) = 0. It follows that θ_{n}(O(k)) = 0.

4. Reflexivity

The goal of this section is to prove Proposition 4.8, which we will use to prove Theorems 1.6 and 1.8. The arguments in this section apply when 𝕂 is not a field but just a commutative ring, in which case “Vect” means the symmetric monoidal category M_{𝕂} of all 𝕂-modules.

In this section we denote by (I, ≤) a generic poset which is ℵ_{0}-directed, which means that any two i, j ∈ I are dominated by some k ∈ I. As always, we regard I as a category with an arrow i → j for i ≤ j.

4.1. Definition. An I-indexed pro-object (or just pro-object, when I is understood) in a category T is a functor I^{op} → T . An I-indexed pro-algebra (or just pro-algebra) is an I-indexed pro-object in the category Alg of 𝕂-algebras and homomorphisms. The morphisms between an I-indexed pro-algebra A = {A_{i}}_{i∈I^{op}} and a J -indexed pro-algebra B = {B_{j}}_{j∈J^{op}} are

hom(A, B) = \underleftarrow{lim}_{j∈J} \underrightarrow{lim}_{i∈I} hom(A_{i}, B_{j}).

An acceptable I-indexed pro-algebra is a pro-algebra whose indexing poset I has countable cofinality (i.e. it has a countable cofinal subset), and for which the transition maps A_{j} → A_{i} for i ≤ j are onto.

4.2. Remark. Given a homomorphism A → B of associative algebras, both the scalar restriction functor M_{B} → M_{A} and the scalar extension functor (−) ⊗_{A} B : M_{A} → M_{B} are cocontinuous, and so we have 2-functors Restrict : Alg^{op} → Pres_{𝕂} and Extend : Alg → Pres_{𝕂}. Thus any I^{op}-indexed diagram A = {A_{i}}_{i∈I^{op}} defines, by using scalar restriction, an I-indexed diagram {M_{A_{i}}}_{i∈I} in Pres_{𝕂} as well as, by using scalar extension, an I^{op}-indexed diagram \{M_{A_{i}}\}_{i∈I^{op}}. Whenever we write \underrightarrow{lim}_{i∈I} M_{A_{i}}, we will mean the colimit of the former; when we write \underleftarrow{lim}_{i∈I^{op}} M_{A_{i}}, we will mean the limit of the latter.

The notions of “pro-object” and “ind-object” are not restricted to categories — they also make sense in any bicategory. Since (2-)functors take pro- and ind-objects to proand ind-objects, for each pro-algebra {A_{i}}_{i∈I}, we get, via either restricting or extending, an ind-object {M_{A_{i}}}_{i∈I} in Pres_{𝕂} and a pro-object {M_{A_{i}}}_{i∈I^{op}} in Pres_{𝕂}, both varying functorially for morphisms of pro-algebras. Moreover, just as in the case of 1-categories, the bicategorical \underrightarrow{lim} and \underleftarrow{lim} are functors from ind- and pro-objects, respectively, to objects. It follows in particular that the assignments

\{A_{i}\}_{i∈I^{op}} ↦ \underrightarrow{lim}_{i} M_{A_{i}}

and

\{A_{i}\}_{i∈I^{op}} ↦ \underleftarrow{lim}_{i} M_{A_{i}}

take isomorphic pro-algebras to equivalent categories.

4.3. Remark. The cofinal countability of I for acceptable pro-algebras will come up in a number of ways:

First, it allows us to assume that (I, ≤) is ℕ = {0, 1, …} with the usual order, as pro-algebras indexed by varying posets form a category, and every pro-object whose indexing poset has countable cofinality is isomorphic in this category to an ℕ-indexed one. Since Remark 4.2 assures that the discussion below is invariant under isomorphism in the category of pro-algebras, we will make such a substitution in the sequel.

Second, for an acceptable ℕ-indexed pro-algebra {A_{i}} the map from the limit Â := \underleftarrow{lim}_{i} A_{i} to each A_{j} is onto. In general, when I has uncountable cofinality, it is possible for each A_{i} to be nontrivial (i.e. not the ground field) and all transition maps A_{j} → A_{i} to be onto but nevertheless for \underleftarrow{lim} A_{i} to be trivial; see e.g. [Ber, Corollary 8].

4.4. Definition. A module V over a filtered limit Â = \underleftarrow{lim}_{i} A_{i} of algebras is called discrete if for every v ∈ V there is some i ∈ ℕ for which v is killed by ker(Â ↠ A_{i}).

