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Home , Disk (mathematics), Homotopy theory, Real projective plane, Real projective space, Sphere

The real projective in type theory

Ulrik Buchholtz Egbert Rijke Technische Universität Darmstadt Carnegie Mellon University Email: [email protected] Email: [email protected]

Abstract—Homotopy type theory is a version of Martin- and homotopy theory developed in homotopy Löf type theory taking advantage of its homotopical models. type theory (homotopy groups, including the fundamen- In particular, we can use and construct objects of homotopy tal of the , the Hopf fibration, the Freuden- theory and reason about them using higher inductive types. In this article, we construct the real projective spaces, key thal theorem and the van Kampen theorem, players in homotopy theory, as certain higher inductive types for example). Here we give an elementary construction in . The classical definition of RPn, in homotopy type theory of the real projective spaces as the quotient identifying antipodal points of the RPn and we develop some of their basic properties. n-, does not translate directly to homotopy type theory. R n In classical homotopy theory the real Instead, we define P by induction on n simultaneously n with its of 2-element sets. As the base RP is either defined as the space of lines through the + case, we take RP−1 to be the empty type. In the inductive origin in Rn 1 or as the quotient by the antipodal action step, we take RPn+1 to be the mapping of the of the 2-element group on the sphere Sn [4]. It has a n of the tautological bundle of RP , and we use its simple cell complex description with one cell in each universal property and the univalence to define the i ≤ n. Real projective spaces are frequently tautological bundle on RPn+1. By showing that the total space of the tautological bundle used as examples or as bases for computations. Each of RPn is the n-sphere Sn, we retrieve the classical descrip- RPn has as universal the sphere Sn, and tion of RPn+1 as RPn with an (n + 1)- attached to it. The its is Z/2Z for n > 1. R ∞ infinite dimensional P , defined as the In homotopy type theory, we would like to have access sequential colimit of RPn with the canonical inclusion , is equivalent to the Eilenberg-MacLane space K(Z/2Z, 1), to these spaces and their basic properties, but the above which here arises as the subtype of the universe consisting of definitions do not directly carry over to this setting. 2-element types. Indeed, the infinite dimensional projective We instead give an elementary inductive construction of space classifies the 0-sphere bundles, which one can think the spaces RPn together with their universal coverings of as synthetic bundles. covn : RPn → U. (We write U for the first universe These constructions in homotopy type theory further illustrate the utility of homotopy type theory, including the of types.) Indeed, the universal coverings are fibrations n interplay of type theoretic and homotopy theoretic ideas. where the fibers are 2-element sets, so the maps cov Keywords. Real projective spaces, Homotopy Type Theory, factor through the sub-type of U consisting of 2-element Univalence axiom, Higher inductive types. sets, US0 (see SectionII below). To illustrate how our approach matches traditional constructions, note that we I.INTRODUCTION get the real projective by attaching a 2-cell to Homotopy type theory emerged from the discovery the circle along the map of degree of 2. Geometrically, arXiv:1704.05770v1 [math.AT] 19 Apr 2017 of a homotopical interpretation of Martin-Löf’s identity this amounts to gluing a disk to Möbius strip along its types [1] and in particular from the construction of , cf. Fig.1. the model in simplicial sets [6] and the proposal of In a companion article, we shall describe another con- the univalence axiom [13, 14]. We refer to the book on struction of both real and complex projective spaces as homotopy type theory for details [12]. homotopy quotients of (∞-)group actions (of the 2-element Homotopy type theory allows us to reason synthet- group O(1) and the circle group U(1), respectively). ically about the objects of (spaces, Our construction (like other constructions in homo- paths, , etc.) analogously to how the setting topy type theory) gives the homotopy types of real projec- of allows us to reason synthetically tive spaces, as opposed to more refined structure such about points, lines, , etc. (as opposed to analyti- as that of smooth or real algebraic varieties. cally in terms of elements and subsets of the Cartesian A benefit of having a construction in homotopy type plane R2). Already in [12] we find a portion of algebraic theory is that it applies in any model thereof, and not constructors

inl : A → A +C B

inr : B → A +C B and a constructor

glue : ∏(c:C) inl( f (c)) = inr(g(c)).

