Covering Spaces in Homotopy Type Theory Favonia Carnegie Mellon University favonia@cmu.edu This material is based upon work supported by the This material is based upon work supported by the 1 National Science Foundation under Grant No. 1116703. National Science Foundation under Grant No. 1116703.
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Why bother? Fundamental Groups! 3
[ computer checked ] This work is covered by Agda 4
Covered Topics Classification Universality 5
Part 0 Definition 6
Definition of Covering Spaces cover covering projection base 7
Definition of Covering Spaces cover covering projection base 7
Definition of Covering Spaces cover covering projection base 7
Definition of Covering Spaces exact cover copies covering projection base 7
Definition of Covering Spaces Set 8
Definition of Covering Spaces Set HoTT True Facts #28 Continuity is free! 8
Definition of Covering Spaces Set HoTT True Facts #28 Continuity is free! 8
Definition of Covering Spaces A → Set Cover over A 9
Definition of Covering Spaces fiber over a A → Set Cover over A a 9
Definition of Covering Spaces fiber over a A → Set Cover over A It is a functor! a 9
Definition of Covering Spaces set ₁ a ₁ fiber a ₂ set ₂ over a A → Set Cover over A It is a functor! a 9
Definition of Covering Spaces set ₁ a ₁ iso fiber a ₂ set ₂ over a A → Set Cover over A It is a functor! a 9
Definition of Covering Spaces fiber over a A → Set Cover over A path-connected? a 10 10
Definition of Covering Spaces fiber over a A → Set Cover over A path-connected? pointed? a 10 10
circle 11 11
? circle 12 12
? circle 13 13
Part 1 Classification 14 14
Goal Find representations of covering spaces 15 15
path-connected 16 16
transport path-connected 16 16
transport a p a p path-connected 16 16
For example… Green part: "fixable" Yellow + Blue: inherent twists p r q 17 17
For example… Green part: p "fixable" Yellow + Blue: inherent twists p r q 17 17
For example… Green part: p "fixable" q Yellow + Blue: inherent twists p r q 17 17
For example… Green part: p "fixable" r q Yellow + Blue: inherent twists p r q 17 17
For example… Green part: p "fixed" r q Yellow + Blue: inherent twists p r q 18 18
Green part: Green part: p "fixed" "fixed" r q Yellow + Blue: inherent twists p r q Loops 19 19
Automorphisms Green part: Green part: p "fixed" "fixed" r q Yellow + Blue: inherent twists p r q Loops 19 19
For circles… 1 → 2 2 → 1 1 → 1 1 → 1 1 → 1 2 → 3 2 → 2 3 → 2 loop It is su ff icient to check the generator loop 20 20
X t 2. Automorphisms by e S . 1 di ff erent loops loop loop 21 21
X t 2. Automorphisms by e S . 1 di ff erent loops loop loop 21 21
Fundamental Group Sets of loops based at a point = = = id 22 22
X t 2. Automorphisms by e S . 1 di ff erent loops loop elements in fundamental group loop 23 23
Fix G = fundamental group A G-set is a set X with an action of G map from G to automorphisms of X 24 24
Fix G = fundamental group A G-set is a set X with an action of G map from G to automorphisms of X with functoriality… id g 1 g 1 g 2 g 2 24 24
Fix G = fundamental group G-set A set X equipped with an action, a map from G to automorphisms Classification Theorem G-sets and covering spaces are equivalent. 25 25
For circles… loop It is su ff icient to check the generator loop 26 26
loop successor 27 27
Proof Cover G-Set 28 28
1. Cover → G-set Set loop transport (restricted to loops) as the action loop 29 29
2. Cover → G-set → Cover Given a G-set = a set X and an action Given a G-set = a set X and an action Construct a cover such that such that 1. Every fiber is isomorphic to X 2. Transport is the action (restricted to loops) (restricted to loops) 30 30
