Grothendieck Categories As a Bilocalization of Linear Sites Toposes in Como

Grothendieck categories as a bilocalization of linear sites Toposes in Como

Julia Ramos González June 27, 2018

University of Antwerp Motivation Noncommutative Algebraic Geometry

Algebra Geometry

Grothendieck categories as a bilocalization of linear sites 1 Noncommutative Algebraic Geometry

Algebra Geometry

A = k[x, y]/⟨y − x2⟩

Grothendieck categories as a bilocalization of linear sites 1 Noncommutative Algebraic Geometry

Algebra Geometry Category Theory

Qch(Sp( (A)) =) 2 A = k[x, y]/⟨y − x ⟩ k[x,y] Mod ⟨y−x2⟩

Grothendieck categories as a bilocalization of linear sites 1 Noncommutative Algebraic Geometry

Algebra Geometry Category Theory

Qch(Sp( (A)) =) 2 A = k[x, y]/⟨y − x ⟩ k[x,y] Mod ⟨y−x2⟩

Noncommutative multiplication

Grothendieck categories as a bilocalization of linear sites 1 Noncommutative Algebraic Geometry

Algebra Geometry Category Theory

Qch(Sp( (A)) =) 2 A = k[x, y]/⟨y − x ⟩ k[x,y] Mod ⟨y−x2⟩

Noncommutative ????? multiplication

Grothendieck categories as a bilocalization of linear sites 1 Noncommutative Algebraic Geometry

Algebra Geometry Category Theory

Qch(Sp( (A)) =) 2 A = k[x, y]/⟨y − x ⟩ k[x,y] Mod ⟨y−x2⟩

Categorical Noncommutative ????? information is still multiplication available!

Grothendieck categories as a bilocalization of linear sites 1 Noncommutative Algebraic Geometry

Algebra Geometry Category Theory

Qch(Sp( (A)) =) 2 A = k[x, y]/⟨y − x ⟩ k[x,y] Mod ⟨y−x2⟩

Categorical Noncommutative ????? information is still multiplication available!

Mod(A) - afﬁne NC A NC ring → space

Grothendieck categories as a bilocalization of linear sites 1 Noncommutative Algebraic Geometry

Algebra Geometry Category Theory

Qch(Sp( (A)) =) 2 A = k[x, y]/⟨y − x ⟩ k[x,y] Mod ⟨y−x2⟩

Categorical Noncommutative ????? information is still multiplication available!

Mod(A) - afﬁne NC A NC ring → space Qgr(A) - projective NC A NC graded ring + ... → space (Serre’s thm)

Grothendieck categories as a bilocalization of linear sites 1 k − commutative ring Deﬁnition A k-linear Grothendieck category is a cocomplete k-linear abelian category with a generator and exact ﬁltered colimits.

Motivation Theorem (Gabber) Let X be a scheme. Qch(X) is a Grothendieck abelian category.

Theorem (Gabriel-Rosenberg) We can recover the geometry of a scheme X from Qch(X).

Grothendieck categories

Which categories? Grothendieck categories will be our models for NC spaces

Grothendieck categories as a bilocalization of linear sites 2 Motivation Theorem (Gabber) Let X be a scheme. Qch(X) is a Grothendieck abelian category.

Theorem (Gabriel-Rosenberg) We can recover the geometry of a scheme X from Qch(X).

Grothendieck categories

k − commutative ring Deﬁnition A k-linear Grothendieck category is a cocomplete k-linear abelian category with a generator and exact ﬁltered colimits.

Which categories? Grothendieck categories will be our models for NC spaces

Grothendieck categories as a bilocalization of linear sites 2 Grothendieck categories

k − commutative ring Deﬁnition A k-linear Grothendieck category is a cocomplete k-linear abelian category with a generator and exact ﬁltered colimits.

Which categories? Grothendieck categories will be our models for NC spaces

Motivation Theorem (Gabber) Let X be a scheme. Qch(X) is a Grothendieck abelian category.