Figure 1. Möbius strip. The real is obtained by gluing The elimination principle of A +C B provides a way of on a disk along the boundary. (TikZ code adapted from [5].) defining sections of type families P : A +C B → Type (not necessarily small). If we have p : ∏ P(inl(a)) just in the standard one of infinity , ∞Gpd. A (a:A) For example, it is conjectured (and proved in many pB : ∏(b:B) P(inr(b)) cases) that Grothendieck ∞-toposes provide models. As as well as paths the real projective spaces are just certain colimits built P from the unit type, they sit in the discrete part of any pC : ∏(c:C) pA( f (c)) =glue(c) pB(g(c)), Grothendieck ∞-topos, i.e., in the inverse image of the f P(x) then we get a : ∏(x:A+C B) . More- geometric to ∞Gpd. In the settings of real- over, the section f we obtain this way satisfies the or smooth-cohesive homotopy type theory, they should computation rules f (inl(a)) ≡ pA(a) : P(inl(a)) and be the of the incarnations of the real projective f (inr(b)) ≡ pB(b) : P(inr(b)). We also have a wit- spaces as topological or smooth manifolds, respectively, ness for propositional equality for map on paths, w : cf. the similar situation for the circle discussed in [10]. ∏(c:C) apd f (glue(c)) = pC(c) (we refer to the homotopy It is also expected that homotopy type theory can be P type theory book for details on dependent paths a =p b modeled in elementary ∞-toposes [11], which do not in and the map on paths operations ap f and apd f ). general admit a geometric morphism to ∞Gpd, and From pushouts, it is easy to derive also homotopy hence something like our construction would be needed coequalizers of maps f , g : A ⇒ B for A, B : U. to access the homotopy types of the real projective spaces From these we get sequential colimits of diagrams in such models. f0 f1 f2 In the remainder of this introduction we describe in A0 A1 A2 ··· more detail the precise setting of homotopy type theory A N → U f A → A we are working in as well as the techniques we use. consisting of : and : ∏(n:N) n n+1. We also get the suspension ΣA of a type A as the We work in an intensional dependent type theory with pushout a univalent universe U containing the type of natural A 1 numbers, the unit type and the empty type, and we assume that U is closed under homotopy pushouts. S Furthermore, to use the universal property of pushouts 1 ΣA. N to map into arbitrary types regardless of their size, we n assume function extensionality for all dependent types. The S are defined by recursion on n : N−1, (This is automatic if every type belongs to a univalent by iteratively suspending the (−1)-sphere, which is the universe, but we do not assume more than one universe.) empty type. Thus, for each n : N−1, the (n + 1)-sphere The types A : U are called small types. has a north pole and a south pole, and there is a n Let us briefly recall how homotopy pushouts are homotopy glue : S → (N = S) which suspends the obtained as higher inductive types: if we are given n-sphere between the poles. In particular, the 0-sphere 0 A, B, C : U and f : C → A and g : C → B, then we S is a 2-element type. form the pushout of f and g The pushout of the product projections A × B → A and A × B → B gives the A ∗ B. Lemma 8.5.10 in [12] Σ ' S0 ∗ Sn+1 ' g establishes that A A, so it follows that C B S0 ∗ Sn. f inr With pushouts, it is also possible to define for any type A : U its propositional truncation kAk, which is the A A + B inl C reflection of A into the type of mere propositions. This means we have a function |−| : A → kAk, precompo- as the higher inductive type A +C B : U with sition with which gives an equivalence (kAk → B) '

