Free Higher Groups in Homotopy Type Theory

by   Nicolai Kraus, et al.

Given a type A in homotopy type theory (HoTT), we can define the free infinity-group on A as the loop space of the suspension of A+1. Equivalently, this free higher group can be defined as a higher inductive type F(A) with constructors unit : F(A), cons : A -> F(A) -> F(A), and conditions saying that every cons(a) is an auto-equivalence on F(A). Assuming that A is a set (i.e. satisfies the principle of unique identity proofs), we are interested in the question whether F(A) is a set as well, which is very much related to an open problem in the HoTT book. We show an approximation to the question, namely that the fundamental groups of F(A) are trivial, i.e. that the 1-truncation of F(A) is a set.



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1. Introduction

An important feature of Martin-Löf type theory (MLTT) is the identity type which makes it possible to express equality inside type theory. More precisely, if is a type (in any context), and are terms, then is a type whose inhabitants represent proofs that and are equal. As it is common nowadays, we write or instead of , and call its elements simply equalities. Homotopy type theory (HoTT) embraces the fact that may come with interesting structure. This means that, in many cases, we do not only care about the question whether and are equal, but also how or in which ways they are equal. For example, due to Voevodsky’s univalence axiom, is equivalent to . Here, is the type of booleans, is a type universe, and the two equalities stem from the two ways in which is equivalent to itself. Another non-trivial example is which turns out to be equivalent to the type of integers [17], where is the “circle” in HoTT.

A type where each such type of equalities can have at most one inhabitant is called a set, and further said to satisfy the principle of unique identity proofs (UIP). An example for a set is the type of natural numbers: is equivalent to the empty type , while is equivalent to the unit type , and so on. Many algebraic structures can be implemented straightforwardly if we are happy to do everything with sets; for example, [22, Def 6.11.1] defines a set-level group to be a tuple , where is a set, together with a multiplication operation , a unit element , an inversion operator , and equalities expressing the usual laws. The book then further shows how one can construct the free set-level group over a given set (see [22, Chp 6.11]).

More interesting and challenging is it to define higher-level structures, not restricted to sets. Since we use HoTT (rather than, say, set theory) as our foundation, it is natural to attempt this. For example, it has been known for some time that externally, every type carries the structure of an -groupoid [18, 24]

. Internally, we can play a trick and use the following definition, which probably can be considered “HoTT folklore”:

Definition 1.

An -group is a type which is equivalent to a loop space. More precisely, is a group means that we have a connected type and a point together with an equivalence . If is an -group represented by and is a second -group represented by , then a homomorphism is a pointed function .

In other words, an -group is a type that admits a delooping. Clearly, the unit element of this group is , and composition is given by composition of equalities. Some theory of higher groups in homotopy type theory has very recently been developed by Buchholtz, van Doorn and Rijke [10].

It is worth noting that Definition 1 makes use of the fact that a suitable notion of an (untruncated) group already naturally exists in HoTT, which is not the case for many other interesting structures. Defining untruncated algebraic structures in general and directly is an important open problem in HoTT. To see why this is hard, let us start from the set-level definition above and remove the condition that is a set. The equalities which guarantee that the multiplication is associative, the unit is neutral, and inverses cancel, are not sufficient anymore; they would not give a well-behaved definition of a higher group. For example, one may ask oneself how one would prove that equals : there are two canonical ways, and these should coincide (“MacLane’s pentagon”), but any such rule that we add would have to satisfy its own coherences. It is currently unknown whether it is possible to complete this sort of definition in a satisfactory way. In a nutshell, the problem is that the usual definitions would, if expressed internally in type theory, amount to infinite structures of coherences. In classical homotopy theory, these conditions are often organised in the form of an operad [21, 1], but a representation of such structures that can be written down in type theory has not been discovered so far. This is certainly not for a lack of trying; cf. the much-discussed open problem of defining semisimplicial types [23].

In this paper, we study the free -group over a type . It is folklore in homotopy type theory that a suitable definition of the free -group over is given by the loop space of a “wedge of -many circles”, or put differently, the suspension . For us, it will however be more helpful to give a more explicit construction of this free higher group which we call . We define to be the higher inductive type (HIT) [22, Chp 6] which as constructors has a neutral element and a multiplication operation , together with conditions ensuring that each is an equivalence. This definition encodes the a priori infinite tower of coherence condition suitably and it will turn out that it is equivalent to the loop space of .