Theorem (Gabriel-Rosenberg) We can recover the geometry of a scheme X from Qch(X).

Grothendieck categories as a bilocalization of linear sites 2 Linear sites and Grothendieck categories: the objects Deﬁnition A k-linear Grothendieck topology T on a is a Mod(k)-enriched version of the classical notion of Grothendieck topology, i.e. for every A ∈ a the covering sieves are submodules R ⊆ a(−, A) fulﬁlling the usual axioms of a Grothendieck topology in the enriched setup.

Deﬁnition A k-linear site is a pair (a, T ) as above. This is one instance of the enriched sheaf theory introduced by Borceux and Quinteiro in [BQ96]. This particular example has been analysed later on by Lowen in [Low16] with deformation theory purposes.

Linear sites

a − small k-linear category ≡ enriched over Mod(k)

Grothendieck categories as a bilocalization of linear sites 3 Deﬁnition A k-linear site is a pair (a, T ) as above. This is one instance of the enriched sheaf theory introduced by Borceux and Quinteiro in [BQ96]. This particular example has been analysed later on by Lowen in [Low16] with deformation theory purposes.

Linear sites

a − small k-linear category ≡ enriched over Mod(k) Deﬁnition A k-linear Grothendieck topology T on a is a Mod(k)-enriched version of the classical notion of Grothendieck topology, i.e. for every A ∈ a the covering sieves are submodules R ⊆ a(−, A) fulﬁlling the usual axioms of a Grothendieck topology in the enriched setup.

Grothendieck categories as a bilocalization of linear sites 3 This is one instance of the enriched sheaf theory introduced by Borceux and Quinteiro in [BQ96]. This particular example has been analysed later on by Lowen in [Low16] with deformation theory purposes.

Linear sites

Deﬁnition A k-linear site is a pair (a, T ) as above.

Grothendieck categories as a bilocalization of linear sites 3 Linear sites

Grothendieck categories as a bilocalization of linear sites 3 Deﬁnition

op • k-linear presheaves: Mod(a) := Funk(a , Mod(k)) • k-linear sheaves: Sh(a, T ) ⊆ Mod(a) the full subcategory of k-linear presheaves F such that the restriction ∼ Mod(a)(a(−, A), F) −→= Mod(a)(R, F),

for all A ∈ a and all R ∈ T (A).

Proposition (Borceux-Quinteiro) The inclusion Sh(a, T ) ⊆ Mod(a) is a localization functor. Its k-linear exact left adjoint #: Mod(a) −→ Sh(a, T ) is called sheaﬁﬁcation.

Linear sheaves and presheaves

(a, T ) − k-linear site

Grothendieck categories as a bilocalization of linear sites 4 Proposition (Borceux-Quinteiro) The inclusion Sh(a, T ) ⊆ Mod(a) is a localization functor. Its k-linear exact left adjoint #: Mod(a) −→ Sh(a, T ) is called sheaﬁﬁcation.

Linear sheaves and presheaves

(a, T ) − k-linear site Deﬁnition

for all A ∈ a and all R ∈ T (A).

Grothendieck categories as a bilocalization of linear sites 4 Linear sheaves and presheaves

(a, T ) − k-linear site Deﬁnition

for all A ∈ a and all R ∈ T (A).

Proposition (Borceux-Quinteiro) The inclusion Sh(a, T ) ⊆ Mod(a) is a localization functor. Its k-linear exact left adjoint #: Mod(a) −→ Sh(a, T ) is called sheaﬁﬁcation.

Grothendieck categories as a bilocalization of linear sites 4 Remark For each Grothendieck category C, there exist multiple choices of ∼ linear sites (a, T ) such that C = Sh(a, T ).

The Gabriel-Popescu theorem

Theorem (Gabriel-Popescu, generalization by Lowen) k-Linear Grothendieck categories are the categories of sheaves over k-linear sites (k-linear Giraud theorem).