4.5. Lemma. If {A_{i}} is an acceptable pro-algebra, the canonical functor

(\underrightarrow{lim}_{i} M_{A_{i}}) ⊠ M_{B} ≃ \underrightarrow{lim}_{i}(M_{A_{i}} ⊠ M_{B}) → MÂ ⊠ M_{B} ≃ M_{Â⊗B},

obtained by restricting scalars along Â → A_{i}, is full and faithful. Its essential image consists of those (Â ⊗ B)-modules V whose underlying Â-modules are discrete.

Proof. According to Lemma 2.5 and Remark 2.6, an object of \underrightarrow{lim}_{i}(M_{A_{i}} ⊠ M_{B}) consists of a sequence {M_{i}}_{i∈ℕ}, where, for each i, M_{i} is an A_{i}-module in M_{B}, together with compatible isomorphisms

M_{i} ≅ hom_{A_{j}}(A_{i}, M_{j}) ≤ M_{j}

for each i ≤ j, realizing M_{i} as the maximal submodule of M_{j} on which the A_{j} action factors through A_{j} → A_{i}.

The functor \underrightarrow{lim}_{i}(M_{A_{i}} ⊠ M_{B}) → MÂ ⊠ M_{B} takes such a sequence to the vector space M = \underrightarrow{lim}_{i} M_{i} along with an action by Â ⊗ B that factors through A_{i} ⊗ B when acting on the subvector space M_{i}. Note that, for each i, M_{i} can be recovered from M as the maximal subspace on which the Â-action factors through A_{i}:

M_{i} ≅ homÂ(A_{i}, M)

Conversely, any Â⊗B-module V determines a filtered vector space whose ith filtered piece V_{i} is the maximal subspace of V on which the Â-action factors through A_{i}. Discreteness of V is equivalent to the canonical map \underrightarrow{lim} V_{i} → V being an isomorphism. Thus the discrete modules are precisely the ones in the essential image of \underrightarrow{lim}_{i}(M_{A_{i}} ⊠ M_{B}) → MÂ ⊠ M_{B}.

Suppose that M = \underrightarrow{lim} M_{i} and N = \underrightarrow{lim} N_{i} are discrete, and that f : M → N is an Â ⊗ B-linear map. Then f restricts to an A_{i} ⊗ B-linear map f_{i} : M_{i} → N_{i} for each i, and the sequence {f_{i}}_{i∈ℕ} defines a morphism {M_{i}}_{i∈ℕ} → {N_{i}}_{i∈ℕ}, whose image under the functor \underrightarrow{lim}_{i} M_{A_{i}} → MÂ is f. Full faithfulness follows.

Note that we have used surjectivity of Â → A_{i} from Remark 4.3; Lemma 4.5 can fail for I uncountable.

Since (−)^{∗} = Hom(−, Vect) turns colimits into limits, we have

(\underrightarrow{lim} M_{A_{i}})^{∗} ≃ \underleftarrow{lim}(M_{A_{i}})^{∗} ≃ \underleftarrow{lim}(_{A_{i}}M)

(where the second equivalence uses the Eilenberg–Watts theorem). An object of this limit is a sequence {M_{i}}_{i∈ℕ} whose ith entry is a left A_{i}-module, along with isomorphisms M_{i} ≅ A_{i} ⊗_{A_{j}} M_{j} for j > i, compatible for triples k > j > i. Notice that this implies that the corresponding maps M_{j} → M_{i} of A_{j}-modules are onto. An example is the regular module A = {A_{i}}_{i∈ℕ} with the canonical isomorphisms, which when thought of as an object of (\underrightarrow{lim} M_{A_{i}})^{∗} is nothing but the forgetful functor \underrightarrow{lim} M_{A_{i}} → Vect. It enjoys hom(A, A) ≅ Â.

More generally, any M ∈ \underleftarrow{lim}(_{A_{i}}M) has an underlying pro-vector space M = \{M_{i}\}_{i∈ℕ^{op}}, and hom(A, M) ≅ M̂ = \underleftarrow{lim} M_{i} is its limit in Vect. It consists of compatible systems of elements {m_{i} ∈ M_{i}}_{i∈ℕ}, i.e. m_{j} is the image of m_{i} under the projection M_{i} ↠ A_{j} ⊗_{A_{i}} M_{i} ≅ M_{j}.

For the remainder of this section we let M = \underleftarrow{lim}_{i} _{A_{i}}M.

