The Simplicial Model of Univalent Foundations (after Voevodsky)

We present Voevodsky's construction of a model of univalent type theory in the category of simplicial sets. To this end, we first give a general technique for constructing categorical models of dependent type theory, using universes to obtain coherence. We then construct a (weakly) universal Kan fibration, and use it to exhibit a model in simplicial sets. Lastly, we introduce the Univalence Axiom, in several equivalent formulations, and show that it holds in our model. As a corollary, we conclude that Martin-L\"of type theory with one univalent universe (formulated in terms of contextual categories) is at least as consistent as ZFC with two inaccessible cardinals.


Introduction
The Univalent Foundations programme is a new proposed approach to foundations of mathematics, originally suggested by the third-named author in [Voe06] (closely related to independent work of Awodey, Warren, and collaborators [AW09]), building on the systems of dependent type theory developed by Martin-Löf and others.
A major motivation for earlier work with such logical systems has been their well-suitedness to computer implementation. One notable example is the Coq proof assistant, based on the Calculus of Inductive Constructions (a closely related dependent type theory), which has shown itself feasible for large-scale formal verification of mathematics, with developments including formal proofs of the Four-Colour Theorem [Gon08] and the Feit-Thompson (Odd Order) Theorem [GAA + 13].
One feature of dependent type theory which has previously remained comparatively unexploited, however, is its richer treatment of equality. In traditional foundations, equality carries no information beyond its truth-value: if two things are equal, they are equal in at most one way. This is fine for equality between elements of discrete sets; but it is unnatural for objects of categories (or higher-dimensional categories), or points of spaces. In particular, it is at odds with the informal mathematical practice of treating isomorphic (and sometimes more weakly equivalent) objects as equal; which is why this usage must be so often disclaimed as an abuse of language, and kept rigorously away from formal statements, even though it is so appealing.
In dependent type theory, equalities can carry information: two things may be equal in multiple ways. So the basic objects-the types-may behave not just like discrete sets, but more generally like higher groupoids (with equalities being morphisms in the groupoid), or spaces (with equalities being paths in the space). And, crucially, this is the only equality one can talk about within the logical system: one cannot ask whether elements of a type are "equal on the nose", in the classical sense. The logical language only allows one to talk about properties and constructions which respect its equality.
The Univalence Axiom, introduced by the third-named author, strengthens this characteristic. In classical foundations one has sets of sets, or classes of sets, and uses these to quantify over classes of structures. Similarly, in type theory, types of types-universes-are a key feature of the language. The Univalence Axiom states that equality between types, as elements of a universe, is the same as equivalence between them, as types. It formalises the practice of treating equivalent structures as completely interchangeable; it ensures that one can only talk about properties of types, or more general structures, that respect such equivalence. In sum, it helps solidify the idea of types as some kind of spaces, in the homotopy-theoretic sense; and more practically-its original motivation-it provides for free many theorems (transfer along equivalences, naturality with respect to these, and so on) which must otherwise be re-proved by hand for each new construction.
The main goal of this paper is to justify the intuition outlined above, of types as spaces. Specifically, we focus on simplicial sets, a well-studied model for spaces in homotopy theory; we construct a model of type theory in the category sSet of simplicial sets, and show that it satisfies the Univalence Axiom. (For comparison with other familiar notions: simplicial sets present the same homotopy theory as topological spaces, and form the basis of one of the most-studied models for higher groupoids.) It also follows from this model that the Univalent Foundations are consistent, provided that the classical foundations we use are (precisely, ZFC together with the existence of two strongly inaccessible cardinals, or equivalently two Grothendieck universes).
This paper therefore includes a mixture of logical and homotopy-theoretic ingredients; however, we have aimed to separate the two wherever possible. Good background references for the logical parts include [NPS90], a general introduction to the type theory; [Hof97], for the categorical semantics; and [ML84], the locus classicus for the logical rules. For the homotopytheoretic aspects, [GJ09] and [Hov99] are both excellent and sufficient references. Finally, for the category-theoretic language used throughout, [ML98] is canonical.
Organisation. In Section 1 we consider general techniques for constructing models of type theory. After setting out (in Section 1.1) the specific type theory that we will consider, we review (Section 1.2) some fundamental facts about its semantics in contextual categories, following [Str91]. In Section 1.3, we use universes to construct contextual categories, and hence models of (the structural core of) type theory; and in Section 1.4, we use categorical constructions on the universe to model the logical constructions of type theory. Together, these present a new solution to the coherence problem for modelling type theory (cf. [Hof95b]).
In Section 2, we turn towards constructing a model in the category of simplicial sets. Sections 2.1 and 2.2 are dedicated to the construction and investigation of a (weakly) universal Kan fibration (a "universe of Kan complexes"); in Section 2.3 we use this universe to apply the techniques of Section 1, giving a model of the full type theory in simplicial sets. Section 3 is devoted to the Univalence Axiom. We formulate univalence first in type theory (Section 3.1), then directly in homotopy-theoretic terms (Section 3.2), and show that these definitions correspond under the simplicial model (Section 3.3). In Section 3.4, we show that the universal Kan fibration is univalent, and hence that the Univalence Axiom holds in the simplicial model. Finally, in Section 3.5 we discuss an alternative formulation of univalence, shedding further light on the universal property of the universe.
We should mention here that this paper is based in large part on the ongoing unpublished manuscript [Voe12] of the third-named author.
Related work. While the present paper discusses just models of the Univalent Foundations, the major motivation for this is the actual development of mathematics within these foundations. The work on this so far exists mainly in unpublished but publicly-available form: notably the computerformalizations [HoTa] and [V + ] (for an introduction to which, see [PW14]), but also [AGMLV11], [HoTb], and most recently the HoTT Book, [Uni13].
Earlier work on homotopy-theoretic models of type theory can be found in [HS98], [AW09], [War11]. Other current and recent work on such models includes [GvdB11], [AK11], and [Shu14]. Other general coherence theorems, for comparison with the results of Section 1, can be found in [Hof95b] and [LW12]. Univalence in homotopy-theoretic settings is also considered in [Moe11] and [GK12]. (These references are, of course, far from exhaustive.) Acknowledgements. First and foremost we are grateful to Michael Warren, whose illuminating seminars and discussions heavily influenced our understanding and presentation of the material. We also thank Ieke Moerdijk, Mike Shulman, and Karol Szumi lo, for helpful correspondence, conversations, and corrections to drafts; and Steve Awodey, without whose constant support and encouragement in many ways this paper would not exist.
The first-named author was financially supported during this work by the NSF, Grant DMS-1001191 (P.I. Steve Awodey), and by a grant from the Benter Foundation (P.I. Thomas C. Hales); the second-named author, by an AARMS postdoctoral fellowship at Dalhousie University, and grants from NSERC (P.I.'s Peter Selinger, Robert Dawson, and Dorette Pronk).

Models from Universes
In this section, we set up the machinery which we will use, in later sections, to model type theory in simplicial sets. The type theory we consider, and some of the technical machinery we use, are standard; the main original contribution is a new technique for solving the so-called coherence problem, using universes.
1.1. The type theory under consideration. Formally, the type theory we will work with is a slight variant of Martin-Löf's Intensional Type Theory, as presented in e.g. [ML84]. The rules of this theory are given in full in Appendix A; briefly, it is a dependent type theory, taking as basic constructors Π-, Σ-, Id-, and W-types, 0, 1, +, and one universeà la Tarski.
A related theory of particular interest is the Calculus of Inductive Constructions, on which the Coq proof assistant is based ( [Wer94]). CIC differs from Martin-Löf type theory most notably in its very general scheme for inductive definitions, and in its treatment of universes. We do not pursue the question of how our model might be adapted to CIC, but for some discussion and comparison of the two systems, see [PM96], [Bar12], and [Voe12, 6.2].