Grothendieck categories as a bilocalization of linear sites 5 The Gabriel-Popescu theorem

Theorem (Gabriel-Popescu, generalization by Lowen) k-Linear Grothendieck categories are the categories of sheaves over k-linear sites (k-linear Giraud theorem).

Remark For each Grothendieck category C, there exist multiple choices of ∼ linear sites (a, T ) such that C = Sh(a, T ).

Grothendieck categories as a bilocalization of linear sites 5 Grothendieck categories and linear sites: the (1-)categories For classical topos theory interests:

• The (1-)category Toposk with: { } • Obj(Toposk) = k-linear Grothendieck categories A B { ∗ B A } • Toposk( , ) = geometric k-linear functors F : : F∗

The (1)-categories of Grothendieck categories

For geometric interests:

• The (1-)category Grt with: • Obj(Grt) = {k-linear Grothendieck categories} • Grt(A, B) = {colimit preserving k-linear functors A → B}

Grothendieck categories as a bilocalization of linear sites 6 The (1)-categories of Grothendieck categories

For geometric interests:

• The (1-)category Grt with: • Obj(Grt) = {k-linear Grothendieck categories} • Grt(A, B) = {colimit preserving k-linear functors A → B}

For classical topos theory interests:

• The (1-)category Toposk with: { } • Obj(Toposk) = k-linear Grothendieck categories A B { ∗ B A } • Toposk( , ) = geometric k-linear functors F : : F∗

Grothendieck categories as a bilocalization of linear sites 6 The (1-)categories of linear sites

For geometric interests:

• The (1-)category Sitek,cont with: • Obj(Sitek,cont) = {k-linear sites}

• Sitek,cont((a, Ta), (b, Tb)) = {continuous k-linear functors f :(a, Ta) → (b, Tb)} = ∗ {k-linear f : a → b | f : Mod(b) → Mod(a): M 7→ M ◦ f

restricts to a map fs : Sh(b, Tb) → Sh(a, Ta)}

Grothendieck categories as a bilocalization of linear sites 7 The (1-)categories of linear sites

For geometric interests:

• The (1-)category Sitek,cont with: • Obj(Sitek,cont) = {k-linear sites}

• Sitek,cont((a, Ta), (b, Tb)) = {continuous k-linear functors f :(a, Ta) → (b, Tb)} = ∗ {k-linear f : a → b | f : Mod(b) → Mod(a): M 7→ M ◦ f

restricts to a map fs : Sh(b, Tb) → Sh(a, Ta)}

Recall: Proposition (SGA 4)

If f :(a, Ta) −→ (b, Tb) is a continuous morphism, there exists a s s functor f : Sh(a, Ta) −→ Sh(b, Tb) such that f ⊣ fs, and in particular, it is colimit preserving.

Grothendieck categories as a bilocalization of linear sites 7 The (1-)categories of linear sites

For geometric interests:

• The (1-)category Sitek,cont with: • Obj(Sitek,cont) = {k-linear sites}

• Sitek,cont((a, Ta), (b, Tb)) = {continuous k-linear functors f :(a, Ta) → (b, Tb)} = ∗ {k-linear f : a → b | f : Mod(b) → Mod(a): M 7→ M ◦ f

restricts to a map fs : Sh(b, Tb) → Sh(a, Ta)}

For classical topos theory interests:

• The (1-)category Sitek with:

• Obj(Sitek) = {k-linear sites}

• Sitek((a, Ta), (b, Tb)) = {k-linear morphisms of sites f :(a, Ta) → (b, Tb)} =

{continuous k-linear functors f :(a, Ta) → (b, Tb) such that s f : Sh(a, Ta) → Sh(b, Tb) is exact}

Grothendieck categories as a bilocalization of linear sites 7 −→ op • ψ : Sitek Toposk given by:

ψ(a, Ta) := Sh(a, Ta)

s ψ [f :(a, Ta) → (b, Tb)] := [f : Sh(a, Ta) Sh(b, Tb): fs]

