A Functorial Excursion Between Algebraic Geometry and ...mellies/papers/Mellies20submitted.pdf · The language of Algebraic Geometry combines two comple-mentary and dependent levels

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A Functorial Excursion BetweenAlgebraic Geometry and Linear Logic

Paul-André MellièsInstitut de Recherche en Informatique Fondamentale (IRIF)

Université de Paris, [email protected]

AbstractThe language of Algebraic Geometry combines two comple-mentary and dependent levels of discourse: on the geometricside, schemes define spaces of the same cohesive nature asmanifolds ; on the vectorial side, every scheme X comesequipped with a symmetric monoidal category of quasico-herent modules, which may be seen as generalised vectorbundles on the scheme X . In this paper, we use the func-tor of points approach to Algebraic Geometry developedby Grothendieck in the 1970s to establish that every co-variant presheaf X on the category of commutative rings —and in particular every scheme X — comes equipped “aboveit” with a symmetric monoidal closed category PshModXof presheaves of modules. This category PshModX definesmoreover a model of intuitionistic linear logic, whose ex-ponential modality is obtained by glueing together in anappropriate way the Sweedler dual construction on ringalgebras. The purpose of this work is to establish on firmmathematical foundations the idea that linear logic shouldbe understood as a logic of generalised vector bundles, inthe same way as dependent type theory is understood todayas a logic of spaces up to homotopy.

Keywords algebraic geometry, functor of points, linearlogic, presheaves of modules, Sweedler dual constructionACM Reference Format:Paul-André Melliès. 2020. A Functorial Excursion Between Alge-braic Geometry and Linear Logic. In Proceedings of IRIF . ACM, NewYork, NY, USA, 16 pages.

1 IntroductionThe first calculus ever designed in human history is probablyelementary arithmetic with addition, subtraction and multi-plication. Beautiful but still somewhat rudimentary, the corecalculus becomes much more intricate and challenging whenone extends it with division. The critical novelty of divisionwith respect to the other operations is that, indeed, it is nota total function, because one needs to check that the denom-inator y is not equal to zero before performing the fractionxy of a value x by the value y. Understood with program-ming languages in mind, elementary arithmetic extendedwith division thus provides the basic example of a language

IRIF, Paris, France2020.

admitting “syntax errors” and meaningless expressions suchas 30 or 00. In an extraordinarily fruitful insight, Descartesunderstood that elementary arithmetic is intrinsically relatedto geometry, and that every system of polynomial equations(constructed with addition, subtraction and multiplication)describes an algebraic variety defined as the set of solutionsof the system of equations. Typically, the circle C of radius 1with center positioned at the origin may be described as theset of coordinates px ,yq P RˆR in the cartesian plane, satis-fying the well-known quadric equation x2 ` y2 “ 1. Hence,the domain of definition D of the two-variable polynomial fwith division

f : px ,yq ÞÑ 1x 2`y2´1 : D R (1)

is precisely defined as the complement set D “ R2zC of thecircle C in the plane R2. It should be noted that the algebraiccurve C defines a closed set in the usual euclidian topologyas well as in the Zariski topology, and that the domain D ofdefinition thus defines an open set in both topologies. Therelies a basic lesson and important principle of continuity:every time an operation such as f obtained by dividing twopolynomials can be performed on a given point x of thespace, there exists a “sufficiently small” neighborhoodU ofthe point x such that the operation f can be performed onevery point y P U of that neighborhood.

Basic Principles of Algebraic Geometry. A number of ex-ceptionally talented mathematicians were able to turn, gener-ations after generations, the study of this basic and primitivecalculus of polynomials with division into a sophisticatedand flourishing field of investigation — called Algebraic Ge-ometry — at the converging point of algebra, geometry andlogic, see for instance [6, 8, 9, 19, 22]. Thanks to the vision-ary ideas by Grothendieck, who played a defining role inthis specific turn, the prevailing point of view of AlgebraicGeometry today is operational and functorial at the sametime. Operational, because the notion of geometric spaceelaborated in the theory is not primary but secondary, andderived from the intuition that every space X should de-fine a topological space pX ,ΩX q equipped for every openset U with a set OX pU q of local operations, called regularfunctions. These regular functions are typically defined aspolynomials with division, in the same fashion as (1). This

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IRIF, Paris, France Paul-André Melliès

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set OX pU q of regular functions is closed under addition, sub-traction and multiplication, and thus defines a commutativering. Functorial, because every space in Algebraic Geometrygives rise to a ringed space [11, 14] defined as a topologicalspace pX ,ΩX q equipped with a contravariant functor

OX : ΩopX Ring

from the category ΩX of open sets of X ordered by inclusionto the category Ring of commutative rings. The purpose ofthe functorial action of OX is to describe the restrictionfunctions

OX pV Ď U q : OX pU q OX pV q.

which take an operation f P OX pU q defined on an opensetU and restrict it to an operation

f |V :“ OX pV Ď U qpf q

on an open subset V Ď U . This operational and functorial(rather than directly geometric) understanding of the notionof space is supported by the fundamental observation thatevery commutative ring R defines a ringed space

SpecR “ pSpecR,OSpecRq (2)

on the set of prime ideals p of the commutative ring R. Thisringed space SpecR is called the affine scheme associatedto the commutative ring R. Every space X of the theory,called a scheme, is then defined as a patchwork of affineschemes SpecRi carefully glued together using the functoriallanguage of presheaf and sheaf theory.

Modules and Vector Bundles. One distinctive feature ofAlgebraic Geometry is that every scheme X comes equippedwith a symmetric monoidal category qcModX of quasico-herent sheaf ofOX -modules. By a presheaf ofOX -modulesM of a ringed space pX ,OX q, one simply means a contravari-ant functor

M : ΩopX Ab (3)

to the category Ab of abelian groups, and such that eachabelian group MpU q defines a OX pU q-module in the usualalgebraic sense. An important result of Algebraic Geometrycalled the Serre-Swann theorem states that a vector bun-dle on a suitable ringed space pX ,OX q can be equivalentlyencoded as its sheaf of sections, which defines a locallyfree and projective sheaf of OX -modules, see [26, 31]. Ac-cordingly, a sheaf of OX -modules is called quasicoherentwhen it is locally presentable – in the expected sense that itis locally the cokernel of a morphism of free modules. Thecategory qcModX extends the category of vector bundlesin order to define an abelian category where kernels anddirect images can be computed. Consequently, a sheaf ofOS -modules A over a scheme S should be understood assome kind of very liberal notion of vector bundle over S .

This leads us to the foundational dichotomy betweenschemes and sheaves of modules which lies at the heartof contemporary Algebraic Geometry:

manifolds „ schemesvector bundles „ sheaves of modules

Linear logic. Our main purpose in the paper is to associateto every scheme X a specific model of intuitionistic linearlogic, where formulas are interpreted as generalised vectorbundles, and where proofs are interpreted as suitable vec-tor fields. In order to achieve that aim, we will construct amodel of intuitionistic linear logic, which we find convenientto formulate directly as a linear-non-linear adjunction.Recall that in this formulation of the categorical semanticsof intuitionistic linear logic, the formulas of the logic areinterpreted as the objects of a symmetric monoidal closedcategory pLinear,b,⊸, Iq equipped with an adjunction

Multiple Linear

Lin

J

Exp

(4)

where (1) the category Multiple has finite products withcartesian product ˆ and terminal object 1 and (2) the functorLinear is symmetric monoidal (in the strong sense) frompMultiple,ˆ,1q to pLinear,b, Iq. The resulting comonad

! :“ Lin ˝ Exp (5)

on the category Linear interprets the exponential modalityof linear logic, whose effect is to relax the linearity constrainton the formulas of the form !Awhich may be thus duplicatedand erased. Note that the category Multiple itself is notrequired to be cartesian closed. The reason is that for everyobject K of the category Multiple and for every object Lof the category Linear, the object K ñ L defined in thecategory Linear in the following way:

K ñ L :“ LinpKq ⊸ L

comes equipped with a natural bijection:

MultiplepK ˆK 1,ExppLqq – MultiplepK 1,ExppK ñ Lqq.

