a functorial excursion between algebraic geometry and ...mellies/papers/mellies20submitted.pdf ·...
TRANSCRIPT
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
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
1
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
IRIF, Paris, France Paul-André Melliès
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
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
2
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
A Functorial Excursion Between Algebraic Geometry and Linear Logic IRIF, Paris, France
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
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
m
3
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
IRIF, Paris, France Paul-André Melliès
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
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
4
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
A Functorial Excursion Between Algebraic Geometry and Linear Logic IRIF, Paris, France
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
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
5
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
IRIF, Paris, France Paul-André Melliès
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
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
6
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
A Functorial Excursion Between Algebraic Geometry and Linear Logic IRIF, Paris, France
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
homomorphism f : M Ñ N making the diagrams commute:
A bR A B bS B
A B
f bu f
mA mB
f
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
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
IRIF, Paris, France Paul-André Melliès
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
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
making the diagram below commute:
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
PointspX q PointspY q
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
PointspX q PointspY q
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
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
A Functorial Excursion Between Algebraic Geometry and Linear Logic IRIF, Paris, France
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
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:
PointspX q PointspY q
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:
PointspX q PointspY q
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
9
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
IRIF, Paris, France Paul-André Melliès
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
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
PointspX q PointspY 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
10
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
A Functorial Excursion Between Algebraic Geometry and Linear Logic IRIF, Paris, France
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
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
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
IRIF, Paris, France Paul-André Melliès
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
making the diagram below commute:
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:
PointspX q PointspY q
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
12
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
A Functorial Excursion Between Algebraic Geometry and Linear Logic IRIF, Paris, France
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
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.
13
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
IRIF, Paris, France Paul-André Melliès
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
[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
PointspX q PointspY q
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:
PointspX q PointspY q
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
PointspX q PointspY q
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:
PointspX q PointspY q
Mod
Ring
Pointspf q
M
πX
N
πYπ
λν
In particular, the natural transformation
φ “ ν ˝ λ : M N ˝ Pointspf q
14
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
A Functorial Excursion Between Algebraic Geometry and Linear Logic IRIF, Paris, France
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
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
f ˚ : PshModY PshModX
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
PointspX q PointspY q
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:
PointspX q PointspY q
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
PointspX q PointspY q
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:
PointspX q PointspY q
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
f ˚ : PshModY PshModX
formulated in (24).
15
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
IRIF, Paris, France Paul-André Melliès
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
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
pu :RÑS,xPX pSqqPPointspyRˆX q
resu´
rMx ,Pf px,Y puqpyqqsS
¯ ı
R
–
ż
pR,yqPPointspY q
ż
pu :RÑS,xPX pSqqPPointspyRˆX q
”
Ny , resu´
rMx ,Pf px,Y puqpyqqsS
¯ ı
R
–
ż
pR,yqPPointspY q
ż
pu :RÑS,xPX pSqqPPointspyRˆX q
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