nLab free loop space of classifying space

Idea

For $\mathcal{G}$ a topological group or simplicial group, or generally an ∞-group, the homotopy type of the free loop space of its classifying space/delooping is equivalent to the homotopy quotient of the conjugation ∞-action of $\mathcal{G}$ on iself:

$$\mathcal{L} \mathbf{B}\mathcal{G} \;\; \simeq \;\; \mathcal{G} \sslash_{ad} \mathcal{G} \,.$$

For discrete groups (in particular for finite groups) this is seen by elementary inspection (Example below) and as such is familiar from many related constructions, such as that of inertia groupoids.

In more generality the statement is folklore (as witnessed by parts of the MO discussion) but rarely argued in detail.

For topological groups of the homotopy type of a CW-complex there is a point-set argument (Gruher 2007) with continuous paths in the universal principal bundle which, with some care, largely mimics the naive example .

More generally, such as for simplicial groups, there is a more abstract computation of the defining homotopy pullback by fibrant resolution, which is spelled out as Prop. below, following an analogous argument for topological groups indicated in Klein, Schochet & Smith 2009.

Finally, since, in the full generality of ∞-groups internal to any (∞,1)-topos, hence for ∞-group ∞-stacks $\mathcal{G}$, the homotopy quotient of an action, regarded in the slice over the delooping/moduli stack $\mathbf{B}\mathcal{G}$, in fact characterizes and defines the action as an ∞-action

$$\mathcal{G} \,\in\, Groups(\mathbf{H}) \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; \mathcal{G}Actions(\mathbf{H}) \;\; \underoverset {\;\;\;\;\simeq\;\;\;\;} {(-)\sslash \mathcal{G}} {\longrightarrow} \;\; \mathbf{H}_{/\mathbf{B}\mathcal{G}}$$

one may turn this around (NSS 12a, Exp. 4.13 (3), SS 20, Exp. 2.82) and define the adjoint action of an ∞-group ∞-stack to be that whose homotopy quotient over $\mathbf{B}\mathcal{G}$ is the left vertical morphism (1) in the defining homotopy pullback-construction of the free loop space object of $\mathbf{B}\mathcal{G}$.

Proof

The statement is folklore, but complete proofs in the literature are rare.

Recall that the free loop space object of $\mathbf{B}\mathcal{G}$ (in particular) is defined to be homotopy pullback/homotopy fiber product of the diagonal morphism on $\mathbf{B}\mathcal{G}$ along/with itself:

(1)$$\array{ \mathcal{L} \mathbf{B}\mathcal{G} &\longrightarrow& \mathbf{B}\mathcal{G} \\ \big\downarrow &\swArrow_{\mathrlap{pb}}& \big\downarrow {}^{\mathrlap{diag}} \\ \mathbf{B}\mathcal{G} &\xrightarrow{\;\;\;diag\;\;\;}& \mathbf{B}\mathcal{G} \times \mathbf{B} \mathcal{G} }$$

For simplicial sets

We discuss the presentation of the phenomenon in the classical model structure on simplicial sets.

Let

• $\mathcal{G}$ be a simplicial group,

and write

• $\mathcal{G}\Actions(sSet)$ for the category of $\mathcal{G}$-action objects internal to SimplicialSetsl

• $W \mathcal{G} \in \mathcal{G}Actions(sSet)$ for its universal principal simplicial complex;

• $\overline{W}\mathcal{G} \,=\, \frac{W \mathcal{G}}{\mathcal{G}} \in sSet$ for the simplicial classifying space;

• $\mathcal{G}_{ad} \in \mathcal{G}Actions(sSet)$ for the adjoint action of $\mathcal{G}$ on itself:

  (2)$$\array{ \mathcal{G}_{ad} \times \mathcal{G} &\xrightarrow{\;\;\;}& \mathcal{G}_{ad} \\ (g_k,h_k) &\mapsto& h_k \cdot g_k \cdot h_k^{-1} }$$

  which we may understand as the restriction along the diagonal morphism $\mathcal{G} \xrightarrow{diag} \mathcal{G} \times \mathcal{G}$ of the following action of the direct product group:

  $$\array{ \mathcal{G}_{ad} \times (\mathcal{G} \times \mathcal{G}) &\xrightarrow{\;\;\;}& \mathcal{G}_{ad} \\ (g_k, (h'_k, h_k)) &\mapsto& h'_k \cdot g_k \cdot h^{-1}_k \mathrlap{\,.} }$$