This form of currification is sufficient to interpret the simply-typed λ-calculus, in the category Multiple, using the hierar-chy of types of the form ExppLq for L in the category Linear,see [3, 23] for details. The reader will notice here impor-tant and fascinating connections with the properties of mo-noidal adjunctions in Algebraic Geometry, see for instance[2, 10, 20].

The functor of points approach. In order to associate amodel of linear logic (4) to every scheme X , one thus needsto define a symmetric monoidal closed category Linear. Inorder to keep the construction simple, and to avoid usingsheafification [19] and quasi-coherators [16, 18, 34], we take

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A Functorial Excursion Between Algebraic Geometry and Linear Logic IRIF, Paris, France

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the radical step to define Linear as the category PshModXof presheaves of OX -modules, and module homomorphismsbetween them. The category

Linear “ PshModX

is symmetric monoidal closed [13] and thus comes with atensor product bX , a tensorial unit OX (the structure sheaf)and an internal hom⊸X . Following a well-established tradi-tion in linear logic, we then define Multiple as the cartesiancategory

Multiple “ PshCoAlgXof commutative comonoids in the category PshModX ,which are called commutative coalgebras in that context.In order to work in a convenient setting, we will make greatusage of the functor of points approach to Algebraic Ge-ometry, developed by Grothendieck and his school in the1970s [5, 13, 18]. This approach is based on the observationthat the Spec construction (2) defines a functor

Spec : Ringop Ñ Scheme (6)

into the category Scheme of schemes. The functor (6) inducesin turn a nerve functor

nerve : Scheme rRing,Sets

where the presheaf nervepX q associates to every commuta-tive ring R the set of R-points in the scheme X , simplydefined as

SchemepSpecR,X q

The nerve functor is fully faithful and enables one to seeevery scheme X as a specific covariant presheaf

nervepX q : Ring Set

over the category Ring of commutative rings. We find con-venient to work in that functorial setting, and to extend ourinquiry from schemes to Ring-spaces, defined as covariantpresheaves over the category Ring of commutative rings.So, to every such Ring-space X , we will associate a spe-cific model of intuitionistic linear logic, where formulas areinterpreted in PshModX as presheaves of OX -module, un-derstood as generalised vector bundles on the Ring-space X .The construction of the linear-non-linear adjunction (4) andof the exponential modality (5) will be then performed usingthe Sweedler dual construction, which reveals a fascinatingduality between algebras and coalgebras [1, 4, 15, 27, 30].

Cardinality issues. In order to deal with the cardinality is-sues traditionally associated to the functor of point approach,see [5, 13, 18], we suppose given two universes U, V suchthatU P V. The elements ofV are called setswhile every setin the universeV in bijection with an element ofU is called asmall set. In general, a category has a class of objects and ofmorphisms, and a category is thus called a small categorywhen its class of objects and of morphisms are sets, that is,elements of the universe V. Note that the universe V itself

is a class. In the same way as in [5], we suppose that thecategory Basis introduced in §2 contains a full subcategorynoted Basis which is small. The small category Basis istypically defined in examples as the subcategory of objectswhose support is an element of U. We suppose moreoverthat this full subcategory Basis is closed under tensor prod-uct. The category Ring of commutative rings in this smallcategory Basis is also small, and a full subcategory of Ring.The Ring-spaces X are then defined as covariant presheaves

X : Ring Set

of commutative rings, that is, as functors from Ring to thecategory Set of all sets, understood as elements of the uni-verseV. What matters is that the category PointspX q associ-ated to the Ring-spaces X is a small category.

Outline of the paper. We start by introducing the cate-goryRing of commutative rings in §2 and the categoryModRof R-modules in §3. The categoriesMod andMod of mod-ules are formulated in §4 and §5. We then equip in §6 everycategoryModR with a tensor product noted bR , and deducein §7 that Mod is symmetric monoidal as a ringed category.We then construct in §8 and §9 the categoriesAlg andCoAlgof commutative algebras and coalgebras. The cofree com-mutative comonoid computed by Sweedler double dual isdescribed in §10. We then introduce the functor of points ap-proach in §11 and express in §12 the notion of presheaf mod-ules in that language. The categories PshMod and PshMod

of presheaves of modules are introduced in §13 and §14. Ourmain technical result comes in §15 with the observation thatPshMod is symmetric monoidal closed above the cartesianclosed category rRing,Sets. From this, we deduce in §16 thatthe category PshModX is symmetric monoidal closed. Afterintroducing in §17 the category PshCoAlg of presheaves ofcommutative coalgebras, we construct the linear-non-linearadjunction in §18 and conclude in §19.

2 The category Ring of commutative ringsWe suppose given a symmetricmonoidal category pBasis,b,1q

with reflexive coequalizers, preserved by the tensor producton each component. The basic example we have in mind isthe category Basis “ Ab of abelian groups and linear mapsbetween them. For that reason, we choose to use the terminol-ogy of Algebraic Geometry, and define a ring as a monoidobject pR,m ,eq in the monoidal category pBasis,b,1q. Inother words, a ring pR,m ,eq is a triple consisting of an ob-ject R of the category Basis and two mapsm : R b R Ñ Rand e : 1 Ñ R making the diagrams below commute:

R b R b R R b R

R b R R

mbR

Rbm m

m

R b R

R R

R b R

mebR

Rbe

idR

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A commutative ring is a ring pR,m ,eq such that the dia-gram below commutes

R b R R b R

Rm

γR,R

m

where γA,B : A b B Ñ B b A denotes the symmetry mapat instance A,B of the symmetric monoidal category Basis.Note that a commutative ring is just the same thing as acommutative monoid object pR,m ,eq in the symmetricmonoidal category Basis. Given two rings R and S , a ringhomorphism

u : pR,mR ,eRq pS ,mS ,eS q

is a map u : R Ñ S of the category Basis, making the dia-grams below commute:

R b R S b S

R S

mR

ubu

mS

u

1

R S

eR eS

u

The category Ring has the commutative rings as objects, andthe ring homomorphisms u : R Ñ S between them as maps.It is worth mentioning that the category Ring has finite sums,defined by the tensor product R,S ÞÑ RbS of the underlyingcategory Basis, together with the initial object defined asthe monoidal unit 1 seen as a commutative monoid.

3 The categoryModR of R-modulesGiven a commutative ring R defined in §2 as a commutativemonoid object in the category Basis, an R-module pM ,actqis a pair consisting of an objectM and of a map

act : R b M Ñ M

in the category Basis, making the diagrams below commute:

R b R b M R b M

R b M M

mRbM

Rbact act

act

R b M

M M

acteRbM

idM

Equivalently, an R-module is an Eilenberg-Moore algebrafor the monad A ÞÑ R b A associated to the commutativeringR in the category Basis. AR-module homomorphismf : M Ñ N between two R-modules M and N is a mapf : M Ñ N making the diagram below commute:

R b M R b N

M N

Rbf

actM actNf

(7)

We writeModR for the category of R-modules and R-modulehomomorphisms f : M Ñ N between them.