One abuse of notation that we should mention: we will sometimes write e.g. A(x) or t(x, y) to indicate free variables on which a term or type may depend, so that we can later write A(g(z)) to denote the substitution [g(z)/x]A more readably. Note however that the variables explicitly shown need not actually appear; and there may also always be other free variables in the term, not explicitly displayed.
1.2. Contextual categories. Rather than constructing an interpretation of the syntax directly, we work via the intermediary notion of contextual categories, a class of algebraic objects abstracting the key structure carried by the syntax. 1 The plain definition of a contextual category coresponds to the structural core of the syntax; further syntactic rules (logical constructors, etc.) correspond to extra algebraic structure that contextual categories may carry. Essentially, contextual categories provide a completely equivalent alternative to the syntactic presentation of type theory.
Why is this duplication of notions desirable? The trouble with the syntax is that it is mathematically tricky to handle. Any full presentation must account for (among other things) variable binding, capture-free substitution, and the possibility of multiple derivations of a judgement; and so a direct interpretation must deal with all of these, at the same time as tackling the details of the particular model in question. By passing to contextual categories, one deals with these subtleties and bureaucracy once and for all, and obtains a clear framework for subsequently constructing models.
Conversely, why not work only with contextual categories, dispensing with syntax entirely? The trouble on this side is that working with higher-order logical structure in contextual categories quickly becomes unreadable: compare, for instance, the statements of functional extensionality in Sections A.4 and B.3.
The reader preferring to take contextual categories as primary may regard the syntax as essentially a notation for working within them: a powerful, flexible, and intuitive notation, but one whose validity requires non-trivial work to establish. (The situation is comparable to that of string diagrams, as used in monoidal and more elaborately structured categories [Sel11].) A contextual category C consists of the following data: (1) a category C; (2) a grading of objects as Ob C = n:N Ob n C; (3) an object 1 ∈ Ob 0 C; (4) maps ft n : Ob n+1 C G G Ob n C (whose subscripts we usually suppress); (5) for each X ∈ Ob n+1 C, a map p X : X G G ft X (the canonical projection from X); 1 Contextual categories are not the only option; the closely related notions of categories with attributes [Car78,Mog91,Pit00], categories with families [Dyb96,Hof97], and comprehension categories [Jac93] would all also serve our purposes.
(6) for each X ∈ Ob n+1 C and f : Y G G ft (X), an object f * (X) and a map q(f, X) : f * (X) G G X; such that: (1) 1 is the unique object in Ob 0 (C); (2) 1 is a terminal object in C; (3) for each n > 0, X ∈ Ob n C, and f : Y G G ft(X), we have ft(f * X) = Y , and the square is a pullback (the canonical pullback of X along f ); and (4) these canonical pullbacks are strictly functorial: that is, for X ∈ Ob n+1 C, 1 * ft X X = X and q(1 ft X , X) = 1 X ; and for X ∈ Ob n+1 C, Remark 1.2.2. Note that these may be seen as models of a multi-sorted essentially algebraic theory ([AR94, 3.34]), with sorts indexed by N + N × N.
This definition is best understood in terms of its prototypical example: Example 1.2.3. Let T be any type theory. Then there is a contextual category C(T), described as follows: • Ob n C(T) consists of the contexts [x 1 :A 1 , . . . , x n :A n ] of length n, up to definitional equality and renaming of free variables; • maps of C(T) are context morphisms, or substitutions, considered up to definitional equality and renaming of free variables. That is, a map f : [x 1 :A 1 , . . . , x n :A n ] G G [y 1 :B 1 , . . . , y m :B m (y 1 , . . . , y m−1 )] is represented by a sequence of terms f 1 , . . . , f m such that . .
and two such maps [f i ], [g i ] are equal exactly if for each i, • composition is given by substitution, and the identity Γ G G Γ by the variables of Γ, considered as terms; • 1 is the empty context [ ]; • ft[x 1 :A 1 , . . . , x n+1 :A n+1 ] = [x 1 :A 1 , . . . , x n :A n ]; • for Γ = [x 1 :A 1 , . . . , x n+1 :A n+1 ], the map p Γ : Γ G G ft Γ is the dependent projection context morphism simply forgetting the last variable of Γ; • for contexts G G Γ. For this reason, when working with contextual categories, we will often write just "sections" to refer to sections of dependent projections.
We will also use several other notations deserving of particular comment. For an object Γ, we will write e.g. (Γ, A) to denote an arbitrary object with ft(Γ, A) = Γ, and will then write the dependent projection p (Γ,A) simply as p A ; similarly, (Γ, A, B), and so on. Similarly, we will write f * not only for the canonical pullbacks of appropriate objects, but also the pullbacks of maps between them.
As mentioned above, Definition 1.2.1 alone corresponds precisely to the basic judgements and structural rules of dependent type theory. Similarly, each logical rule or type-or term-constructor corresponds to certain extra structure on a contextual category. We make this correspondence precise in Theorem 1.2.9 below, once we have set up the appropriate definitions.
Definition 1.2.4. A Π-type structure on a contextual category C consists of the following data: (1) for each (Γ, A, B) ∈ Ob n+2 C, an object (Γ, Π(A, B)) ∈ Ob n+1 C; . . and such that for a : (5) and moreover, all the above operations are stable under substitution: for any morphism f : These are direct translations of the rules for Π-types given in A.2. Similarly, all the other logical rules of Appendix A may be routinely translated into structure on a contextual category; see Appendix B and [Hof97, 3.3] for more details and discussion.
Example 1.2.5. If T is a type theory with Π-types, then C(T) carries an evident Π-type structure; similarly for Σ-types and the other constructors of Sections A.2 and A.3.
Remark 1.2.6. Note that all of these structures, like the definition of contextual categories themselves, are essentially algebraic in nature.
Definition 1.2.7. A map F : C G G D of contextual categories, or contextual functor, consists of a functor C G G D between underlying categories, respecting the gradings, and preserving (on the nose) all the structure of a contextual category.
Similarly, a map of contextual categories with Π-type structure, Σ-type structure, etc., is a contextual functor preserving the additional structure.
Remark 1.2.8. These are exactly the maps given by considering contextual categories as essentially algebraic structures.
We are now equipped to state precisely the sense in which the structures defined above correspond to the appropriate syntactic rules: Theorem 1.2.9. Let T be the type theory given by the structural rules of Section A.1, plus any combination of the logical rules of Sections A.2, A.3. Then C(T) is initial among contextual categories with the appropriate extra structure.
(Unfortunately, there is (to our knowledge) no general presentation of this theorem in the existing literature. The Correctness Theorem of [Str91, Ch. 3, p. 181] treats it in detail for a specific set of logical constructors; the fact that it holds for other selections from among the standard rules is wellknown in folklore. Given this slightly unsatisfactory situation, we present our main results in a form not dependent on this theorem, and explicitly note all points where we do make use of it.) In other words, if C is a contextual category with structure corresponding to the logical rules of a type theory T, then there is a functor C(T) G G C interpreting the syntax of T in C. This justifies the definition: Definition 1.2.10. A model of dependent type theory with any selection of the logical rules of Section A.2 is a contextual category equipped with the structure corresponding to the chosen rules.
1.3. Contextual categories from universes. The major difficulty in constructing models of type theory is the so-called coherence problem: the requirement for pullback to be strictly functorial, and for the logical structure to commute strictly with it. In most natural categorical situations, operations on objects commute with pullback only up to isomorphism, or even more weakly; and for constructors with weak universal properties, operations on maps (corresponding for example to the Id-elim rule) may also fail to commute with pullback. Hofmann [Hof95b] gives a construction which solves the issue for Π-and Σ-types, but Id-types in particular remain problematic with this method. Other methods exist for certain specific categories ( [HS98], [War08]), but are not applicable to the present case.
