Visualizing Geometric Morphisms An application of the Logic for - PowerPoint PPT Presentation

1 Visualizing Geometric Morphisms An application of the “Logic for Children” project to Category Theory (talk @ “Logic and Categories” workshop, UniLog 2018) By: Eduardo Ochs (UFF, Brazil) Selana Ochs 2018vichy-vgms-slides June 22, 2018 11:28

2 Logic / categories / toposes for children ( Very short version; for the long version see the resources for the “Logic for Children” workshop) Many years ago... Non-Standard Analysis For Children: using “internal views” and examples with fjnite objects that are easy to draw → Johnstone’s “Topos Theory” → FAR too abstract for me → I NEED A VERSION FOR CHILDREN OF THIS Heyting Algebras that are subset of Z 2 (paper) Presheaves that can be drawn on a subset of Z 2 (new)

3 Planar Heyting Algebras for Children http://angg.twu.net/math-b.html ) Main defjnition: between a “left” and a “right wall”. Main theorems: every ZHA is a Heyting Algebra every ZHA is a topology in disguise ( ↑ paper submitted in 2017 — A ZHA is a fjnite subset of Z 2 made of all even points ( x + y = 2 k ) between (0 , 0) and ⊤ (The “Z” in ZHA means “ ⊂ Z 2 ”) ↑ a ZHA

4 The paper PHAfC shows how using a formula with four cases. can be calculated quickly It uses LR-coordinates and to visualize this (on ZHAs). Planar Heyting Algebras for Children an intuitionistic logic. than two truth-values and Most toposes have more Lots of fun! Go read it!) ( ↑ Very good paper! No prerequisites! 44 43 34 42 33 24 41 32 23 14 40 31 22 13 04 30 21 12 03 20 11 02 10 01 00 ⊤ · · · · · shows how the ‘ → ’ on ZHAs · ( ∨ ) · ( → ) · P · · · · · Q · · ( ∧ ) · · · ⊥

5 Planar Heyting Algebras for Children 2: Local Operators The second paper in the series. Sheaves correspond to local operators on HAs. A local operator on a ZHA corresponds to slashing the ZHA by diagonal cuts and blurring the distinction between the truth-values in each region. PHAfC doesn’t mention categories. 46 45 36 35 26 34 25 24 23 14 22 13 04 12 03 11 02 01 00 PHAfC2 doesn’t mention categories yet.

6 � � � � � ZCategories � � how to draw it. Add a fjnite set of arrows. (Optional step: rename its points.) Use this set as the set of objects of a category. Choose a fjnite subset of Z 2 . 1 1 1 ❖ ❖ ✴ ❄ ❖ ❄ ✴ ❖ ❄ ❖ ❖ ✴ ❄ ❖ ❄ ✴ ❖ ❄ ❖ ✴ ❖ ❖ 2 2 2 3 3 3 ✴ ✴ ✴ ✴ ✴ ✴ This is a ZCategory. The Z 2 -coordinates tell 4 4 4 5 5

7 ZPresheaves and ZToposes � � � � � � � � � � � � � (“Positional notations”) � A ZPresheaf is a functor F : A → Set , where A is a ZCategory. (Obs: not F : A op → Set !) A ZPresheaf F inherits its drawing instructions from A .     1 1 1 F 1 F 1 F 1 ❖ ❖ ✴ ❄ ❖ ❖ ✴ ❄ ❖ ❖ ❄ ❖ ✴ ✴ ❄ ❖  ❄ ❖    ❄ ❖ ❖ ✴ ❄ ✴ ❖ ❖ ❄    ❖  ✴ ❄ ❖ ❄ ✴ ❖ ❖ ❖ ❄  ✴ ❖   ✴ ❖  A = 2 2 2 ❖ 3 3 3 F = F 2 F 2 F 2 F 3 F 3 F 3 ✴ ✴     ✴ ✴    ✴  ✴ ✴ ✴     ✴ ✴    ✴  ✴ 4 4 4 5 5 F 4 F 4 F 4 F 5 F 5 A ZTopos is a category Set A where A is a ZCategory.

8 � ✤ � � ✤ � Internal views � ✤ � ✤ ✤ � ✤ (Part 1: functions) The internal view of the function √ : N → R is: − 1 0 0 1 1 √ 2 2 √ 3 3 4 2 √ n n √ N R (‘ �→ ’s take elements of a blob-set to another blob-set)

9 ✤ ✤ � Internal views ✤ � � � ✤ ✤ � ✤ � � � � � � � (Part 2: functors) instead of blob-sets. Compare: ✤ ✤ � ✤ Internal views of functors have blob-categories − 1 0 0 A A FA FA 1 1 √ 2 2 √ g F g 3 3 4 2 B B FB FB √ n n F C D √ N R

10 � � � � ✤ � ✤ � ✤ � Internal views � � � � ✤ � ✤ ✤ (Part 3: omitting the blobs) A A FA FA A A FA FA g F g g F g B B FB FB B B FB FB F C D F C D

11 � � � ✤ � � Internal views ✤ � � � � � � ✤ � � ✤ � � � � � � � � (Part 4: adjunctions) � ✤ ✤ � � Left: generic adjunction L ⊣ R Middle: generic geometric morphism f ∗ ⊣ f ∗ Right: g.m. between toposes Set A and Set B f ∗ F f ∗ F f ∗ F f ∗ F LC LC C C F F F F D D RD RD G G f ∗ G f ∗ G G G f ∗ G f ∗ G f ∗ � f ∗ L Set A Set A Set B Set B D D C C E E F F R f ∗ f ∗ f A B