4 The categoryMod of modules andmodule homomorphisms

In the same spirit, a module is defined as a pair pR,Mq con-sisting of a commutative ring R and of an R-moduleM . Now,a module homomorphism,

pu, f q : pR,Mq Ñ pS ,N q

is a pair consisting of a ring homomorphism u : R Ñ S andof a map f : M Ñ N making the diagram below commute:

R b M S b N

M N

ubf

actM actNf

(8)

The category Mod has the modules pR,Mq as objects, andthe module homomorphisms pu, f q : pR,Mq Ñ pS ,N q asmorphisms. There is an obvious functor

π : Mod Ring (9)

which transports every module pR,Mq to its underlying com-mutative ring R, and every module homomorphism pu, f q :pR,Mq Ñ pS ,N q to its underlying ring homomorphism u :R Ñ S . For that reason, we often find convenient to write

u : R S |ù f : M N

for a module homomorphism pu, f q : pR,Mq Ñ pS ,N q as de-fined in (8). The notation is inspired by [24] and the intuitionthat every ring homomorphism u : R Ñ S induces a “fiber”consisting of all the module homomorphisms of the formpu, f q : pR,Mq Ñ pS ,N q living “above” and refining the ringhomomorphismu : R Ñ S . Note moreover that the fiber of πof a commutative ring R coincides with the category ModRdefined above.As it is well-known, the functor π : Mod Ñ Ring de-

fines a Grothendieck fibration. The reason is that every pairconsisting of a ring homomorphism u : R Ñ S and of aS-module pN ,actN q induces an R-module noted

resuN “ pN ,act1N q

with same underlying object N as the original S-module, andwith action map act1N : RbN Ñ N defined as the composite:

act1N “ R b N S b N NubN actN

The S-module pN ,actN q comes moreover with a modulehomomorphism

u : R S |ù idN : resuN N (10)

which is weakly cartesian (or cartesian in the original senseby Grothendieck [12], Exposé VI, Def. 5.1) in the sense thatevery module homomorphism

u : R S |ù f : M N

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factors uniquely as

M resuN NpidR ,hq pu,idN q

for a map h : M Ñ resuN in the categoryModR . Note thatthe unique solution h is in fact equal to the original mapf : M Ñ N , seen this time as a map between R-modules.Given a ring homomorphism u : R Ñ S , the existence of aweakly cartesian map of the form (10) for every S-module Nensures the existence of a functor

resu : ModS ModR (11)

called restriction of scalar along u. This added to the factthat the composite of two weakly cartesian maps is a weaklycartesian map, establishes that the functor π : Mod Ñ Ringis a Grothendieck fibration.At this stage, we use the fact that the category Basis has

reflexive coequalizers, preserved by the tensor product com-ponentwise, in order to show that the functor (11) has a leftadjoint noted

extu : ModR ModS .

The functor extu is constructed as follows. Every ring homo-morphism u : R Ñ S induces a R b S-module noted R bu Sand defined as the reflexive coequalizer of the diagram:

R b R b S R b S

mRbS

pRbmS q˝pRbubSq

RbeRbS

computed in the category Basis. Now, given three commuta-tive rings R, S1 and S2, we define the functor

⊛R : ModS1bR ˆ ModRbS2 ModS1bS2 (12)

which transports a pair pM ,N q consisting of a S1bR-moduleMand a RbS2-moduleN to the S1bS2-moduleMbRN definedas the reflexive coequalizer of the diagram:

M b R b N M b N

actM b N

M b actN

M b eR b N

computed in the category Basis. Here, the two maps actM :MbR Ñ M and actN : RbN Ñ N are deduced by restrictionof scalar from the S1bR-module structure ofM and theRbS2-module structure of N . The left adjoint functor extu is thendefined in the following way

extu : M ÞÑ M ⊛R pR bu Sq

by applying the construction (12) to the R-moduleM and tothe R b S-module R bu S defined earlier, in order to obtainthe S-moduleM bR pR bu Sq.

5 The categoryMod of modules andmodule retromorphism

Here, we make the extra assumption that the category Basisis symmetric monoidal closed, with coreflexive equalizers.The internal hom-object in Basis is noted rM ,N s. Amoduleretromorphism

pu, f q : pS ,N q Ñ pR,Mq

is a pair pu, f q consisting of a ring homomorphism u : R Ñ

S and of a map f : N Ñ M making the diagram belowcommute:

R b M R b N S b N

M N

actM

Rbf ubN

actN

f

(13)

The categoryMod has ring modules as objects and moduleretromorphisms pu, f q : pS ,N q Ñ pR,Mq as morphisms.There is an obvious functor

π : Mod Ring (14)

which transports every module retromorphism pu, f q to theunderlying ring homomorphism u : R Ñ S . Note that thefunctor π is a Grothendieck fibration, which coincides infact with the opposite Grothendieck fibration of π definedin (9), see [17, 32]. It turns out that the functor π is in facta bifibration. Given a ring homomorphism u : R Ñ S , thefunctor defined by coextension of scalar along u

coextu : ModR ModS

transports every R-module pM ,actM q to the S-module

rS ,Msu (15)

defined as the coreflexive equalizer, in the category A , ofthe diagram below:

rS ,Ms rR b S ,Ms

rubS,Ms˝rmS ,Ms

rRbS,actM s˝rRb ,Rb´s

reRbS,Ms

Note that the coreflexive equalizer rS ,Msu provides an in-ternal description, in the category Basis, of the set of mapsf : S Ñ M making the diagram below commute:

R b M R b S

S b S

M S

actM

Rbf

ubS

mS

f

(16)

or equivalently, as the set of R-module homomorphismsf : resuS Ñ M . In summary, putting together the results and

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constructions of §3, §4 and §5, every ring homomorphism u :R Ñ S induces three functors

ModR ModScoextu

extu

resu

organized into a sequence of adjunctions

extu % resu % coextu

where extension of scalar extu is left adjoint, and coextensionof scalar coextu , right adjoint to restriction of scalar resu .

6 The categoryModR is symmetricmonoidal closed

A well-known fact of algebra is that the categoryModR ofR-modules is symmetric monoidal closed. Its tensor product

bR : ModR ˆ ModR ModR

transports every pair of R-modulesM , N into the reflexivecoequalizer of the diagram below

M b R b N M b N

actM b N

M b actN

M b eR b N

computed in the category Basis, and its tensorial unit is thecommutative ring 1R “ R itself, seen as an R-module. Theinternal hom of the category ModR noted

r´,´sR : ModopR ˆ ModR ModR

transports every pair of R-modulesM , N to the coreflexiveequalizer of the diagram below:

rM ,N s rR b M ,N s

ractM ,Ms

rRbM,actN s˝rRb ,Rb´s

reRbM,N s

Note that the definitions of the tensor product bR and ofthe internal hom r´,´sR are mild variations of the construc-tions (12) and (15) described previously in §4 and §5.

