1 Introduction
The homotopy hypothesis is the statement that homotopy types (topological spaces with trivial homotopy groups above level ) correspond to groupoids for via the fundamental groupoid construction. In Grothendieck’s original version in Pursuing Stacks [Grothendieck1983] this was a conjecture about a particular model of groupoids. It is also a theorem for many particular models of groupoids, for example the Kan simplicial sets, but it is now mostly taken to be a property defining groupoids up to equivalence.
In this paper, we investigate the homotopy hypothesis in the context of homotopy type theory (HoTT). HoTT refers to the homotopical interpretation of MartinLöf’s dependent type theory [AwodeyWarren2009, Voevodsky06]. In this homotopical interpretation, every typetheoretical construction corresponds to a homotopyinvariant construction on spaces.
In HoTT, every type has a path space given by the identity type. For a pointed type we can construct the loop space, which has the structure of an group. Moreover, if the type is truncated, then we can retreive the usual notion of groups, 2groups and higher groups. This allows us to define a higher group internally in the language of type theory as a type that is the loop space of a pointed connected type, its delooping.
We also investigate groups that can be delooped more than once, which gives groups with additional coherences. The full family of groups we consider is in Table 1, which we will explain in detail in section 3.
Our approach is additionally validated by the corresponding observation in topos theory, where it is a theorem that the category of pointed, connected objects in is equivalent to the category of higher group objects in , for any topos [LurieHTT, Lemma 7.2.2.11(1)].
We have formalized most of our results in the HoTT library [vDvRB2017HoTTLean] of the Lean Theorem Prover [Moura2015]. The formalized results can be found in the file https://github.com/cmuphil/Spectral/blob/master/higher_groups.hlean. We will indicate the major formalized results in this paper by referring to the name in the formalization inside square brackets. For more information about the formalization, see section 8.
We are indebted to Michael Shulman for writing a blog post [Shulmanclassifying]
on classifying spaces from a univalent perspective.
2 Preliminaries
In this paper we will work in the type theory of the HoTT book [TheBook], although all arguments will also hold in a cubical type theory, such as [CCHM2016, chtt]. In this section we briefly introduce the concepts we need for the rest of the paper.
The type theory contains dependent function types , which are more traditionally denoted as and dependent pair types , which are traditionally denoted as . We choose to use this Agdainspired notation because we often deal with deeply nested dependent sum types.
Within a type we have the identity type or path type . We have various operations on paths, such as concatenation and inversion of paths. The functorial action of a function on a path is denoted . The constant path is denoted .
A type can be truncated, denoted , which is defined by recursion on :
For any type we write for its truncation, i.e., is an truncated type equipped with a map such that for any truncated type the precomposition map
is an equivalence. Then we define being connected as . Properties of truncations and connected maps are established in Chapter 7 of [TheBook].
The type of pointed types is . The type of truncated types is and for connected types it is . We will combine these notations as needed.
Given we define the loop space , which is pointed with basepoint . The homotopy groups of are defined to be . These are group in the usual sense when , with neutral element and group operation induced by path concatenation.
Given the type of pointed maps from to is . Given we write for the first projection and for the second projection. The fiber of a pointed map is defined by , which is pointed with basepoint .
In HoTT we can use higher inductive types to construct EilenbergMacLane spaces [FinsterLicata2014]. For a group we define as the following HIT.
HIT
;
;
;
.
(Using the univalent universe , other direct definitions are
also possible, for instance, is equivalent to the type of
small torsors.)
Let denote the suspension of , i.e., the homotopy pushout of . For an abelian group can now inductively define . Then we have the following result [FinsterLicata2014].
Theorem 1.
Let be a group and , and assume that is abelian when . The space is connected and truncated and there is a group isomorphism .
In some of our informal arguments we use the descent theorem for pushouts,^{1}^{1}1Recall from [LurieHTT, §6.1.3], following ideas from Charles Rezk, that we can define the toposes among locally cartesian closed categories as those whose colimits are van Kampen, viz., satisfying descent. which states that for a commuting cube of types
(1) 
if the bottom square is a pushout and the vertical squares are pullbacks, then the top square is also a pushout. We will use the following slight generalization.
Theorem 2.
Consider a commuting cube of types as in (1), and suppose the vertical squares are pullback squares. Then the square
is a pullback square.
Proof.
It suffices to show that the pullback
has the universal property of the pushout. This follows by the descent theorem, since by the pasting lemma for pullbacks we also have that the vertical squares in the cube
are pullback squares. ∎
In the formalization, arguments using descent are more conveniently done via the equivalent principle captured formally as the flattening lemma [TheBook, §6.12].
3 Higher groups
Recall that types in HoTT may be viewed as groupoids: elements are objects, paths are morphisms, higher paths are higher morphisms, etc.