2 (A → B) for any mere proposition B. There are several and let PX : X → U and PY : Y → U be type families that are constructions of the propositional truncation in terms of compatible in the sense that there is a fiberwise equivalence pushouts and colimits due to Van Doorn [3], Kraus [7], e : ∏(a:A) PX( f (a)) ' PY(g(a)). and the second-named author [9]. Let Ptot : X +A Y → U be the unique type family, defined As a final special case of a pushout we shall need, we via the universal property of X +A Y, for which there are have the Cf of a map f : A → B. This is equivalences just the pushout of f and the unique map A → 1. In ( ) ' (inl( )) case A ≡ Sn is a sphere, we think of f as describing αinl :∏(x:X) PX x Ptot x an n-sphere in B which is bounded by an (n + 1)-cell αinr :∏(y:Y) PY(y) ' Ptot(inr(y)). in C . We also say that C is obtained by attaching an f f Then the square (n + 1)-cell to B via the attaching map f . The inclusion of the unique element of 1 can be thought of as a hub ha,pi7→hg(a),e(p)i ∑ P ( f (a)) ∑ P (y) and the glue paths parametrized by elements of Sn as (a:A) X (y:Y) Y spokes. (In Section 6.7 of [12] the term ‘hubs-and-spokes ha,pi7→h f (a), hy,pi7→hinr(y),αinr(p)i method’ is used for this way of attaching cells.) P (x) P (z) ∑(x:X) X ∑(z:X+AY) tot A model for our setup is given by the cubical sets hx,pi7→hinl(x),αinl(p)i of [2] (they verify the rules for suspensions—the same of which commutativity is witnessed by argument works for pushouts), and this model can itself be interpreted in extensional Martin-Löf type theory λha, pi. hglue(a), –i, with one universe containing 0, 1, N and closed under disjoint union and dependent sums and products.1 is a pushout square. The remainder of this paper is organized as follows. In As a corollary, we obtain the fiber SectionII we study the type of 2-element sets, US0 , which is a (large) model of RP∞. In particular, we show that S0 Sn RPn. 0 (S = A) ' A for any A : US0 , and that US0 classifies the S0-bundles over any type. To prove these facts, we use By the fiber sequence notation F ,→ E  B we mean Licata’s encode-decode method, in the following form. that we have a dependent type over B, such that the fiber over the base point of B is (equivalent to) F, and Lemma I.1 (Encode-decode method). Let A be a pointed the total space is (equivalent to) E. type with base point a0, and let B : A → U be a type family Finally, in SectionIV we recover RP∞ as the sequential with a point b0 : B(a0). Then the following are equivalent: colimit of the RPn, and this is now a small type. We show (i) The type ∑(x:A) B(x) is contractible. that the tautological bundle of RP∞ is an equivalence ∞ (ii) The fiberwise map into US0 , so we see that US0 is indeed a model for RP . enc ( = ) → ( ) SectionVI concludes. a0,b0 : ∏{x:A} a0 x B x enc (refl ) ≡ II.THETYPEOF 2-ELEMENTSETS defined by a0,b0 a0 : b0 is an equivalence. Definition II.1. For any type X, we define the connected Proof. Since the total space ∑(x:A) a0 = x is contractible, this is a direct corollary of Theorem 4.7.7 of [12]. component of X in U to be the type

Then in Section III we use this to give definitions of UX :≡ ∑(A:U) kA = Xk. RPn for finite n, together with its tautological S0-bundle covn RPn U RPn+1 In particular, we have the type S0 : → S0 . To recover the description of RPn ( + ) 0 as with an n 1 -cell attached to it, we show that US0 ≡ ∑(A:U) kA = S k the total space of the tautological bundle of RPn is the n-sphere. Here we shall need the flattening lemma for of 2-element sets. pushouts, cf. [12, Lemma 8.5.3]. An X-bundle over a type A is defined to be a type family B : A → UX. Lemma I.2 ( Lemma). Consider a pushout square U g A term of type X is formally a pair of a small type A Y A : U together with a term of type kA = Xk, but since f inr the latter is a mere proposition we usually omit it, and consider the term itself as a small type. X X + Y inl A Theorem II.2. The type 1The model has been formalized in Nuprl by Mark Bickford: http: //www.nuprl.org/wip/Mathematics/cubical!type!theory/ ∑ A (A:US0 )