4.6. Lemma. Let M = {M_{i}}_{i∈ℕ} and N = {N_{i}}_{i∈ℕ} be objects in M, and f : M → N be an epimorphism (i.e. all components f_{i} : M_{i} → N_{i} are onto). Let also m_{j} ∈ ker(f_{j}) be an element for some fixed j ∈ ℕ. Then m_{j} can be expanded to a compatible system of elements m_{i} ∈ ker(f_{i}), i ∈ ℕ.

Proof. We may as well assume j = 0. Choose any preimage m′_{1} ∈ M_{1}, and consider its image n′_{1} ∈ N_{1}. It lies in ker(N_{1} → N_{0}) = ker(A_{1} → A_{0}) · N_{1}, which thus lifts to some element in ker(A_{1} → A_{0}) · M_{1}. Subtracting this element from m′_{1}, we get our desired lift m_{1} ∈ M_{1}. Now repeat to construct m_{2}, etc.

4.7. Lemma. The objects of M admit functorial free resolutions of the form

N ↦ (A^{⊕S_{1}(N)} → A^{⊕S_{0}(N)} → N → 0),

where S_{i} are functors M → Set.

Proof. Define S_{0}(N) = N̂, and the natural epimorphism A^{⊕S_{0}(N)} → N in the obvious way: For n ∈ N̂ ≅ hom_{M}(A, N), the map from the summand of A^{⊕N̂} indexed by n ∈ N̂ to N is simply n.

The resulting map A^{S_{0}(N)} → N is surjective at each level j ∈ ℕ, because any n_{j} ∈ N_{j} can be lifted to a compatible system of elements n_{i} ∈ N_{i}, i ∈ ℕ.

Next, let S_{1}(N) be the set of compatible systems of elements a_{i} ∈ ker(A_{i}^{⊕S_{0}(N)} → N_{i}) and define A^{⊕S_{1}(N)} → A^{⊕S_{0}(N)} similarly as before. Taking f to be our morphism A^{⊕S_{0}(N)} → N, Lemma 4.6 ensures that

A^{⊕S_{1}(N)} → A^{⊕S_{0}(N)} → N → 0

is exact at each level, i.e. that its components in the categories M_{A_{i}} are exact. This implies that the diagram is a cokernel in M, as desired.

We are now equipped to prove the following proposition, of which Theorems 1.6 and 1.8 are consequences.

4.8. Proposition. Let A = {A_{i}}_{i∈I^{op}} be an acceptable pro-algebra and B be any 𝕂algebra. The functor

η : Hom(\underleftarrow{lim}(_{A_{i}}M), M_{B}) → M_{Â⊗B}, F ↦ F (A)

is fully faithful. Its essential image consists of those Â ⊗ B-modules which are discrete over Â in the sense of Definition 4.4.

Proof. As before, we will abbreviate M := \underleftarrow{lim}(_{A_{i}}M), and use Remark 4.3 to set I = ℕ. Step 1: The essential image of η contains all Â-discrete Â ⊗ B-modules. Given some Â-discrete Â ⊗ B-module V, let V_{i} := {v ∈ V : ker(Â → A_{i}) · v = 0} be

the largest A_{i}-submodule of V. Then i ≤ j implies V_{i} ⊆ V_{j} and we have V = \underrightarrow{lim}_{i} V_{i}. If M ∈ M, we obtain a B-linear map

V_{i} ⊗_{A_{i}} M_{i} ≅ V_{i} ⊗_{A_{i}} (A_{i} ⊗_{A_{j}} M_{j}) ≅ V_{i} ⊗_{A_{j}} M_{j} → V_{j} ⊗_{A_{j}} M_{j}.

Define the functor F : M → M_{B} by F (M) := \underrightarrow{lim}_{i} V_{i} ⊗_{A_{i}} M_{i}. Clearly, F is cocontinuous and satisfies F (A) ≅ V.

Step 2: The essential image of η contains only Â-discrete Â ⊗ B-modules. Let F ∈ Hom(M, M_{B}). We want to show that for every v ∈ F (A), there is an i ∈ ℕ

such that v is killed by ker(Â ↠ A_{i}).

