Fibration of Toposes PSSL 101, Leeds

Transcription

1 Fibration of Tooses PSSL 101, Leeds Sina Hazratour Setember 2017

2 AUs AUs as finitary aroximation of Grothendieck tooses Pretooses 1 finite limits 2 stable finite disjoint coroducts 3 stable effective quotients Giraud s theorem Arithmetic Universes arametrized list objects 1 ε List(A) cons A List(A) Grothendieck tooses set-indexed coroducts + small hom-sets + a set of generators

3 AUs There is no reason we should restrict ourselves to Set-tooses. If S is any elementary toos, any bounded geometric morhism : E S is equivalent to Sh S (C, J) S where (C, J) is an internal site in S and Sh S (C, J) is the toos of S-valued sheaves. This is sometimes known as relativized Giraud s theorem. So, we have More details: [(Elehant, 2002) B3.3.4, C2.4] A Grothendieck toos over Set S := A bounded geometric morhism : E S an internal site in S Note: we always assume base toos S has n.n.o (same with an AU)

4 AUs AUs versus Grothendieck tooses Suose a geometric theory T can be exressed in an "arithmetic way". AUs Grothendieck tooses Classifying sace AU T S[T] T 2 T 1 AU T 2 AU T 1 S[T 1 ] S[T 2 ] Base Base indeendent Base deendent Infinities Intrinsic; rovided by List Extrinsic; from S e.g. infinite e.g. N = List(1) coroducts in the category of sheaves Results A single result in AUs a family of results for tooses arametrized by base S

5 Outline of the talk Outline of the talk Steven Vickers (2016). Sketches for arithmetic universes. In: URL: htts://arxiv.org/abs/ develoes a theory of AUs and resents them by a 2-category Con of contexts. Con has finite strict PIE limits. It also ossesses strict ullbacks along certain class of mas called context extension.

6 Outline of the talk Outline of the talk Steven Vickers (2016). Sketches for arithmetic universes. In: URL: htts://arxiv.org/abs/ develoes a theory of AUs and resents them by a 2-category Con of contexts. Con has finite strict PIE limits. It also ossesses strict ullbacks along certain class of mas called context extension. (O)fibrations in 2-categories: Ross Street (1974). Fibrations and Yoneda s lemma in a 2-category. In: Lecture Notes in Math., Sringer, Berlin Vol.420, : internal to reresentable 2-categories (with strict ullbacks and cotensor with 2). Relies on existence of comma objects. Generalizes notion of Grothendieck (o)fibrations in Cat. Peter Johnstone (1993). Fibrations and artial roducts in a 2-category. In: Alied Categorical Structures Vol.1, : internal to 2-categories with bi-ullbacks without use of comma objects in definition.

7 Outline of the talk Outline of the talk Using classifying tooses of contexts, we rove that (o)fibrations of contexts give rise to (o)fibrations of tooses. at the level of syntax (contexts) we need strict constructions and the level of semantics we need lax constructions. Street-Style (o)fibrations of contexts in Con Syntactic level (strict) models Johnstone-Style (o)fibrations of tooses in To Semantic level (non-strict)

8 A very brief into contexts 2-category Con The 2-category Con of contexts which is develoed in (Vickers, 2016). We start with structure of sketches: An AU-sketch is a structure with sorts and oerations as shown in this diagram. U b Λ 2 U list Λ 0 U 1 Γ 1 Γ 2 G 2 Γ 1 Γ 2 d i (i=0,1,2) c e G 1 d i (i=0,1) s G 0 tm i U o U 0

9 A very brief into contexts A morhism of AU-sketches is a family of carriers for each sort that also reserves oerators. Some of this morhism deserve the name extension, which are in fact, finite sequence of simle extensions. A simle extension consist of adding fresh nodes, edges and commutativities for universals which have been freshly added.

10 A very brief into contexts A morhism of AU-sketches is a family of carriers for each sort that also reserves oerators. Some of this morhism deserve the name extension, which are in fact, finite sequence of simle extensions. A simle extension consist of adding fresh nodes, edges and commutativities for universals which have been freshly added. The next fundamental concet is the notion of equivalence extension. When we have a sketch morhism, we may get some derived edges and commutativities. The idea of equivalence extension is to add them at this stage. The added elements are indeed uniquely determined by elements of the original, so the resented AUs are isomorhic as a result of an equivalence extension.

