nLab delooping

Idea

The delooping of an object $A$ is, if it exists, a uniquely pointed object $\mathbf{B} A$ such that $A$ is the loop space object of $\mathbf{B} A$:

$$A \simeq \Omega(\mathbf{B} A)$$

In particular, if $A = G$ is a group then its delooping

• in the context Top is the classifying space $\mathcal{B}G$

• in the context ∞-Grpd is the one-object groupoid $\mathbf{B}G$.

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

$$|\mathbf{B}G| \simeq \mathcal{B}G \,.$$

See looping and delooping.

Definition

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

$$\array{ \Omega X &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& X } \,.$$

Conversely, if $A$ is given and a homotopy pullback diagram

$$\array{ A &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& \mathbf{B}A }$$

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

$$A = \Omega \mathbf{B} A$$

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

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

Remarks

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

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 $\mathbf{C}$ is given as a homotopy simplicial object, i.e. a (infinity,1)-functor

$$C : \Delta^{op} \to \mathbf{C}$$

$$\cdots C_2 \stackrel{\to}\rightrightarrows C_1 \rightrightarrows C_0$$

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

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

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 $C =$ 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 $G$ be a group regarded as a discrete groupoid in the (∞,1)-topos ∞Grpd of ∞-groupoids.

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

• with a single object $\bullet$,

• with $Hom_{\mathbf{B} G}(\bullet, \bullet) = G$, or equivalently $Aut_{\mathbf{B}G}(\bullet) = G$,

• and with composition of morphisms in $\mathbf{B} G$ being given by the product operation in the group.

More informally but more suggestively we may write

$$\mathbf{B} G = \{ \bullet \stackrel{g}{\to} \bullet | g \in G\}$$

or

$$\mathbf{B}G = \{ \bullet \righttoleftarrow g \;|\; g \in G \}$$

to emphasize that there is really only a single object.

Notice how the homotopy pullback works in this simple case:

the universal 2-cell $\eta$

$$\array{ G &\to& {*} \\ \downarrow &\Downarrow^{\eta}& \downarrow \\ {*} &\to& \mathbf{B}G }$$

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

$$G \to {*} \to \mathbf{B}G$$

to itself, whose component map

$$\eta : Obj(G) \to Mor(\mathbf{B}G)$$

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

Deloopings of higher categorical structures

There is also a notion of delooping which takes a pointed $(n, k+1)$-category $C$ to a pointed $(n+1, k)$-category $\mathbf{B} C$ in which $\mathbf{B} C$ has a single $0$-cell $\bullet$, and where $\hom(\bullet, \bullet) = C$. This is a tautological construction if one accepts the delooping hypothesis, which views a $(n, k+1)$-category $C$ as a special type of $(n+k+1)$-category, namely a pointed $k$-connected $(n+k+1)$-category: by viewing such as a fortiori a pointed $(k-1)$-connected $(n+k+1)$-category, we get the delooping $\mathbf{B} C$.

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

• $\mathbf{B} M$ has a single $0$-cell $\bullet$,

• the $1$-cells $\bullet \to \bullet$ of $\mathbf{B} M$ are named by objects of $M$, and the composite of $\bullet \stackrel{a}{\to} \bullet \stackrel{b}{\to} \bullet$ is $\bullet \stackrel{a \otimes b}{\to} \bullet$ (using the monoidal product $\otimes$ of $M$),

• the $2$-cells of $\mathbf{B} M$ are similarly named by morphisms of $M$; the vertical composition of $2$-cells in $\mathbf{B} M$ is given by composition of morphisms of $M$, and the horizontal composition of $2$-cells in $\mathbf{B} M$ is given by taking the monoidal product of the morphisms that name them in $M$.

Along similar lines, the delooping of a braided monoidal category produces a monoidal bicategory, and delooping of that is a tricategory or (weak) $3$-category. See delooping hypothesis for more.

Related concepts

• 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]

• David Wärn, Eilenberg-MacLane spaces and stabilisation in homotopy type theory [arXiv:2301.03685]