7 The categoryMod as a symmetricmonoidal ringed category

The category Mod defined in §4 comes with a functor π :Mod Ñ Ringwhose fibers are precisely the categoriesModRof R-modules. For that reason, the family of tensor prod-ucts bR described in §6 induces a fibrewise (or vertical)tensor product on Mod above Ring, which may be conve-niently described in the followingway. Consider the categoryMod ˆRing Mod defined by the pullback diagram below:

Mod ˆRing Mod Mod

Mod Ring

π

π

The fibrewise tensor product

bMod : Mod ˆRing Mod Mod (17)

transports every pair of modules pR,Mq and pR,N q on thesame commutative monoid R to the R-module pR,M bR N q,and every pair of module homomorphisms

u : R S |ù h1 : M1 N1

u : R S |ù h2 : M2 N2

above the same ring homomorphismu : R Ñ S to themodulehomomorphism

u : R S |ù h1 bu h2 : M1 bR M2 N1 bS N2

where h1 bu h2 is defined as the unique map making thediagram below commute:

M1 b R b M2 N1 b S b N2

M1 b M2 N1 b N2

M1 bR M2 N1 bS N2

h1bubh2

actM1bM2 M1bactM2 actN1bN2 N1bactN2

h1bh2

quotient map quotient map

h1buh2

The fibrewise unit is defined as the functor

1Mod : Ring Mod (18)

which transports every commutative ring R into itself, seenas an R-module. The categorical situation may be under-stood in the following way. A ringed category is definedas a pair pC ,πq consisting of a category C and of a functorπ : C Ñ Ring to the category of commutative rings. Theslice 2-category CatRing has ringed categories as objects,fibrewise functors and natural tranformations as 1-cells and2-cells. The 2-category CatRing is cartesian, with carte-sian product defined by the expected pullback above Ring.Using that setting, one observes that the fibrewise tensorproduct (17) and tensor unit (18) define a symmetric pseu-domonoid structure on the ringed category pMod,πq in the2-category CatRing.

8 The category Alg of commutativealgebras

Given a commutative ring R in the category Basis, a com-mutative R-algebra A is defined as a commutative monoidin the symmetric monoidal category pModR ,bR ,1Rq. A com-mutative algebra is defined as a pair pR,Aq consisting of acommutative ring R and of a commutative R-algebra A. Analgebra map pu, f q : pR,Aq Ñ pS ,Bq is a pair pu, f q consist-ing of a ring homomorphism u : R Ñ S and of a module

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homomorphism f : M Ñ N making the diagrams commute:

A bR A B bS B

A B

f bu f

mA mB

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1R 1S

A B

u

eA eB

f

We suppose that for every commutative monoid R, the sym-metric monoidal category ModR has a free commutativemonoids. This means that the forgetful functor

ForgetR : AlgR ModR

has a left adjoint, which we note

SymR : ModR AlgR

By glueing together the fibers ModR and AlgR , we obtain inthis way an adjunction

Mod AlgSym

J

Forget

which is moreover vertical (or fibrewise) above Ring.

9 The category CoAlg of commutativecoalgebras

Suppose given a commutative ring R. A commutative R-coalgebra K is defined as a commutative comonoid in thesymmetric monoidal category pModR ,bR ,1Rq. In the sameway as in §8, we suppose that for every commutativemonoidR,the symmetric monoidal categoryModR has a cofree com-mutative comonoids. This means that the forgetful functor

ForgetR : CoAlgR ModR

has a right adjoint, which we note

CoFreeR : ModR CoAlgR

By glueing together the fibersModR and CoAlgR , we obtainin this way an adjunction

CoAlg ModForget

J

CoFree

(19)

which is moreover vertical (or fibrewise) above Ring.

10 The cofree construction for Basis “ AbOne remarkable aspect of the adjunction (19) in §9 is that itcan be defined at a purely formal level, without the need foran explicit description of the cofree construction CoFreeRperformed in each fiber ModR . This is important becauseMurfet has given [4, 27] an explicit description of the con-struction in the case when R “ k of an algebraically closedfield of characteristic 0, but the finite dual construction bySweedler [1, 15, 29, 30, 33] remains somewhat mysteriousin the case of a general commutative ring R. Let us give a

brief description of the construction here. First of all, thefree commutative algebra SymR can be computed as followswith enough colimits in the categoryModR :

SymR : M ÞÑà

nPN

M bsymR ¨ ¨ ¨ b

symR M

whereM bsymR ¨ ¨ ¨ b

symR M denotes the symmetrized tensor

product of M with itself, taken n times. In the case whenR “ k is a field, the Sweedler construction transports everyk-vector space V into the commutative k-algebra defined as

CoFreek pV q “ pSymk pV ˚qq˝

where the vector space V ˚ denotes the dual of the vectorspace V , and A ÞÑ A˝ denotes Sweedler’s finite dual con-struction, which transports everyR-algebraA (not necessar-ily commutative) to a R-coalgebra A˝. In the general case ofa commutative ring R, one needs to apply instead the formalconstruction designed in [29, 30] and adapted to the construc-tion of the cofree commutative R-coalgebra CoFreeRpMq

generated by an R-moduleM in the category ModR .

11 Functors of points and Ring-spacesWeworkwith covariant presheavesX ,Y on the categoryRingof commutative rings, which we call Ring-spaces. To everysuch Ring-space

X : Ring Set

we associate its Grothendieck category PointspX q whoseobjects are the pairs pR,xq with x P X pRq and whose mapspR,xq Ñ pS ,yq are ring homomorphisms u : R Ñ S trans-porting the element x P X pRq to the element y P X pSq, inthe sense that

X puqpxq “ y.

The category PointspX q comes with an obvious functor

πX : PointspX q Ring Ring

This functor, called the functor of point ofX , defines in fact adiscrete Grothendieck opfibration above the category Ring.A map f : X Ñ Y between Ring-spaces may be equivalentlydefined as a functor

f : PointspX q PointspY q

making the diagram below commute:

PointspX q PointspY q

Ring

f

πX πY

Note that the functor f is itself necessarily a discrete opfibra-tion: this follows from the fact that discrete fibrations definea right orthogonality class of a factorization system on Cat,with cofinal functors as elements of the left orthogonalityclass. Note also that the Ring-space

SpecZ : R ÞÑ t˚Ru (20)7

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is the terminal object of the category rRing,Sets, and thatits Grothendieck category is isomorphic to category Ringitself.

12 Presheaves of modulesWe suppose given a Ring-space

X : Ring Set

The following definition is adapted from [5, 18].

Definition 12.1. A presheaf of modulesM on the Ring-space X or more simply, an OX -moduleM , consists of thefollowing data:

‚ for each point pR,xq P PointspX q, a module Mx overthe commutative ring R,

‚ for each map u : pR,xq Ñ pS ,yq in PointspX q, a mod-ule homomorphism

u : R S |ù θpu,xq : Mx Ny

living over the ring homomorphism u : R Ñ S .The map θ is moreover required to satisfy the followingfunctorial properties: first of all, the identity on the pointpR,xq in the category PointspX q is transported to the identitymap on the associated R-module

idR |ù θpidpR,xqq “ idMx

and given two maps

pu,xq : pR,xq Ñ pS ,yq pv,yq : pS ,yq Ñ pT ,zq

in the category PointspX q, one has:

v ˝ u |ù θppv,yq ˝ pu,xqq “ θpv,yq ˝ θpu,xq

where composition is computed in the discrete opfibrationPointspX q Ñ Ring.

In the sequel, we will use the following equivalent formu-lation of presheaves of modules:

Proposition 12.1. An OX -moduleM is the same thing as afunctor

M : PointspX q Mod


PointspX q Mod

Ring

M

πX π

Note that this specific formulation of presheaves of mod-ules is the one used by Kontsevich and Rosenberg in theirwork on noncommutative geometry [18]. See also the discus-sion on the nLab entry [28]. It should be noted moreover thatevery Ring-space X comes equipped with a specific presheaf

of module, called the structure presheaf of modules of X ,and defined as the composite

OX : PointspX q Ring ModπX 1Mod (21)

where the functor O : Ring Ñ Mod denotes the sectionof π : Mod Ñ Ring which transports every commutativering R to itself, seen as an R-module.