In order to obtain coherence for our model, we thus give a new construction based on universes (not necessarily the same as universes in the type-theoretic sense, though the two may sometimes coincide). Definition 1.3.1. Let C be a category. A universe in C is an object U together with a morphism p :Ũ G G U , and for each map f : X G G U a choice of pullback square The intuition here is that the map p represents the generic family of types over the universe U .
By abuse of notation, we often refer to the universe simply as U , with p and the chosen pullbacks understood.
Given a map f : Y G G X, we will often write f (or Y , if f is understood) for a map X G G U such that f ∼ = P (X, f ) . Also, for a sequence of maps f 1 : X G G U , f 2 : (X; f 1 ) G G U , etc., we write (X; f 1 , . . . , f n ) for ((. . . (X; f 1 ); . . .); f n ).
(2) This contextual category is well-defined up to canonical isomorphism given just C and p :Ũ G G U , independently of the choice of pullbacks and terminal object.
Justified by the second part of this proposition, we will not explicitly consider the choices of pullbacks and terminal object when we construct the universe in the category sSet of simplicial sets.
As an aside, let us note that every small contextual category arises in this way: Proposition 1.3.4. Let C be a small contextual category. Consider the universe U in the presheaf category [C op , Set] given by with the evident projection map, and any choice of pullbacks.
Then [C op , Set] U is isomorphic, as a contextual category, to C.
Proof. Straightforward, with liberal use of the Yoneda lemma.
1.4. Logical structure on universes. Given a universe U in a category C, we want to know how to equip C U with various logical structure-Π-types, Σ-types, and so on. For general C, this is rather fiddly; but when C is locally cartesian closed (as in our case of interest), it is more straightforward, since local cartesian closedness allows us to construct and manipulate "objects of U -contexts", and hence to construct objects representing the premises of each rule.
In working with locally cartesian closed structure, given a map f : A G G B, we will follow topos-theoretic convention and write Σ f and Π f respectively for the left and right adjoints to the pullback functor f * : Also, the intended map A G G B is often clearly determined by the objects A and B, as some sort of associated projection; in such a case, we will write Σ A→B , Π A→B for the functors arising from this map.
An alternative notation for locally cartesian closed categories is their internal logic, extensional dependent type theory [See84], [Hof95b]. While this language is convenient and powerful, we avoid it due to the difficulties of working clearly with two logical languages in parallel.
Returning to the question at hand, first consider Π-types. We know that dependent products exist in C; so informally, we need only to ensure that U (considered as a universe of types) is closed under such products. Specifically, given a type A in U over some base X (that is, a map A : X G G U ), and a dependent family of types B over A, again from U (i.e. a map B : A := (X; A ) G G U ), the product Π A→X B of this families in the slice C/X should again "live in U "; that is, there should be a map Π(A, B) : X G G U such that (X; Π(A, B) ) ∼ = Π A→X B. Moreover, we need this construction to be strictly natural in X.
Due to the strict naturality requirement, we cannot simply provide this structure for each X and A, B individually. Instead, we construct an object U Π representing such pairs (A, B), and a generic such pair (A gen , B gen ) based on U Π . It is sufficient to define Π in this generic case X = U Π ; the construction then extends to other X by precomposition, and as such, is automatically strictly natural in X. Precisely: Definition 1.4.1. Given a universe U in an lccc C, define (This definition can be expressed in several ways, according to one's preferred notation. In the internal language of C as an LCCC, it can be written , showing it more explicitly as an internalisation of the premises of the Π-form rule. Using a more traditional internal-hom notation, it could alternatively be written as Hom U (Ũ , U ×Ũ ).) Pulling backŨ along the projection U Π G G U induces an object A gen = U × U U Π , along with a projection map α gen : A gen G G U Π . Similarly, pulling backŨ along the counit (the evaluation map of the internal hom) induces an object (B gen , β gen ) over A gen : Moreover, the universal properties of the LCCC structure ensure that for induces the given sequence via precomposition and pullback: Definition 1.4.2. A Π-structure on a universe U in a lccc C consists of a map Π : U Π G G U. whose realisation is a dependent product for the generic dependent family of types; that is, it is equipped with an isomorphism Π * Ũ ∼ = Π αgen B gen over U Π , or equivalently with a mapΠ : Σ U Π →1 Π αgen B gen G GŨ making the square The approach used here gives a template which we follow for all the other constructors, with extra subtleties entering the picture just in the cases of Id-types and (type-theoretic) universes, since these structures do not arise from strict category-theoretic constructions.
Definition 1.4.3. Take U Σ to be the object representing the premises of the Σ-form rule: Since these are the same as the premises of the Π-form rule, we have in this case that U Σ = U Π ; and we have again the generic family of types Definition 1.4.4. A Σ-structure on a universe U in a lccc C consists of a map Σ : U Σ G G U whose realisation is a dependent sum for the generic dependent family of types; that is, it is equipped with an isomorphism Σ * Ũ ∼ = Σ αgen B gen over U Σ (or again equivalently with a mapΣ : Σ U Σ →1 Σ αgen B gen G GŨ making the appropriate square a pullback).
Id-structure requires a few auxiliary definitions. 2 Recall first the classical notion of weak orthogonality of maps: has a section. Say i is moreover stably orthogonal to f if for every object C of C, C × i is orthogonal to f .
In a cartesian closed category, this notion has an internal analogue: The following proposition connects the classical and internal notions: Given i, f as above, there exists a lifting operation for i against f if and only if i is stably orthogonal to f .
Proof. If i is stably orthogonal to f , then a lifting operation may be obtained as (the exponential transpose of) a filler for the canonical square Conversely, any square from ; composing this with a lifting operation provides a map C G G Y B , whose transpose is a filler for the square.
As shown in [AW09] and [GG08], the rules for Id-types can be understood roughly as follows. In a model where dependent types are interpreted as fibrations, the identity type over a type A (in any slice C/Γ) is a factorisation of the diagonal ∆ A : A G G A × Γ A as a stable trivial cofibration, followed by a fibration. (Here, by a stable trivial cofibration, we mean a map which is stably orthogonal to fibrations, in C/Γ.) Additionally, choices of all data (including liftings) must be given which commute with pullbacks in the base Γ.
In our case, the "fibrations" are just the pullbacks of p; so it suffices to consider orthogonality between the first map of the factorisation and p itself. Moreover, as for Π-and Σ-structure above, we demand the structure just in the universal case where A isŨ , in the slice C/U . Finally, an internal lifting operation turns out to be exactly the structure required to give chosen lifts commuting with pullbacks. We therefore define: Definition 1.4.8. Id-structure on a universe consists of maps commutes, together with a lifting operation J for r against p × U in C/U .
Remark 1.4.9. By virtue of Proposition 1.4.7, we could instead simply stipulate that r be stably orthogonal to p × U . We choose the current version since it provides exactly the structure required for Theorem 1.4.15, without requiring any arbitrary choices.
Definition 1.4.10. W-structure on a universe consists of a map W : such that W * Ũ is an initial algebra for the polynomial endofunctor of C/U W specified by β gen : B gen G G A gen , i.e. the endofunctor (For details on polynomial endofunctors and their algebras, see [MP00], [GH04].) Definition 1.4.11. 0-structure on U consists of a map 0 : (By analogy with the preceding definitions, one might refer to 1 here as U 0 , and similarly in the next two definitions. We choose not to do so simply for the sake of readability.) Definition 1.4.12. 1-structure on U consists of a map 1 : 1 G G U such that 1 * Ũ ∼ = 1.
Definition 1.4.13. +-structure on U consists of a map + : . Finally, we consider the structure on U needed to give a universe (in the type-theoretic sense) in C U . Here, for the first time, we need to consider a nested pair of universes, since the internal universe of C U must be some smaller universe U 0 in C.