3 of pointed 2-element sets is contractible. Hence we can complete the proof by constructing a term of type Since the theorem is a statement about pointed 0 e(N) = a 2-element sets, we will invoke the following general ∑(e:S 'A) . (II.5) lemma which computes equality of pointed types. It is time for a little trick. Instead of constructing Lemma II.3. For any A, B : U and any a : A and b : B, we a term the type in Eq. (II.5), we will show that this have an equivalence of type type is contractible. Since being contractible is a mere proposition, this allows us to eliminate the assumption     0 0 hA, ai = hB, bi ' ∑(e:A'B) e(a) = b . kS = Ak into the assumption p : S = A. Note that the end point of p is free. Therefore we eliminate p into Construction. By Theorem 2.7.2 of [12], the type on the reflS0 . Thus, we see that it suffices to show that the type left hand side is equivalent to the type ∑(p:A=B) p∗(a) = b. (As in [12], p∗ denotes the transport function relative ∑(e:S0'S0) e(N) = a to a fibration.) By the univalence axiom, the map is contractible for any a : S0. 0 idtoequivA,B : (A = B) → (A ' B) This can be done by case analysis on a : S . Since we have the equivalence neg : S0 ' S0 that swaps N and S, is an equivalence for each B : U. Therefore, we have an it follows that ∑(e:S0'S0) e(N) = N is contractible if and equivalence of type only if ∑(e:S0'S0) e(N) = S is contractible. Therefore, we     only need to show that the type ∑(p:A=B) p∗(a) = b ' ∑(e:A'B) equivtoid(e)∗(a) = b ∑(e:S0'S0) e(N) = N Moreover, by equivalence induction (the analogue of path induction for equivalences), we can compute the is contractible. For the center of contraction we take transport: hidS0 , reflNi. It remains to construct a term of type equivtoid(e)∗(a) = e(a). ∏(e:S0'S0) ∏(p:e(N)=N) he, pi = hidS0 , reflNi. It follows that (equivtoid(e) (a) = b) ' (e(a) = b). ∗ Let e : S0 ' S0 and p : e(N) = N. By Lemma II.4, we Furthermore, we will invoke the following general have an equivalence of type lemma which computes equality of pointed equiva-     he, pi = hid 0 , reflNi ' ∑ p = h(N) . lences. S (h:e∼idS0 )

Lemma II.4. For any he, pi, h f , qi : ∑(e:A'B) e(a) = b, we Hence it suffices to construct a term of the type on the have an equivalence of type right hand side.     We define a homotopy h : e ∼ idS0 by case analysis: we he, pi = h f , qi ' ∑(h:e∼ f ) p = h(a)  q . take h(N) :≡ p. To define h(S), note that the type fibe(S) is contractible. Therefore, we have a center of contraction Construction. The type he, pi = h f , qi is equivalent to the 0 hx, qi : fibe(S). Recall that equality on S is decidable, type ∑(h:e= f ) h∗(p) = q. Note that by the principle of so we have a term of type (x = N) + (x = S). Since function extensionality, the map idtohtpy : (e = f ) → e(N) = N, it follows that ¬(x = N). Therefore we have (e ∼ f ) is an equivalence. Furthermore, it follows by x = S and e(x) = S. It follows that e(S) = S, which we homotopy induction that for any h : e ∼ f we have an use to define h(S). equivalence of type The main application we have in mind for Theo- (htpytoid(h)∗(p) = q) ' (p = h(a)  q). rem II.2, is a computation of the identity type of the type of 2-element sets, via the encode-decode method, We are now ready to prove that the type of pointed Lemma I.1. 2-element sets is contractible. Corollary II.6. The canonical map Proof of Theorem II.2. We take hS0, Ni as the center of 0 contraction. We need to define an identification of type enc 0 (S = A) → A S ,N : ∏{A:U 0 } 0 S hS , Ni = hA, ai, for any A : US0 and a : A. is an equivalence. Let A : US0 and a : A. By Lemma II.3, we have an equivalence of type Another way of stating the following theorem, is by → U S0  0    saying that the map 1 S0 classifies the -bundles. hS , Ni = hA, ai ' ∑(e:S0'A) e(N) = a .