12 particularize The diagrams for the general case and for a particular case “for adults” case general � (hard) generalize Working in two languages in parallel (easy) “for children” case particular between the two approaches... Ideas: do things “for children” and “for adults” have the same shape!!! in parallel, fjnd ways to transfer knowledge      �   

13 Working in two universes in parallel In Non-Standard Analysis we have transfer theorems universe � � Non-Standard universe (ultrapower)   � Standard �  

14 � ✤ � � Our fjrst geometric morphism � � � � � � � � � � � � � (for children; inclusion, sheaf) ✤ (for adults)     F 1 F 1 ւ ւ ց ց � � � � F 2 F 2 F 3 F 3 F 2 F 2 F 3 F 3     ց ց ւ ւ ց ց ց ց ւ ւ ց ց     F 4 F 4 F 5 F 5 F 4 F 4 F 5 F 5     ց ց ւ ւ F 6 F 6 f ∗ F f ∗ F F F   G 2 × G 4 G 3 G 2 × G 4 G 3   ւ ւ ց ց � � � � G 2 G 2 G 3 G 3 G 2 G 2 G 3 G 3     ց ց ւ ւ ց ց ց ց ւ ւ ց ց     G 4 G 4 G 5 G 5 G 4 G 4 G 5 G 5 G G f ∗ G f ∗ G     ց ց ւ ւ 1 1 f ∗ f ∗ F F E E Set A Set A Set B Set B f ∗ f ∗  1  ւ ց f � � � 2 3 2 3   ց ւ ց ց ւ ց   4 5 4 5   ց ւ 6

15 followed by a closed g.m. . � inclusion A factorization any � Any inclusion factors as a dense g.m. Each ‘ Any g.m. factors as a surjection followed by an inclusion. Section A4: Geometric Morphisms Elephant = Bible � ’ below is a g.m. (an adjunction) A D surjection � B A B D dense � C close � B C D The Elephant constructs the toposes B , C and the maps.

16 inclusion ✤ ✤ ✤ ✤ ✤ ✤ ✤ A factorization: version using ZPresheaves � ✤ This would be a nicer theorem — that if we start any through ZToposes... � with ZToposes Set A and Set D the factorization can be Set A Set A Set D Set D Set A Set A surjection � B Set D Set D Set D B B dense � C closed � Set B Set B Set D Set D C C Set C

17 � � That factorization, for children how the Elephant defjnes sujection, inclusion, etc... that only has ZToposes, and we use it to understand We start with a particular case, with a factorization ( s is not an inclusion, i is not a surjection, and so on) F G H I g (any) Set A Set A Set D Set D s (surjection) � Set B i (inclusion) Set A Set A Set B Set B Set D Set D Set D d (dense) � Set C c (closed) � Set B Set B Set C Set D Set D

18 � (monic) � � The surjection-inclusion factorization for children � ✤ � � � � ✤ � � ✤ � � � ✤ � � � � ✤ � � � ✤ � g ∗ I g ∗ I I I F F g ∗ F g ∗ F g (any) Set A Set D s ∗ G s ∗ G G G G i ∗ i ∗ G i ∗ I i ∗ I I I ηG ǫG (iso) � F F s ∗ F s ∗ F s ∗ s ∗ G G G G i ∗ G i ∗ G s (surjection) � i (inclusion) Set A Set B Set B Set D

19 � � � � (monic) � ✤ � ✤ � The dense-closed factorization for children � � ✤ � � � � � ✤ � � � ✤ � � ✤ � i ∗ i ∗ G i ∗ I i ∗ I I I ǫG (iso) � G G G i ∗ G i ∗ G i (inclusion) Set B Set D d ∗ H d ∗ H c ∗ I c ∗ I H H K I I ηK d ∗ d ∗ K G G d ∗ G d ∗ G H H c ∗ H c ∗ H d (dense) � c (closed) � Set B Set C Set C Set D ( K is a constant ZPresheaf in Set C )

20 � ✤✤ ✤ ✤ ✤ ✤ ✤✤ ✤✤ ✤ ✤ ✤ ✤ Acoording to the Elephant... ✤✤ � A4.2.7, 4.2.10: comonads and coalgebras A4.5.9, A4.5.20: (can’t be!) a (any) Set A Set A Set D Set D to build B we need s (surjection) � B i (inclusion) Set A Set A Set D Set D Set D B B ( Set B ) G ( Set B ) G d (dense) � C c (closed) � Set B Set B Set D Set D C C C = sh ¬¬ ( Set D ) sh ¬¬ ( Set D ) sh ¬¬ ( Set D ) (???) (???) Set C

21 Another strategy (or by Kan extensions) Start with a functor g : A → D . It induces a geometric morphism g ∗ ⊣ g ∗ . g ∗ is trivial to build. g ∗ can be found by guess-and-test. The functor g can: collapse objects, (1 2) → (1) create objects, ( ) → (3) collapse arrows, (4 �� 5) → (4 → 5) create arrows, (6 7) → (6 → 7) Try to factor it. Example: if g just collapses objects...