Definition 1.4.14. An internal universe (U 0 , i) in U consists of arrows Given these, i induces by pullback a universe structure (p 0 ,Ũ 0 , . . .) on U 0 . We say that U 0 is closed under Π-types in U if U 0 carries a Π-structure Π 0 , commuting with i in the sense that the square Similarly, we say that U 0 is closed under Σ-types (resp. Id-types, etc.) if it carries Σ-structure Σ 0 (resp. an Id-structure (Id 0 , r 0 ), etc.) commuting with i.
With these structures defined, we can now prove that they are fit for purpose: Theorem 1.4.15. Π-structure (resp. Σ-structure, etc.) structure on a universe U induces Π-type structure (resp. Σ-type structure, etc.) on C U . Moreover, an internal universe (U 0 , i) in U closed under any combination of Π-types, Σ-types, etc., induces a universeà la Tarski in C U closed under the corresponding constructors.
Proof. This proof is esentially a routine verification; we give the case of Π-types in full, and leave the rest mostly to the reader.
In a nutshell, the constructor Π is induced by the map Π; and the constructors λ and app are induced by the corresponding lccc structure in C.
Precisely, we treat the rules of Π-types (corresponding to the components of the desired Π-type structure) one at a time.
(Π-form): The premises . By construction, this is stable under substitution along any map f : ∆ G G Γ, since substitution in C U is again just composition in C. (Π-intro): Besides Γ, A, B as before, we have an additional premise This is by definition a map 1 A G G B in C/A, corresponding by adjunction to a mapt : sot corresponds to a section of Π(A, B) over Γ, which we take as λ(t). Stability under substitution follows by the uniqueness in the universal property of Π A→Γ B.
We could alternatively have defined λ more analogously to Π, by representing the premises as a single map ( A , B , t then taking the transpose of the generic term t gen over U Π-intro ; and then pulling this back along ( A , B , t). In fact, thanks to the uniqueness in the universal property of Π Agen→U Π B gen , that would give the same result as the present, more straightforward, definitition. However, the alternative definition has the advantage that its stability under substitution follows simply from properties of pullbacks; this becomes important for Id-types, whose universal property lacks a uniqueness condition.
(Π-app): The premises now are Together, these give a section over Γ of Π A→Γ B × Γ A; so composing this with the evaluation map ev A,B of Π A→Γ B gives a map Γ G G B lifting a, which we take to be app(f, a).
(Π-comp): here, we have premises Γ, A, B, t as in Π-intro, and a as in Π-app; and we have formed app(λ(t), a) as prescribed above. So, unwinding as desired, by the usual rules of LCCCs. This completes the proof for Π-structures. As indicated above, the remaining constructors are for the most part entirely analogous; the only subtlety is in the case for the Id-elim rule. In this case, there are two ways that one could define the appropriate structure: one can either pull back to each specific context and then choose liftings, or choose a lifting in the universal context and then pull it back (as discussed following the Π-intro case above). The second of these is the correct choice: the first is not automatically stable under substitution. (For other constructors, this distinction does not arise, since their strict categorical universal properties canonically determine the maps involved.) And, in fact, the "universal lifting" required is precisely the internal lifting operation provided by the Id-structure on U .

The Simplicial Model
In this section, we construct (for any regular cardinal α) a Kan fibration p α : U α G G U α , weakly universal among Kan fibrations with α-small fibers, and investigate the key properties of U α and p α . In particular, we show that U α is a Kan complex, and (when α is inaccessible) carries the various logical structures defined in Section 1.4. Together, these yield our first main goal: a model of type theory in sSet, with an internal universe.
2.1. A universe of Kan complexes. In constructing a universe U α intended to represent α-small Kan fibrations, one might expect (by the Yoneda lemma) to simply define (U α ) n as the set of α-small fibrations over ∆[n]. This definition has two problems: firstly, it gives not sets, but proper classes; and secondly, it is not strictly functorial, since pullback is functorial only up to isomorphism.
Some extra technical device is therefore needed to resolve these issues. Several possible solutions exist; we take the approach of passing to isomorphism classes, having first added well-orderings to the mix so that fibrations have no non-trivial automorphisms (without which the crucial Lemmas 2.1.4, 2.1.5 would fail). We emphasise, however, that this is the sole reason for introducing the well-orderings: they are of no intrinsic interest or significance.
Definition 2.1.1. A well-ordered morphism of simplicial sets consists of an ordinary map of simplicial sets f : Y G G X, together with a function assigning to each simplex x ∈ X n a well-ordering on the fiber Y are well-ordered morphisms into a common base X, an isomorphism of well-ordered morphisms from f to f ′ is an isomorphism Y ∼ = Y ′ over X preserving the well-orderings on the fibers.
Proposition 2.1.2. Given two well-ordered sets, there is at most one isomorphism between them. Given two well-ordered morphisms over a common base, there is at most one isomorphism between them.
Proof. The first statement is classical (and immediate by induction); the second follows from the first, applied in each fiber.
Definition 2.1.3. Fix (for the remainder of this and the following section) a regular cardinal α. Say a map of simplicial sets f : Y G G X is α-small if each of its fibers Y x has cardinality < α.
Given a simplicial set X, define W α (X) to be the set of isomorphism classes of α-small well-ordered morphisms Y G G X; together with the pullback action W α (f ) := f * : W α (X) G G W α (X ′ ), for f : X ′ G G X, this gives a contravariant functor W α : sSet op G G Set.
Lemma 2.1.4. W α preserves all limits: For each x ∈ X n , choose some i andx ∈ F (i) with ν(x) = x, and set Y x := (Y i )x. By Proposition 2.1.2, this is well-defined up to canonical isomorphism, independent of the choices of representatives i,x, Y i , f i . The total space of these fibers then defines a well-ordered morphism f : Y G G X, with fibers of size < α, and with pullbacks isomorphic to f i as required.
For injectivity, suppose f, f ′ are well-ordered morphisms over X, and ν * i f ∼ = ν * i f ′ for each i. By Proposition 2.1.2, these isomorphisms must agree on each fiber, so together give an isomorphism f ∼ = f ′ .
Define the simplicial set W α by where y denotes the Yoneda embedding ∆ G G sSet.
Lemma 2.1.5. The functor W α is representable, represented by W α . Proof. The functors W α , Hom(−, W α ) agree up to isomorphism on the standard simplices (by the Yoneda lemma), and send colimits in sSet to limits; but every simplicial set is canonically a colimit of standard simplices.
Notation 2.1.6. Given an α-small well-ordered map f : Y G G X, the corresponding map X G G W α will be denoted by f .
Applying the natural isomorphism above to the identity map W α G G W α yields a universal α-small well-ordered simplicial set W α G G W α . Explicitly, n-simplices of W α are classes of pairs i.e. the fiber of W α over an n-simplex f ∈ W α is exactly (an isomorphic copy of) the main fiber of f . So, by construction: Proposition 2.1.7. The canonical projection W α G G W α is strictly universal for α-small well-ordered morphisms; that is, any such morphism can be expressed uniquely as a pullback of this projection.
Corollary 2.1.8. The canonical projection W α G G W α is weakly universal for α-small morphisms of simplicial sets: any such morphism can be given, not necessarily uniquely, as a pullback of this projection.
Proof. By the well-ordering principle and the axiom of choice, one can wellorder the fibers, and then use the universal property of W α .
Definition 2.1.9. Let U α ⊆ W α (respectively, U α ⊆ W α ) be the subobject consisting of (isomorphism classes of) α-small well-ordered fibrations 3 ; and define p α : U α G G U α as the pullback: Proof. Consider a horn to be filled x G G U α 3 Here and throughout, by "fibration" we always mean "Kan fibration".
for some 0 ≤ k ≤ n. It factors through the pullback where by the definition of U α and U α , x is a fibration. Thus the left square admits a diagonal filler, and hence so does the outer rectangle.