13 The category PshMod of presheaves ofmodules and forward morphisms

We construct the category PshMod of presheaves of mod-ules and forward morphisms, in the following way. Apresheaf of modules pX ,Mq is defined as a pair consistingof a Ring-space

X : Ring Set

together with a presheaf OX -module M , as formulated in§12. A forward morphism between presheaf of modules

pf ,φq : pX ,Mq pY ,N q

is defined as a morphism (= natural transformation) of Ring-spaces f : X Ñ Y together with a natural transformation


Mod

Pointspf q

M N

φ (22)

The natural transformation is moreover required to be verti-cal (or fibrewise) above Ring, in the sense that the naturaltransformation obtained by composing

π ˝ M π ˝ N ˝ Pointspf qπ˝φ

is equal to the identity natural transformation. Equivalently,and expressed in a somewhat more fundamental way, oneasks that the natural transformation

πX π ˝ M π ˝ N ˝ Pointspf q πY ˝ Pointspf qid π˝φ id

obtained by composing the three natural transformationsdepicted below


Mod

Ring

Pointspf q

M

πX

N

πYπ

φ

idid

coincides with the identity natural transformation from πXto πY ˝ f . There is an obvious functor

p : PshMod rRing,Sets (23)8

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which transports every presheaf of modules pX ,Mq to its un-derlying Ring-space X , and every forward morphism pf ,φq :pX ,Mq Ñ pY ,N q to its underlying morphism f : X Ñ Ybetween Ring-spaces. We thus find convenient to write

f : X Y |ù φ : M N

for a forward morphism pf ,φq : pX ,Mq Ñ pY ,N q betweenpresheaves of modules.The functor p is a Grothendieck fibration because every

morphism f : X Ñ Y between Ring-spaces X and Y inducesa functor

f ˚ : PshModY PshModX (24)

which transports every OY -module N into the OX -moduleN ˝ Pointspf q obtained by precomposition with the discretefibration Pointspf q, as depicted below:


Mod

Ring

Pointspf q

πX πY

N

π

In fact, it turns out that the functor p is also a Grothendieckbifibration, but for less immediate reasons. In order to es-tablish the property, we make the extra assumption that thecategory Ring as well as every category ModR associatedto a commutative ring R has small colimits. Note that theproperty holds in the case of the category Basis “ Ab ofabelian groups.

Proposition 13.1. For every morphism f : X Ñ Y betweenRing-spaces, there exists a functor

f ! : PshModX PshModY

left adjoint to the functor f ˚.

A proof of the statement based on a purely 2-categorical con-struction of the OY -module f !pMq appears in the Appendix,§A. It is worth noting that the OY -module f !pMq can be alsodescribed more directly with an explicit formula:

f !pMq : y P Y pRq ÞÑà

txPX pRq,f x“yu

Mx P ModR .

The adjunction f ! % f ˚ gives rise to a sequence of naturalbijections, formulated in the type-theoretic fashion of [24]:

idX : X Ñ X |ù M Ñ f ˚pN q

f : X Ñ Y |ù M Ñ N

idY : Y Ñ Y |ù f !pMq Ñ N

14 The category PshMod of presheaf ofmodules and backward morphisms

We construct the category PshMod of presheaves of mod-ules and backward morphisms, in the following way. Abackward morphism between presheaves of modules

pf ,ψ q : pX ,Mq pY ,N q

is defined as a morphism f : X Ñ Y between Ring-spacestogether with a natural transformation

ψ : N ˝ f M : PointspX q PointspY q

which is moreover vertical in the sense that the diagrambelow commutes:


Mod

Ring

f

M

πX

N

πYπ

ψ

The category PshMod has presheaves ofmodules as objects,and backward morphism as morphisms. There is an obviousfunctor

p : PshModrRing,Sets

We thus find convenient to write

f : X Y |ùop ψ : M N

for such a backward morphism pf ,ψ q : pX ,Mq Ñ pY ,N q

between presheaves of modules. As the opposite of the fibra-tion p, the functor p is also a Grothendieck fibration withthe functor

pf ˚qop : PshModopY PshModop

X

as pullback functor. In order to establish the following prop-erty, we make the extra assumption that the category Ringas well as every categoryModR associated to a commutativering R has small limits.

Proposition 14.1. For every morphism f : X Ñ Y betweenRing-spaces, there exists a functor called direct image

f˚ : PshModX PshModY .

right adjoint to the functor f ˚.

Note that, accordingly, the functor pf˚qop is left adjoint tothe functor pf ˚qop . Quite interestingly, the proof of the state-ment works just as in the case of Prop. 13.1, and relies on apurely 2-categorical construction of the OY -module f˚pMq,dual to the construction of the OY -module f !pMq, see theAppendix, §B for details. The adjunction f ˚ % f˚ gives

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rise to a sequence of natural bijections, formulated in thetype-theoretic fashion of [24]:

idX : X Ñ X |ùop M Ñ f ˚pN q

f : X Ñ Y |ùop M Ñ N

idY : Y Ñ Y |ùop f˚pMq Ñ N

In summary, we obtain that every morphism f : X Ñ Ybetween Ring-spaces X and Y induces three functors

PshModX PshModY

f˚

f !

f ˚ (25)

organized into a sequence of adjunctions

f ! % f ˚ % f˚.

15 The category PshMod is symmetricmonoidal closed above rRing,Sets

We establish in this section one of the main conceptual andtechnical contributions of the paper, directly inspired by thework by Mellies and Zeilberger on refinement type systems[24, 25]. As a presheaf category, the category rRing,Sets ofRing-spaces is cartesian closed. We exhibit here a symmet-ric monoidal closed structure on the category PshMod ofpresheaf of modules, designed in such a way that the functorp is symmetric monoidal closed. The construction is new, asfar as we know. The result is important from a methodologi-cal point of view, since it establishes the general presheavesof modules (instead of the more traditional notion of quasi-coherent sheaf of modules) as an appropriate foundation fora connection between algebraic geometry and formal logic.Suppose given a pair of Ring-spaces

X ,Y : Ring Set

and a pair of presheaves of modulesM and N over them:

M P PshModX N P PshModY .

Recall that the cartesian product X ˆ Y of Ring-spaces isdefined pointwise:

X ˆ Y : R ÞÑ X pRq ˆ Y pRq.

The tensor product

M b N P PshModXˆY

is defined using the isomorphism:

PointspX ˆ Y q – PointspX q ˆRing PointspY q

as the presheaf of modules

PointspX ˆ Y q Mod ˆRing Mod ModpM,N q b

where the functor pM ,N q is defined by universality. The unitof the tensor product just defined is the structure presheafof modules

pSpecZ,OSpecZq : pR,˚Rq ÞÑ R P ModR

on the terminal object SpecZ of the category rRing,Sets,where ˚R denotes the unique element of the singleton setSpecZpRq, see (20). Before explaining the definition of theinternal hom

M ⊸ N : PointspX ñ Y q Mod

on presheaves of modules (27), we recall that the internalhom

X ñ Y : Ring Set

is defined as the covariant presheaf which associates to everycommutative ring R P Ring the set

X ñ Y : R ÞÑ prRing,SetsyRqpyR ˆ X ,yR ˆ Y q

of natural transformations making the diagram commute:

yR ˆ X yR ˆ Y

yR

f

πR,X πR,Y(26)

Here, yR P rRing,Sets denotes the Yoneda presheaf gener-ated by the commutative ring R, in the following way:

yR : S ÞÑ RingpR,Sq : Ring Set

while πR,X and πR,Y denote the first projections. The maincontribution of the section comes now, with the followingconstruction. The presheaf of modules

M ⊸ N P PshModXñY (27)

is constructed in the following way. To every element

f P pX ñ Y qpRq

we associate the R-module

pM ⊸ N qf

consisting of all natural transformations φ making the dia-gram commute:

PointspyR ˆ X q PointspyR ˆ Y q


Mod

Ring

f

PointspπR,X q PointspπR,Y q

πX

M N

πYπ

φ

(28)Such a natural transformation φ is a family of module homo-morphisms

idS : S S |ù φx,u : Mx Nf px,uq

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for u : R Ñ S and x P X pSq, natural in u and x in thesense that for every ring homomorphism v : S Ñ S 1 withX pvqpxq “ x 1, the diagram should commute:

Mx Nf px,uq

Mx 1 Nf px 1,v˝uq

φx ,u

Mv Nf pv,vq

φx 1,v˝u

The R-module of such natural transformations can be com-puted in the categoryModR using the following end formula:

ż

pu :RÑS,xPX pSqqPPointspyRˆX q

resu´

rMx ,Nf px,uqsS

¯

One establishes that

Theorem 15.1. The tensor productM ,N ÞÑ M b N and theimplication M ,N ÞÑ M ⊸ N equip the category PshModwith the structure of a symmetric monoidal category. Thisstructure is moreover transported by the functor p in (23) to thecartesian closed structure of the presheaf category rRing,Sets.

