nLab
delooping

Idea

The delooping of an object $A$ is, if it exists, a uniquely pointed object $BA$ such that $A$ is the loop space object of $BA$:

$$A\simeq \Omega (BA)$$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 $ℬG$

• in the context ∞-Grpd is the one-object groupoid $BG$.

Under the homotopy hypothesis these two objects are identified: the geometric realization of the groupoid $BG$ is the classifying space $ℬG$:

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

Definition

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

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

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

$$\begin{array}{ccc}A& \to & *\\ \downarrow & & \downarrow \\ *& \to & BA\end{array}$$\array{ A &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& \mathbf{B}A }

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

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

and we say that $BA$ is the delooping of $A$.

Remarks

If $B$ 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 $C$ is given as a homotopy simplicial objects, i.e. a (infinity,1)-functor

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

$$\cdots C_2\overrightarrow{\rightrightarrows }{C}_1\rightrightarrows C_0$$\cdots C_2 \stackrel{\to}\rightrightarrows C_1 \rightrightarrows C_0

satisfying certain conditions (prop. 6.1.2.6) which are such that if $C_0=*$ is the point we have an internal group in a homtopical 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 $(\mathrm{\infty }\mathrm{,1})$-category Top to arbitrary $(\mathrm{\infty }\mathrm{,1})$-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 $BC$.

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 H-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 $BG$ exists and is, up to equivalence, the groupoid

• with a single object $•$,

• with ${\mathrm{Hom}}_{BG}(•,•)=G$, or equivalently ${\mathrm{Aut}}_{BG}(•)=G$,

• and with composition of morphisms in $BG$ being given by the product operation in the group.

More informally but more suggestively we may write

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

or

$$BG=\{•⟲g\mid g\in G\}$$\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 $\eta$

$$\begin{array}{ccc}G& \to & *\\ \downarrow & {\Downarrow }^{\eta }& \downarrow \\ *& \to & BG\end{array}$$\array{ G &\to& {*} \\ \downarrow &\Downarrow^{\eta}& \downarrow \\ {*} &\to& \mathbf{B}G }

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

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

to itself, whose component map

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

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

Discussion

Eric: When the two arguments coincide in ${\mathrm{Hom}}_{BG}(•,•)$, is there another notation, e.g. maybe ${\mathrm{Aut}}_{BG}(•)$ or something?

Urs: yes. In general for $C$ a category and $c\in C$ an object one writes ${\mathrm{End}}_C(c):={\mathrm{Hom}}_C(c,c)$ (for “endomorphisms”). One writes ${\mathrm{Aut}}_C(c)\subset {\mathrm{End}}_C(c)$ for the subset of all endomorphims that are invertible (are “automorphisms”). So for $C$ a groupoid, we have alsways ${\mathrm{End}}_C(c)={\mathrm{Aut}}_C(c)$.

Eric: Thanks. Maybe I read too much into choices of notation, but I can sort of see why you prefer $•\to •$ if you are accustomed to thinking in terms of $\mathrm{Hom}(•,•)$. They look the same! My brain is wired to think of $\mathrm{Aut}(•)$, so I prefer $•⟲$ (because they look the same). Maybe there is no connection. I know it is irrelevant, but just a random observation :)

Urs: Well, the main reason is that one doesn’t get very far with any computation or any nontrivial statement when not allowing oneself to write several $•$s, even though they all denote the same object. Try drawing the naturality squares of the natural transformations that appear in the above discussion without using several copies of the point!

And notice that in every other context, you wouldn’t hesitate to use several copies of one symbol that denotes the same variable. Try writing an equation with many $x$s in it by writing all the $x$s on the same spot. All you’d get is an unreadable mess. Nobody would ever complain that in $x=\ln (x)$ the two $x$s are really the same and should hence be drawn on the same spot.

Remarkably, I should add that I had this discussion before with professional pure mathematicians. Once they stopped a talk I gave when I wrote $•\stackrel{g}{\to }•$ to the board, complaining that Ii drew two copies of that bullet. If it’s just about saying quickly what $BG$ is like that’s fine with me, but for doing anything nontrivial with it it becomes useless. Try drawing $BG$ for $G$ a 2-group or a 3-group! Not to speak of drawing the transformations between functors between these.

Eric: Here is an attempt to convert one of your diagrams above to a single $•$ version.

I’m not saying one way is better than the other. I’m just making a statement about my inability to process some of the diagrams. Some diagrams are not even amenable to these “3d projections”, but for those that can be, it would help me to convert them.