Lemma 2.1.11. An α-small well-ordered morphism f : Proof. For '⇒', assume that f : Y G G X is a fibration. Then the pullback of f to any representable is certainly a fibration: Conversely, suppose f factors through U α . Then we obtain: where the lower composite is f , and the outer rectangle and the right square are by construction pullbacks. Hence so is the left square; so by Lemma 2.1.10 f is a fibration.
Corollary 2.1.12. The functor U α is representable, represented by U α . The map p α : U α G G U α is strictly universal for α-small well-ordered fibrations, and weakly universal for α-small fibrations.
In Section 3.5, we will strengthen this universal property, showing that while the representation of a fibration as a pullback of p α may not be strictly unique, it is unique up to homotopy: precisely, the space of such representations is contractible.
2.2. Kan fibrancy of the universe. The previous section provides the main ingredients needed to use U α as a universe in the sense of Section 1, and hence to give a model of the core type theory. However, to give additionally a type-theoretic universe within that model, we need to show that each U α itself can be seen as a type of the model; in other words, that it is Kan. The main goal of this section is therefore to prove the following theorem: Theorem 2.2.1. The simplicial set U α is a Kan complex.
Before proceeding with the proof we will gather four useful lemmas. The first two concern minimal fibrations, which for the present purposes are a technical device whose details, beyond these two lemmas, are unimportant.
Lemma 2.2.2 (Quillen's Lemma, [Qui68]). Any fibration f : Y G G X may be factored as f = pg, where p is a minimal fibration and g is a trivial fibration.
Lemma 2.2.3 ([BGM59, III.5.6]; see also [May67,Cor. 11.7]). Suppose X is contractible, with x 0 ∈ X, and p : Y G G X is a minimal fibration with fiber F := Y x 0 . Then there is an isomorphism over X: For Lemma 2.2.5, the proof we give is due to André Joyal; we include details here since the original [Joy11] is not currently publicly available. For this, and again for Theorem 3.4.1 below, we make crucial use of exponentiation along cofibrations; so we pause first to establish some facts about this.

For any n-simplex x : ∆[n]
G G B, we have (Π i p) x ∼ = Hom sSet/B (x, Π i p) ∼ = Hom sSet/B (i * x, p). As a subobject of ∆[n], i * x has only finitely many non-degenerate simplices, so (Π i p) x injects into a finite product of fibers of p and is thus of size < α.

Lemma 2.2.5 ([Joy11, Lemma 0.2]). Trivial fibrations extend along cofibrations. That is, if t : Y
G G X is a trivial fibration and j : X G G X ′ is a cofibration, then there exists a trivial fibration t ′ : Y ′ G G X ′ and a pullback square of the form: Moreover, if t is α-small, then t ′ may be chosen to also be.
Proof. Take t ′ := Π j t. By part 1 of Lemma 2.2.4, this is a trivial fibration; by part 2, j * Y ′ ∼ = Y ; and by part 3, it is α-small.
We are now ready to prove that U α is a Kan complex.
Proof of Theorem 2.2.1. We need to show that we can extend any horn in U α to a simplex: y Corollary 2.1.12, such a horn corresponds to an α-small well-ordered fibration q : To extend q to a simplex, we just need to construct an α-small fibration Y ′ over ∆[n] which restricts on the horn to Y : By the axiom of choice one can then extend the well-ordering of q to q ′ , so the map q ′ : ∆[n] G G U α gives the desired simplex. By Quillen's Lemma, we can factor q as where q t is a trivial fibration and q m is a minimal fibration. Both are still α-small: each fiber of q t is a subset of a fiber of q, and since a trivial fibration is onto, each fiber of q m is a quotient of a fiber of q. By Lemma 2.2.3, we have an isomorphism Y 0 ∼ = F × Λ k [n], and hence a pullback diagram: By Lemma 2.2.5, we can then complete the upper square in the following diagram, with both right-hand vertical maps α-small fibrations: Since α is regular, the composite of the right-hand side is again α-small; so we are done.

2.3.
Modelling type theory in simplicial sets. To prove that U α carries the structure to model type theory, we will need a couple of further lemmas; firstly, that taking dependent products preserves fibrations: Then the dependent product Π p q is a fibration over X.
Proof. The pullback functor p * : sSet/X G G sSet/Y preserves trivial cofibrations (since sSet is right proper and cofibrations are monomorphisms); so its right adjoint Π p preserves fibrant objects. Secondly, to model Id-types, we will require well-behaved fibered path objects. The construction below may be found in [War08, Thm. 2.25]; we recall it in more elementary terms, which will be useful to us later. Definition 2.3.2. Given a fibration p : E G G B, define the fibered path object P B (E) as the pullback the object of paths in E that are constant in B.
The "constant path" map c : E G G E ∆[1] factors through P B (E); call the resulting map r p : E G G P B (E). There are also evident source and taget maps s p , t p : P B (E) G G E. (On all of these maps, we will omit the subscripts when they are clear from context.)

Proposition 2.3.3. For any fibration
give a factorisation of the diagonal map ∆ p : E G G E × B E over B as a (trivial cofibration, fibration); and this is stable over B in that the pullback along any B ′ G G B is again such a factorisation.
Proof. It is clear that these maps give a factorisation of ∆ p over B. To see that they are a trivial cofibration and a fibration respectively, consider the pullback construction of P B (E) via two intermediate stages: Now (s, t) is certainly a fibration, since it is a pullback of the map , which is one by the monoidal model category axioms. But s is a retraction of r, so r is a weak equivalence (by 2-out-of-3) and a monomorphism, so is a trivial cofibration as desired.
Finally, stability of these properties under pullback follows immediately from the stability (up to isomorphism) of the construction itself: for any f : B ′ G G B, there is a canonical isomorphism P B ′ (f * E) ∼ = f * P B (E), commuting with the maps r, s, t.
Moreover, if β < α is also inaccessible, then U β gives an internal universe in U α closed under all these constructors.
Proof. (Π-structure): Given a pair of α-small fibrations Z q G G Y p G G X, the dependent product Π p q in sSet/X is again a fibration, by Lemma 2.3.1; it is also α-small, since α is inaccessible. Hence by Corollary 2.1.12, the universal dependent product over U α

Π-form
is representable as the pullback of U α along some map Π : U α Π-form G G U α , giving the desired Π-structure.
(Σ-structure): Similarly, given α-small fibrations Z q G G Y p G G X, the composite p · q is again an α-small fibration. So the universal dependent sum over U α Σ-form is representable by some map Σ : U α Σ-form G G U α . (Id-structure): Given any α-small fibration p : Y G G X, consider the fac- The fibration (s, t) is easily seen to be α-small; and by Proposition 2.3.3, r is stably orthogonal to (s, t) over X.
Applying this construction to p α : U α G G U α itself yields, via Proposition 1.4.7, the desired Id-structure on U α .
(W-structure): Given α-small fibrations Z q G G Y p G G X, the initial algebra W q G G X for the induced polynomial endofunctor on sSet/X may be obtained as a transfinite colimit of iterations of the endofunctor; it can be shown from this description that it is again an α-small fibration [MvdB13, Thm. 3.4].
(Internal universe.) Since β < α, U β is itself α-small; and by Theorem 2.2.1, it is Kan. So U β is representable as the pullback of U α along some u β : 1 G G U α . Moreover, there is a natural inclusion i : U β G G U α , with U α [β] ∼ = i * U α by construction. Together these give the desired internal universe (u β , i).
Finally, to see that (u β , i) is closed under the appropriate constructors in i, note that for each of Π, Σ, and Id as constructed above, the image of the composite with i lies again in U β , and hence factors through i; for instance, in the case of Π, (Note that while we do already have Π-structure (and so on) on U β as constructed in the first parts of this theorem, those choices of the structure do not automatically commute with i.) Corollary 2.3.5. Let β < α be inaccessible cardinals. Then there is a model of dependent type theory in sSet Uα with all the logical constructors of Section A.2, and a universe (given by U β ) closed under these constructors.