The reader will find in the Appendix, §C, the central argu-ment for the proof of Thm. 15.1.

16 The category PshModX is symmetricmonoidal closed

We illustrate the benefits of Thm. 15.1 by establishing, in thespirit of [24, 25], that the category PshModX associated to agiven Ring-space

X : Ring Set

is symmetric monoidal closed. The tensor productM bX Nof a pair of OX -modulesM ,N is defined as

M bX N :“ ∆˚pM b N q

where we use the notation

∆ : X X ˆ X

to denote the diagonal map coming from the cartesian struc-ture of the category rRing,Sets of covariant presheaves. Thetensorial unit is defined as the structure presheaf of modulesOX defined in (21). The internal homM ⊸X N of a pair ofOX -modulesM ,N is defined as

M ⊸X N :“ curry˚pM ⊸ ∆˚pN qq

wherecurry : X X ñ pX ˆ X q

is the map obtained by currifying the identity map

idXˆX : X ˆ X X ˆ X

on the second component X . One obtains that

Proposition 16.1. The category PshModX equipped withbX and⊸X defines a symmetric monoidal closed category.

idX : X Ñ X |ù pM bX N q Ñ P

idX : X Ñ X |ù ∆˚pM b N q Ñ P

idX : X Ñ X |ùop P Ñ ∆˚pM b N q

∆ : X Ñ X ˆ X |ùop P Ñ M b N

idXˆX : X ˆ X Ñ X ˆ X |ùop ∆˚pPq Ñ M b N

idXˆX : X ˆ X Ñ X ˆ X |ù M b N Ñ ∆˚pPq

curry : X Ñ X ñ pX ˆ X q |ù N Ñ M ⊸ ∆˚pPq

idX : X Ñ X |ù N Ñ curry˚pM ⊸ ∆˚pPqq

idX : X Ñ X |ù N Ñ pM ⊸X Pq

Figure 1. Sequence of natural bijections establishing thatpM bX ´q is left adjoint to pM ⊸X ´q forM P PshModX .

The proof that pM bX ´q is left adjoint to pM ⊸X ´q

can be decomposed in a sequence of elementary naturalbijections, as described in Fig. 1.Moreover, given a morphism X Ñ Y in rRing,Sets and

twoOY -modulesM andN , the fact that∆Y ˝ f “ pf ˆ f q˝∆Xand the isomorphism

pf ˆ f q˚

pM b N q – f ˚pMq b f ˚pN q

imply that

f ˚ : PshModY PshModX

defines a strongly monoidal functor, in the sense thatthere exists a family of isomorphisms in PshModX

mX ,M,Y ,N : f ˚pMq bX f ˚pN q f ˚pM bY N q„

mX ,Y : OX f ˚pOY q„

making the expected coherence diagrams commute. Fromthis follows that its right adjoint functor f˚ as well as theadjunction f ˚ % f˚ are lax symmetric monoidal ; and that itsleft adjoint functor f ! as well as the adjunction f ! % f ˚ areoplax symmetric monoidal. In particular, the two functors f˚

and f ! come equipped with natural families of morphisms:

f˚pMq bN f˚pY q f˚pM bX N q OY f˚pOX q

f !pM bX N q f !pMq bY f !pN q f !pOX q OY

parametrized by OX -modulesM and N . See for instance thediscussion in [23], Section 5.15.

17 The category PshCoAlg of presheaves ofcommutative coalgebras

A presheaf pX ,Kq of commutative coalgebras is definedas a pair consisting of a Ring-space X : Ring Ñ Set and ofa functor

K : PointspX q CoAlg (29)11

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CoAlg Mod

PointspX q Ring

Forget

πK

πX

A morphism pf ,φq : pX ,Kq Ñ pY ,Lq between two suchpresheaves of commutative coalgebras is defined as a pairconsisting of a map f : X Ñ Y between presheaves and of anatural transformation φ : K Ñ L ˝ f making the diagrambelow commute:


CoAlg

Mod

Ring

f

K

πX

L

πY

Forget

π

φ

The resulting category PshCoAlg of presheaves of commu-tative coalgebras comes equipped with an obvious forgetfulfunctor

Lin : PshCoAlg PshMod (30)

and thus with a composite functor

q “ p ˝ Lin : PshCoAlg rRing,Sets

We write PshCoAlgX for the fiber category of the func-tor q above a given Ring-space X P rRing,Sets. By con-struction, PshCoAlgX is the category of commutative OX -coalgebras, defined as the presheaves of commutative coal-gebras of the form pX ,Kq ; and of morphisms of the formpidX ,φq : pX ,Kq Ñ pX ,Lq between them. An important ob-servation for the construction of the model of linear logicwhich comes next is that

Proposition 17.1. The category PshCoAlgX coincides withthe category of commutative comonoids in the symmetric mo-noidal category pPshModX ,bX ,OX q.

Another important property to notice at this stage is that thefunctor q is a bifibration. This essentially comes from thefact that the adjunction f ! % f ˚ on presheaves of modulesdescribed in (25) is in fact an oplax monoidal adjunction (seethe discussion in §16) and thus lifts to an adjunction

PshCoAlgX PshCoAlgYf !

f ˚

between the categories of commutative coalgebras.

18 The linear-non-linear adjunction onPshModX

We have established in §16 that the category PshModX issymmetric monoidal closed for every Ring-space X . In orderto obtain a model of intuitionistic linear logic, we constructbelow a linear-non-linear adjunction

PshCoAlgX PshModX

LinX

J

ExpX

(31)

The functor Lin defined in (30) is a functor of categoriesfibered above Ring, and the functor LinX is thus obtainedby restricting it to the fiber of X :

LinX : PshCoAlgX PshModX

By a general and well-known property of categories of com-mutative comonoids, together with Prop. 17.1, the categoryPshCoAlgX is cartesian, with the cartesian product

K ˆ L “ K bX L

and the terminal object defined as the structure presheaf ofmodules OX and tensorial unit of PshModX , see for instance[23], Section 6.5, for a discussion. From this follows that thefunctor LinX is strong symmetric monoidal.

