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Higher Structures in Homotopy Type Theory Abstract The intended model of the homotopy type theories used in Univalent Foundations is the [MATH] -category of homotopy types, also known as [MATH] -groupoids. The problem of higher structures is that of constructing the homotopy types needed for mathematics, especially tho... |
Introduction Homotopy type theory is at the same time a foundational endeavor, in which the aim is to provide a new foundation for mathematics, and an area of mathematics and logic, in which the aim is to provide tools for the mathematical analysis of homotopical (higher dimensional) structures. Let us call the former ... |
In the present chapter we use the issue of higher structures as a lens with which to study both of these aims and their relations to other foundational approaches. |
To motivate the problem of higher structures, we need to recall that the intended universe of UF, and the principal model of HoTT, is the realm of [MATH] -groupoids, a homotopical kind of algebraic structures that have elements, identifications, identifications between identifications, etc. ad infinitum, and these iden... |
[MATH] -groupoids are the same as homotopy types , so we shall use these terms interchangeably, with a slight preference for the latter, as then homotopy type theories are both theories of homotopy types as well as homotopical type theories. |
A common misconception is that higher homotopy types only occur in, or only are relevant to, homotopy theory. That is very far from the case, as even the type of sets, as used in most of mathematical practice, is [MATH] -type. And higher structures now feature prominently in many areas ranging from geometry, algebra, a... |
It is a key point of difference between UF and earlier approaches to foundations inspired by category theory that the former takes |
[MATH] -groupoids rather than various notions of higher categories to be the basic objects of mathematics, from which the rest are obtained by adding further structure. This insight was due to Voevodsky who remarked that many natural constructions are not functorial in the sense of category theory. (Think for example o... |
Because UF aims to be a foundation for all of mathematics, it is necessary that its language, in the shape of the HoTT, provide the means of construction for all the homotopy types that are used in mathematics. For the construction of sets , this is not such a big problem, as most of the sets that occur in mathematics ... |
The main problems appear when it comes to higher dimensional homotopy types. We discuss some positive results (structures that have already been constructed) as well as some open problems (structures that have not already been constructed) in Sect. |
We remark that although we expect some actual negative results (i.e., impossibility proofs) for some of the open problems with respect to some particular homotopy type theories, these have yet to appear. But anticipating that further means of construction will be necessary, we discuss potential solutions in Sect. |
For the remainder of this Introduction, we shall consider an analogy. Martin-Löf type theory can be considered as a formal system for making constructions. In fact, a variant with an impredicative universe was called the Calculus of Constructions (CoC), and a further extension, the Calculus of Inductive Constructions |
(CIC) is the basis for the proof-assistant Coq. And we shall be concerned with the question of the limits of the methods of construction available in constructive type theories. An obvious analogy presents itself, namely with euclidean geometry and the limits of the methods of geometric constructions using ruler and co... |
Obviously we can include among the list of further means of construction such well-known principles as the law of excluded middle (LEM), Markov’s principle (MP), the axiom of choice (AC), various kinds of transfinite induction (TI), as well as principles of impredicativity. Some of these, as well as weaker versions of ... |
constructive taboos because admitting them is contrary to certain philosophical outlooks inspired by constructivism or intuitionism, and also because they cannot be mechanically executed at all, or only with greatly increased computational complexity. |
A further aspect of the constructive taboos is that they reduce the number of models in which we can interpret the constructions. It is well-known that constructive systems admit many useful models, indeed, this is one reason why classical mathematicians may be interested in such systems. Non-homotopical constructive s... |
[MATH] -toposes, which can be seen either as generalizations of Kripke models, as generalized spaces, or indeed as generalized worlds of sets. |
It is suspected that HoTT can be modeled in higher toposes, more precisely, [MATH] -toposes . These dramatically extend the usefulness of HoTT, for instance as explained in Schreiber’s Chapter. Earlier extensions of Martin-Löf type theory often imposed axioms, such as the uniqueness of identity proofs |
(UIP), that rule out higher dimensional models. These contradict the univalence axiom and may be called homotopical taboos . More refined axioms may hold in [MATH] -toposes corresponding to |