Eric: Here is another attempt in case that one is not quite right…

Toby: I think that perhaps part of the problem is that people see ‘$•$’ as a generic placeholder like ‘$-$’; notice that the two dashes in ‘$[-,-]$’ below do not refer to the same thing! (With ‘$x$’, you know that the two copies refer to the same thing, else one of them would be ‘$y$’ instead.) So perhaps people need to be told that the bullet is like a letter and not like a dash.

Along those lines, sometimes it's nice to use ‘$G$’ itself in place of the bullet. Ultimately, you can think of this as Cayley's Theorem: every group acts on itself (somewhat ambiguously on the left or on the right). Then with $\{G⟲g\mid g\in G\}$, you have a picture of $BG$ as a subcategory of Set, consisting of the underlying set of $G$ as the only object and the functions given by the action of the elements of $G$ on that set as the morphisms.

Urs: I have to admit that I don’t quite understand Eric’s pictures. Here is what I meant concerning the transformation:

let $f_1,f_2:G\to H$ be two group homomorphisms and $\eta :B{f}_1\Rightarrow B{f}_2$ be a natural transformation between the corresponding functors $B{f}_1,B{f}_2:BG\to BH$. Then $\eta$ has a single component $\eta :*\to H$ to be denoted ${\eta }_{*}\in H$, which is required to make the naturality square

$$\begin{array}{ccc}*& \stackrel{{\eta }_{*}}{\to }& *\\ {\downarrow }^{f_1(g)}& & {\downarrow }^{f_2(g)}\\ *& \stackrel{{\eta }_{*}}{\to }& *\end{array}$$\array{ {*} &\stackrel{\eta_*}{\to}& {*} \\ \downarrow^{f_1(g)} && \downarrow^{f_2(g)} \\ {*} &\stackrel{\eta_*}{\to}& {*} }

commute, for all $g\in G$. You can’t draw such diagrams when insisting that all the points $*$ appearing here are to be drawn on the same spot.

And, yes, as Toby indicates, we could give that point any other name, which maybe makes that more manifest. Let’s call it $x$, then we have

$$\begin{array}{ccc}x& \stackrel{{\eta }_{*}}{\to }& x\\ {\downarrow }^{f_1(g)}& & {\downarrow }^{f_2(g)}\\ x& \stackrel{{\eta }_{*}}{\to }& x\end{array}$$\array{ x &\stackrel{\eta_*}{\to}& x \\ \downarrow^{f_1(g)} && \downarrow^{f_2(g)} \\ x &\stackrel{\eta_*}{\to}& x }

and the desire to draw all $x$s on the same spot should diminish yet a bit more.

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Eric: Is there a difference between $\mathrm{Hom}$ and $\mathrm{hom}$? For example, hom-set says $\mathrm{hom}$, but internal hom has $\mathrm{Hom}$, $\mathrm{hom}$, and $\mathrm{HOM}$.

Urs: I have added some links regarding this point at Notation. There is no really universally adopted convention here, but people do use the different capitalizations to indicate different things.

Usually “Hom” is the ordinary hom-set, while some variant of this is usually chosen for the internal hom. Sometimes “HOM”, yes. Many people like an underlined “Hom” for the internal Hom. But also $\mathrm{hom}$ and $[-,-]$ may denote internal homs. The last one is the standard choice in Kelly’s standard book on enriched category theory (though concerning just monoidal category here), so I like that one.

Eric: Is $[-,-]$ also functor category? I see you answered this at Notation. Handy :)

Urs: yes, you see by itself one often write $\mathrm{Func}(-,-)$ for the functor category. But the category Cat is an enriched category over itself, with the internal hom being $[-,-]:=\mathrm{Func}(-,-)$. So I like writing $[-,-]$ for the functor category. But in many other places you’ll find $\mathrm{Func}(-,-)$ or other things.

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The following discussion originally took place at Dijkgraaf-Witten theory.

Eric: This notation seems to cause some initial confusion. At least until you realize both $•$’s are the same, so the morphism is really a loop. Why not just represent it as a loop? I like this notation:

$$BG=\{•⟲g\mid g\in G\}$$\mathbf{B}G = \{\bullet\righttoleftarrow g | g\in G\}

What do you think?

Or better yet

$$BG=•⟲G.$$\mathbf{B}G = \bullet\righttoleftarrow G.

Toby: I like your first suggestion, so I implemented it; but I think that I only understand the second suggestion since I already know what it means.

Urs: added link to delooping above so that we have one page where this is treated discussed, since it appears in lots of other entries, too.