11 A very brief into contexts A morhism of AU-sketches is a family of carriers for each sort that also reserves oerators. Some of this morhism deserve the name extension, which are in fact, finite sequence of simle extensions. A simle extension consist of adding fresh nodes, edges and commutativities for universals which have been freshly added. The next fundamental concet is the notion of equivalence extension. When we have a sketch morhism, we may get some derived edges and commutativities. The idea of equivalence extension is to add them at this stage. The added elements are indeed uniquely determined by elements of the original, so the resented AUs are isomorhic as a result of an equivalence extension. Contexts are a restricted form of sketches for arithmetic universes. Every 0-cells, 1-cells, and 2-cells in Con are introduced in finite number of stes. e.g. 1, O, T 1 T 2, T, T, etc.

12 A very brief into contexts Finally, Con(T 0, T 1 ) consists of all osans (E, F ) from T 0 to T 1 : T 0 E F T 0 T 1 where F is a sketch extension morhism and E an sketch equivalence. 2-cells are given as context mas (e, α) from T 0 to T 1 where T 0 and T 1 are themselves contexts.

13 A very brief into contexts A central issue for models of sketches is that of strictness. The standard sketch-theoretic notion is non-strict: for a universal, such as a ullback of some given osan, the ullback cone can be interreted as any ullback of the osan. Contexts give us good handle over strictness: Proosition (Vickers, 2017) Let U : T 1 T 0 be an extension ma in Con, that is to say one deriving from an extension T 0 T 1. Suose in some AU A we have a model M 1 of T 1, a strict model M 0 of T 0, and an isomorhism φ 0 : M 0 = M 1U. T 1 M 1 φ 1 = M 1 U T 0 M 0 φ 0 = M 1U Then there is a unique model M 1 of T 1 and isomorhism φ 1 : M 1 = M 1 such that 1 M 1 is strict, 2 M 1U = M 0, 3 φ 1U = φ 0, and 4 φ 1 is equality on all the rimitive nodes for the extension T 0 T 1.

14 Fibration 2-monad of Street Street-style fibrations in strict 2-categories For a reresentable 2-category K, and for each 0-cell B in K. Street defines a strict KZ-monad on strict slice 2-category K/B. exand 1-cells : E B in K which suort the structure a seudo-algebra w.r.t to this 2-monad are called fibrations. 1-cells suorting the structure of an algebra are called slit fibrations. For ofibration there is a similar story using another KZ-monad. exand

15 Fibration 2-monad of Street Street-style fibrations in strict 2-categories Chevalley criteria: Proosition (Street) : E B is a (slit) fibration if and only if has a right adjoint with (identity) isomorhism counit. E e 1 e 0 E B/ E R() Φ B 1 B

16 A brief summary of Con Con has ullbacks along context extensions and also cotensor with 2 and that makes ossible to use Street s definition to define context fibrations. Definition Context extension U is said to be an extension ma with fibration roerty if it is a Street-style fibration in the 2-category Con, that is γ 1 : T 1 cod (T 1) has a right adjoint γ 1 λ 1 with co-unit of adjunction given by strict equality, that is γ 1 λ 1 = id cod T 1. λ 1 T 1 cod (T 1 ) γ 1 ι c γ 0 dom (T 1 ) T 1 ι d U U 0 T 0 cod U α T 0 T 0 dom

17 Elehant s Definition of Fibration Elehant s Definition of Fibration For 2-category of elementary tooses we can not use Street s definition, since this 2-category does not have strict ullbacks, but only bi-ullbacks along bounded geometric morhisms. One remedy is to look at Section B of (Johnstone, 2002) which rovides a definition of fibration for 1-cells in any 2-category with bi-ullbacks. However, one only needs existence of bi-ullbacks of the class of 1-cells that one would like to define as fibrations. This definition can be very well used in 2-category of elementary tooses to define certain bounded geometric morhisms as fibrations. However, Elehant s definition is comlicated and difficult to use for uroses of our work. We introduce a 2-category GTo and utilize it to simlify Elehant s definition (with slight modification). Essentially, we wra u the information of iso 2-cells involved in Elehant s definition as art of structure of 1-cells in GTo.

18 GTo We construct a 2-category GTo secified by the following data: 0-cells are of the form E S E, S : elementary tooses, and : bounded geometric morhism. 1-cells from q to are of the form f = f, f, f, where q F A f f f E S f : f f q: isomorhism geometric transformation.