Consider the direct sum A^{⊕ℕ} of ℵ_{0}-many copies of A in M. The object A, although a generator of M, is not compact when the pro-algebra A does not “stabilize” (i.e. when for infinitely many i the maps A_{i+1} → A_{i} are not isomorphisms). A manifestation of this is that hom(A, A^{⊕ℕ}) is not Â^{⊕ℕ} but rather the projective limit \underleftarrow{lim}_{i}(A_{i}^{⊕ℕ}) as computed in Vect. In other words, a homomorphism A → A^{⊕ℕ} is the same as a system of homomorphisms {f_{j} : A → A}_{j∈ℕ} with the property that for all i ∈ ℕ, all but finitely many of the f_{j}s belong to ker(Â ↠ A_{i}).

Suppose now that v ∈ F (A) is such that for every i ∈ ℕ, there is some f_{i} ∈ ker(Â → A_{i}) which does not annihilate v. These f_{i}s define a homomorphism f : A → A^{⊕ℕ}. Since F is cocontinuous, we get a linear map

F (f) : F (A) → F (A^{⊕ℕ}) ≅ F (A)^{⊕ℕ}

for which every component of F (f)(v) is non-zero. This is absurd, since F (f)(v) is supposed to sit inside the direct sum F (A)^{⊕ℕ}.

Step 3: The functor η is faithful.

This follows immediately from Lemma 4.7 and the fact that colimit-preserving functors preserve epimorphisms.

Step 4: The functor η is full.

Let F, G ∈ Hom(M → M_{B}). We have to show that any Â⊗B-morphism ϕ : F (A) → G(A) is the A-component θ_{A} of a natural transformation θ : F → G.

Recall from Lemma 4.6 that we have functorial free resolutions for objects in M. We claim that to prove fullness of η, it suffices to check, for all sets S and T and all morphisms f : A^{⊕S} → A^{⊕T}, the commutativity of

\begin{matrix}F (A)^{⊕S} \\ F (A)^{⊕T} \\ G(A)^{⊕S} \\ G(A)^{⊕T} \end{matrix} F (f) G(f) ϕ^{⊕S} ϕ^{⊕T}  (§)

Indeed, we can then set θ_{A^{⊕S}} = ϕ^{⊕S} for every set S and extend this to a natural transformation θ via the functorial resolutions.

The right hand vertical arrow in (§) embeds into

\begin{matrix}F (A)^{×T} \\ G(A)^{×T} \end{matrix}_{ϕ^{×T}}

(direct product rather than direct sum). For each s ∈ S and each t ∈ T, let ι_{s} : A → A^{⊕S} denote the inclusions onto the sth summand, and π_{t} : A^{⊕T} → A the projection onto the tth summand (the latter exists in any additive category). Then the (t, s)th matrix entry

in the composition F (A)^{⊕S} ⟶^{F (f)} F (A)^{⊕T} → F (A)^{×T} is just F (f′) for f′ = π_{t} ∘ f ∘ ι_{s}; the (t, s)th matrix entry in G(A)^{⊕S} → G(A)^{×T} is G(f′) for the same f′ ∈ hom_{M}(A, A). Thus, by replacing f with f′ in (§), we can assume S and T are singletons.

Finally, the commutativity of

\begin{matrix}F (A) \\ F (A) \\ G(A) \\ G(A) \end{matrix} F (f′) G(f′) ϕ ϕ

follows from the fact that F (f′) and G(f′) are nothing but the actions of f′ ∈ hom_{M}(A, A) ≅ Â on F (A) and G(A), while ϕ is by definition a morphism of Â-modules.

4.9. Remark. Actually Proposition 4.8 can be proven in a more general setting and provides a partial universal property of \underleftarrow{lim}(_{A_{i}}M): If C is a locally finitely presentable 𝕂-linear category (i.e. locally presentable and strongly generated by its compact objects), then Hom(\underleftarrow{lim}(_{A_{i}}M), C) is equivalent to the category of discrete Â-module objects in C. These are objects T ∈ C equipped with a homomorphism of 𝕂-algebras Â → End_{C}(T) such that T = \underrightarrow{lim}_{i} T_{i}, where T_{i} := ker(T → ∏_{a∈ker(Â→A_{i})} T ) is the largest A_{i}-submodule object of T. The same proof as before works. Only in Step 2 we have to replace v by a morphism P → F (A), where P is any compact object.

Proof of Theorem 1.6. Lemma 2.5 identifies the category C from the statement of Theorem 1.6 with the colimit \underrightarrow{lim}_{i} M_{A_{i}} in Pres_{𝕂}. Consider the triangle

(\underrightarrow{lim}_{i} M_{A_{i}}) ⊠ M_{B} M_{Â⊗B} Hom((\underrightarrow{lim}_{i} M_{A_{i}})^{∗}, M_{B}) dd inclusion η

of functors. First, we claim that it commutes (up to canonical natural isomorphism).