The most basic properties that one would expect from a free higher group are easy to prove. More intriguing is the question what the free -group has to do with the free set-based group. Clearly, we would want the former to be a generalisation of the latter. The most obvious way of interpreting this is to ask whether, for a set , the free higher group coincides with the construction of the set-based higher group. It turns out that the central question is the following:

Question 2.

If is a set, is a set as well?

One reason why we believe that Question 2 is hard is that a slight generalisation of it is a known open problem in HoTT which has been recorded in the book (see [22, Ex. 8.2]). To be precise, the open problem asks whether, for a set , the suspension is a -type; our question is equivalent to asking whether is a -type. A positive answer to the open problem would imply a positive answer to our Question 2, but we do not expect that our question is fundamentally easier. (Recall from the book [22] that the suspension is the HIT with constructors and , for “north”, “south”, “meridian”.)

The core of our paper consists of a proof of a weakened version, a first approximation, of Question 2. Our main result can be phrased as follows:

Theorem 3.

If is a set, then all fundamental groups of are trivial. In other words, is a set.

Our strategy to prove this is to define a simple reduction system together with a non-recursive approximation, written , to the free -group. These are both based on the usual construction of the free monoid on , that is, . The proof of Theorem 3, with all the tools and strategies that we need to develop, constitutes the main part of the paper.

The reason why our strategies are not sufficient to provide a full answer to Question 2 is that is really only an approximation. Defining in a non-recursive way (i.e. without using some sort of induction that quantifies over elements of itself) seems to, as least as far as we can see, correspond exactly to expressing the infinite coherence tower “directly”. We would not be surprised if it turned out that Question 2 was in fact independent of “standard HoTT” (the type theory developed in the book [22]), and if the status of Question 2 was related to the status of semisimplicial types. We will come back to this in the conclusions of the paper.


The type theory that we consider in this paper is the standard homotopy type theory developed in the book [22]. This means that we have univalent universes , function extensionality, and higher inductive types (HITs). Regarding notation, we strive to stay close to [22], with the exception that we write instead of . All performed constructions will preserve the universe level, hence there is no risk at omitting it. Uncurrying is done implicitly, allowing us to write instead of .


We give the precise definition of the free -group in Section 2, together with some simple observations. The statements in this section (apart from the definition of and its universal property) are not important for the main part of the paper, the proof of Theorem 3, which is given in Section 3. As a corollary of the constructions, we get that coincides with the set-based construction of the free group, and does under the assumption that Question 2 has a positive answer. In Section 4, we make some concluding remarks and discuss related open problems in homotopy type theory.

2. Free -Groups

2.1. Definition and First Properties

Let us start with an explicit construction of the free higher group as a higher inductive type , since this is the central concept of the paper. We use a point constructor for the neutral element, and a constructor which “multiplies” an element of with any other group element. We write instead of since, most of the time, we regard as a fixed variable. The trick which completes the definition in an elegant way, due to Paolo Capriotti, is to add the condition that is an equivalence for every . This cannot be done directly (at least not according to the usual intuitive rules for presentations of higher inductive types), but it can be “unfolded” and expressed via a suitable collection of constructors. Let us show the concrete definition.

Definition 4.

The free -group over a given type is the following higher inductive type :

At first glance, the above HIT appears complicated and rather unappealing. Due to the “unfolding”, the underlying idea that we have discussed above is somewhat hidden. The constructors and are the standard ones that one would use to define the type of lists over , , or, in other words, the free monoid over . The remaining four constructors simply say that, for every , the function is a half-adjoint equivalence as defined in [22, Chp 4]. This is the “unfolding” mentioned before; note that we could equally well have used other definition of equivalences, such as the “bi-invertible” or “contractible fibre” constructions. In any case, this means that we can think of as being fully described as a triple , with .

To make use of the type , we need to know an elimination principle for it. This can be stated as an induction (dependent elimination) principle, which is how it is done in the book [22]. More concise, and (we would say) conceptually clearer, is the approach of phrasing it using a universal property, in other words, a recursion (non-dependent) elimination principle with a uniqueness property. The equivalence between these approaches for inductive types has been discussed by Awodey, Gambino, and Sojakova [8], for some HITs, by Sojakova [20], and a restricted version for set-truncated HITs can be found in [2]. For concrete HITs, such as our , it is straightforward to derive the various elimination principles from each other. We state the universal property using the presentation as a homotopy-initial algebras [8]:

Principle 5.

We say that an -algebra structure on a type consists of a point , a map , and a proof . We say that the type of -algebra morphisms between and consists of triples , where , , and . Then, the induction principle of is equivalent to saying that is homotopy initial, i.e. that for any , the type of morphisms from to is contractible.