4 0 n Theorem II.7. Let B : A → US0 be a S -bundle. Then the can be described as a map RP → US0 . We perform square this construction in Definition III.1 using the properties of the type of 2-element types developed in §II, and ∑(x:A) B(x) 1 in Theorem III.4 we show that the total space of the 0 pr1 S tautological bundle on RPn is the n-sphere.

A U 0 B S Definition III.1. We define simultaneously for each n : n N−1, the n-dimensional real projective space RP , and commutes via a homotopy RA,B : ∏(x:A) ∏(y:B(x)) B(x) ' n n the tautological bundle cov : RP → U 0 . S0, and is a pullback square. S0 S Construction. The construction is by induction on n : Construction. Since B(x) : US0 for any x : A, we have by −1 N−1. For the base case n :≡ −1, we take RP :≡ 0. Corollary II.6 the fiberwise equivalence −1 Then there is a unique map of type RP → US0 , which enc S0 cov−1 S0,N(B(x)) : ( = B(x)) ' B(x) we take as our definition of S0 . For the inductive step, suppose RPn and covn are indexed by x : A. Hence it follows by Theorem 4.7.7 S0 defined. Then we define RPn+1 to be the pushout of [12] that the induced map of total spaces is an n equivalence. It follows that the diagram n cov ∑(x:RP ) S0 (x) 1 ( ) base ∑(x:A) B x pr1 ' RPn RPn+1 incl ∑ (S0 = B(x)) 1 (x:A) In other words, RPn+1 is the mapping cone of the tauto- pr1 0 pr1 λ . S logical bundle, when we view the tautological bundle as n n pr n cov RP the projection 1 : (∑(x:RP ) S0 (x)) → . A US0 B covn+1 RPn+1 → U To define S0 : S0 we use the universal commutes. Since the inner square is a pullback square, property of RPn+1. Therefore, it suffices to show that the it follows that the outer square is a pullback square. outer square in the diagram

n III.FINITEDIMENSIONALREALPROJECTIVESPACES n cov ∑(x:RP ) S0 (x) 1 Classically, the (n + 1)-st real projective space can be base pr1 obtained by attaching an (n + 1)-cell to the n-th real S0 projective space. This suggests a way of defining the real RPn RPn+1 (III.2) incl projective spaces that involves simultaneously defining n n n RP and an attaching map αn : S → RP . Then we n+1 covn U 0 obtain RP as the mapping cone of αn, i.e., as a S0 S pushout commutes. Indeed, in Theorem II.7 we have constructed Sn αn RPn a homotopy

n 0 Rn :≡ RRPn,covn : ∏(x:RPn) ∏(y:covn ) covS0 (x) ' S , 1 RPn+1, S0 S0 and in fact, this square is a pullback. and we have to somehow find a way to define the n+1 n −1 0 1 attaching map αn+1 : S → RP to continue the Example III.3. We have RP = 0, RP = 1, and RP = inductive procedure. However, it is somewhat tricky to S1. obtain these attaching maps directly, and we have chosen Theorem III.4. For each n : N− , there is an equivalence to follow a closely related path towards the definition 1 n n of the real projective spaces that takes advantage of the en : S ' ∑(x:RPn) covS0 (x). machinery of dependent type theory. + n n In other words, RPn 1 is obtained from RPn by at- Observe that the attaching map αn : S → RP is just the tautological bundle (or the quotient map taching a single (n + 1)-disk, i.e., as a pushout that identifies the antipodal points). This suggests that Sn 1 we may proceed by defining simultaneously the real RPn covn pr1◦en projective space and its tautological bundle S0 . n 0 The tautological bundle on RP is an S -bundle, so it RPn RPn+1.