From linear sites to Grothendieck categories

We can deﬁne (pseudo)functors: −→ • ϕ : Sitek,cont Grt given by:

ϕ(a, Ta) := Sh(a, Ta)

s ϕ [f :(a, Ta) → (b, Tb)] := [f : Sh(a, Ta) → Sh(b, Tb)]

Grothendieck categories as a bilocalization of linear sites 8 From linear sites to Grothendieck categories

We can deﬁne (pseudo)functors: −→ • ϕ : Sitek,cont Grt given by:

ϕ(a, Ta) := Sh(a, Ta)

s ϕ [f :(a, Ta) → (b, Tb)] := [f : Sh(a, Ta) → Sh(b, Tb)]

−→ op • ψ : Sitek Toposk given by:

ψ(a, Ta) := Sh(a, Ta)

s ψ [f :(a, Ta) → (b, Tb)] := [f : Sh(a, Ta) Sh(b, Tb): fs]

Grothendieck categories as a bilocalization of linear sites 8 The localization intuition { → } T (G) uA : f(A) B uA generates a covering of B in (b, b)

(F) f is “full up to covers in (a, Ta)”

(FF) f is “faithful up to covers in (a, Ta)”

s Then fs : Sh(b, Tb) → Sh(a, Ta) is an equivalence, and hence so is f .A continuous f with those properties is called an LC morphism.

Remark Observe that every LC morphism is in particular a k-linear morphism of sites, hence we have:

T T ⊆ T T ⊆ T T LC((a, a), (b, b)) Sitek((a, a), (b, b)) Sitek,cont((a, a), (b, b))

Lemme de comparaison (LC) morphisms

Theorem (SGA 4, Lowen, ...)

Let f :(a, Ta) −→ (b, Tb) be a continuous k-linear morphism such −1 that f Tb = Ta and satisfying:

Grothendieck categories as a bilocalization of linear sites 9 (F) f is “full up to covers in (a, Ta)”

(FF) f is “faithful up to covers in (a, Ta)”

s Then fs : Sh(b, Tb) → Sh(a, Ta) is an equivalence, and hence so is f .A continuous f with those properties is called an LC morphism.

Remark Observe that every LC morphism is in particular a k-linear morphism of sites, hence we have:

T T ⊆ T T ⊆ T T LC((a, a), (b, b)) Sitek((a, a), (b, b)) Sitek,cont((a, a), (b, b))

Lemme de comparaison (LC) morphisms

Theorem (SGA 4, Lowen, ...)

Let f :(a, Ta) −→ (b, Tb) be a continuous k-linear morphism such −1 that f Tb = Ta and satisfying: { → } T (G) uA : f(A) B uA generates a covering of B in (b, b)

Grothendieck categories as a bilocalization of linear sites 9 (FF) f is “faithful up to covers in (a, Ta)”

s Then fs : Sh(b, Tb) → Sh(a, Ta) is an equivalence, and hence so is f .A continuous f with those properties is called an LC morphism.

Remark Observe that every LC morphism is in particular a k-linear morphism of sites, hence we have:

T T ⊆ T T ⊆ T T LC((a, a), (b, b)) Sitek((a, a), (b, b)) Sitek,cont((a, a), (b, b))

Lemme de comparaison (LC) morphisms

Theorem (SGA 4, Lowen, ...)

Let f :(a, Ta) −→ (b, Tb) be a continuous k-linear morphism such −1 that f Tb = Ta and satisfying: { → } T (G) uA : f(A) B uA generates a covering of B in (b, b)

(F) f is “full up to covers in (a, Ta)”

Grothendieck categories as a bilocalization of linear sites 9 s Then fs : Sh(b, Tb) → Sh(a, Ta) is an equivalence, and hence so is f .A continuous f with those properties is called an LC morphism.