It follows that pointed connected types may be viewed as higher groups, with carrier . The neutral element is the identity path, the group operation is given by path composition, and higher paths witness the unit and associativity laws. Of course, these higher paths are themselves subject to further laws, etc., but the beauty of the typetheoretic definition is that we don’t have to worry about that: all the (higher) laws follow from the rules of the identity types.
Writing for the carrier, it is common to write for the pointed connected type such that . We call the delooping of . Let us write
for the type of higher groups, or groups. Note that for we also have using the first projection as a coercion. Using the last definition, this is the loop space map, and not the usual coercion!
We recover the ordinary setlevel groups by requiring that is a type, or equivalently, that is a type. This leads us to introduce
for the type of groupal (grouplike) groupoids, also known as groups. For a setlevel group, we have .
For example, the integers as an additive group are from this perspective represented by their delooping , i.e., the circle.
Of course, double loop spaces are even better behaved than mere loop spaces (e.g., they are commutative up to homotopy by the EckmannHilton argument [TheBook, Theorem 2.1.6]). Say a type is tuply groupal if we have a fold delooping, , such that .
Mixing the two directions, let us introduce the type
[GType_equiv] 
for the type of tuply groupal groupoids.^{2}^{2}2This is called in [BaezDolan1998], but here we give equal billing to and , and we add the “G” to indicate groupstructure. (We allow taking in which case the truncation requirement is simply dropped. [InfGType_equiv])
Note that . This shift in indexing is slightly annoying, but we keep it to stay consistent with the literature.
Since there are forgetful maps
given by we can also allow to be infinite, by setting
In section 6 we prove the stabilization theorem (Theorem 6), from which it follows that for .
When , this is the type of stably groupal groups, also known as connective spectra. If we also relax the connectivity requirement, we get the type of all spectra, and we can think of a spectrum as a kind of groupoid with morphisms for all .
4 Elementary theory
Given any type of objects , any has an automorphism group with (the connected component of at ). Clearly, if is truncated, then so is and so is truncated, and hence an group.
Moving across the homotopy hypothesis, for every pointed type we have the fundamental group of , . Its truncation (an instance of decategorification, see section 6) is the fundamental group of , , with corresponding delooping .
If we take , we get the usual symmetric groups , where is a set with elements. (Note that is the type of all element sets.) We give further constructions related to ordinary groups in section 7.
4.1 Homomorphisms and conjugation
A homomorphism between higher groups is any function that can be suitably delooped. For , we define
For (connective) spectra we need pointed maps between all the deloopings and pointed homotopies showing they cohere.
Note that if are homomorphisms between setlevel groups, then and are conjugate if are freely homotopic (i.e., equal as maps ).
Also observe that for , that is, for , so this suggests that is truncated. (The calculation verifies this for the identity component.) To prove this, we need to use an induction using the definition of truncated. If , then its selfidentity type is equivalent to . This type is no longer a type of pointed maps, but rather a type of pointed sections of a fibration of pointed types.
Definition 1.
If and , then we introduce the type of pointed sections,
This type is itself pointed by the trivial section .
Theorem 3.
Let be an connected, pointed type for some , and let be a fibration of truncated, pointed types for some . Then the type of pointed sections, , is truncated. [is_trunc_ppi_of_is_conn]
Proof.
The proof is by induction on .
For the base case we have to show that the type of pointed sections is a mere proposition. Since it is pointed, it must in fact be contractible. The center of contraction is the trivial section . If is another section, then we get a pointed homotopy from to from the elimination principle for pointed, connected types [TheBook, Lemma 7.5.7], since the types are truncated.
To show the result for , taking the case as the induction hypothesis, it suffices to show for any pointed section that its selfidentity type is truncated. But this type is equivalent to , which is again a type of pointed sections, and here we can apply the induction hypothesis. ∎
Corollary 1.
Let and . If is connected, and is truncated, then the type of pointed maps is truncated. In particular, is an type for .
Corollary 2.
The type is truncated. [is_trunc_GType]
Proof.
This follows immediately from the preceding corollary, as the type of equivalences is a subtype of the homomorphisms from to . ∎
If (so we’re in the stable range), then becomes a stably groupal groupoid. This generalizes the fact that the homomorphisms between abelian groups form an abelian group.
The automorphism group of a higher group is in . This is equivalently the automorphism group of the pointed type . But we can also forget the basepoint and consider the automorphism group of . This now allows for (higher) conjugations. We define the generalized center of to be (generalizing the center of a setlevel group, see below in subsection 4.3).
4.2 Group actions
In this section we consider a fixed group with delooping . An action of on some object of type is simply a function . The object of the action is , and it can be convenient to consider evaluation at to be a coercion from actions of type to . To equip with a action is to give an action with . The trivial action is the constant function at . Clearly, an action of on is the same as a homomorphism .
If is a universe of types, then we have actions on types . These types are thus simply types in the context of . A map of types from to is just a function .
If is a type, then we can form the
 invariants