5 Proof. For n ≡ −1, we have RP−1 ≡ 0 and the Corollary III.6. We obtain the fiber sequence cov−1 unique tautological bundle S0 . Therefore the type −1 S0 Sn RPn − cov ( ) . ∑(x:RP 1) S0 x is equivalent to the empty type, − which is S 1 by definition. This gives the base case. n n Hence, for each k ≥ 2 we have πk(S ) = πk(RP ). n Now assume that we have an equivalence en : S ' n n n Proof. Since we have the double cover cov : RP → U 0 ∑ n cov (x). Our goal is to construct the equiva- S0 S (x:RP ) S0 n lence with total space S , we obtain the long exact sequence n+1 n+1 en+1 : S ' ∑(x:RPn+1) cov 0 (x). n n S ··· πk+1(S ) πk+1(RP ) such that the square

n inl n+1 0 n n S S πk(S ) πk(S ) πk(RP )

en en+1 (III.5)

n n+1 n n cov ( ) + cov ( ) 0 ∑(x:RP ) S0 x ∑(x:RPn 1) S0 x πk−1(S ) πk−1(S ) ···

0 commutes. By the functoriality of the join (or equiva- Since πk(S ) = 0 for k ≥ 1, we get the desired isomor- lently, by equivalence induction on en), it suffices to find phisms. an equivalence IV. THE INFINITE DIMENSIONAL REAL PROJECTIVE  n  0 n+1 n cov ( ) ∗ S ' + cov ( ) SPACE α : ∑(x:RP ) S0 x ∑(x:RPn 1) S0 x , Observe that from the definition of RPn and its tauto- such that the bottom in the diagram logical cover, we obtain a commutative diagram of the Sn inl Sn+1 form: ∗id −1 incl 0 incl 1 incl en en S0 RP RP RP ··· 0 1 cov cov 0 n inl  n  0 S0 S ∑ n cov (x) ∑ n cov (x) ∗ S (x:RP ) S0 (x:RP ) S0 cov−1 cov2 S0 S0 α US0

n+1 + cov ( ) ∑(x:RPn 1) S0 x Using this sequence, we define the infinite dimensional real projective space and its tautological cover: commutes. We construct this equivalence using the flat- Definition IV.1. We define the infinite real projective tening lemma, Lemma I.2, from which we get a pushout space RP∞ to be the sequential colimit of the finite real square: projective spaces. The double covers on RPn define a 0 cocone on the type sequence of real projective spaces, so ∑(x:RPn) ∑(y:covn (x)) covS0 (x) ∑(t:1) S S0 cov∞ P∞ U we also obtain S0 : R → S0 . cov∞ Theorem IV.2. The double cover S0 is an equivalence from n n+1 ∞ n cov ( ) + cov ( ) RP to U 0 . ∑(x:RP ) S0 x ∑(x:RPn 1) S0 x S Proof. We have to show that the fibers of cov∞ are con- We can calculate this pushout by constructing a natural S0 tractible. Since being contractible is a mere proposition, transformation of spans (diagrams in U of the form and since the type U 0 is connected, it suffices to show · ← · → ·), as indicated by the diagram in Fig.2. S that the fiber To show that the map u in Fig.2 is an equivalence, it 0 ∞ suffices to show that Rn(x, y, z) = Rn(x, y, z) for any x, ∑(x:RP∞) S = covS0 (x) y, and z, because then it follows that u is homotopic to cov∞ S0 U the total map of a fiberwise equivalence. More generally, of S0 at : S0 is contractible. By Corollary II.6 we it suffices to show that R (X, x, y) = R (X, y, x), have an equivalence of type US0 ,T US0 ,T where T is the tautological bundle on US0 . Since US0 0 ∞ ∞ (S = covS0 (x)) ' covS0 (x), is connected and since our goal is a mere proposition, we only need to verify the claim at the base point S0 for every x : RP∞. Therefore it is equivalent to show that of US0 . This boils down to verifying that the group the type ∞ multiplication of Z/2Z is indeed commutative. ∑(x:RP∞) covS0 (x)