Proposition

The free loop space object of the simplicial classifying space $\overline{W} \mathcal{G}$ is isomorphic in the classical homotopy category to the Borel construction of the adjoint action (2):

$$\mathcal{L} \big( \overline{W}\mathcal{G} \big) \;\; \simeq \;\; \mathcal{G}_{ad} \sslash \mathcal{G} \;\;\;\;\;\; \in \;\; Ho\big( sSet_{Qu} \big)$$

Proof

Consider the following commuting diagram of SimplicialSets:

Here:

• all objects are fibrant objects (Kan complexes), since simplicial classifying spaces $\overline{W}\mathcal{G}$ are Kan complexes, by this Prop., which, recall, comes about as follows:

  • simplicial groups have underlying Kan complexes (this Thm., which we use below to know that $\mathcal{G}_{ad} \to \ast$ is a Kan fibration), hence are fibrant objects in the model structure on simplicial groups;

  • the simplicial classifying space-functor $W(-)/(-)$ is a right Quillen functor to the model structure on reduced simplicial sets (this Prop.), hence takes values in fibrant objects;

  • fibrant reduced simplicial sets are Kan complexes (this Prop.);

• the right vertical morphism is a fibration (Kan fibration), since it is the image of the Kan fibration $\mathcal{G}_{ad} \to \ast$ under the right Quillen functor $(W \mathcal{G} \times (-)) /\mathcal{G}$ on the Borel model structure (this Prop.), see this Example;

• the top right morphism is a simplicial weak equivalence, by two-out-of-three, as it is a right inverse of the triangular morphisms, which are manifestly a simplicial weak eqivalence since $W \mathcal{G}$ is contractible (by this Prop.).

One verifies by inspection that the commuting square shown is an ordinary pullback in SimplicialSets.

Finally notice that the simplicial classifying space-construction is indeed a model for the delooping

$$\overline{W}\mathcal{G} \;\; \simeq \;\; \mathbf{B}\mathcal{G} \;\;\; \in \; Ho(\infty Groupoids)$$

(essentially by its Quillen adjunction to the simplicial loop space-functor (here) and the May recognition theorem, see NSS 12a, Cor. 3.33).

In summary this means (by this Prop.) that this ordinary pullback-square represents the homotopy pullback (1) that defines the free loop space object.

For topological spaces

A point-set argument for topological spaces/topological groups is spelled out in Gruher 2007, App. A. Discussion of the presentation of the phenomenon in the classical model structure on topological spaces in the abstract style above is indicated in Klein, Schochet & Smith 2009.

In homotopy type theory

Recall that in homotopy type theory, an ∞-group $G$ is represented by its delooping type, a pointed connected type $\mathbf{B} G$.

(The categorical semantics of this is a groupal A-∞ ∞-stack, in some (∞,1)-topos, so that the following is actually about free loop spaces in the generality of inertia ∞-stacks, not though in the yet further generality of free loop ∞-stacks.)

An ∞-action of $G$ on a term $a$ of type $A$ is given by a group homomorphism $G \to Aut_A(a)$, represented by a morphism of pointed homotopy types $\mathbf{B} G \to_* (A,a)$. (Since $\mathbf{B} G$ is connected, it doesn’t matter whether we restrict the codomain to the connected component of $A$ at $a$.)

We thus see that the type of all ∞-actions of $G$ on objects of $A$ is the function type $\mathbf{B} G \to A$. In particular, the type of $U$-small $G$-types, where $U$ is a type universe, is $\mathbf{B} G \to U$.

By adjointness, the homotopy orbit type of a $G$-type $X \colon \mathbf{B} G \to U$ is given by the dependent sum-type,

$$X \!\sslash\! G \;\coloneqq\; \sum_{t : \mathbf{B} G} X(t)$$

(The homotopy fixed points are given by the dependent product-type, as discussed at ∞-action.)