, also known as the homotopy fixed points, and the
 coinvariants

, which is also known as homotopy orbit space or the homotopy quotient .
It is easy to see that these constructions are respectively the right and left adjoints of the functor that sends a type to the trivial action on , , which is just the constant family at . Indeed, the adjunctions are just the usual argument swap and (un)currying equivalences, for ,
If we think of an action as a typevalued diagram on , this means that the homotopy fixed points and the homotopy orbit space form the homotopy limit and homotopy colimit of this diagram, respectively.
Proposition 1.
Let be a homomorphism of higher groups with delooping , and let be a map of types. By composing with we can also view and as types, in which case we get a homotopy pullback square:
Proof.
The vertical maps are induced by , and the horizontal maps are induced by . The homotopy pullback corner type is calculated as
and under this equivalence the top and the left maps are the canonical ones. ∎
Every group carries two canonical actions on itself:
 the right action

, , and the
 the adjoint action

, (by conjugation).
We have , and , the free loop space of . Recalling that , we see that , i.e., the conjugacy classes of homomorphisms from to . Since the integers are the free (higher) group on one generator, this is just the conjugacy classes of elements of . But that is exactly what we should get for the homotopy orbits of under the conjugation action.
The above proposition has an interesting corollary:
Corollary 3.
If is a homomorphism of higher groups, then is equivalent to the homotopy fiber of the delooping , where acts on via the induced right action.
Proof.
We apply Proposition 1 with being the canonical map from the right action of to the action of on the unit type. Then the square becomes:
By definition, classifies principal bundles: pullbacks of the right action of . That is, a principal bundle over a type is a family represented by a map such that for all .
For example, for every higher group we have the corresponding Hopf fibration represented by the map corresponding under the loopsuspension adjunction to the identity map on . (This particular fibration can be defined using only the induced space structure on .)
This perspective underlies the construction of the first and the third named author of the real projective spaces in homotopy type theory [realprojective]. The fiber sequences are principal bundles for the elements group with delooping , the type of element types.
4.3 Back to the center
We mentioned the generalized center above and claimed that it generalized the usual notion of the center of a group. Indeed, if is a setlevel group, then an element of corresponds to an element of , or equivalently, a map from the sphere to sending the basepoint to . By the universal property of as a HIT, this again corresponds to a homotopy from the identity on to itself, . This is precisely a homotopy fixed point of the adjoint action of on itself, i.e., a central element.
4.4 Equivariant homotopy theory
Fix a group . Suppose that is actually the (homotopy) type of a topological group. Consider the type of (small) types with a action. Naively, one might think that this represents equivariant homotopy types, i.e., sufficiently nice^{3}^{3}3Sufficiently nice means the CWspaces. The same homotopy category arises by taking all spaces with a action, but then the weak equivalences are the maps that induce weak equivalences on fixed point spaces for all closed subgroups of . topological spaces with a action considered up to equivariant homotopy equivalence. But this is not so.
By Elmendorf’s theorem [Elmendorf1983], this homotopy theory is rather that of presheaves of (ordinary) homotopy types on the orbit category of . This is the full subcategory of the category of spaces spanned by the homogeneous spaces , where ranges over the closed subgroups of .
Inside the orbit category we find a copy of the group , namely as the endomorphisms of the object corresponding to the trivial subgroup . Hence, a equivariant homotopy type gives rise to type with a action by restriction along the inclusion . (Here we consider as a (pointed and connected) topological groupoid on one object.)
As remarked by Shulman [ShulmanEI], when is a compact Lie group, then is an inverse EI category, and hence we know how to model type theory in the presheaf topos over . And in certain simple cases we can even define this model internally. For instance, if is a cyclic group of prime order, then a small equivariant type consists of a type with a action, together with another type family , where gives for each homotopy fixed point a type of proofs or “special reasons” why that point should be considered fixed [ShulmanEI, 7.6]. Hence the total space of is the type of actual fixed points, and the projection to implements the map from actual fixed points to homotopy fixed points.
Even without going to the orbit category, we can say something about topological groups through their classifying types in type theory. For example [Camarena], if is injective, then the homotopy fiber of is by Corollary 3 is the homotopy orbit space , which in this case is just the coset space , and hence in type theory represents the homotopy type of this coset space. And if