To see this, note first that dd factors through (\underrightarrow{lim}_{i} M_{A_{i}})^{∗∗} ⊠ M_{B} and the triangle is obtained simply by applying − ⊠ M_{B} to the case B = 𝕂. There is hence no loss of generality if we just assume B = 𝕂 for the purpose of verifying the claim.

By definition, η ∘ dd sends an object {M_{i}} as in Lemma 4.5 to its image through A = \{A_{i}\} regarded as an object of (\underrightarrow{lim}_{i} M_{A_{i}})^{∗} ≃ \underleftarrow{lim}_{i} _{A_{i}}M. Since \{M_{i}\} is the colimit in \underrightarrow{lim}_{i} M_{A_{i}} of M_{i}, this image is \underrightarrow{lim}(M_{i} ⊗_{A_{i}} A_{i}) ≅ \underrightarrow{lim} M_{i}.

In other words, the composition η ∘ dd simply assembles all M_{i}s together via the identifications M_{i} ≤ M_{j} into a filtered vector space with compatible right actions by A_{i} on the pieces of the filtration. But this is exactly the identification of \underrightarrow{lim}_{i} M_{A_{i}} with a subcategory of MÂ from Lemma 4.5. This concludes the justification of the claim.

Now, according to Lemma 4.5 and Proposition 4.8, both vertical functors are fully faithful with essential image the category of Â-discrete Â ⊗ B-modules. It follows that dd is an equivalence of categories.

We conclude this section with a proof of Theorem 1.8. Recall that for a stack X over 𝕂, the category Qcoh(X) is not just 𝕂-linear and locally presentable, but also symmetric monoidal in the sense of Definition 2.9.

Following [Bra11, HR14], call a stack X over 𝕂 tensorial if for any affine scheme Spec(B) over 𝕂, the functor

hom(Spec(B), X) → Hom_{⊗}(Qcoh(X), M_{B}), f ↦ f^{∗}

is an equivalence. Such stacks are called “2-affine” in [CJF13, BC14], where symmetric monoidal locally presentable (resp. cocomplete) categories are called “commutative 2rings.”

In particular, we can talk about tensorial schemes, a notion which we now extend to ind-schemes in the obvious manner. Recall that an ind-scheme X = \underrightarrow{lim}_{i} X_{i} is a formal directed colimit of schemes X_{i} where the transition morphisms X_{i} → X_{j} (i ≤ j) are closed immersions. One defines Qcoh(X) := \underleftarrow{lim}_{i} Qcoh(X_{i}). It inherits a symmetric monoidal structure from the symmetric monoidal structures on the Qcoh(X_{i})s.

4.10. Definition. An ind-scheme X = \underrightarrow{lim}_{i∈I} X_{i} is tensorial if for every commutative algebra B the canonical functor

\underrightarrow{lim}_{i} hom(Spec(B), X_{i}) = hom(Spec(B), X) → Hom_{⊗}(Qcoh(X), M_{B}), f ↦ f^{∗}

is an equivalence of categories. Unpacking this, given an acceptable commutative proalgebra \{A_{i}\}_{i∈I^{op}}, the corresponding affine ind-scheme \underrightarrow{lim} Spec(A_{i}) is tensorial if for every commutative algebra B the canonical functor

\underrightarrow{lim}_{i} hom(A_{i}, B) → Hom_{⊗}( \underleftarrow{lim}_{i} M_{A_{i}}, M_{B})

sending an algebra homomorphism ϕ : A_{i} → B to the composition

\begin{matrix}M \\ M_{A_{i}} \\ M_{B}, \end{matrix}_{−⊗_{A_{i}}B}

is an equivalence of categories.

Proof of Theorem 1.8. Let \{A_{i}\}_{i∈I^{op}} and B be as above and M := \underleftarrow{lim}_{i} M_{A_{i}}. We tackle the full faithfulness and essential surjectivity of \underrightarrow{lim} hom(A_{i}, B) → Hom_{⊗}(M, M_{B}) separately.