We will come back to -algebras later.

An obvious question is whether really deserves to be called the free -group on . There are two points: first, we need to check that it is a higher group in the sense of Definition 1, and second, we have to justify the attribute free.

For the first point, note that the suspension has an equivalent description which can be obtained by essentially collapsing the point with the path given by the unit type: it is the HIT with a single point constructor and a family of loops indexed over , as in , a wedge of -many circles. Note that is automatically connected. A further side remark is that is canonically equivalent to the circle . We then observe:

Lemma 6.

The free -group is an -group, with as its delooping. The canonical equivalence maps the structure as one would expect, i.e. we have and .

Note that this statement is completely independent from the rest of the paper.


This is a relatively straightforward generalisation of the proof that the loop space of is equivalent to the type of integers. The proof is an application of [22, Lem 8.9.1] and does not provide much insight, which is why we choose to omit it. For a detailed argument, one can easily adapt the proof given by Brunerie (for an only slightly different statement) in [9, Sec 6]. ∎

From the above lemma, we can in particular observe that is a presentation of the type of integers.

Next, we need to justify why we call the free higher group. The following presentation of the argument was suggested by Paolo Capriotti. Let us consider the following diagram:


Here, is the universe of pointed types. The function maps a type to , while projection simply forgets the point. As in [22, Chp 6.5], we regard the suspension as a function , mapping to , and is the loop space. For and , it is easy to see that there is a canonical equivalence


and by [22, Lem 6.5.4], we have


The above diagram (1) should for our purpose only be regarded as an illustration of these two equivalences. Talking about the adjunctions more precisely is difficult since the correct notions would be -categorical. This leads into a territory that is vastly unexplored in homotopy type theory [12], although higher adjunctions can be represented using only a finite amount of data [19]; here, we do not go further into this.

Let be a given -group, represented by . This means that we have . We can then calculate:

by Def 1 and Lem 6
by (3)
by (2)

Thus, is “morally” left adjoint to the forgetful functor which returns the underlying type of a higher group.

2.2. On Alternative Constructions

As a preparation for the development in Section 3, and to better understand the difficulties with , let us attempt to construct in a different way. Let us write for ; we call the type of elements of with a sign, and we think of as and as . For , we write for the element we get by changing the sign. Of course, this means that .

Elements of the free group are, at least intuitively, lists over . The difficulty is that different lists may represent the same group element. This happens, for example, for (i.e. ) and the empty list, both of which represent the unit of the group. We can avoid this problem by quotienting to identify the list with the list . This quotient will be a set by definition of the quotient (set quotient) operation. If we are happy to work only with sets, and to set-truncate everything, then this is entirely possible, and in fact, it is a construction of the set-based free group given in [22, Thm 6.11.7]. If, like in this paper, we do not want to restrict ourselves to sets, we might think of taking a HIT which has path constructors for each such pair of lists, without set-truncating. The problem is that we need coherences: if we use a path constructor to reduce one redex and then a second, we should get the same equality as if we reduce the second redex and then the first. When looking at three redexes, we need to express that these equalities “fit together”, and so on. This is an instance of the problem of infinite coherences which seem to be hard and possibly impossible to express in HoTT. In Section 3, we will perform a finite approximation of this construction in order to show Theorem 3, although we will see that a couple of additional arguments are required to complete the proof.

Alternatively, we could think to only consider lists over in normal form, i.e. lists which come together with a proof that they do not contain a redex. The type of lists over in normal form is a set (assuming that is a set), and the presentation is indeed fully coherent. The trouble is that we are in general unable to define a suitable binary operation on this set, i.e. we are lacking a group operation. If we have two lists in normal form, their concatenation might not be in normal form, and for arbitrary types, we have no way of calculating a normal form or even checking whether we already have a normal form.

Unsurprisingly, the approach with normal forms works if has decidable equality:

Proposition 7.

If has decidable equality, in the sense that


then has decidable equality as well. Moreover, is in this case canonically equivalent to the set-truncated construction of the free group as given in [22, Chp 6.11].

Proof sketch.