is a short exact sequence of topological groups, then is a fibration sequence, i.e., we can recover the delooping of as the homotopy fiber of the map .
4.5 Some elementary constructions
If we are given a homomorphism , represented by a pointed map where is the type of pointed types merely equivalent to , we can build a new group, the semidirect product, with classifying type . The type is indeed pointed (by the pair of the basepoint in and the basepoint in the pointed type ), and connected, and hence presents a higher group . An element of is given by a pair of an element and an identification in . But since the action is via pointed maps, the second component is equivalently an identification in , i.e., an element of . Under this equivalence, the product of and is indeed .
As a special case we obtain the direct product when is the trivial action. Here, .
As another special case we obtain the wreath products of a group and a symmetric group . Here, acts on the direct power by permuting the factors. Indeed, using the representation of as the type of element types, the map is simply . Hence the delooping of the wreath product is just .
5 Setlevel groups
In this section we give a proof that the column of Table 1 is correct. Note that for the homtypes are sets, which means that forms a 1category. Let be the category of ordinary setlevel groups (a set with multiplication, inverse and unit satisfying the group laws) and the category of abelian groups.
Theorem 4.
We have the following equivalences of categories (for ):
[cGType_equivalence_Grp]  
[cGType_equivalence_AbGrp] 
Since this theorem has been formalized we will not give all details of the proof.
Proof.
Let and be a group which is abelian if and let . If we have a group homomorphism we get a map . For this follows directly from the induction principle of . For we can define the group homomorphism as the composite , and apply the induction hypothesis to get a map . By the adjunction we get a pointed map , and by the elimination principle of the truncation we get a map .
We can now show that is the expected map, that is, the following diagram commutes, but we omit this proof here.
Now if is a group isomorphism, by Whitehead’s Theorem for truncated types [TheBook, Theorem 8.8.3] we know that is an equivalence, since it induces an equivalence on all homotopy groups (trivially on the levels other than ). We can also show that is natural in .
Note that if we have a group homomorphism , we also get a group homomorphism , and by the above construction we get a pointed map . This is functorial, which follows from naturality of .
Finally, we can construct the equivalence explicitly. We have a functor which sends to . Conversely, we have the functor . We have natural isomorphisms by Theorem 1 and by the application of Whitehead described above. The construction is exactly the same for after replacing by . ∎
6 Stabilization
In this section we discuss some constructions with higher groups [BaezDolan1998]. We will give the actions on the carriers and the deloopings, but we omit the third component, the pointed equivalence, for readability. We recommend keeping Table 1 in mind during these constructions.
 decategorification

 discrete categorification

These functors make a reflective subcategory of . That is, there is an adjunction [Decat_adjoint_Disc]^{4}^{4}4In the formalization the naturality of the adjunction is a separate statement, [Decat_adjoint_Disc_natural]. This is also true for the other adjunctions. such that the counit induces an isomorphism [Decat_Disc]. These properties are straightforward consequences of the universal property of truncation.
There are also iterated versions of these functors.
 decategorification

 discrete categorification

These functors satisfy the same properties: [InfDecat_adjoint_InfDisc] such that the counit induces an isomorphism [InfDecat_InfDisc].
For the next constructions, we need the following properties.
Definition 2.
For we define the connected cover of to be . We have the projection .
Lemma 1.
The universal property of the connected cover states the following. For any connected pointed type , the pointed map
given by postcomposition with , is an equivalence. [connect_intro_pequiv]
Proof.
Given a map , we can form a map . First note that for the type is truncated and inhabited for . Since is connected, the universal property for connected types shows that we can construct a for all such that . Then we can define the map . Now is pointed, because .
Now we show that this is indeed an inverse to the given map. On the one hand, we need to show that if , then . The underlying functions are equal because they both send to . They respect points in the same way, because . The proof that the other composite is the identity follows from a computation using fibers and connectivity, which we omit here, but can be found in the formalization. ∎
The next reflective subcategory is formed by looping and delooping.
 looping