19 GTo 2-cells between any two 1-cells f and g are of the form α = α, α where α : f g and α : f g are geometric transformations q F A f g f g f α α g E S in such a way that the obvious diagram of 2-cells commutes.

20 GTo Comosition of 1-cells k : r q and f : q is given by asting. more exlicitly, f k = f k, (f k ) ( f k), f k. Vertical and horizontal comosition of 2-cells is defined comonent-wise. Identity 1-cells and 2-cells are defined trivially.

21 Reformulating Fibration of Tooses in GTo Suose E and S are elementary tooses and : E S is a bounded geometric morhisms. We call a fibration in 2-category To whenever for any geometric transformation α : f g : A S, we have a 1-cell l(α): g f and a 2-cell α: f l(α) g in GTo, and moreover the following axioms are satisfied: Unack

22 Reformulating Fibration of Tooses in GTo Suose E and S are elementary tooses and : E S is a bounded geometric morhisms. We call a fibration in 2-category To whenever for any geometric transformation α : f g : A S, we have a 1-cell l(α): g f and a 2-cell α: f l(α) g in GTo, and moreover the following axioms are satisfied: Unack 1 If α = id f, then there exists an isomorhism 2-cell ι 0 : id f l(α) in GTo with ι 0 = id ida and α (f τ 0 ) = id f. Unack

23 Reformulating Fibration of Tooses in GTo Suose E and S are elementary tooses and : E S is a bounded geometric morhisms. We call a fibration in 2-category To whenever for any geometric transformation α : f g : A S, we have a 1-cell l(α): g f and a 2-cell α: f l(α) g in GTo, and moreover the following axioms are satisfied: Unack 1 If α = id f, then there exists an isomorhism 2-cell ι 0 : id f l(α) in GTo with ι 0 = id ida and α (f τ 0 ) = id f. Unack 2 If β : g h is another geometric transformation, then there exists an isomorhism 2-cell ι α,β : l(α) l(β) l(βα) in such a way that the following diagram of 2-cells in GTo commutes: f l(α) l(β) f ι α,β α l(β) = g l(β) β f l(βα) βα h Unack

24 Reformulating Fibration of Tooses in GTo 3 Lifting of α is comatible with left whiskering; That is, given any geometric morhism k : B A of tooses, we require l(α k) to fit into the following bi-ullback square in GTo: (gk) k g g l(α k) = κ l(α) (f k) f k f where k f and k g are ullback 1-cells over k. We also require asting of 2-cells α and κ to be equal to 2-cell α k. Unack

25 Reformulating Fibration of Tooses in GTo 4 For any 1-cells x = x, x, id where x : D f E, and y = y, id (g ) y, id A where y : D g E, any 2-cell β = β, α : f x g y in GTo is uniquely factored through α, that is there is a unique 2-cell µ in GTo with roerty (α y) (f µ) = β, that is to say the two asting diagrams in below are equal: g y y x µ f l(α) f = g y y x β f f g α g g g Unack

26 Reformulating Fibration of Tooses in GTo Fix an elementary toos S. Every context T gives rise to an indexed category over T : BTo/S, where T(E): = T -Mod-(E) = category of models of T in E

27 Reformulating Fibration of Tooses in GTo Fix an elementary toos S. Every context T gives rise to an indexed category over T : BTo/S, where T(E): = T -Mod-(E) = category of models of T in E Note that T encasulates data of all the models in all Grothendieck tooses (with base S). Vickers (2017) calls them "elehant theories" after (Elehant, 2002), and also to convey their big structure.

28 Reformulating Fibration of Tooses in GTo Fix an elementary toos S. Every context T gives rise to an indexed category over T : BTo/S, where T(E): = T -Mod-(E) = category of models of T in E Note that T encasulates data of all the models in all Grothendieck tooses (with base S). Vickers (2017) calls them "elehant theories" after (Elehant, 2002), and also to convey their big structure. Of course not all elehant theories arise from contexts. For instance if U is a context extension and M is a strict model of context T in base toos S, then T 1 /M is an elehant theory but not a context. T 1 /M(E): = strict models of T 1 in E which reduce to M via U

29 Reformulating Fibration of Tooses in GTo Fix an elementary toos S. Every context T gives rise to an indexed category over T : BTo/S, where T(E): = T -Mod-(E) = category of models of T in E Note that T encasulates data of all the models in all Grothendieck tooses (with base S). Vickers (2017) calls them "elehant theories" after (Elehant, 2002), and also to convey their big structure. Of course not all elehant theories arise from contexts. For instance if U is a context extension and M is a strict model of context T in base toos S, then T 1 /M is an elehant theory but not a context. T 1 /M(E): = strict models of T 1 in E which reduce to M via U Certain elehant theories are geometric and have classifying tooses. T and T 1 /M are such examles.

