ALGANT Master's Thesis Kan Complexes as a Univalent Model of Type Theory Gabriele Lobbia Supervisor: Prof. Denis-Charles Cisinski Academic Year 2017-2018 Do not worry about your difficulties in Mathematics. I can assure you mine are still greater. Albert Einstein A mathematician is a device for turning coffee into theorems. Therefore a comathematician is a device for turning cotheorems into ffee. Acknowledgements First of all I would like to express my full gratitude to my supervisor Prof. Denis-Charles Cisisnki, for accepting me as his students, for beeing always available for everything I needed, for helping me in the search for my future academic occupation and for teaching me so many aspects of mathematics. Every meeting has been enlightening math-wise and enjoyable on a personal level. I also have to thanks all the ALGaNT Consortium for the opportunity they gave me. In these two years I was able to meet fantastic people that helped me growing as a mathematician and as a person, and the credit is firstly theirs. The involment in this program has been crucial in the search for a PhD, and I am really grateful for this too. Speaking of people I have met, a special thanks goes to every friend that helped making this experience unique and amazing. Thanks to all my "mathematics" friends that shared with me this incredible passion of mine and adviced me whenever I was struggling. Thanks to all my friends from Milan. They have always been there for me and I am sure they will always be, even though I might be away from home for sometime. Thanks to my "Erasmus" friends, that beared me during all the writing of this thesis and helped me live the experience abroad in all his different parts. Last but not least, grazie alla mia famiglia. Ai miei genitori, ai miei fratelli, ai miei zii, ai miei nonni, a mia cugina e alla mia zia Lella. Senza nes- suno di loro sarei diventato la persona che sono, anche se in effetti potrebbe essere una colpa e non un merito. Ma sopratutto, senza il loro incondizion- ato supporto e aiuto non sarei mai riuscito a raggiungere nessun obbiettivo, quale Laurea o altro. ii Contents Introduction 1 1 Kan Complexes 2 1.1 Preliminary Definitions . 2 1.2 Nerve of a Category . 8 1.3 Fibrations and Complexes . 12 1.4 Homotopy Theory . 18 1.5 Minimal Complexes . 23 2 Type Theory 28 2.1 λ-Calculus . 28 2.2 Simple Type Theory . 33 2.3 Dependent Type Theory . 39 2.4 Homotopy Type Theory . 44 3 A Univalent Model 55 3.1 Categorical Setting . 55 3.2 The Simplicial Model . 58 3.3 The Univalence Axiom . 68 3.4 Simplicial Univalence . 73 A Yoneda Lemma 78 A.1 The Statement . 78 A.2 Important Consequences . 79 B Model Categories 84 B.1 Main Definitions . 84 B.2 Some Properties . 86 iii Introduction The aim of this Thesis is to illustrate an example of a model of Martin-L¨of Dependent Type Theory where Voedvosky's Univalence Axiom holds. The main results are explained in [KL16]. The first chapter will handle the construction of a model category in the enviroment of simplicial sets (sSet). In order to do that we will investigate the definitions of Kan complexes and Kan fibrations. Secondly we will give a rough overview of type theory in general. In particular, we will start giving an idea of what Simple and Dependent Type Theory mean. Then, finally, we will se a natural model of Homotopy Type Theory in category theory, thanks to [Awo16]. The last chapter will deal with the Univalence Axiom itself and the theo- rems that explain why Kan complexes can be seen as a model of type theory where it holds. 