Now, the adjoint action of $G$ on itself is given by the morphism

$$\array{ G^{ad} \colon & \mathbf{B} G &\to& U \\ & t &\mapsto& (t =_{\mathbf{B} G} t) \,. }$$

Indeed, given any path $p : t =_{\mathbf{B} G} u$ in $\mathbf{B} G$ and any element $q : G^{ad}(t)$, the transport of $q$ along $p$ in the family $G^{ad}$ is equal to the conjugate $p^{-1} \cdot q \cdot p : G^{ad}(u)$, as proven by path induction. (Here, we write path composition in diagrammatic order.)

Putting this together, we get that the homotopy orbits of the adjoint action are

$$\begin{aligned} G^{ad} \!\sslash\! G & \;=\; \sum_{t: \mathbf{B} G} (t =_{\mathbf{B} G} t) \\ & \;\simeq\; (ʃS^1 \to \mathbf{B} G), \end{aligned}$$

where in the last step we used the universal property of the homotopy type (shape) of the circle type, $ʃS^1 \simeq \mathbf{B}\mathbb{Z}$, defined as a higher inductive type (here) with a point constructor $base : ʃS^1$ and a path constructor $loop : base =_{ʃS^1} base$.

The function type $ʃS^1 \to B G$ is the representation of the free loop type $\mathcal{L}(\mathbf{B} G)$ of $\mathbf{B} G$, completing the argument.

Examples

For discrete (finite) groups

Example

(free loop space of classifying space of discrete groups)

The archetypical example has $\mathcal{G} = G \in Groups(Sets)$ a discrete group, such as (but not necessarily) a finite group. In this case the classifying space $B \mathcal{G} \simeq K(G,1)$ is an Eilenberg-MacLane space whose homotopy type is represented in the classical homotopy category simply by the delooping groupoid $G \rightrightarrows \ast$ of $G$. With this, the analysis of its free loop space follows from elementary inspection:

The hom-groupoid

$$\big[ \mathbf{B}\mathbb{Z} ,\, \mathbf{B}G \big] \;\simeq\; \big( G \times G \underoverset {pr_2} { Ad } {\rightrightarrows} G \big)$$

has

• as objects the functors $\mathbf{B}\mathbb{Z} \longrightarrow \mathbf{B}G$, hence equivalently group homomorphisms $\mathbb{Z} \to G$, hence equivalently elements $g \in G$;

• as morphisms the natural transformations between these, whose single component $h \in G$ must make the naturality square commute

$$\array{ \mathbf{B}\mathbb{Z} &\xrightarrow{\;\;\;}& \mathbf{B}G \\ \\ \bullet && \bullet &\xrightarrow{\;\;\;h\;\;\;}& \bullet \\ \big\downarrow {}^{\mathrlap{1}} && \big\downarrow {}^{\mathrlap{g}} && \big\downarrow {}^{\mathrlap{g'}} \\ \bullet && \bullet &\xrightarrow{\;\;\;h\;\;\;}& \bullet }$$

The condition that this commutes means equivalently that $g'$ is the image of $g$ under the conjugation action by $h$:

$$h \cdot g' \;=\; g \cdot h \;\;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\;\; g' = h^{-1} \cdot g \cdot h \;=\; Ad_h(g) \,.$$

This shows that the hom-groupoid is the action groupoid of the conjugation action

$$\mathcal{L} \mathbf{B}G \;\simeq\; \big[ \mathbf{B}\mathbb{Z} ,\, \mathbf{B}G \big] \;\simeq\; G \sslash_{ad} G \,.$$

This simple example essentially re-appears in the discussion of inertia groupoids.