Remark Observe that every LC morphism is in particular a k-linear morphism of sites, hence we have:

T T ⊆ T T ⊆ T T LC((a, a), (b, b)) Sitek((a, a), (b, b)) Sitek,cont((a, a), (b, b))

Lemme de comparaison (LC) morphisms

Theorem (SGA 4, Lowen, ...)

Let f :(a, Ta) −→ (b, Tb) be a continuous k-linear morphism such −1 that f Tb = Ta and satisfying: { → } T (G) uA : f(A) B uA generates a covering of B in (b, b)

(F) f is “full up to covers in (a, Ta)”

(FF) f is “faithful up to covers in (a, Ta)”

Grothendieck categories as a bilocalization of linear sites 9 Remark Observe that every LC morphism is in particular a k-linear morphism of sites, hence we have:

T T ⊆ T T ⊆ T T LC((a, a), (b, b)) Sitek((a, a), (b, b)) Sitek,cont((a, a), (b, b))

Lemme de comparaison (LC) morphisms

Theorem (SGA 4, Lowen, ...)

Let f :(a, Ta) −→ (b, Tb) be a continuous k-linear morphism such −1 that f Tb = Ta and satisfying: { → } T (G) uA : f(A) B uA generates a covering of B in (b, b)

(F) f is “full up to covers in (a, Ta)”

(FF) f is “faithful up to covers in (a, Ta)”

s Then fs : Sh(b, Tb) → Sh(a, Ta) is an equivalence, and hence so is f .A continuous f with those properties is called an LC morphism.

Grothendieck categories as a bilocalization of linear sites 9 Lemme de comparaison (LC) morphisms

Theorem (SGA 4, Lowen, ...)

Let f :(a, Ta) −→ (b, Tb) be a continuous k-linear morphism such −1 that f Tb = Ta and satisfying: { → } T (G) uA : f(A) B uA generates a covering of B in (b, b)

(F) f is “full up to covers in (a, Ta)”

(FF) f is “faithful up to covers in (a, Ta)”

s Then fs : Sh(b, Tb) → Sh(a, Ta) is an equivalence, and hence so is f .A continuous f with those properties is called an LC morphism.

Remark Observe that every LC morphism is in particular a k-linear morphism of sites, hence we have:

T T ⊆ T T ⊆ T T LC((a, a), (b, b)) Sitek((a, a), (b, b)) Sitek,cont((a, a), (b, b))

Grothendieck categories as a bilocalization of linear sites 9 The roof theorem

Theorem (Stacks Project + RG)

Given F : Sh(a, Ta) → Sh(b, Tb) colimit preserving, there exists a subcanonical site (c, Tc) and continuous morphisms f, u as in

(c, Tc) f u

(a, Ta) (b, Tb),

s with u an LC morphism, such that F = us ◦ f .

Grothendieck categories as a bilocalization of linear sites 10 The roof theorem

Theorem (Stacks Project + RG)

Given F : Sh(a, Ta) → Sh(b, Tb) colimit preserving, there exists a subcanonical site (c, Tc) and continuous morphisms f, u as in

(c, Tc) f u

(a, Ta) (b, Tb),

s with u an LC morphism, such that F = us ◦ f .

GABRIEL-ZISMAN LOCALIZATION: Morphisms in Grt are obtained in-

verting LC morphisms in Sitek,cont.

Grothendieck categories as a bilocalization of linear sites 10 The roof theorem

Theorem (Stacks Project + RG) ∗ Given F : Sh(a, Ta) Sh(b, Tb): F∗ geometric morphism, there exists a subcanonical site (c, Tc) and morphisms of sites f, u as in

(c, Tc) f u

(a, Ta) (b, Tb),

∗ s s with u an LC morphism, such that F = us ◦ f and F∗ = fs ◦ u .