30 Reformulating Fibration of Tooses in GTo Theorem (Vickers,2017) Suose U : T 1 T 0 is a context extension. For any model M of T 0 in a (base) toos S, S[T 1/M] is an S-toos, and moreover, for any geometric (not necessarily bounded) morhism f : A S, the classifying toos A[T 1/f M] is got by bi-ullback of S[T 1/M] along f : A[T 1/f M] f A f f f S[T 1/M] S

31 Reformulating Fibration of Tooses in GTo Theorem (Vickers,2017) Suose U : T 1 T 0 is a context extension. For any model M of T 0 in a (base) toos S, S[T 1/M] is an S-toos, and moreover, for any geometric (not necessarily bounded) morhism f : A S, the classifying toos A[T 1/f M] is got by bi-ullback of S[T 1/M] along f : A[T 1/f M] f A f f f S[T 1/M] S Theorem (S.H., 2017) If U : T 1 T 0 is an extension ma of contexts with fibration roerty, and M is any model of T 0 in an elementary toos S, then : S[T 1/M] S is a fibration in the 2-category To.

32 Reformulating Fibration of Tooses in GTo l(α) A[T 1 /g M] α g S[T 1 /M] A[T 1 /f M] f g A α f g S[T 1 /M] S A f S finding l(α): equivalent to finding a model of T 1 /f M in A[T 1 /g M]. g := (G T 1 g M, g α M ) cod (T 1 ) -Mod- A[T 1 /g M] Model N : = g (λ 1 ; γ 0 ; δ d ) corresonds to l(α).

33 Reformulating Fibration of Tooses in GTo g λ 1 : induces a 1-cell v : u in GTo with v = id. We have θ = (θ, α) and f l(α) = vq 0 and v q 1 = g. Pasting all these 2-cells in GTo defines α = (α, α). g q 1 v g θ u q 0 l(α) f f

34 Reformulating Fibration of Tooses in GTo Local homeomorhism of tooses as ofibration Definition A geometric morhism F E is a local homeomorhism whenever F E/A for some object A of E. For S a bounded S 0 toos, and T 0 = O and T 1 the extended context of T 0 with a fresh edge from terminal to the unique node of T 0 : S/M S[T 1 /M] S 0 [X, x] = S 0 [X ][T 1 /X ] M S S 0 [X ] M

35 Reformulating Fibration of Tooses in GTo References Elehant (2002). Sketches of an elehant: A toos theory comendium. In: Oxford Logic Guides Vol.2.no.44, Oxford University Press. Johnstone, Peter (1993). Fibrations and artial roducts in a 2-category. In: Alied Categorical Structures Vol.1, (2002). Sketches of an elehant: A toos theory comendium. In: Oxford Logic Guides Vol.1.no.44, Oxford University Press. Street, Ross (1974). Fibrations and Yoneda s lemma in a 2-category. In: Lecture Notes in Math., Sringer, Berlin Vol.420, Vickers, Steven (2016). Sketches for arithmetic universes. In: URL: htts://arxiv.org/abs/ (2017). Arithmetic universes and classifying tooses. In: URL: htts://arxiv.org/abs/

36 Reformulating Fibration of Tooses in GTo End! Thank you for your attention!

37 Sulement A 2-monad of Street Let K be a reresentable 2-category. Define K/B to be the (strict) slice 2-category over B. (Street, 1974) constructs a 2-monad R : K/B K/B which takes an object (E, ) to (B/, R()) where B/ e E is a comma square. R() B Φ 1 B Back to resentation

38 Sulement A 2-monad of Street Remark Φ can be decomosed as follows: B/ e E B/ ˆ ˆ d 1 E R() Φ = B d 1 B B 1 B Φ d 0 1 B 1 B

39 Sulement A 2-monad of Street If f : E E is a 1-cell in K/B, then define B/f to be the unique 1-cell with f ˆd 1 = ˆd 1 B/f and ˆ B/f = ˆ. ˆd 1 B/ E B/f f ˆ B/ ˆ ˆd 1 E B d 1 B