The functor ExpX is defined in the following way: it trans-ports an OX -moduleM defined by a functor

PointspX q ModM

to the commutative OX -algebra defined by postcomposition

PointspX q Mod CoAlgM CoFree

We claim that the functor ExpX is right adjoint to the forget-ful functorLinX . Indeed, given a commutativeOX -coalgebraKand an OX -moduleM , there is a family of bijections

PshModX pLinX pKq,Mq – PshCoAlgX pK ,ExpX pMqq

natural in K and M , derived from the fact that there is aone-to-one relationship between the OX -module homomor-phisms

φ : LinX pKq M

defined as the (fibrewise) natural transformations of the form

PointspX q

CoAlg Mod

K M

Forget

φ

and the commutative OX -coalgebra morphisms

ψ : K ExpX pMq

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defined as the (fibrewise) natural transformations of the form

PointspX q

CoAlg Mod

K M

CoFree

ψ

The bijection itself comes from the fact that the functorCoFree is right adjoint to Forget in the 2-category CatRingof ringed categories, see for instance [23], section 5.11, for adiscussion. We conclude with the main result of the article:

Theorem 18.1. The adjunction (31) is a linear-non-linearadjunction, and thus defines for every Ring-space X a modelof intuitionistic linear logic on the symmetric monoidal closedcategory PshModX of presheaves of OX -modules.

19 Conclusion and future worksOur main technical contribution in this work is to resolve anold open question in the field of mathematical logic, whichis to construct a model of linear logic — including the expo-nential modality — in the functorial language of AlgebraicGeometry. By performing this construction in the presentpaper, we hope to integrate linear logic as a basic and verynatural component in the current process of geometrizationof type theory. The guiding idea here is that linear logicshould be seen as the logic of generalised vector bundles,in the same way as Martin-Löf type theory with identity isseen today as the logic of spaces up to homotopy, formulatedin the language of 8-topos theory. One would thus obtainthe following dictionary:

dependent types „ spaces up to homotopylinear types „ vector bundles

The idea was already implicit in Ehrhard’s differential linearlogic [7] and it is thus very good news to see this founda-tional intuition confirmed by our construction. In a niceand inspiring series of recent works, Murfet and his studentClift [4, 27] have established thatModR equipped with theSweedler exponential modality defines a model of differentiallinear logic whenever the commutative ring R “ k is an alge-braically closed field of characteristic 0. One important ques-tion which we leave for future work is to understand whetherthe vector bundle semantics of linear logic just constructedin PshModX extends as it stands (or as a slight variant) to amodel of differential linear logic. Another important researchdirection in good harmony with homotopy type theory willbe to shift to Derived Algebraic Geometry in the style ofToën and Lurie [21, 35] by building our constructions on thesymmetric monoidal category Basis “ dgAb of differentialgraded abelian groups, with its category Ring “ dgAlg ofcommutative differential graded algebras considered up toquasi-isomorphisms. We leave that important and fascinat-ing question for future work.

References[1] Matthieu Anel and André Joyal. 2013. Sweedler theory

of (co)algebras and the bar-cobar constructions. (2013).https://arxiv.org/abs/1309.6952.

[2] Paul Balmer, Ivo Dell’Ambogio, and Beren Sanders. 2016.Grothendieck-Neeman duality and the Wirthmüller isomorphism.Compositio Mathematica 8 (2016), 1740–1776.

[3] Nick Benton. 1994. A Mixed Linear and Non-Linear Logic: Proofs,Terms and Models. In Computer Science Logic, 8th International Work-shop, CSL ’94 (Lecture Notes in Computer Science), Vol. 933. SpringerVerlag.

[4] James Clift and Daniel Murfet. 2017. Cofree coalgebras and differentiallinear logic. (2017). https://arxiv.org/abs/1701.01285.

[5] Michel Demazure and Peter Gabriel. 1980. Introduction to AlgebraicGeometry and Algebraic Group. Mathematics Studies, Vol. 39. NorthHolland.

[6] Jean Dieudonné. 1972. The Historical Development of Algebraic Ge-ometry. The American Mathematical Monthly 79, 8 (October 1972),827–866.

[7] Thomas Ehrhard. 2018. An introduction to Differential Linear Logic:proof-nets, models and antiderivatives. Mathematical Structures inComputer Science 28, 7 (2018), 995–1060.

[8] David Eisenbud and Joe Harris. 1991. The Geometry of Schemes. Num-ber 197 in Graduate Texts in Mathematics. Springer Verlag.

[9] Barbara Fantechi, Lothar Götsche, Luc Illusie, Steven L. Kleiman, NitinNitsure, and Angelo Vistoli. 2005. Fundamental Algebraic Geometry.Mathematical Surveys and Monographs, Vol. 123. American Mathe-matical Society.

[10] Halvard Fausk, Po Hu, and J. Peter May. 2003. Isomorphisms betweenleft and right adjoints. Theory and Applications of Categories 11 (2003),107–131.

[11] Alexander Grothendieck. 1960. Eléments de Géométrie Algébrique.Publications de l’Institut des Hautes Etudes Scientifiques 4 (1960), 5–228.

[12] Alexander Grothendieck. 1971. Revêtements étales et Groupe Fonda-mental. Lecture Notes in Mathematics, Vol. 224. Springer Verla.

[13] Alexander Grothendieck. 1972. Théorie des topos et cohomologie étaledes schémas. Vol. 269. Springer Verlag.

[14] Robin Hartshorne. 1977. Algebraic Geometry. Graduate Texts in Math-ematics, Vol. 1977. Springer Verlag.

[15] Martin Hyland and Andrea Schalk. 2003. Glueing and orthogonalityfor models of linear logic. Theoretical Computer Science 294 (2003),183–231.

[16] Luc Illusie. 1971. Théorie des Intersections et Théorème de Riemann-Roch.Vol. 225. Springer Verlag, Chapter Existence de résolutions globales,160–222.

[17] Anders Kock. 2015. The dual fibration in elementary terms. (2015).https://arxiv.org/abs/1501.01947.

[18] Maxim Kontsevich and Alexander Rosenberg. 2004. Noncommutativestacks. Technical Report. Max-Planck Institute for Mathematics.

[19] Saunders Mac Lane and Ieke Moerdijk. 1992. Sheaves in Geometry andLogic. Springer Verlag.

[20] Joseph Lipman. 2009. Notes on Derived Functors and GrothendieckDuality. In Foundations of Grothendieck Duality for Diagrams of Schemes(Lecture Notes in Mathematics), Vol. 1960. Springer Verlag, 1–259.

[21] Jacob Lurie. 2004. Derived algebraic geometry. Ph.D. Dissertation.Massachusetts Institute of Technology.

[22] PhilippeMalbos. 2020. Two algebraic byways from differential equations:Gröbner bases and quivers. Vol. 28. Springer, Chapter Noncommutativelinear rewriting: applications and generalizations.

[23] Paul-André Melliès. 2009. Categorical semantics of linear logic. Num-ber 27 in Panoramas et Synthèses. Société Mathématique de France, 1– 196.

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[24] Paul-André Melliès and Noam Zeilberger. 2015. Functors are TypeRefinement Systems. In Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL2015, Mumbai, India, January 15-17, 2015.

[25] Paul-André Melliès and Noam Zeilberger. 2016. A bifibrational recon-struction of Lawvere’s presheaf hyperdoctrine. In Proceedings of the31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS2016, New York, NY, USA, July 5-8, 2016.

[26] Archana S. Morye. 2012. Note on the Serre-Swan theorem. Mathema-tische Nachrichten (2012).

[27] Daniel Murfet. 2015. On Sweedler’s cofree cocommutative coalgebra.Journal of Pure and Applied Algebra 219, 12 (2015).

[28] nLab entry on modules. 2020. https://ncatlab.org/nlab/show/module[29] Hans-E. Porst. 2015. The Formal Theory of Hopf Algebras, Part I.