Step 1: Full faithfulness. Any functor whose domain is equivalent to a set is automatically faithful, and any functor whose codomain is equivalent to a set is automatically full. Since \underrightarrow{lim} hom(A_{i}, B) is already a set, it thus suffices to check that Hom_{⊗}(M, M_{B}) is equivalent to a set. But a morphism F → G of cocontinuous symmetric monoidal functors M → M_{B} is unique, if it exists, and in that case an isomorphism, because, by Proposition 4.8, it is completely determined by F (A) → G(A), and this has to be the composition of the distinguished isomorphisms F (A) ≅ B and B ≅ G(A).

Step 2: Essential surjectivity. We want to show that every F ∈ Hom_{⊗}(M, M_{B})

factors as M → M_{A_{i}} ⟶^{−⊗_{A_{i}}B} M_{B} as a symmetric monoidal functor. Proposition 4.8 identifies the category Hom_{⊗}(M, M_{B}) with the category of Â-discrete Â⊗ B-modules V together with a symmetric monoidal structure on the corresponding cocontinuous functor F : M → M_{B} with F (A) = V.

Symmetric monoidality provides a distinguished isomorphism of B-modules F (A) ≅ B, so that we may assume that the underlying B-module of V is just B. The Â-module structure on B comes from the algebra homomorphism Â = End(A) → End(F (A)) ≅ End(B) ≅ B. Discreteness means that every element of B is killed by some ker(Â → A_{i}), but it clearly suffices to demand this for 1 ∈ B. In other words, Â → B factors through some A_{i}.

4.11. Corollary. The functor from acceptable commutative pro-algebras to symmetric monoidal locally presentable categories, mapping \{A_{i}\} to \underleftarrow{lim}_{i} M_{A_{i}}, is fully faithful.

Following [CJF13, BC14], this may be interpreted as the statement that affine indschemes (indexed by countable posets) are 2-affine.

Proof. If A = {A_{i}}_{i∈I^{op}} and B = {B_{j}}_{j∈J^{op}} are acceptable commutative pro-algebras then

hom(A, B) := \underleftarrow{lim}_{j} \underrightarrow{lim}_{i} hom(A_{i}, B_{j}) ≃ \underleftarrow{lim}_{j} Hom_{⊗}(\underleftarrow{lim}_{i} M_{A_{i}}, M_{B_{j}})≃ Hom_{⊗}(\underleftarrow{lim}_{i} M_{A_{i}}, \underleftarrow{lim}_{j} M_{B_{j}}).

The first equivalence uses Theorem 1.8.

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Mathematisches Institut, Universität Münster Einsteinstr. 62, D-48149 Münster, GERMANY

Department of Mathematics, University of Washington Box 354350, Seattle, WA 98195-4350, USA

Mathematics Department, Northwestern University 2033 Sheridan Road, Evanston, IL 60208, USA

Email: brandenburg@uni-muenster.de
      chirva@math.washington.edu
      theojf@math.northwestern.edu
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THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods. Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication. Full text of the journal is freely available in .dvi, Postscript and PDF from the journal’s server at http://www.tac.mta.ca/tac/ and by ftp. It is archived electronically and in printed paper format.

Subscription information Individual subscribers receive abstracts of articles by e-mail as they are published. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. For institutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh, rrosebrugh@mta.ca.

Information for authors The typesetting language of the journal is TEX, and L^{A}TEX2e is strongly encouraged. Articles should be submitted by e-mail directly to a Transmitting Editor. Please obtain detailed information on submission format and style files at http://www.tac.mta.ca/tac/.

Managing editor. Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca

TEXnical editor. Michael Barr, McGill University: barr@math.mcgill.ca

Assistant TEX editor. Gavin Seal, Ecole Polytechnique Fédérale de Lausanne:

gavin seal@fastmail.fm

Transmitting editors.

Clemens Berger, Université de Nice-Sophia Antipolis: cberger@math.unice.fr Richard Blute, Université d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: ronnie.profbrown(at)btinternet.com

Valeria de Paiva: valeria.depaiva@gmail.com

Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Kathryn Hess, Ecole Polytechnique Fédérale de Lausanne: kathryn.hess@epfl.ch Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, Macquarie University: steve.lack@mq.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@buffalo.edu Tom Leinster, University of Edinburgh: Tom.Leinster@ed.ac.uk Ieke Moerdijk, Radboud University Nijmegen: i.moerdijk@math.ru.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Giuseppe Rosolini, Università di Genova: rosolini@disi.unige.it Alex Simpson, University of Edinburgh: Alex.Simpson@ed.ac.uk James Stasheff, University of North Carolina: jds@math.upenn.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca