nLab
free loop space of classifying space

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

For 𝒢\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 𝒢\mathcal{G} on iself:

B𝒢𝒢 ad𝒢. \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 𝒢\mathcal{G}, the homotopy quotient of an action, regarded in the slice over the delooping/moduli stack B𝒢\mathbf{B}\mathcal{G}, in fact characterizes and defines the action as an ∞-action

𝒢Groups(H)𝒢Actions(H)()𝒢H /B𝒢 \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 B𝒢\mathbf{B}\mathcal{G} is the left vertical morphism (1) in the defining homotopy pullback-construction of the free loop space object of B𝒢\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 B𝒢\mathbf{B}\mathcal{G} (in particular) is defined to be homotopy pullback/homotopy fiber product of the diagonal morphism on B𝒢\mathbf{B}\mathcal{G} along/with itself:

(1)B𝒢 B𝒢 pb diag B𝒢 diag B𝒢×B𝒢 \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

and write

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

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

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

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

    (2)𝒢 ad×𝒢 𝒢 ad (g k,h k) h kg kh k 1 \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 𝒢diag𝒢×𝒢\mathcal{G} \xrightarrow{diag} \mathcal{G} \times \mathcal{G} of the following action of the direct product group:

    𝒢 ad×(𝒢×𝒢) 𝒢 ad (g k,(h k,h k)) h kg kh k 1. \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 W¯𝒢\overline{W} \mathcal{G} is isomorphic in the classical homotopy category to the Borel construction of the adjoint action (2):

(W¯𝒢)𝒢 ad𝒢Ho(sSet Qu) \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:

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

W¯𝒢B𝒢Ho(Groupoids) \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 GG is represented by its delooping type, a pointed connected type BG\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 GG on a term aa of type AA is given by a group homomorphism GAut A(a)G \to Aut_A(a), represented by a morphism of pointed homotopy types BG *(A,a)\mathbf{B} G \to_* (A,a). (Since BG\mathbf{B} G is connected, it doesn’t matter whether we restrict the codomain to the connected component of AA at aa.)

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

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

XG t:BGX(t) 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 GG on itself is given by the morphism

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

Indeed, given any path p:t= BGup : t =_{\mathbf{B} G} u in BG\mathbf{B} G and any element q:G ad(t)q : G^{ad}(t), the transport of qq along pp in the family G adG^{ad} is equal to the conjugate p 1qp:G ad(u)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

G adG = t:BG(t= BGt) (ʃS 1BG), \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 1BʃS^1 \simeq \mathbf{B}\mathbb{Z}, defined as a higher inductive type (here) with a point constructor base:ʃS 1base : ʃS^1 and a path constructor loop:base= ʃS 1baseloop : base =_{ʃS^1} base.

The function type ʃS 1BGʃS^1 \to B G is the representation of the free loop type (BG)\mathcal{L}(\mathbf{B} G) of BG\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 𝒢=GGroups(Sets)\mathcal{G} = G \in Groups(Sets) a discrete group, such as (but not necessarily) a finite group. In this case the classifying space B𝒢K(G,1)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*G \rightrightarrows \ast of GG. With this, the analysis of its free loop space follows from elementary inspection:

The hom-groupoid

[B,BG](G×Gpr 2AdG) \big[ \mathbf{B}\mathbb{Z} ,\, \mathbf{B}G \big] \;\simeq\; \big( G \times G \underoverset {pr_2} { Ad } {\rightrightarrows} G \big)

has

B BG h 1 g g h \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 gg' is the image of gg under the conjugation action by hh:

hg=ghg=h 1gh=Ad h(g). 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

BG[B,BG]G adG. \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 𝒜AbGroup(sSet)\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 𝒜\mathcal{A} with its delooping:

[B,B𝒜]𝒜×B𝒜. \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):

[B,B𝒜] 𝒜 ad×W𝒜𝒜 𝒜 triv×W𝒜𝒜 𝒜×W𝒜𝒜 𝒜×W¯𝒜 𝒜×B𝒜 \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:

As an immediate special cases of Prop. we have:

Example

(free loop space of Eilenberg-MacLane spaces)

For nn \in \mathbb{N} and

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

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

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

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

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

where the nn-fold delooping

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

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

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.

References

A point-set proof in TopologicalSpaces is given in

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

See also:

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

and formulated more in the language of homotopy type theory:

Last revised on July 11, 2021 at 15:53:14. See the history of this page for a list of all contributions to it.