Quaestiones Mathematicae 38 (2015), 631–682.[30] Hans-E. Porst and Ross Street. 2016. Generalizations of the Sweedler

Dual. Applied Categorical Structures 24 (2016), 619–647.[31] Jean-Pierre Serre. 1955. Faisceaux algébriques cohérents". Annals of

Mathematics 61, 2 (1955), 197–278.[32] Thomas Streicher. 1999. Fibered Categories à la Jean Bénabou. (1999).

https://arxiv.org/pdf/1801.02927.pdf.[33] Moss E. Sweedler. 1969. Hopf Algebras. W. A. Benjamin, Inc., New

York.[34] Leovigildo Alonso Tarrio, Ana Jeremias Lòpez, Marta Pérez Rodríguez,

and Maria J. Vale Gonsalves. 2008. The derived category of quasi-coherent sheaves and axiomatic stable homotopy. Advances in Mathe-matics 218, 4 (2008), 1224–1252.

[35] Bertrand Toën. 2014. Derived Algebraic Geometry. EMS Surveys inMathematical Sciences 1, 2 (2014), 153–240.

A Proof of Prop. 13.1We use the fact that the functor π : Mod Ñ Ring is a bifibra-tion, and more specifically a Grothendieck opfibration. Fromthis follows that the categoryMod has small colimits, whichare moreover preserved by the functor π . Now, suppose thatwe are in the following situation


Mod

Ring

Pointspf q

M

πX πYπ

Here, we use the fact that the category PointspX q is a smallcategory. This enables us to compute the left Kan extensionof the functorM along the discrete fibration

Pointspf q : PointspX q PointspY q

One obtains in this way a functor M 1 : PointspY q Ñ Modand a natural transformation λ which exhibitsM 1 as the left

Kan extension ofM along Pointspf q, as depicted below:


Mod

Ring

Pointspf q

M

πX

M 1

π

λ

The left Kan extension is a pointwise Kan extension, com-puted by small colimits. As already mentioned, the functor πpreserves small colimits, and thus pointwise left Kan exten-sions. This establishes that the natural transformation π ˝ λexhibits the functor π ˝ M 1 as the left Kan extension of thecomposite functor π ˝ M along Pointspf q. The universalityproperty of left Kan extensions ensures the existence of aunique natural transformation µ : π ˝ N 1 ñ πY as depictedbelow


Mod

Ring

Pointspf q

M

πX πYπ

λ

µ

such that the composite

π ˝ M π ˝ M 1 ˝ Pointspf q πY ˝ Pointspf qπ˝λ µ˝Pointspf q

is equal to the identity natural transformation on the func-tor πX “ πY ˝ Pointspf q. The functor π is a Grothendieckbifibration. From this follows that the associated postcompo-sition functor

CatpPointspY q,Modq CatpPointspY q,Ringq

is also a Grothendieck bifibration. In particular, there existsfor that reason a functor N : PointspY q Ñ Ring and a natu-ral transformation ν : M 1 Ñ N which is cocartesian abovethe natural transformation µ : π ˝ M 1 Ñ π ˝ N , as depictedin the diagram below:


Mod

Ring

Pointspf q

M

πX

N

πYπ

λν

In particular, the natural transformation

φ “ ν ˝ λ : M N ˝ Pointspf q

14

https://ncatlab.org/nlab/show/module

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obtained by composing ν and λ is vertical, in the sense thatπ ˝ φ is equal to the identity, and N ˝ π “ πY . The OY -module f !pMq associated to the OX -module N is simplydefined as

N : PointspY q Ring.

We obtain in this way a functor

f ! : PshModX PshModY

left adjoint to the inverse image functor


formulated in (24).

B Proof of Prop. 14.1We proceed exactly as in the proof of Prop. 13.1 in the previ-ous section, except that the orientation of the natural trans-formations is reversed. We thus use the fact in the proofthat the functor π : Mod Ñ Ring is a bifibration, and morespecifically a Grothendieck fibration. From this follows thatthe category Mod has small limits, which are moreover pre-served by the functor π . Now, suppose that we are in thefollowing situation


Mod

Ring

Pointspf q

M

πX πYπ

In the same way as we did in the proof of Prop. 13.1, weuse the fact that the category PointspX q is a small category.This enables us to compute the right Kan extension of thefunctorM along the discrete fibration

Pointspf q : PointspX q PointspY q

We obtain in this way a functorM 1 : PointspY q Ñ Mod anda natural transformation ρ which exhibits M 1 as the rightKan extension ofM along Pointspf q, as depicted below:


Mod

Ring

Pointspf q

M

πX

M 1

π

ρ

The right Kan extension is a pointwise Kan extension, com-puted by small limits. The functor π preserves small limits,and thus pointwise right Kan extensions. This establishesthat the natural transformation π ˝ ρ exhibits the functorπ ˝ M 1 as the right Kan extension of the composite functor

π ˝ M along Pointspf q. The universality property of rightKan extensions ensures the existence of a unique naturaltransformation µ : πY ñ π ˝ N 1 as depicted below


Mod

Ring

Pointspf q

M

πX πYπ

ρ

µ

such that the composite

πY ˝ Pointspf q π ˝ M 1 ˝ Pointspf q π ˝ Mµ˝Pointspf q π˝ρ

is equal to the identity natural transformation on the func-tor πX “ πY ˝ Pointspf q. The functor π is a Grothendieckbifibration. From this follows that the associated postcompo-sition functor

CatpPointspY q,Modq CatpPointspY q,Ringq

is also a Grothendieck bifibration. In particular, there existsfor that reason a functor N : PointspY q Ñ Ring and a natu-ral transformation ν : N Ñ M 1 which is cartesian above thenatural transformation µ : π ˝ N Ñ π ˝ M 1, as depicted inthe diagram below:


Mod

Ring

Pointspf q

M

πX

N

πYπ

ρ

ν

In particular, the natural transformation

ψ “ ρ ˝ ν : N ˝ Pointspf q M

obtained by composing ρ and ν is vertical, in the sense thatπ ˝ ψ is equal to the identity, and N ˝ π “ πY . The OY -module f˚pMq associated to the OX -module N is simplydefined as

N : PointspY q Ring.

We obtain in this way a functor

f˚ : PshModX PshModY

right adjoint to the inverse image functor


formulated in (24).

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C Proof of Thm. 15.1We suppose given a triple of Ring-spaces X , Y , Z together with a morphism

f : X ˆ Y Z

The category rRing,Sets is cartesian closed, and the morphism f thus gives rise by currification to a morphism noted

д : Y X ñ Z

We establish now a bijection between the setPshModf pM b N ,Pq

of forward module morphisms of the form

f : X ˆ Y Z |ù φ : M b N P

and the setPshModдpN ,M ⊸ Pq

of forward module morphisms of the form

д : Y X ñ Z |ù ψ : N M ⊸ P

The bijection is established by a series of elementary bijection applied to the enriched end formulas:

PshModдpN ,M ⊸ Pq

–

ż

pR,yqPPointspY q

”

Ny ,

ż


resu´

rMx ,Pf px,Y puqpyqqsS

¯ ı

R

–

ż

pR,yqPPointspY q

ż


”

Ny , resu´

rMx ,Pf px,Y puqpyqqsS

¯ ı

R

–

ż

pR,yqPPointspY q

ż


resu”

extuNy , rMx ,Pf px,Y puqpyqqsS

ı

S

–

ż

pyPY pRq,u :RÑS,xPX pSqq

resu rextuNy , rMx ,Pf px,Y puqpyqqsS sS

–

ż

pS,yqPPointspY q,pS,xqPX pSq

rNy , rMx ,Pf px,yqsS sS

–

ż

pS,x,yqPPointspXˆY q

rMx bS Ny ,Pf px,yqsS

– PshModf pM b N ,Pq

16

A Functorial Excursion Between Algebraic Geometry and ...mellies/papers/Mellies20submitted.pdf · The language of Algebraic Geometry combines two comple-mentary and dependent levels

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