[MATH] -toposes (the [MATH] -localic [MATH] -toposes LurieHTT, , Sect. 6.4) ), but not in more general [MATH] -toposes. These are called constructive-homotopical taboos |
Infinity groupoids and the homotopy hypothesis The types in UF are supposed to be homotopy types , so let us dwell a bit on what they are, both from an intuitive point of view, and from the perspective of mathematics developed in set-theoretic foundations. |
Intuitions are always hard to convey, and in the case of the notion of homotopy type, even more so. Intuition is, after all, best developed through practice and familiarity. One way to build an intuition for homotopy types is through working in a homotopy type theory, either on paper or with the help of a proof assista... |
As a first approximation we can say that types [MATH] are collections of objects together with for each pair of objects [MATH] , a type of |
identifications [MATH] , together with meaningful operations on these identifications, such as the ability to compose and invert them. And there should also be higher order operations that produce identifications between identifications, such as an identification [MATH] for any [MATH] . This description is meant to cap... |
[MATH] -groupoids , and on this view, two types [MATH] can be identified if there is a (weak) functor [MATH] that is an equivalence of [MATH] -groupoids. |
Another intuition comes from describing types as (nice) topological spaces up to homotopy equivalence. The objects are the points of the space, and the identifications are the paths between points. |
The homotopy hypothesis is the idea that these separate intuitions capture the same underlying concept. It grew out of Grothendieck’s homotopy hypothesis concerning a particular definition of [MATH] -groupoids Grothendieck1983 . The modern terminology is due to Baez Baez2007 |
In order to explain the subtlety of the situation, let us turn to the most common implementation of the idea of [MATH] -groupoids in the context of set-theoretic mathematics. Here these are represented by |
simplicial sets satisfying a certain filling condition. These simplicial sets are called Kan complexes in honor of KanIII . A simplicial set is a functor [MATH] , where [MATH] is the category of non-empty finite ordinals and order-preserving functions. This means concretely that a simplicial set consists of a set of [M... |
simplices as points, the [MATH] -simplices as lines between points, [MATH] -simplices as triangles, etc. The Kan filling condition says that if we are given [MATH] compatible |
[MATH] -simplices in [MATH] in the sense that they could be [MATH] of the [MATH] faces of an [MATH] -simplex, then there exists some such |
[MATH] -simplex. This condition is illustrated in Fig. in some low-dimensional cases. In each case, we can think of the given data as a map from a horn , a sub-simplicial set [MATH] of the standard |
[MATH] -simplex [MATH] consisting of the union of all the faces opposite the [MATH] th vertex, into [MATH] . A lift is some extension of this to a map from [MATH] to [MATH] , or equivalently, an [MATH] -simplex in |
[MATH] with the requisite faces. For example, in Fig. 1(b) , if we are given two [MATH] -simplices [MATH] and [MATH] in [MATH] with a common endpoint, then there exists some [MATH] -simplex representing both a composite of [MATH] and [MATH] |
(the third face [MATH] together with the interior representing the fact that [MATH] is the composite of [MATH] and [MATH] Note that Kan complexes give a non-algebraic notion of |
[MATH] -groupoid: there exists composites and higher simplicial identifications, but there are no operations singling out a particular composite. |
Here we come to a potential pitfall: we cannot say that homotopy types are Kan complexes, for they have different criteria of identity: In usual mathematical practice we identify two simplicial sets if they are isomorphic (this is already a weaker notion of identity than that provided by set theory!), whereas an identi... |
considered as homotopy types should be a homotopy equivalence And this is perhaps an appropriate point at which to give a type-theoretic take on Quine’s Quine1969 famous slogans: |
1. To be is to be the value of a variable, and 2. No entity without identity. In we require moreover that all variables be typed, so we say rather that to be an [MATH] is to be the value of a variable of type |
[MATH] and more importantly, to be is to be an element of a type and in we do not require any notion of identity between entities of different types, but we do require as an essential part of giving a type [MATH] that the identity type [MATH] , for [MATH] , is meaningful and correctly expresses the means of identifying... |
The discrepancy between the notion of identity between the model objects (here Kan complexes) and the desired notion of identity (here homotopy equivalence) is usually addressed using relative categories as a tool. A relative category consists of a category equipped with a wide subcategory of weak equivalences . This i... |
The category of simplicial sets [MATH] can be equipped with the structure of a Quillen model category in which the fibrant objects are the Kan complexes (these are also cofibrant as all objects are cofibrant) and the weak equivalences between Kan complexes are the homotopy equivalences. The category of topological spac... |