Assuming Theorem 1.2.9, we can now interpret the syntax of type theory as an internal language in sSet, writing [[J]] for the interpretation of any judgement J. In doing so, we will make several systematic abuses of notation. Firstly, referring in the syntax to fibrations, we will write E rather than E , and so on, whenever some choice of name E : B G G U α for the fibration is understood; and conversely, referring to the interpretation of a type Γ ⊢ T type, we use [[T ]] to refer to the fibration over [ [Γ]

] given by pulling back U α along the literal interpretation [[Γ ⊢ T type]] : [[Γ]]
G G U α . Readers preferring not to rely on Theorem 1.2.9 may instead read each interpreted syntactic judgement in the following as the corresponding expression in the language of contextual categories, since we use only individual instances of the interpretation, not the entire function.
As a first characteristic of the model, we note that both of the extra principles on equality of functions hold.
Proposition 2.3.6. The η-rule and functional extensionality rules of Section A.4 hold in the simplicial model.
Proof. The η-rule follows immediately from our use of categorical exponentials to interpret Π-types, by the uniqueness in the categorical universal property.
For functional extensionality, Garner [Gar09,Sec. 5] shows that it holds just if each product of identity types, app(f, x), app(g, x)) type admits the structure given by the rules for the identity type on the corresponding product types, So it is enough to show that for any pair of (α-small, well-ordered) fibra- the interpretation of the product of identity types gives a suitably stable path object for the interpretation of the product types, For this, it is clear that Π p (s, t) : is a cofibration since Π p preserves monomorphisms; and it is a weak equivalence, since Π p preserves trivial fibrations (Lemma 2.2.5), and so the retraction Π p s q : Π p (P Y Z) G G Π p Z is again a trivial fibration. Finally, the by the Beck-Chevalley condition in an LCCC, the entire construction is stable under pullback in X, as required.
It now remains only to show that the Univalence Axiom holds in this model.

Univalence
In this section, we will introduce the Univalence Axiom, and show that it holds in the simplicial model.
The proof of this involves both simplicial and type-theoretic components; we keep these separate, as far as possible. First of all (Section 3.1), we define univalence type-theoretically and state the Univalence Axiom; next, we define an analogous simplicial concept of univalence (Section 3.2); we then show that via the simplicial model, the two notions coincide (Section 3.3).
Finally, in Section 3.4, we prove our main theorem: that U α is univalent (using the simplicial sense), and hence that the Univalence Axiom holds in the simplicial model of type theory. Lastly, in Section 3.5, we discuss an alternative formulation of simplicial univalence, and so obtain an up-to-homotopy uniqueness statement for the weak universal property of U α .
Once again, the reader who wishes to avoid dependence on Theorem 1.2.9 in the following may simply translate each individual syntactic expression used into the language of contextual categories.
3.1. Type-theoretic univalence. To state the univalence axiom, we first need to define a few basic notions in the type theory. • A left homotopy inverse for f is a function g : B G G A, together with a homotopy g · f ≃ 1 A . Formally, we define the type LInv(f ) of left homotopy inverses to f : • Analogously, we define the type RInv(f ) of right homotopy inverses: (g(y)), y) type • We say f : A G G B is a homotopy isomorphism (or more briefly, an h-isomorphism) if it is equipped with both a left and a right inverse: It may perhaps be surprising that we use homotopy isomorphisms rather than the more familiar homotopy equivalences, with a single two-sided homotopy inverse. The reason is that while a map carries either structure if and only if it carries the other, the type, or object, of such structures on a map is different. In particular, the analogue of Lemma 3.3.4 for homotopy equivalences does not hold; for further discussion of these issues, see [Uni13,Ch. 4]. Suppose now that A is any type, and x : A ⊢ B(x) type a family of types over A. By the identity elimination rule, we can derive x, y:A, u:Id A (x, y) ⊢ w x,y,u : HIso(B(x), B(y)).
This can equivalently be seen as a map x, y:A ⊢ w x,y : [Id A (x, y), HIso(B(x), B(y))]. Definition 3.1.3. We say the family B(x) is univalent if for each x, y, the map w x,y is itself a homotopy isomorphism: ⊢ isUnivalent(x:A.B(x)) := Π x,y:A isHIso(w x,y ).
Axiom 3.1.4. The Univalence Axiom, for a given type-theoretic universe U , is the statement that the canonical family El of types over U is univalent.
Informally, the Univalence Axiom says that just as elements of the universe correspond to types, so equalities in the universe correspond to equivalences between types. In particular, since every statement or construction must respect propositional equality, the Univalence Axiom stipulates that the language can never distinguish between equivalent types.

Simplicial univalence.
To define a simplicial notion of univalence, we first need to construct the object of weak equivalences between fibrations p 1 : E 1 G G B and p 2 : E 2 G G B over a common base. In other words, we want an object representing the functor sending (X, f ) ∈ sSet/B to the set Eq X (f * E 1 , f * E 2 ). As we did for U α , we proceed in two steps, first exhibiting it as a subfunctor of a functor more easily seen (or already known) to be representable.
For the remainder of the section, fix fibrations E 1 , E 2 as above over a base B. Since sSet is locally Cartesian closed, we can construct the exponential object between them: Then for any X, a map X G G Hom B (E 1 , E 2 ) corresponds to a map f : X G G B, together with a map u : f * E 1 G G f * E 2 over X. Together with the Yoneda lemma, this implies the explicit description: an n-simplex of Hom B (E 1 , E 2 ) is a pair Proof. Follows immediately from Lemma 2.3.1, since the exponential is a special case of dependent products.
Within Hom B (E 1 , E 2 ), we now want to construct the subobject of weak equivalences. G G E 2 be a weak equivalence over B, and suppose g : B ′ G G B. Then the induced map between pullbacks g * E 1 G G g * E 2 is a weak equivalence.
Proof. The pullback functor g * : sSet/B G G sSet/B ′ preserves trivial fibrations; so by Ken Brown's Lemma [Hov99, Lemma 1.1.12], it preserves all weak equivalences between fibrant objects. Thus, weak equivalences from E 1 to E 2 form a subfunctor of the functor of maps from E 1 to E 2 . To show that this is representable, we need just to show: G G F 1 , and any n ≥ 1, we have by the long exact sequence for a fibration: Each π n (f b ) is an isomorphism, so by the Five Lemma, so is π n (f ), for n ≥ 1. The case n = 0 is the same in spirit, but requires a little more work since the Five Lemma is unavailable. We have a square with both horizontal arrows surjections, and π 0 (f b ) an isomorphism. To show that π 0 (f ) is an isomorphism, it therefore suffices to show that for each x ∈ π 0 (E 1 ), the restriction of f b to a map of fibers is again an isomorphism. But this follows from the continuation of the long exact sequence to an exact sequence of pointed sets: where e is any point of F 1 such that [e] = x.
Thus π n (f ) is an isomorphism for each n ≥ 0 and basepoint e ∈ E 1 ; so f is a weak equivalence.
Definition 3.2.5. Take Eq B (E 1 , E 2 ) to be the subobject of Hom B (E 1 , E 2 ) consisting of all n-simplices such that w is a weak equivalence. (By Lemma 3.2.3, this indeed defines a simplicial subset.) From Lemma 3.2.4, we immediately have: E 2 ). Then u is a weak equivalence if and only if (f, u) factors through Eq B (E 1 , E 2 ).
Thus, maps X G G Eq B (E 1 , E 2 ) correspond to pairs of maps where w is a weak equivalence.
While Lemma 3.2.4 was stated just as required by representability, its proof actually gives a slightly stronger statement: Proof. Suppose we wish to fill a square: By the universal property of Eq B (E 1 , E 2 ) this corresponds to showing that we can extend a weak equivalence w : . By Lemma 3.2.2, we can certainly find some map w extending w. But then since ∆[n] is connected, Lemma 3.2.7 implies that w is a weak equivalence.