Thanks to [22, Thm 6.11.7], we can take set-quotiented lists (as described above) as the definition of the set-truncated free group. Using decidable equality of , it is easy to see that this quotient is equivalent to the type of lists in normal form; let us write for the latter type. An element of is a list together with a propositional property, and we have an embedding . What is left to do is to compare with . Note that is a set without being explicitly set-truncated. There is a canonical -algebra structure on , giving rise to a map . Further, one can construct a function , by induction on the list. The empty list is mapped to , translates into an application of , and becomes . These functions give rise to an equivalence , and since has decidable equality, enjoys the same property. ∎

3. The fundamental group of the free -group

In this section, the core of the paper, we develop a couple of techniques that, when combined, allow us to prove Theorem 3. For the whole section, let us assume that is a given set. Given lists , we write for their concatenation, i.e. the list we get by simply joining the two lists as in . Since this operation is associative (up to a canonical and fully coherent equality), we omit brackets and write for both and . Given , we regard as a one-element list and allow ourselves to write e.g.  or .

3.1. A simple reduction system in type theory

As discussed in Section 2.2 above, we can think of elements of as lists over , and the main problem is that different lists represent the same group element. This motivates the development of a system of reductions.

Definition 8.

The type family , which expresses that a list represents the same group element as the empty list (i.e. the neutral element of the group ), is defined as follows. We first define an auxiliary family by induction on the natural numbers:

Using this, we set .

If we have indexed inductive families in the theory, we can alternatively define directly as such a family generated by

The two definitions are essentially the same, only represented in different ways. In both cases, given , we say that witnesses that can be reduced to the empty list and we call a reduction sequence. We view it as a sequence consisting of steps, each of which removes a single redex . An example of a reduction sequence could be pictured as follows, where each step is represented by an arrow annotated with the redex it reduces:

Remark 9.

There are a couple of points that we want to point out explicitly.

  1. In the above example and in the discussions to come, is already positive or negative, which means that every redex is of the form ; the possibility is already covered.

  2. The number of steps of is simply half of the length of the list , which means that all elements of have the same number of steps. In particular, it is easy to prove that is empty if

    is odd.

  3. For a given list , there is no way to compute a reduction sequence, since we do not know whether an occurring pair forms a redex. A reduction encodes equalities which guarantee that all redexes that it reduces are really redexes. Deciding whether is a redex would require decidable equality on (but of course, we can always check whether an element of is positive or negative, and this analysis might give us that is definitely not a redex).

  4. For a given , equality on is decidable. This is because a sequence encodes the positions of the redexes that it reduces, and positions are decidable, while the (in general undecidable) equalities on are propositions. Similarly, if we have , we can say in which step is reduced, since this is encoded by a position.

Let us remind ourselves that the goal of the paper is to show that has trivial fundamental groups. This is a statement about equalities between equalities. If we think of a reduction sequence as a proof that a list represents the neutral group element, i.e. as something giving rise to an equality proof, it is hopefully intuitive that we now want to discuss the relationship between different reduction sequences. In a nutshell, we want to give a criterion which guarantees that two reduction sequences give rise to equal equalities. To do so, we consider transformations:

Definition 10.

Let be a list and a reduction sequence. We consider the following two operations, each of which allows us to create a new reduction sequence in from :

  1. Swap two consecutive independent steps in . More precisely, if is a sequence of the form


    we can change it to


    Analogously, we can change (8) into (7).

  2. If a step reduces a redex in a list of the form , we can change this step to remove the redex instead, or vice versa. This means that


    can be changed to


    or vice versa.

We say that can be transformed into if there is a finite chain of these operations that changes into .

After what we said in the paragraph before Definition 10, the best we could hope for is that any reduction sequence can be transformed into any other reduction sequence (of the same list ). Indeed, this is what we will show. We start with a technical lemma which will not only help us to prove what we just said (Corollary 12), but also another useful consequence (Corollary 13).

Lemma 11.

Assume we are given a list of the form , i.e. a list in with an explicitly given redex . Assume further that we have a reduction sequence . It is possible to transform into a reduction sequence which reduces the redex in the first step, i.e. starts with .


Let , , , and be given. Let us write for the number of the step in which is reduced, and for the number of the step in which is reduced. There are three cases:

  • If , then the redex is reduced in step . If , there is nothing to do. Otherwise, we can swap this step with step , since the two steps will be independent of each other. Swapping a further times, we can move the step reducing to the beginning of the sequence.

  • If , then are not reduced together, but is reduced with some to its right instead. Note that . Before step , the list thus has to be of the form , and step consists of reducing . We define to be the reduction sequence which is identical to in every step expect in step where it reduces ; this is the second of the two possible operations in Definition 10. We are now in case one ().

  • The case is analogous to the case . ∎

Corollary 12.

Any reduction sequence can be transformed into any other reduction sequence. More precisely, for and , we can transform into .