2. Cover → G-set → Cover Given a G-set = a set X and an action Given a G-set = a set X and an action Construct a cover such that such that 1. Every fiber is isomorphic to X 2. Transport is the action (restricted to loops) (restricted to loops) Magic: Higher inductive types 30 30
2. Cover → G-set → Cover X was a fiber X Other fibers missing 31 31
β α 2. Cover → G-set → Cover Base path p would induce an isomorphism (by “transport” ) p 32 32
β α 2. Cover → G-set → Cover Base path p would induce an isomorphism (by “transport” ) Fake it with p a formal one! Point β is p α 32 32
α β 2. Cover → G-set → Cover data R (a : A) : Set : ∀ p α → R a formal transport p Point β is p α 33 33
2. Cover → G-set → Cover Di ff erent q’s give di ff erent copies Needs a way to merge copies from di ff erent base paths p q 34 34
α 2. Cover → G-set → Cover If it will be some cover… ? p q 35 35
α 2. Cover → G-set → Cover If it will be some cover… q must be (q p ⁻ ¹) p = ? loop p q loop 35 35
α 2. Cover → G-set → Cover If it will be some cover… q must be (q p ⁻ ¹) p = loop q α = (loop p) α p = p (loop α ) Key: functoriality q loop 35 35
α 2. Cover → G-set → Cover Going back to the construction… We mimic functoriality q α = (loop p) α = p (loop α ) action is transport for loops p q α = (loop p) α q loop = p (loop α ) 36 36
α 2. Cover → G-set → Cover We mimic functoriality q α = (loop p) α = p (loop α ) data R (a : A) : Set : ∀ p α → R a p : ∀ l p α → (l p) α q = p (l α ) loop 37 37
2. Cover → G-set → Cover data R (a : A) : Set : ∀ p α → R a : ∀ l p α → (l p) α = p (l α ) Theorem Theorem R is equivalent to the original cover R is equivalent to the original cover Acknowledgements: Thanks to Guillaume Brunerie, Daniel Grayson and Chris Kapulkin for helping me state and prove this. 38 38
[ recap ] [ recap ] Classification Theorem If G is the fundamental group G-sets and covering spaces are equivalent. 39 39
Technical Notes WARNING: NASTY MATH AHEAD All truncations were omi ed. You want this lemma: Given a constant (pointwise-equal) function f : A → B where B is a set find a g : ||A|| → B such that f = g · | - | f B A ||A|| | - | g 40 40
Part 2 Universality covers that cover every cover covers that cover every cover 41 41
Universality Universal Assumption: Everything is path-connected and pointed 42 42
Universality e u q i n u Universal Assumption: Everything is path-connected and pointed 42 42
A simple universal cover set of paths with one end fixed pointed Assumption: Everything is pointed and path-connected. 43 43
Technical Notes WARNING: NASTY MATH AHEAD The simple universal cover is λ x . || = x || ₀ x set of paths with one end fixed 44 44
Technical Notes WARNING: NASTY MATH AHEAD The simple universal cover is λ x . || = x || ₀ x set of paths with one end fixed Path induction! 44 44
Theorem It is inital. It is inital. It is equivalent to any simply connected cover. Simply Connected one and only one 45 45
circle 46 46
circle 46 46
Z 2 Z 1 Z 3 Group quotients! Z circle 46 46
Theorem It is inital. It is equivalent to any simply connected cover. Simply Connected one and only one 47 47
Weak Initiality transport q p one to many p pointed q 48 48
Weak Initiality transport q p = quotient one to many p pointed q 48 48
Strong Initiality cover ₁ p cover ₂ =? Su ff icient to consider p = identity path ( and collide) pointed 49 49
Theorem It is inital. It is equivalent to any simply connected cover. 50 50
If li ed p and q are the same… p q simply one to one connected p pointed q 51 51
If li ed p and q are the same… p s.c. identifies = q li ed paths simply one to one connected p pointed q 51 51
If li ed p and q are the same… p s.c. identifies = q li ed paths projection is simply = one to one connected retraction of = li ing p pointed q 51 51
Theorem It is inital. It is equivalent to any simply connected cover. 52 52
fiber over = fundamental group 53 53
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