While on the subject, we collect a proposition which is not required for the definition of univalence, but which will be useful later: 1 , E 2 , E ′ 2 are fibrations over a common base B, and w 1 : Proof. As weak equivalences between fibrations, w 1 and w 2 are fibered homotopy equivalences over B. Choosing fibered homotopy inverses v 1 , v 2 for w 1 and w 2 respectively gives a homotopy inverse Hom B (v 1 , v 2 ) for Hom B (w 1 , w 2 ) : . But by Lemma 3.2.7, the image of a homotopy in Hom whose endpoints lie in Eq must lie entirely in Eq; so the restriction Eq B (v 1 , v 2 ) gives a homotopy inverse for Eq B (w 1 , w 2 ), as desired.
We are now ready to define univalence. Let p : E G G B be a fibration. We then have two fibrations over B × B, given by pulling back E along the projections. Call the object of weak equivalences between these Eq(E) := Eq B×B (π * 1 E, π * 2 E). Concretely, simplices of Eq(E) are triples There are also source and target maps s, t : Eq(E) G G B, given by the sending (b 1 , b 2 , w) to b 1 and b 2 respectively. These are both retractions of δ; and by Corollary 3.2.8, if B is fibrant then they are moreover fibrations.
Since δ E is always a monomorphism (thanks to its retractions), this is equivalent to saying that B G G Eq(E) G G B × B is a (trivial cofibration, fibration) factorisation of the diagonal ∆ B : B G G B × B, i.e. that Eq(E) is a path object for B.
We conclude this section with a few examples, and non-examples, of univalent fibrations.
(1) The canonical map X G G 1 is univalent if and only if the space of homotopy auto-equivalences of X is contractible.
(2) The identity map X G G X is univalent if and only if X is either empty or contractible. In particular, the identity map 1 + 1 G G 1 + 1 is not univalent: it has two fibers which are equivalent, over points that are not connected by any path.

3.3.
Equivalence of type-theoretic and simplicial univalence. Having defined the type-theoretic and simplicial notions of univalence, we now wish to show that they coincide. As ever, we make essential use of representability; in particular, we work with the interpretations of type-theoretic notions entirely via their universal properties. With this in view, we need to define what are represented by the interpretations of LInv, isHIso, etc.
Definition 3.3.1. Let p 1 : E 1 G G B, p 2 : E 2 G G B be fibrations over a common base (as in Definition 3.2.1).
Define HomLInv B (E 1 , E 2 ) to be the set of maps with a left homotopy inverse from X to Y , i.e. triples (f, g, H), where f : E 1 G G E 2 and g : E 2 G G E 1 are maps over B, and H is a fibred homotopy from g · f to 1 E 1 , defined using the fibred path space P B (E 1 ) (as used for the Id-structure in the proof of Theorem 2.3.4).
Similarly, define HomRInv B (E 1 , E 2 ) to consist of triples (f, g, H), where f, g are as before, and H is now a fibred homotopy from f · g to 1 E 2 , defined using P B (E 2 ).
Finally, these both come with evident projections to Hom B (E 1 , E 2 ); define   E 2 ) : is thus a pullback of the dependent product of the universal pair of fibrations over U α Π-form , and so by the Beck-Chevalley condition is a dependent product for the pullbacks of these fibrations along (E 1 , E 2 ) . But these pullbacks are isomorphic to E 1 , E 1 × B E 2 , by the two pullbacks lemma and the construction of A gen , B gen as pullbacks of U α G G U α .
; but this is exactly the usual construction of exponentials in slices from dependent products [Joh02, A1.5.2].
We also note, from the proof of the preceding lemma: There is a natural isomorphism over B: Following this, we take HIso B (E 1 , E 2 ) := [[HIso B (E 1 , E 2 )]], and define HomLInv, HomRInv similarly. G G Hom B (E 1 , E 2 ) factors through Eq B (E 1 , E 2 ); and the resulting map HIso B (E 1 , E 2 ) G G Eq B (E 1 , E 2 ) is a trivial fibration.
Proof. The given map HIso B (E 1 , E 2 ) G G [E 1 , E 2 ] ∼ = Hom B (E 1 , E 2 ) corresponds, under the isomorphisms of Lemma 3.3.2, to the maps on hom-sets where the middle map just forgets the chosen homotopy inverses of an hisomorphism. But since any map admitting both homotopy inverses is a weak equivalence, the natural map Thus, we obtain the desired map HIso B (E 1 , E 2 ) G G Eq B (E 1 , E 2 ), corresponding to the forgetful function HIso Combining this with the left-hand pullback square in Lemma 3.3.2, we can consider HIso B (E 1 , E 2 ) as the pullback: where EqLInv, EqRInv are defined by the pullbacks above, and represent weak equivalences equipped with a left (resp. right) homotopy inverse. To show that the map HIso B (E 1 , E 2 ) G G Eq B (E 1 , E 2 ) is a trivial fibration, it thus suffices to show that the maps are each trivial fibrations.
Lemma 3.3.5. For B, E 1 , E 2 as above, the map is a trivial fibration. Equivalently, left homotopy inverses to equivalences between fibrant objects extend along cofibrations.
Proof. For EqLInv B (E 1 , E 2 ) G G Eq B (E 1 , E 2 ), we need to find a filler for any diagram of the form Writing f for the induced map X G G B and F i for f * E i , this square corresponds (by the universal properties of Eq and EqLInv) to a weak equivalencew : F 1 G G F 2 , and a fibered left homtopy inverse to w := i * w ; that is, l : i * F 2 G G i * F 1 , and a homotopy H : l · w ≃ 1 i * F 1 , all fibered over Y : A filler then corresponds to a fibered left homotopy inverse (l,H) tow, extending (l, H). These data and desiderata may be summed up in a single commuting diagram: Replacing the sub-diagrams on the left by their colimits, we see that we seek precisely a diagonal filler for an associated square: G G X is a fibration, we just need to show that the left-hand map of pushouts, induced by is a trivial cofibration. For convenience, call this map t.
To see that t is a weak equivalence, consider it in the square The top map is a trivial cofibration by the pushout-product property; the vertical maps are pushouts of w andw along cofibrations, so are also weak equivalences; and so by 2-out-of-3, t is a weak equivalence.
On that other hand, to see that t is a cofibration, consider it as induced by maps t 0 , t 1 as in: Here t 0 is isomorphic to the inclusion (since pulling back preserves products and pushouts), so is mono. Next, i 0 and i 1 have disjoint images, so t 1 is also mono. Finally, the intersection of the images of t 0 and t 1 is exactly the image of i * F 1 ; so t, as the induced map from (i Thus t is a trivial cofibration, completing the proof of the lemma.
Lemma 3.3.6. For B, E 1 , E 2 as above, the map is a trivial fibration. Equivalently, right homotopy inverses to equivalences between fibrant objects extend along cofibrations.
Proof. We must provide lifts against any cofibration i : Y ֒−→ X: Analogously to the previous lemma, and again writing f : X G G B, F i := f * E 1 , the square corresponds to a weak equivalencew : F 1 G G F 2 over X together with a fibered right homotopy inverse to w := i * w , i.e. r : i * F 2 G G i * F 1 and a homotopy H : w · r ≃ 1 i * F 2 over Y ; and a filler corresponds to a fibered right homotopy inverse (r,H) forw, extending (r, H).
Again, putting these conditions together, we see that they correspond to filling another square: where the pullbacks are just the fibered mapping path spaces.
Now i * F 2 ֒−→ F 2 is certainly a cofibration; so to provide the filler, it suffices to show that the right-hand map is a trivial fibration. As the target map from a mapping path space, it is certainly a fibration. To see that it is a weak equivalence, consider the triangle The top map is the inclusion of a deformation retraction, so is a weak equivalence; so by 2-out-of-3, the source map ev 0 is a weak equivalence. But ev 1 is homotopic to ev 0 , so is also a weak equivalence, as required.