40 Sulement A 2-monad of Street If f : E E is a 1-cell in K/B, then define B/f to be the unique 1-cell with f ˆd 1 = ˆd 1 B/f and ˆ B/f = ˆ. ˆd 1 B/ E B/f f ˆ B/ ˆ ˆd 1 E B Similarly if σ : f g is a 2-cell then we have a unique induced 2-cell B/σ : B/f B/g with ˆd 1 B/σ = σ ˆd 1 and ˆ B/σ = id ˆ. d 1 B

41 Sulement A 2-monad of Street Proosition R is a KZ monad.

42 Sulement A 2-monad of Street Unit of monad i : id R at (E, ) is given by the unique arrow i : E B/ with roerty that R() i = and ˆd 1 i = 1 E, and moreover Φ i = id, all inferred by universal roerty of comma object B/. i E B/ E ˆ ˆd 1 B d 1 B d 0 Φ id It follows that ˆd 1 i with identity counit. B B B 1 1

43 Sulement A 2-monad of Street Multilication of monad m : R 2 R at (E, ) is given by the unique arrow m : B/R() B/ B/R() B/ E d ˆd 1 1 ˆ B B B d d 1 1 Φ d 0 B B B B m ˆ B d 0 1 d 1 1 d 0 1 Φ 1 with roerty that R() m = R 2 () and ˆd 1 m = ˆd 1 d 1, and moreover Φ m = (Φ d 1 ) (Φ d 0 ˆ), all inferred by universal roerty of comma object B/.

44 Sulement A 2-monad of Street Examle When K = Cat, unit i takes an object N of E to the object (N, 1 M ) of B/, where M = (N). N M 1 M N M Multilication m takes an object (N 2, f : M 0 M 1, g : M 1 M 2) of B/R(), where M 2 = (N 2), to the object (N 2, f ; g : M 0 M 2) of B/. N 2 M 0 M 1 M 2 f g N 2 M 0 M 2 f ;g Back to resentation

45 Sulement What does a seudo-algebra of this monad look like? Suose a : (B/, R()) (E, ) is a seudo-algebra for 2-monad R. It involves 1 1-cell a : B/ E such that a = R() 2 invertible 2-cell ζ : 1 a i such that ζ = id. 3 invertible 2-cell θ : a R(a) a m such that θ = id R 2 (). Additionally, ζ and θ satisfy coherence equations of seudo-algebra a.

46 Sulement What does a seudo-algebra of this monad look like? Examle When K = Cat, 1 1-cell a gives us for any object (N 1, f : M 0 M 1) of B/ an object Pull f N 1 of E over N cell ζ gives us an isomorhism between N and Pull 1M N over 1 N, whereby M = (N). 3 2-cell θ gives us as isomorhism between Pull f Pull g N 2 and Pull f ;g N 2 over 1 M0. Additionally, following diagrams commute: Pull f N 1 Pull f Pull g Pull h N 3 Pull f ;g Pull h N 3 a (Rζ) Pull 1M0 Pull f N 1 Pull f N 1 θ Ri Pull f Pull g;h N 3 Pull f ;g;h N 3 Back to resentation

47 Sulement Unacking them yields the following diagram in To: f E l(α) f l(α) g g E A f α g g E E S α A f S where obvious diagram of 2-cells commutes. ack

48 Sulement Unacking τ 0 yields the following diagram in To: l(id) f α f E f E E τ 0 f id f id A A S id id f f ack

49 Sulement Unacking τ α,β yields the following diagram in To: l(β α) h l(β) = E g l(α) E f E = = A Furthermore, we require (β α) (f τ α,β ) = β (α l(β)) l(β α) (f τ α,β ) = l(β) (l(α) l(β)) ack

50 Sulement l(α k) is isomorhic to the bi-ullback of l(α) along k f, which is to say the to left vertical square of the diagram commutes u to an isomorhism. l(α k) (f k) E (gk) E = gk k g E g E E k f E f E E k f = k l(α) S A S B A S k k f k f fk f f g α α g ack

51 Sulement D x l(α) y g E g f E α E f = A g f A α S f ack

52 Sulement With regards to models of a context T, 2-category GTo has a class of very secial objects, namely a classifying toos : S[T] S, for each base toos S, with the classifying roerty given by following equivalence of categories whereby E is an S-toos: Φ : BTo /S (E, S[T]) T -Mod-(E) : Ψ which makes the reresentable object for the index category T.