A reduction sequence is given by a chain of reduction steps, and the number of steps in and are equal (both are ). Thus, it is sufficient to transform into a sequence which consists of the same steps as . By the above lemma, we can transform into a sequence which in the first step reduces whichever redex reduces in the first step. Applying the same argument to the “tail” of the sequences (note that and , each with the first step removed, still reduce the same list), we get a transformation into a sequence which in every step mirrors the reduction of and is thus equal to . ∎

A second easy consequence is that, if a list is reducible, then we cannot “get stuck” while reducing: we can start reducing at an arbitrary position without risking of ending up with an unreducible list. Note that we write for .

Corollary 13.

For any lists and , we have


The direction is immediate, by adding a single reduction step reducing . The direction is an application of Lemma 11. ∎

Remark 14.

Note that Corollary 12 subtly but crucially depends on the assumption that is a set, while Lemma 11, as formulated, would work for arbitrary types . It is true independently of that a reduction sequence is given by a chain of reduction steps. A reduction step encodes the position at which the reduction is taking place (say, the length of the list in Definition 8), together with a proof that the reduction is possible (i.e. a proof that the pair at the position is actually a redex). The second part amounts to an equality in (since “ being a redex” means ); thus, it is a proposition if is a set. In this case, a reduction step is determined by the position, and a reduction sequence is determined by the chain of positions which it encodes. The proof of Corollary 12 relies on this.

Lemma 11 holds even without the requirement of being a set. However, note that the proof of Lemma 11, when it uses the second operation in Definition 10, has to construct a new equality (this is hidden in the sentence “Before step , the list thus has to be of the form ”). Therefore, the new sequence constructed in Lemma 11 will reduce in the first step, but the proof that is indeed a redex could be a nontrivial one.

3.2. A non-recursive approximation to the free -group

We are ready to define a non-recursive approximation to the free group , a HIT that we call . By non-recursive, we mean that constructors of do not use points or paths of in their arguments.

Definition 15.

We define to be the HIT with the following constructors:

We can think of (without the last constructor) as a “wild” quotient of . Recall that we said that lists over correspond to very intentional representations of group elements. The HIT with constructors and can be thought of as a “level approximation” to a fully coherent non-recursive quotient of : we identify some lists which represent the same group element, but the equalities are incoherent. This is partially remedied by the constructors (“swap”) and (“overlap”), ensuring that the equalities generated by satisfy basic coherence. They can be pictured as follows:



and themselves are not directly guaranteed to be coherent; if we omit the constructor , we can think of as a “level approximation”. ensures that all higher equalities hold, by forcing to be -truncated. The statement that is an approximation to the free higher group can then be made by drawing a connection to , which we will do later.

If a list can be reduced, then in , it is indistinguishable from the empty list:

Lemma 16.

We have a function


We need to analyse the element . It encodes a finite number of reduction steps. The first reduction step shows that is of the form , thus the constructor (transported along the equality ) provides us with the equality . Similarly, each of the remaining reduction steps encoded in shows how can be applied, and the concatenation of all these equalities yields .

If is defined as an indexed inductive family, can be constructed by induction on , and the induction step is given by the constructor . ∎

Not only can we show that reducible lists are equal to in , it is also the case that the concrete witness of reducibility does not matter:

Lemma 17.

For any given , the function


is weakly constant, in the following sense:


The constructors and ensure that, if two reduction sequences can be transformed into each other, then they lead to equal proofs of . More precisely, the first operation in Definition 10 is exactly covered by the constructor , while the second operation is covered by . The statement thus follows from Corollary 12. ∎

The point of is that it is easier to reason about than about , thanks to the absence of recursive constructors; one can say that attempts to bridge the gap between and . We first define a property stating that an element of can be reduced. We write for as usual.

Lemma 18.

The family extends to a family as in the following commuting triangle:


We do induction on . Clearly, we have to set . The proof obligation of the constructor is met by Corollary 13. The remaining two constructors are trivial, since they ask for equalities between elements of propositions. ∎

To avoid confusion with elements of , which we call , we use Greek letters for elements of . If is reducible, it is equal to the neutral element:

Lemma 19.

There is a function of type


We do induction on . First, we consider the case , and we want to find . Recall that a weakly constant function into a set (which the codomain here is) factors through the propositional truncation [16], hence since by definition, Lemma 17 gives us a function


such that . We want to extend this function to . Induction on requires us to provide constructions corresponding to , , and . The latter two are contractible, and we do not need to worry about them. The proof obligation for says that, for any , and witnesses , , the triangle