Quillen Quillen1967 proved that these two model categories give rise to equivalent homotopy categories . For this purpose he introduced the notion of (what is now called) a Quillen equivalence between model categories. Given a nice space [MATH] , the corresponding singular Kan complex [MATH] has as [MATH] -simplices th... |
[MATH] given by gluing together topological simplices according to the face and degeneracy maps in [MATH] That the homotopy categories are equivalent is a first step towards getting what we actually want. We would actually like to show that the Quillen model categories of simplicial sets and topological spaces give ris... |
The way this is achieved is by enhancing both [MATH] and [MATH] to simplicially enriched categories (the latter via the singular Kan complex construction on the level of mapping spaces) such that they become simplicial model categories , and then taking the homotopy coherent nerves of the subcategories of homotopy equi... |
Notice that to get a good theory of homotopy types in the classical set-up we seem to need also a good theory of [MATH] -categories, that is, categories (weakly) enriched in homotopy types, in order to also get a good hold on the universe of homotopy types, which is another name of course anticipating the type-theoreti... |
There is another model structure on simplicial sets whose bifibrant objects are the quasi-categories , those that satisfy a weakening of the Kan filling conditions that make them suitable as models of |
[MATH] -categories. This notion was introduced by Boardman and Vogt BoardmanVogt1973 and the resulting theory of [MATH] -categories has been studied extensively by Joyal Joyal2002 and Lurie LurieHTT (see also the appendix of LurieHTT for details on simplicial model categories as discussed above). |
My point in bringing out these technicalities is not only to explain how homotopy types are defined and handled in set-theoretic mathematics, but also to give a sense of the subtleties involved. It has taken many years to give a good account of how to treat higher structure in set-theoretic mathematics (often by workin... |
[MATH] -category-theoretic layer), and there are still many open questions about which constructions and properties are invariant under weak equivalences inside a model category and under Quillen equivalences between model categories. For instance, it was just recently established that a Quillen adjunction always induc... |
[MATH] -groupoids, where composition, inverses, etc., are given by operations rather than merely assumed to exist. It is quite possible that type theory will be influential in this area, see for instance the suggestion of Brunerie Brunerie2016, , Appendix) |
Thus it should come as no surprise that there are still open questions about how to treat higher structures in HoTT/UF, which is a much younger endeavor. These are the matters we shall now turn to. |
Higher Structures in HoTT/UF When Voevodsky proposed using type theory as a foundation for mathematics, he based this on the insight that higher structures in mathematics are not always naturally objects of a higher |
category , but they are always naturally objects of a higher groupoid Among the [MATH] -groupoids we find truncated higher groupoids, those whose structure is concentrated in a finite range of dimensions. At the lowest level (truncation level [MATH] ) we find the contractible types, those that only have one element up ... |
Moving up in the dimensions, we find next the sets, all of whose identity types are propositions, and the [MATH] -groupoids, all of whose identity types are sets, and so on. We recall from Altenkirch’s Chapter that these truncation levels have a natural formalization in HoTT in terms of predicates |
[EQUATION] and that we have corresponding types of [MATH] -truncated types, [MATH] Not all types are truncated. The [MATH] -sphere, for example, has structure in all dimensions, so it’s not an [MATH] -type, for any [MATH] |
The [MATH] -types are related to the universe of all types, [MATH] , via the truncation construction that maps a type [MATH] to its closest |
[MATH] -type [MATH] . There is a construction [MATH] giving rise to an equivalence [EQUATION] for any [MATH] -type [MATH] When we go to discuss higher structure, it is often the untruncated types that are the hardest to construct. The principal reason is that we can often construct truncated types in a top-to-bottom fa... |
I want to emphasize at this point that the sets we discussed above (and in the Chapters of Altenkirch and Ahrens-North) are not the sets of set theory! Following Quine’s dictum, these are different notions because they have different notions of identity. Let us temporarily use subscripts to differentiate, and write set... |
From a type-theoretic point of view, a set is simply a subset of a fixed universal set [MATH] . That is, we have the type [MATH] |
representing the powerset of [MATH] . We have an elementary membership relation, [MATH] and two sets are equal if they have the same elements in this sense. |
This is of course not the set-theorist’s notion of set, according which sets are elements (rather than subsets) of a universe of discourse [MATH] (itself a set ) that is equipped with a membership relation [MATH] satisfying the axiom of extensionality (and preferably many other set-theoretic axioms). |
The naive set-theoretical hope would be to solve the equation [MATH] (as an identification of sets , i.e., an isomorphism). This is impossible because of Cantor’s diagonal argument, but it can be approximated by the cumulative hierarchy [MATH] , a construction that can be performed in HoTT via a higher inductive type H... |
[MATH] , where [MATH] is the type of small subsets of [MATH] . Such sets can be thought of as certain well-founded trees, and their study has a quite combinatorial flavor. |
The default notion of set in HoTT/UF is set given by the [MATH] -type [MATH] , and this seems to be the one most often used in mathematical practice outside of set theory. For instance, in almost all mathematical contexts, each set can be replaced by an isomorphic copy without changing the meaning of anything. Of cours... |
Likewise the notion of category splits into several distinct notions: I will denote by precategory the notion defined in Sect. 4.4 of the Chapter by Ahrens-North, and leave the unadorned term |
category for a univalent precategory. Indeed, in most category-theoretic contexts, each category can be replaced by an equivalent while preserving the meaning. It is also useful to have the term strict category |
HoTTBook, , Sect. 9.6) for a precategory whose type of objects is a set. From the perspective of set-theoretic mathematics, the [MATH] -type of categories arise from a Quillen model category structure on the [MATH] -category of strict categories. |
Most of category theory can be formalized in HoTT/UF using the univalent definition of category. A precategory can be thought of as a category with extra structure, namely equipped with a functor from an |
[MATH] -groupoid. For a strict category, this functor has as domain a [MATH] -dimensional homotopy type. In set-theoretic foundations, it will automatically be the case that every category can be equipped with such a strict structure, but in UF this is an extra assumption, indeed a constructive-homotopical taboo. |
3.1 Analytic and synthetic aspects of HoTT/UF A foundational theory must be synthetic, in that it describes how to construct and reasons with its fundamental objects in terms of postulated rules. It couldn’t be otherwise, for if it described the “fundamental” objects in terms of other, more fundamental, objects, and de... |
Homotopy type theories are synthetic theories of [MATH] -groupoids. The approach is deeply logical , where we think of logic as invariant theory as pioneered by Mautner Mautner1946 and later developed by Tarski in a 1966 lecture Tarski1986 . Both Mautner and Tarski were inspired by the approach to geometry given in Kle... |
Erlangen Program Klein1872 . The idea is that the logical notions are those that are invariant under that maximal notion of symmetries of the universes of discourse. If the universe of discourse is a set, then the corresponding symmetry group is the symmetric group consisting of bijections of the set with itself, but i... |
group consisting of all self-homotopy equivalences. In analogy with the synthetic theories of various notions of geometry (euclidean, affine, projective, etc.), homotopy type theories are synthetic theories of homotopy types (and set theories are synthetic theories of sets ), cf. also Awodey2018 |
The analytic aspect is that all the rest of mathematics, all mathematical objects, their types, and their structure, needs to be developed in terms of homotopy types. And a key criterion for success of a formalized notion is that it satisfies what Ahrens-North call the |
principle of equivalence , and which I linked to Quine’s dictum above: that the identity type captures the intended notion of identifications between the mathematical objects that we are modeling. |
One novel aspect of doing this analytical work in HoTT is when defining a structured object, it can be a challenge already to get the correct carrier type . In set-theoretic foundations, any carrier set of the correct cardinality will do, but in HoTT we are more discerning. |
We do reap some benefits of this extra care. For instance, any construction (which, remember, could be proving a proposition, inhabiting a set, etc.) we perform on a generic category is guaranteed to be invariant under equivalence of categories, and we can use the rules of identity types to transport the construction a... |
Compare this to the situation in set-theoretic foundations: there we have to prove invariance under equivalence for any construction on types of dimension greater than [MATH] . For sets, this is not necessary, because if we are given a set for which the notion of identification between the elements is given by an equiv... |
3.2 Some constructions that are possible Let us finally take a look at some constructions that are possible in HoTT. Many of these are already discussed in HoTTBook references are provided in other cases. I will structure this discussion according to the means of construction used. Firstly, there are those that only us... |