Grothendieck categories as a bilocalization of linear sites 10 The roof theorem

Theorem (Stacks Project + RG) ∗ Given F : Sh(a, Ta) Sh(b, Tb): F∗ geometric morphism, there exists a subcanonical site (c, Tc) and morphisms of sites f, u as in

(c, Tc) f u

(a, Ta) (b, Tb),

∗ s s with u an LC morphism, such that F = us ◦ f and F∗ = fs ◦ u .

op GABRIEL-ZISMAN LOCALIZATION: Morphisms in Toposk are obtained inverting LC morphisms in Sitek.

Grothendieck categories as a bilocalization of linear sites 10 Localization in the bicategorical setup For classical topos theory interests, we consider:

• The 2-category Toposk as before, with 2-morphisms given by the k-linear natural transformations A : F∗ ⇒ G∗ between the right adjoints of the geometric morphisms

• The 2-category Sitek as before, with 2-morphisms given by the k-linear natural trasnformations α : f ⇒ g between morphisms of sites

The 2-categories

For geometric interests, we consider: • The 2-category Grt as before, with 2-morphisms given by the k-linear natural transformations A : F ⇒ G between colimit preserving functors

• The 2-category Sitek,cont as before, with 2-morphisms given by the k-linear natural trasnformations α : f ⇒ g between continous functors

Grothendieck categories as a bilocalization of linear sites 11 The 2-categories

For geometric interests, we consider: • The 2-category Grt as before, with 2-morphisms given by the k-linear natural transformations A : F ⇒ G between colimit preserving functors

• The 2-category Sitek,cont as before, with 2-morphisms given by the k-linear natural trasnformations α : f ⇒ g between continous functors For classical topos theory interests, we consider:

• The 2-category Toposk as before, with 2-morphisms given by the k-linear natural transformations A : F∗ ⇒ G∗ between the right adjoints of the geometric morphisms

• The 2-category Sitek as before, with 2-morphisms given by the k-linear natural trasnformations α : f ⇒ g between morphisms of sites

Grothendieck categories as a bilocalization of linear sites 11 Deﬁnition (Pronk) Let C be a bicategory and W a class of 1-morphisms admitting a left calculus of fractions. A bilocalization of C along W is a pair (C[W−1], Λ) of a bicategory and a pseudofunctor such that:

1. Λ: C → C[W−1] sends elements in W to equivalences; 2. Composition with Λ gives an equivalence of bicategories

−1 Hom(C[W ], D) −→ HomW(C, D)

for each bicategory D.

Bicategories of fractions

Grothendieck categories as a bilocalization of linear sites 12 Bicategories of fractions

Pronk’s bicategories of fractions Pronk introduces in [Pro96] a suitable generalization to the bicategorical setting of the 1-categorical notion of a class of (1-)morphisms admitting a left/right calculus of fractions and deﬁnes bicategories of fractions in this setup. Deﬁnition (Pronk) Let C be a bicategory and W a class of 1-morphisms admitting a left calculus of fractions. A bilocalization of C along W is a pair (C[W−1], Λ) of a bicategory and a pseudofunctor such that:

1. Λ: C → C[W−1] sends elements in W to equivalences; 2. Composition with Λ gives an equivalence of bicategories

−1 Hom(C[W ], D) −→ HomW(C, D)

for each bicategory D. Grothendieck categories as a bilocalization of linear sites 12 Theorem ([Ram18]) There exists a pseudofunctor which sends LC morphisms to equivalences in , such that the pseudofunctor induced by via the universal property of the bilocalization is an equivalence of bicategories.

Main result

Proposition

LC admits a left calculus of fractions in Sitek,cont.

Grothendieck categories as a bilocalization of linear sites 13 Main result

Proposition

LC admits a left calculus of fractions in Sitek,cont. Theorem ([Ram18]) There exists a pseudofunctor

→ Φ: Sitek,cont Grt

which sends LC morphisms to equivalences in Grt, such that the pseudofunctor ˜ −1 → Φ: Sitek,cont[LC ] Grt induced by Φ via the universal property of the bilocalization is an equivalence of bicategories.