Then by the Id-comp rule applied to the definition of w x 1 ,x 2 , the overall com-  Proof. We will show that the target map t : Eq( U α ) G G U α is a trivial fibration. Since t is a retraction of δ Uα , this implies by 2-out-of-3 that δ Uα is a weak equivalence. So, we need to fill a square By the universal properties of U α and Eq( U α ), these data correspond to a weak equivalence w : E 1 G G E 2 between α-small well-ordered fibrations over A, and an extension E 2 of E 2 to an α-small, well-ordered fibration over B; and a filler corresponds to an extension E 1 of E 1 , together with a weak equivalence w extending w: As usual, it is sufficient to construct this first without well-orderings on E 2 ; these can then always be chosen so as to extend those of E 2 .
Recalling Lemmas 2.2.4-2.2.5, we define E 1 and w as the pullback To see that this construction works, it remains to show: (a) i * E 1 ∼ = E 1 in sSet/A, and under this, i * w corrsponds to w; (c) E 1 is a fibration over B, and w is a weak equivalence. For (a), pull the defining diagram of E 1 back to sSet/A; by Lemma 2.2.4 part 2, we get a pullback square in sSet/A, giving the desired isomorphism. For (b), Lemma 2.2.4 part 3 gives that Π i E 1 is α-small over B, so E 1 is a subobject of a pullback of α-small maps.
For (c), note first that by factoring w, we may reduce to the cases where it is either a trivial fibration or a trivial cofibration.
In the former case, by Lemma 2.2.4 part 1 Π i w is also a trivial fibration, and hence so is w; so E 1 is fibrant over E 2 , hence over B.
In the latter case, E 1 is then a deformation retract of E 2 over A; we will show that E 1 is also a deformation retract of E 2 over B. Let H : G G E 2 be a deformation retraction of E 2 onto E 1 . We want some homotopy H : , and 1 E 2 on E 2 × {0}. Since these three maps agree on the intersections of their domains, this is exactly an instance of the homotopy lifting extension property, i.e. a square-filler which exists since the left-hand map is a trivial cofibration.
For H to be a deformation retraction, we need to see that G G E 2 , which factors through E 1 since H was a deformation retraction onto E 1 .
So w embeds E 1 as a deformation retract of E 2 over B; thus E 1 is a fibration over B and w a weak equivalence, as desired.
Putting this together with Corollary 2.3.5, we obtain our main theorem: Theorem 3.4.2. Let β < α be inaccessible cardinals. Then there is a model of dependent type theory in sSet Uα with all the logical constructors of Section A.2, and a universe (given by U β ) closed under these constructors and satisfying the Univalence Axiom.
From this, we can immediately deduce:  3.5. Univalence and pullback representations. We are now ready to give a uniqueness statement for the representation of an α-small fibration as a pullback of p α : U α G G U α : we define the space of such representations, and show that it is contractible.
In fact, we work a bit more generally. For any fibrations q, p, we define a space P q,p of representations of p as a pullback of q; and we show that p is univalent exactly when for any q, P q,p is either empty or contractible.
Let p : E G G B and q : Y G G X be fibrations. We define a functor P q,p : sSet op G G Set, setting P q,p (S) to be the set of pairs of a map f : S × X G G B, and a weak equivalence w : S × E G G f * E over S × X; equivalently, the set of squares such that the induced map S ×Y G G f * E is a weak equivalence. Lemma 3.2.3 ensures that this is functorial in S, by pullback. Lemma 3.5.1. The functor P q,p is representable, represented by the object P q,p := Π X→1 Σ π 1 Eq X×B (π * 1 Y, π * 2 E).
Proof. For any S, we have: Hom(S, Π X→1 Σ π 1 Eq X×B (π * 1 Y, π * 2 E)) ∼ = Hom X (X × S, Σ π 1 Eq X×B (π * 1 Y, π * 2 E)) e. over X × S} ∼ = P q,p (S) Remark 3.5.2. By Yoneda, we see from this that (P q,p ) n ∼ = P q,p (∆[n]). Proof. First, suppose that p is univalent. Take any q such that P q,p is nonempty; then we have some map 1 G G P q,p , corresponding to a square We claim that P q,p G G 1 is a trivial fibration, and hence P q,p is contractible. Π -functors preserve trivial fibrations (since their left adjoints, pullback, preserve cofibrations), so it is enough to show that is a trivial fibration. For this, first note that w, as a weak equivalence between fibrations, is a homotopy equivalence over X, so induces a homotopy equivalence (w · −) : Eq X×B ((π * 1 (f * E), π * 2 E) G G Eq X×B (π * 1 Y, π * 2 E).
So it is enough to show that Eq X×B ((π * 1 (f * E), π * 2 E) G G X × B π 1 G G X is a trivial fibration; but this follows since it is the pullback along f of the source map Eq(E) = Eq B×B (π * 1 E, π * 2 E) G G B × B π 1 G G B, which is a trivial fibration since p is univalent. Conversely, suppose that for every fibration q, P q,p is either empty or contractible; now, we wish to show p univalent. For this, it is enough to show that the source map s : Eq(E) G G B is a trivial fibration, which will hold if each of its fibers is contractible.
So, take some f : 1 G G B, and consider the fiber f * Eq(E). By the universal property of Eq(E), this is isomorphic to P f * p,p ; and it is certainly nonempty, containing the pair (f, 1 f * E ); so by assumption, it is contractible, as desired.
Corollary 3.5.4. For any α-small fibration q, the simplicial set P q,pα of representations of q as a pullback of p α is contractible.
Appendix A. Syntax of Martin-Löf Type Theory We consider the syntax as constructed in two stages. First, one constructs the raw syntax -the set of formulas that are at least parseable, if not necessarily meaningful-as certain strings of symbols, or alternatively, certain labelled trees. On this, one then defines alphaequivalence, i.e. equality up to suitable renaming of bound variables, and the operation of capture-free substitution. This first stage is well standardised in the literature; see e.g. [Hof97] for details.
Secondly, one defines on the raw syntax several multi-place relations-the judgements of the theory. These relations are defined by mutual induction, as the smallest family of relations satisfying a bevy of specified closure conditions, the inference rules of the theory. The details of the judgements and inference rules used vary somewhat; we therefore set our choice out here in full. For the structural rules, our presentation is based largely on [Hof97]; our selection of logical rules, and in particular our treatment of the universe, follows [ML84].
We take as basic the judgement forms We treat contexts as a derived judgement: ⊢ Γ cxt means that Γ is a list (x i :A i ) i<n , with x i distinct variables, and such that for each i < n, (x j :A j ) j<i ⊢ A i type. Definitional equality (also known as syntactic or judgemental equality): A.2. Logical Constructors. In this and subsequent sections, we present rules introducing various type-and term-constructors. For each such constructor, we assume (besides the explicit rules introducing and governing it) a rule stating that it preserves definitional equality in each of its arguments; for instance, along with the Π-intro rule introducing the constructor λ, we assume the rule Again, the special case where B does not depend on x is of particular interest: this gives the cartesian product A × B := Σ x:A B.
Id-types. (Identity types, equality types.) Given W-type structure on C, say (U, El) is closed under W-types if for each a : Γ G G U and b : (Γ, a * El) G G U, we have a map w(a, b) : Γ G G U, such that (Γ, w(a, b) * El) = (Γ, W(a * El, b * El)), and such that for f : Γ ′ G G Γ and a, b as above, f * (w(a, b)) = w(f * a, f * b).
B.3. Extensionality and Univalence. For the following group of rules, let C be a contextual category equipped with chosen Π-and Id-type structure. . Definition B.3.6. Given a universe (U, El) in C, say (U, El) satisfies the Univalence Axiom if C is equipped with a map uvt U,El : 1 G G isUvt(U, El).