Next, there are those that use in addition the univalence axiom. Following that, there are those that can be reduced to one particular higher inductive type, the (homotopy) pushout. |
Finally, we find those constructions that seem to require more advanced higher inductive types, and in the next subsection I shall discuss those for which there is no known construction at the time of writing. |
In Basic Martin-Löf type theory (MLTT) we can already define many important notions such as homotopy fibers and other pullbacks, the predicate [MATH] and the types [MATH] . We have the types of categories and |
[MATH] -categories (cf. HoTTBook, , Sect. 9.7) ), as well as many other types of mathematical objects occurring outside of homotopy theory. But we are severely limited in our ability to construct inhabitants of these types, or prove properties about them. For instance, we cannot prove that [MATH] is valued in propositi... |
With univalence we get function extensionality (as shown by Voevodsky), and we can now prove many structural properties. Besides |
[MATH] landing in [MATH] , we can prove that [MATH] is an [MATH] -type, and we get the equivalence principle for the types of algebraic structures and for categories as mentioned in the Chapter by Ahrens-North. (See also Awodey2014 .) We can also prove that the |
[MATH] th universe is not an [MATH] -type for any external natural number [MATH] KrausSattler2015 At this point we can explore an intermediate route: instead of adding higher inductive types, we can assume the propositional resizing axiom |
HoTTBook, , Axiom 3.5.5) , stating that the inclusion map [MATH] of propositions in the [MATH] th universe into the propositions in the [MATH] st universe is an equivalence. This makes the theory impredicative, and it allows us to mimic many impredicative tricks known from (constructive) set theory. For instance, the p... |
The (homotopy) pushout type is a simple, but versatile example of a higher inductive type. It generalizes the disjoint union. Its inputs are three types [MATH] [MATH] , and [MATH] , together with functions |
[MATH] and [MATH] . (Such a configuration is called a span .) The pushout is a new type [MATH] (often written [MATH] if [MATH] and [MATH] can be deduced from the context) together with injections |
[MATH] and [MATH] fitting together in a square [EQUATION] whose commutativity is given by a constructor [MATH] . (See HoTTBook, , Sect. 6.8) for the elimination and computation rules.) |
Now a quite remarkable phenomenon appears. Most of the higher inductive types that are commonly used can be constructed just from pushouts and the other constructions in MLTT with univalent universes. These include joins and suspensions (and therefore, spheres), cofibers (and thus smash products), sequential colimits, ... |
[MATH] Brunerie2016 , and a formalized proof of the Serre spectral sequence for cohomology Doorn2018 Another important construction enabled by pushouts is the Rezk completion, which turns a precategory into the category it represents. This can in fact be done using univalence alone Ahrens-Kapulkin-Shulman2015 , at the ... |
Because so many things can be developed on the basis of univalence, pushouts, and propositional resizing, Shulman suggested that we define an elementary [MATH] -topos to be a finitely complete and cocomplete, locally cartesian closed [MATH] -category with a subobject classifier and object classifiers Shulman2017 . Let ... |
Finally, let me mention some of the known constructions that seem to require more than the above means, but that can nonetheless be effected via more general higher inductive types. First, there is the cumulative hierarchy as mentioned above HoTTBook, , Sect. 10.5) the Cauchy-complete real numbers HoTTBook, , Sect. 11.... |
structure. For these it is more relevant to mention localizations at a family of maps Rijke-Shulman-Spitters2017 . For example, if we localize at a family of maps of the form [MATH] , for [MATH] , where each type |
[MATH] is a proposition, then we obtain a inner model of type theory in itself, in this case a topological localization Of course, the number of things that have been constructed and proved in HoTT grows every day, so undoubtedly I’ll have left some out. Many of these constructions have already been formalized in proof... |
3.3 Some constructions that seem impossible Because proving propositions is in HoTT/UF a special case of making constructions, any currently open problems count as constructions we don’t yet know how to perform. But for some of these, it is expected that the difficulty is not just that the construction is tricky to per... |
The prime example that I will focus on is that of [MATH] -categories, and the related notions of (semi-)simplicial types. Intuitively, an [MATH] -category [MATH] consists of a type of objects [MATH] , for every pair of objects [MATH] a type of morphisms [MATH] , operations for identities and composition, operations tha... |
ad infinitum . The problem is to come up with a way of specifying all these higher coherence operations in a single type. The basic example of an [MATH] -category from the point of view of type theory is the category of types [MATH] that has as type of objects the universe [MATH] , and as morphisms from [MATH] to [MATH... |
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