Grothendieck categories as a bilocalization of linear sites 13 Main result

Proposition

LC admits a left calculus of fractions in Sitek. Theorem ([Ram18]) There exists a pseudofunctor

→ coop Ψ: Sitek Toposk

coop which sends LC morphisms to equivalences in Toposk , such that the pseudofunctor

˜ −1 → coop Ψ: Sitek[LC ] Toposk

induced by Ψ via the universal property of the bilocalization is an equivalence of bicategories.

Grothendieck categories as a bilocalization of linear sites 13 2. Observe sends LC morphisms to equivalences. 3. One proves that satisﬁes Tommasini’s criterion.

Sketch of the proof

→ 1. Extend ϕ : Sitek,cont Grt between the 1-categories to a → pseudofunctor Φ: Sitek,cont Grt between the 2-categories:

For f, g :(a, Ta) → (b, Tb) continuous functors,

Φ(α : f ⇒ g) := αs : fs ⇒ gs

deﬁned by adjunction from αs : gs ⇒ fs, where

(αs)F(A) := F(αA): gs(F)(A) = F(g(A)) → F(f(A)) = fs(F)(A),

for all F ∈ Sh(b, Tb), all A ∈ a.

Grothendieck categories as a bilocalization of linear sites 14 3. One proves that satisﬁes Tommasini’s criterion.

Sketch of the proof

→ 1. Extend ϕ : Sitek,cont Grt between the 1-categories to a → pseudofunctor Φ: Sitek,cont Grt between the 2-categories:

For f, g :(a, Ta) → (b, Tb) continuous functors,

Φ(α : f ⇒ g) := αs : fs ⇒ gs

deﬁned by adjunction from αs : gs ⇒ fs, where

(αs)F(A) := F(αA): gs(F)(A) = F(g(A)) → F(f(A)) = fs(F)(A),

for all F ∈ Sh(b, Tb), all A ∈ a.

2. Observe Φ sends LC morphisms to equivalences.

Grothendieck categories as a bilocalization of linear sites 14 Sketch of the proof

→ 1. Extend ϕ : Sitek,cont Grt between the 1-categories to a → pseudofunctor Φ: Sitek,cont Grt between the 2-categories:

For f, g :(a, Ta) → (b, Tb) continuous functors,

Φ(α : f ⇒ g) := αs : fs ⇒ gs

deﬁned by adjunction from αs : gs ⇒ fs, where

(αs)F(A) := F(αA): gs(F)(A) = F(g(A)) → F(f(A)) = fs(F)(A),

for all F ∈ Sh(b, Tb), all A ∈ a.

2. Observe Φ sends LC morphisms to equivalences. 3. One proves that Φ satisﬁes Tommasini’s criterion.

Grothendieck categories as a bilocalization of linear sites 14 Sketch of the proof

Tommasini’s criterion [Tom]: Provides necessary and sufﬁcient con- ditions for A : C → D sending a class W of 1-morphisms with a left calculus of fractions in C to equivalences in D, so that its induced pseudofunctor A˜ : C[W−1] → D is an equivalence of bicategories.

Grothendieck categories as a bilocalization of linear sites 14 Sketch of the proof

→ 1. Extend ϕ : Sitek,cont Grt between the 1-categories to a → pseudofunctor Φ: Sitek,cont Grt between the 2-categories:

For f, g :(a, Ta) → (b, Tb) continuous functors,

Φ(α : f ⇒ g) := αs : fs ⇒ gs

deﬁned by adjunction from αs : gs ⇒ fs, where

(αs)F(A) := F(αA): gs(F)(A) = F(g(A)) → F(f(A)) = fs(F)(A),

for all F ∈ Sh(b, Tb), all A ∈ a.