1 Kan Complexes 1.1 Preliminary Definitions We denote ∆ the category with objects the sets [n] = f0; :::; ng for any n 2 N, and morphism the non decreasing functions between them. Definition 1.1.1. We call simplicial set (or complex) a controvariant op functor of the kind X : ∆ ! Set. We write Xn for the set X([n]) and we call its elements the n-simplices of X. Usually the image of a function f :[q] ! [p] through the complex X is denoted by f ∗. From now on we will write sSet for the category whose objects are the simplicial sets and morphisms the natural trasformations between them. A subcomplex will be just a suboject in this category (i.e. a subfunctor). Definition 1.1.2. We write ∆n for the complex defined by the contravariant functor represented by [n], more precisely [q] / ∆([q]; [n]) O f −◦f [p] / ∆([p]; [n]) We call it the standard n-simplex. Definition 1.1.3. Let X be a simplicial set, and x 2 Xm an m-simplex of X. We say that x is degenerate if there exists an epimorphism s :[m] [n] 1 with n < m and a n-simplex y 2 Xn such that x = X(s)(y). Proposition 1.1.4. Let X be a simplicial set. Any morphism a : ∆n ! X has a unique factorization a ∆n / X = b " " ∆m where the first map is an epimorphism and b is not degenerate, i.e. the 2 1 correspondent element in Xm (by the Yoneda Lemma ) is not degenerate. 1[GZ67] Chapter II, x3.1. 2Lemma A.1.1 2 Definition 1.1.5. Let X be a complex. We define SknX the n-skeleton of X to be the subcomplex of X defined as follows 3 n (Sk X)m := fx 2 Xm j x is degenerated from a q-simplex with q ≤ ng Remark 1.1.6. The epimorphisms p :[m] [n] in ∆ are simply the non decreasing surjections. Moreover any epimorphism has a section, i.e. there is a morphism s :[n] ! [m] such that p ◦ s = Id[n]. Similarly the monomorphisms s :[n] [m] are the non decresing injections 4 and they have a retraction p :[m] ! [n] in ∆, i.e. p ◦ s = Id[n]. Definition 1.1.7. We denote with ∆_ n ≡ @∆n := Skn−1∆n the boundary of the standard n-simplex. We can see that any complex X is the union of its skeletons. Moreover there is a filtration ; =: SK−1X ⊆ SK0X ⊆ ::: ⊆ SKnX ⊆ ::: ⊆ X But simplicial sets have even a more important property. In fact we can re- cover SKnX from SKn−1X thanks to a pushout diagram, exactly analogously to the construction of CW-complexes in Topology. We have the more general result: Theorem 1.1.8. Let X,! Y be an inclusion of simplicial sets. Let Σn be the set of non degenerate elements of Yn which do not belong in Xn. Let us denote Y i := SkiY [ X. Then the following diagram is a pushout: ` n n−1 s2Σ @∆ / Y n _ _ ` ∆n / Y n s2Σn where the bottom and top morphism are defined componentwise as the mor- phism ∆n ! Y correspondet to s 2 Y . Moreover we have that Y ∼ lim Y i.5 n = −! Example 1.1.9. Let us define the simplicial set S1 as the pushout of the following diagram @∆1 / ∆1 π ∆0 / S1 3[GZ67] Chapter II, x3.5. 4[GZ67] Chapter II, x2.3. 5Generalization of [GZ67] Chapter II, Proposition 3.8. 3 1 We can see that π is both not injective and not degenerate. In fact #S0 = 1 1 1 0 1 and π corresponds to a different element of S1 than the map ∆ ! ∆ ! S . Remark 1.1.10. (/Alternative definition) We can see that actually [ @∆n = Im(∆q ! ∆n) q<n In fact, we can prove the more general statement: [ SknX = Im(∆q ! X) q≤n That is true because by Yoneda any map ∆q ! X correspond to a unique element of Xq. The condition regarding s to be an epimorphism is satisfied because we take the images of these maps. We recall now an important property of morphisms in ∆, which will be useful to make the definition above easier. First of all we have to give the n definition of the i-th coface map δi :[n − 1] ! [n] ( x if x < i δn(x) = i x + 1 if not Another important class of morphisms is formed by the so called i-th code- n generacy maps σi :[n + 1] ! [n] ( x if x ≤ i σn(x) = i x − 1 if not i n ∗ From now on, for any simplicial set X, we will denote with dn := (δi ) : Xn ! i n ∗ Xn−1 and sn := (σi ) : Xn ! Xn+1 the maps induced on the simplices. By straight forward calculation we can verify that these two classes of maps verify the following properties, which are called simplicial relations6: n+1 n n+1 n 1. δj δi = δi δj−1 for any i < j n n+1 n n+1 2. σj σi = σi σj+1 for any i ≤ j 8 n−1 n−2 >δi σj−1 if i < j n−1 n < 3. σj δi = Id[n−1] if i = j or i = j + 1 > n−1 n−2 :δi−1 σj if i > j + 1 6[GZ67] Chapter II, x2.1. 4 Dually we have that: i j j−1 i 1. dndn+1 = dn dn+1 for any i < j i j j+1 i 2. sn+1sn = sn+1sn for any i ≤ j 8 j−1 i >sn−2dn−1 if i < j i j < 3. dnsn−1 = IdX[n−1] if i = j or i = j + 1 > j i−1 :sn−2dn−1 if i > j + 1 Lemma 1.1.11.