 delooping

We have [Deloop_adjoint_Loop], which follows from Lemma 1 and [Loop_Deloop], which follows from the fact that if is connected.
The last adjoint pair of functors is given by stabilization and forgetting. This does not form a reflective subcategory.
 forgetting

 stabilization

,
where
We have the adjunction [Stabilize_adjoint_Forget] which follows from the suspensionloop adjunction on pointed types.
The next main goal in this section is the stabilization theorem, stating that the ditto marks in Table 1 are justified.
The following corollary is almost [TheBook, Lemma 8.6.2], but proving this in Book HoTT is a bit tricky. See the formalization for details.
Lemma 2 (Wedge connectivity).
If is connected and is connected, then the map is connected. [is_conn_fun_prod_of_wedge]
Let us mention that there is an alternative way to prove the wedge connectivity lemma: Recall that if is connected and is connected, then is connected [joinconstruction, Theorem 6.8]. Hence the wedge connectivity lemma is also a direct consequence of the following lemma.
Lemma 3.
Let and be pointed types. The fiber of the wedge inclusion is equivalent to .
Proof.
Note that the fiber of is , the fiber of is , and of course the fiber of is . We get a commuting cube
in which the vertical squares are pullback squares.
By the descent theorem for pushouts it now follows that is the fiber of the wedge inclusion. ∎
The second main tool we need for the stabilization theorem is:
Theorem 5 (Freudenthal).
If with , then the map is connected.
This is [TheBook, Theorem 8.6.4].
The final building block we need is:
Lemma 4.
There is a pullback square
for any .
Proof.
Note that the pullback of along either inclusion is contractible. So we have a cube
in which the vertical squares are all pullback squares. Therefore, if we pull back along the wedge inclusion, we obtain by the descent theorem for pushouts that the square in the statement is indeed a pullback square. ∎
Theorem 6 (Stabilization).
If , then is an equivalence, and any is an infinite loop space. [stabilization]
Proof.
We show that whenever .
For the first, the unit map of the adjunction factors as
where the first map is connected by Freudenthal, and the second map is connected. Since the domain is truncated, the composite is an equivalence whenever .
For the second, the counit map of the adjunction factors as
where the second map is an equivalence. By the two lemmas above, the first map is connected. ∎
For example, for an abelian group, we have , an EilenbergMacLane space.
The adjunction implies that the free group on a pointed set is . If has decidable equality, is already truncated. It is an open problem whether this is true in general.
Also, the abelianization of a setlevel group is . If is in the stable range (), then .
7 Perspectives on ordinary group theory
In this section we shall indicate how the theory of higher groups can yield a new perspective even on ordinary group theory.
From the symmetric groups , we can get other finite groups using the constructions of subsection 4.5. Other groups can be constructed more directly. For example, , the classifying type of the alternating group, can be taken to be the type of element sets equipped with a sign ordering: this is an equivalence class of an ordering modulo even permutations. Indeed, there are only two possible sign orderings, so this definition corresponds to first considering the short exact sequence
where the last map is the sign map, then realizing the sign map as given by the map that takes an element set to its set of sign orderings, and finally letting be the homotopy fiber of .
Similarly, , the classifying type of the cyclic group on elements, can be taken to be the type of elements sets equipped with a cyclic ordering: an equivalence class of an ordering modulo cyclic permutations. But unlike the above, where we had the coincidence that , this doesn’t corresponds to a short exact sequence. Rather, it corresponds to a sequence
where the delooping of the last map is the map from to that maps an element set to the set of cyclic orderings, of which there are many – since once we fix the position in the ordering of a particular element, we are free to permute the rest.
As another example, consider the map that maps a 4element set to its set of 2by2 partitions, of which there . Using this construction, we can realize some famous semidirect and wreath product identities, such as , , and, for the octahedral group, .
Let us turn to a different way of getting new groups from old, namely via covering space theory.
7.1 groups and covering spaces
The connections between covering spaces of a pointed connected type and sets with an action of the fundamental group of has already been established in homotopy type theory [FavoniaHarper2016]. Let us recall this connection and expand a bit upon it.
For us, a pointed connected type is equivalently an group with delooping . A covering space over is simply a type family that lands in the universe of sets. Hence by our discussion of actions in subsection 4.2 it is precisely a set with a action. Since is a 1type, extends uniquely to a type family , but is the delooping of the fundamental group of , and hence is the uniquely determined choice of a set with an action of the fundamental group.
The universal covering space is the simply connected cover of ,
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