2. Observe Φ sends LC morphisms to equivalences. 3. One proves that Φ satisﬁes Tommasini’s criterion.

Grothendieck categories as a bilocalization of linear sites 14 Sketch of the proof

→ op 1. Extend ψ : Sitek Toposk between the 1-categories to a → coop pseudofunctor Ψ: Sitek Toposk between the 2-categories:

For f, g :(a, Ta) → (b, Tb) morphisms of sites,

Ψ(α : f ⇒ g) := αs : gs ⇒ fs

deﬁned as

(αs)F(A) := F(αA): gs(F)(A) = F(g(A)) → F(f(A)) = fs(F)(A),

for all F ∈ Sh(b, Tb), all A ∈ a.

2. Observe Ψ sends LC morphisms to equivalences. 3. One proves that Ψ satisﬁes Tommasini’s criterion.

Grothendieck categories as a bilocalization of linear sites 14 • We can safely deﬁne constructions in the 2-category of Grothendieck categories at the level of the sites as long as they behave well with respect to LC morphisms

Remarks

• The analogous results can be obtained analogously in the set-theoretical setup.

Grothendieck categories as a bilocalization of linear sites 15 Remarks

• The analogous results can be obtained analogously in the set-theoretical setup. • We can safely deﬁne constructions in the 2-category of Grothendieck categories at the level of the sites as long as they behave well with respect to LC morphisms

Grothendieck categories as a bilocalization of linear sites 15 • In addition, it is also shown in [LRS17] that LC morphisms are closed with respect to this tensor product of sites. • A bicategorical version of monoidal localization à la Day [Day73], developed in unpublished work by Pronk, would immediately provide us with a monoidal structure in Grt, where the tensor product “is given” by the one deﬁned in [LRS17].

One instance: Monoidal structure

Our original motivation: Study of a monoidal structure in Grt

• In [LRS17] we have deﬁned a tensor product of Grothendieck categories based on a tensor product of linear sites, which

deﬁnes a monoidal structure in Sitek,cont.

Grothendieck categories as a bilocalization of linear sites 16 • A bicategorical version of monoidal localization à la Day [Day73], developed in unpublished work by Pronk, would immediately provide us with a monoidal structure in Grt, where the tensor product “is given” by the one deﬁned in [LRS17].

One instance: Monoidal structure

Our original motivation: Study of a monoidal structure in Grt

• In [LRS17] we have deﬁned a tensor product of Grothendieck categories based on a tensor product of linear sites, which

deﬁnes a monoidal structure in Sitek,cont. • In addition, it is also shown in [LRS17] that LC morphisms are closed with respect to this tensor product of sites.

Grothendieck categories as a bilocalization of linear sites 16 One instance: Monoidal structure

Our original motivation: Study of a monoidal structure in Grt

• In [LRS17] we have deﬁned a tensor product of Grothendieck categories based on a tensor product of linear sites, which

deﬁnes a monoidal structure in Sitek,cont. • In addition, it is also shown in [LRS17] that LC morphisms are closed with respect to this tensor product of sites. • A bicategorical version of monoidal localization à la Day [Day73], developed in unpublished work by Pronk, would immediately provide us with a monoidal structure in Grt, where the tensor product “is given” by the one deﬁned in [LRS17].

Grothendieck categories as a bilocalization of linear sites 16 Thank you for your attention

Grothendieck categories as a bilocalization of linear sites 16 References

F. Borceux and C. Quinteiro. A theory of enriched sheaves. Cah. Topol. Géom. Différ. Catég., 37(2):145–162, 1996. B. Day. Note on monoidal localisation. Bull. Austral. Math. Soc., (8):1–16, 1973. W. Lowen. Linearized topologies and deformation theory. Topology Appl., 200:176–211, 2016. W. Lowen, J. Ramos González, and B. Shoikhet. On the tensor product of linear sites and Grothendieck categories. Int. Math. Res. Not. IMRN, 2017. D. Pronk. Etendues and stacks as bicategories of fractions. Compositio Math., (3):24–303, 1996. J. Ramos González. Grothendieck categories as a bilocalization of linear sites. Appl. Categor. Struct., 2018. M. Tommasini. Some insights on bicategories of fractions III - Equivalences of bicategories of fractions. arXiv:1410.6395 [math.CT].