6 hx,zi←hx,y,zi hx,y,zi7→h?,Rn(x,y,z)i n [ 0 ∑(x:RPn) cov 0 (x) ∑(x:RPn) ∑(y:covn (x)) covS0 (x) ∑(t:1) S S S0

id u pr2

n  n  0 0 ∑( RPn) cov 0 (x) ∑( RPn) cov 0 (x) × S S x: S π1 x: S π2

Figure 2. Map of spans used in the proof of Thm III.4. The map u is given by hx, y, zi 7→ hx, z, Rn(x, y, z)i. is contractible. The general version of the flattening VI.CONCLUSION lemma, as stated in Lemma 6.12.2 in [12], can be adapted In the present article, we have constructed the real for sequential colimits, so we can pull the colimit out: it projective spaces, both finite and infinite dimensional, suffices to prove that using only homotopy theoretic methods. We have used 0 n  the univalence axiom in one place, in proving that (S = colimn ∑(x:RPn) covS0 (x) A) ' A for any A : US0 . Otherwise, our methods only is contractible. To do this, observe that the equivalences involve the universal properties of (homotopy) pushouts of Theorem III.4 form a natural equivalence of type and sequential colimits. This suggests that it could be as shown in Fig.3. Indeed, the naturality possible to mimic our construction of the real projective follows from Eq. (III.5). spaces in arbitrary ∞-toposes, even though a description Thus, the argument comes down to showing that of the structure of a model of univalent type theory does ∞ n not (yet) exist for an arbitrary ∞-topos. S :≡ colimn(S ) is contractible. This was first shown in homotopy type theory by Brunerie, and the argument Since the outer square of the diagram in Eq. (III.2) is is basically that the sequential colimit of a type sequence a pullback square, we have a few more remarks about of strongly constant maps (viz., maps factoring through our construction of the real projective spaces. One could 1) is always contractible. define the join of two maps f : A → X and g : B → X with a common codomain X, as the pushout of the Remark IV.3. Note that by our assumption that the uni- pullback. In the case of the real projective spaces, we n verse is closed under pushouts, it follows that each RP are concerned with the map pt : 1 → US0 pointing to the is in U. Since the universe contains a natural numbers type S0. By iteratively joining pt with itself, we obtain object N, it also follows that the universe is closed under the finite dimensional real projective spaces as domains ∗ sequential colimits, and therefore we have RP∞ : U. of the finite join-powers pt n. Indeed, the tautological ∗n Whereas a priori it is not clear that US0 is equivalent bundles are the maps pt . This observation connects to a U-small type, this fact is contained in Theorem IV.2. our construction of the real projective spaces with the so-called ‘join construction’, see [9]. This can be seen V. FORMALIZATION as a procedure of taking ∞-quotients in homotopy type theory. The results of §§II and III have been formalized in 2 Finally, we mention that a special case of the join the Lean proof assistant [8]. The formalization follows construction can also be used to define the complex the informal development very closely, except that we projective spaces in homotopy type theory, and to per- do not use directly the equivalence from Corollary II.6. form Milnor’s construction of the universal bundle of a This is because it takes too much memory in this case to (topological) group. We hope that these methods can also verify the property needed for Theorem III.4. be used to define the higher dimensional Instead, we observe that having decidable equality is a in homotopy type theory. mere proposition, and hence every A : US0 has decidable equality. Thus we can define α : ∏ A → A → S0 Acknowledgments {A:US0 } by setting α(x, y) = N if and only if x = y. It is then easy The authors gratefully acknowledge the support of the to check that each α(x,–) is an involution and hence an Air Force Office of Scientific Research through MURI equivalence, and α is clearly symmetric. grant FA9550-15-1-0053. Any opinions, findings and con- The results of §IV require the flattening lemma for clusions or recommendations expressed in this material sequential colimits, which has not yet been formalized are those of the authors and do not necessarily reflect in Lean. the views of the AFOSR.

2The code is available at: https://github.com/cmu-phil/Spectral/ blob/master/homotopy/realprojective.hlean

7 −1 0 1 − cov ( ) cov ( ) cov ( ) ··· ∑(x:RP 1) S0 x ∑(x:RP0) S0 x ∑(x:RP1) S0 x ' ' '

S−1 S0 S1 ···

Figure 3. Natural equivalence of type sequences for Thm IV.2.

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