For simplicial abelian groups

As a direct corollary of the general statement we have:

Proposition

For $\mathcal{A} \in AbGroup(sSet)$ a simplicial abelian group, the homotopy type of the free loop space of its simplicial classifying space is the homotopy product (Cartesian product) of $\mathcal{A}$ with its delooping:

$$\big[ \mathbf{B}\mathbb{Z}, \, \mathbf{B}\mathcal{A} \big] \;\; \simeq \;\; \mathcal{A} \,\times\, \mathbf{B}\mathcal{A} \,.$$

Proof

This is the composite of the following sequence of equivalences (isomorphisms in the classical homotopy category):

$$\begin{aligned} \big[ \mathbf{B}\mathbb{Z}, \, \mathbf{B}\mathcal{A} \big] & \;\simeq\; \frac{ \mathcal{A}_{\mathrm{ad}} \times W\mathcal{A} }{ \mathcal{A} } \\ & \;\simeq\; \frac{ \mathcal{A}_{\mathrm{triv}} \times W\mathcal{A} }{ \mathcal{A} } \\ & \;\simeq\; \mathcal{A} \times \frac{ W\mathcal{A} }{ \mathcal{A} } \\ & \;\simeq\; \mathcal{A} \times \overline{W}\mathcal{A} \\ & \;\simeq\; \mathcal{A} \times \mathbf{B}\mathcal{A} \end{aligned}$$

Here:

• the first step is Prop. ;

• the second step observes that for abelian groups the adjoint action (2) is actually the trivial action;

• the third step is the evident consequence for the quotient;

• the fourth step is the definition of the simplicial classifying space;

• the last step is the identification of the simplicial classifying space with the delooping as ∞-groups.

As an immediate special cases of Prop. we have:

Example

(free loop space of Eilenberg-MacLane spaces)

For $n \in \mathbb{N}$ and

$$\mathbf{B}^{n+1} \mathbb{Z} \;\simeq\; ʃ \mathbf{B}^n S^1$$

the shape of the circle (n+1)-group, we have

$$\mathcal{L} \mathbf{B}^{n+1}\mathbb{Z} \;\simeq\; \mathbf{B}^n\mathbb{Z} \,\times\, \mathbf{B}^{n+1} \mathbb{Z} \,.$$

More generally, for $A \in AbGroups(Set)$ any discrete abelian group we have

$$\mathcal{L} \mathbf{B}^{n+1}A \;\simeq\; \mathbf{B}^n A \,\times\, \mathbf{B}^{n+1}A \,,$$

where the $n$-fold delooping

$$\mathbf{B}^n A \;\simeq\; K(n,A)$$

is equivalently the homotopy type of the Eilenberg-MacLane space for $A$ in degree $n$ (classifying ordinary cohomology in degree $n$ with coefficients in $A$).

Remark

Example pertains to the discussion of double dimensional reduction of brane charges in ordinary cohomology by the discussion here at geometry of physics – fundamental super p-branes.

Related concepts

• free loop space, free loop stack

• transchromatic character

• double dimensional reduction

• classifying space, simplicial classifying space,

• Sullivan model of classifying space

References

A point-set proof in TopologicalSpaces is given in

• Kate Gruher, A proof of $L B G \simeq Ad(E G)$, Appendix A in: String Topology of Classifying Spaces, 2007 (pdf, proquest:304826261)

An abstract proof in the above style, for topological groups/topological spaces, is indicated in:

• John R. Klein, Claude Schochet, Samuel B. Smith, Lemma 9.1 of: Continuous trace $C^\ast$-algebras, gauge groups and rationalization, Journal of Topology and Analysis Vol. 01, No. 03, pp. 261-288 (2009) (arXiv:0811.0771, doi:10.1142/S179352530900014X)

See also:

• MathOverflow discussion MO/q/20671

Discussion in the generality of ∞-groups in (∞,1)-toposes and defining the adjoint action in this generality:

• Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Example 4.13 (3) in: Principal ∞-bundles – General theory, Journal of Homotopy and Related Structures, Volume 10, Issue 4 (2015), pages 749-801

  (doi:10.1007/s40062-014-0083-6, arXiv:1207.0248)

• Hisham Sati, Urs Schreiber, Example 2.82 in: Proper Orbifold Cohomology (arXiv:2008.01101)

and formulated more in the language of homotopy type theory:

• Ulrik Buchholtz, Floris van Doorn, Egbert Rijke, p. 11 of: Higher Groups in Homotopy Type Theory, LICS ‘18: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (arXiv:1802.04315, doi:10.1145/3209108.3209150)