Under the homotopy hypothesis these two objects are identified: the geometric realization of the groupoid BG\mathbf{B}G is the classifying space ℬG\mathcal{B}G:

|BG|≃ℬG. |\mathbf{B}G| \simeq \mathcal{B}G \,.

See looping and delooping.

Definition

Loop space objects are defined in any (∞,1)-category C\mathbf{C} with homotopy pullbacks: for XX any pointed object of C\mathbf{C} with point *→X{*} \to X, its loop space object is the homotopy pullback ΩX\Omega X of this point along itself:

ΩX → * ↓ ↓ * → X. \array{ \Omega X &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& X } \,.

Conversely, if AA is given and a homotopy pullback diagram

A → * ↓ ↓ * → BA \array{ A &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& \mathbf{B}A }

exists, with the point *→BA{*} \to \mathbf{B} A being essentially unique, by the above AA has been realized as the loop space object of BA\mathbf{B} A

A=ΩBA A = \Omega \mathbf{B} A

and we say that BA\mathbf{B} A is the delooping of AA.

See the section delooping at groupoid object in an (∞,1)-category for more.

If C\mathbf{C} is even a stable (∞,1)-category then all deloopings exist and are then also denoted ΣA\Sigma A and called the suspension of AA.

Characterization of deloopable objects

In section 6.1.3 of

Jacob Lurie, Higher Topos Theory

a definition of groupoid object in an (infinity,1)-category C\mathbf{C} is given as a homotopy simplicial object, i.e. a (infinity,1)-functor

C:Δ op→C C : \Delta^{op} \to \mathbf{C}

⋯C 2⇉→C 1⇉C 0 \cdots C_2 \stackrel{\to}\rightrightarrows C_1 \rightrightarrows C_0

satisfying certain conditions (prop. 6.1.2.6) which are such that if C 0=*C_0 = {*} is the point we have an internal group in a homotopical sense, given by an object C 1C_1 equipped with a coherently associative multiplication operation C 1×C 1→C 1C_1 \times C_1 \to C_1 generalizing that of Stasheff H-space from the (∞,1)(\infty,1)-category Top to arbitrary (∞,1)(\infty,1)-categories.

Lurie calls the groupoid object CC an effective groupoid object in an (infinity,1)-category precisely if it arises as the delooping, in the above sense, of some object BC\mathbf{B}C.

One of the characterizing properties of an (infinity,1)-topos is that every groupoid object in it is effective.

This is the analog of Stasheff’s classical result about H-spaces.

See the remark at the very end of section 6.1.2 in HTT.

Examples

Topological loop spaces

For C=C = Top the (infinity,1)-category of topological spaces, a space is deloopable if it is an A-infinity-space and hence homotopy equivalent to a loop space.

Delooping of a group to a groupoid

Let GG be a group regarded as a discrete groupoid in the (∞,1)-topos ∞Grpd of ∞-groupoids.

Then BG\mathbf{B} G exists and is, up to equivalence, the groupoid

with a single object •\bullet,
with Hom BG(•,•)=GHom_{\mathbf{B} G}(\bullet, \bullet) = G, or equivalently Aut BG(•)=GAut_{\mathbf{B}G}(\bullet) = G,
and with composition of morphisms in BG\mathbf{B} G being given by the product operation in the group.

More informally but more suggestively we may write

BG={•→g•|g∈G} \mathbf{B} G = \{ \bullet \stackrel{g}{\to} \bullet | g \in G\}

BG={•⟲g|g∈G} \mathbf{B}G = \{ \bullet \righttoleftarrow g \;|\; g \in G \}

to emphasize that there is really only a single object.

Notice how the homotopy pullback works in this simple case:

the universal 2-cell η\eta

G → * ↓ ⇓ η ↓ * → BG \array{ G &\to& {*} \\ \downarrow &\Downarrow^{\eta}& \downarrow \\ {*} &\to& \mathbf{B}G }

filling this 2-limit diagram is the natural transformation from the constant functor

G→*→BG G \to {*} \to \mathbf{B}G

to itself, whose component map

η:Obj(G)→Mor(BG) \eta : Obj(G) \to Mor(\mathbf{B}G)

is just the identity map, using that Obj(G)=GObj(G) = G and Mor(BG)=GMor(\mathbf{B}G) = G.

Deloopings of higher categorical structures

There is also a notion of delooping which takes a pointed (n,k+1)(n, k+1)-category CC to a pointed (n+1,k)(n+1, k)-category BC\mathbf{B} C in which BC\mathbf{B} C has a single 00-cell •\bullet, and where hom(•,•)=C\hom(\bullet, \bullet) = C. This is a tautological construction if one accepts the delooping hypothesis, which views a (n,k+1)(n, k+1)-category CC as a special type of (n+k+1)(n+k+1)-category, namely a pointed kk-connected (n+k+1)(n+k+1)-category: by viewing such as a fortiori a pointed (k−1)(k-1)-connected (n+k+1)(n+k+1)-category, we get the delooping BC\mathbf{B} C.

This is just a generalization of the fact that a monoid MM gives rise to a one-object category (which we are denoting BM\mathbf{B} M). For an important example: a monoidal category MM has an associated delooping bicategory BM\mathbf{B} M, where

BM\mathbf{B} M has a single 00-cell •\bullet,
the 11-cells •→•\bullet \to \bullet of BM\mathbf{B} M are named by objects of MM, and the composite of •→a•→b•\bullet \stackrel{a}{\to} \bullet \stackrel{b}{\to} \bullet is •→a⊗b•\bullet \stackrel{a \otimes b}{\to} \bullet (using the monoidal product ⊗\otimes of MM),
the 22-cells of BM\mathbf{B} M are similarly named by morphisms of MM; the vertical composition of 22-cells in BM\mathbf{B} M is given by composition of morphisms of MM, and the horizontal composition of 22-cells in BM\mathbf{B} M is given by taking the monoidal product of the morphisms that name them in MM.

Along similar lines, the delooping of a braided monoidal category produces a monoidal bicategory, and delooping of that is a tricategory or (weak) 33-category. See delooping hypothesis for more.

looping and delooping
May recognition theorem
loop space object, free loop space object,
- delooping
- loop space, free loop space, derived loop space
suspension object
- suspension, reduced suspension

References

Discussion in homotopy type theory:

Ulrik Buchholtz, J. Daniel Christensen, Jarl G. Taxerås Flaten, Egbert Rijke, Central H-spaces and banded types [[arXiv:2301.02636](https://arxiv.org/abs/2301.02636)]
David Wärn, Eilenberg-MacLane spaces and stabilisation in homotopy type theory [[arXiv:2301.03685](https://arxiv.org/abs/2301.03685)]

Last revised on September 19, 2023 at 05:52:43. See the history of this page for a list of all contributions to it.

delooping (changes) in nLab

Context

(∞,1)(\infty,1)-Category theory

Stable homotopy theory

Ingredients

Contents