nLab
discrete infinity-groupoid

Idea

The term discrete $\mathrm{\infty }$-groupoid or bare $\mathrm{\infty }$-groupoid is essentially synonymous to just ∞-groupoid . It is used for emphasis in contexts where one considers $\mathrm{\infty }$-groupoids with extra cohesive structure to indicate that this extra structure is being disregarded, or rather that the special case of discrete such structure is considered.

Definition

Observation

The terminal (∞,1)-sheaf (∞,1)-topos ∞Grpd is trivially a cohesive (∞,1)-topos, where each of the defining four (∞,1)-functors $(\Pi ⊣\mathrm{Disc}⊣\Gamma ⊣\mathrm{coDisc}):\mathrm{\infty }\mathrm{Grpd}\to \mathrm{\infty }\mathrm{Grpd}$ is an equivalence of (∞,1)-categories.

Definition

In the context of cohesive (∞,1)-toposes we say that ∞Grpd defines discrete cohesion and refer to its objects as discrete $\mathrm{\infty }$-groupoids.

More generally, given any other cohesive (∞,1)-topos

$$(\Pi ⊣\mathrm{Disc}⊣\Gamma ⊣\mathrm{codisc}):H\stackrel{\stackrel{\mathrm{Disc}}{\leftarrow }}{\underset{\Gamma }{\to }}\mathrm{\infty }\mathrm{Grpd}$$(\Pi \dashv Disc \dashv \Gamma \dashv codisc) : \mathbf{H} \stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd

the inverse image $\mathrm{Disc}$ of the global section functor is a full and faithful (∞,1)-functor and hence embeds ∞Grpd as a full sub-(∞,1)-category of $H$. A general object in $H$ is a cohesive $\mathrm{\infty }$-groupoid . We say $X\in H$ is a discrete $\mathrm{\infty }$-groupoid if it is in the image of $\mathrm{Disc}$.

Remark

This generalizes the traditional use of the terms discrete space and discrete group:

• a discrete space is equivalently a 0-truncated discrete $\mathrm{\infty }$-groupoid;

• a discrete group is equivalently a 0-truncated group object in discrete $\mathrm{\infty }$-groupoids.

Structures in $\mathrm{Disc}\mathrm{\infty }\mathrm{Grpd}$

We discuss now some of the general abstract structures in a cohesive (∞,1)-topos realized in discrete $\mathrm{\infty }$-groupoids.

Geometric homotopy and Galois theory

We discuss the general absatract notion of geometric homotopy in cohesive $(\mathrm{\infty },1)$-toposes (see here) in the context of discrete cohesion.

By the homotopy hypothesis-theorem the (∞,1)-toposes Top and ∞Grpd are equivalent, hence indistinguishable by general abstract constructions in (∞,1)-topos theory. However, in practice it can be useful to distinguish them as two different presentations for an equivalence class of $(\mathrm{\infty },1)$-toposes.

For that purposes consider the following

Definition

Define the quasi-categories

$$\mathrm{Top}:=N({\mathrm{Top}}_{\mathrm{Quillen}}{)}^{\circ }$$Top := N(Top_{Quillen})^\circ

and

$$\mathrm{\infty }\mathrm{Grpd}:=N({\mathrm{sSet}}_{\mathrm{Quillen}}{)}^{\circ },$$\infty Grpd := N(sSet_{Quillen})^\circ \,,

where on the right we have the standard model structure on topological spaces ${\mathrm{Top}}_{\mathrm{Quillen}}$ and the standard model structure on simplicial sets ${\mathrm{sSet}}_{\mathrm{Quillen}}$ and $N((-{)}^{\circ })$ denotes the homotopy coherent nerve of the simplicial category given by the full sSet-subcategory of these simplicial model categories on fibrant-cofibrant objects.

For

$$(\mid -\mid ⊣\mathrm{Sing}):{\mathrm{Top}}_{\mathrm{Quillen}}\stackrel{\stackrel{\mid -\mid }{\leftarrow }}{\underset{\mathrm{Sing}}{\to }}{\mathrm{sSet}}_{\mathrm{Quillen}}$$({|-| \dashv Sing}) : Top_{Quillen} \stackrel{\overset{{|-|}}{\leftarrow}}{\underset{Sing}{\to}} sSet_{Quillen}

the standard Quillen equivalence of the homotopy hypothesis-theorem given by the singular simplicial complex-functor and geometric realization, write

$$(𝕃\mid -\mid ⊣ℝ\mathrm{Sing}):\mathrm{Top}\stackrel{\stackrel{𝕃\mid -\mid }{\leftarrow }}{\underset{ℝ\mathrm{Sing}}{\to }}\mathrm{\infty }\mathrm{Grpd}$$(\mathbb{L} {|-|} \dashv \mathbb{R}Sing) : Top \stackrel{\overset{\mathbb{L}{|-|}}{\leftarrow}}{\underset{\mathbb{R}Sing}{\to}} \infty Grpd

for the corresponding derived functors (the image under the homotopy coherent nerve of the restriction of $\mid -\mid$ and $\mathrm{Sing}$ to fibrant-cofibrant objects followed by functorial fibrant-cofibrant replacement) that constitute a pair of adjoint (∞,1)-functors modeled as morphisms of quasi-categories.

Since this is an equivalence of (∞,1)-categories either functor serves as the left adjoint and right adjoint and so we have

Observation

Top is exhibited a cohesive (∞,1)-topos over ∞Grpd by setting

$$(\Pi ⊣\mathrm{Disc}⊣\Gamma ⊣\mathrm{coDisc}):\mathrm{Top}\stackrel{\stackrel{ℝ\mathrm{Sing}}{\to }}{\stackrel{\stackrel{𝕃\mid -\mid }{\leftarrow }}{\stackrel{\stackrel{ℝ\mathrm{Sing}}{\to }}{\underset{𝕃\mid -\mid }{\leftarrow }}}}\mathrm{\infty }\mathrm{Grpd}..$$(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : Top \stackrel{\overset{\mathbb{R}Sing}{\to}}{\stackrel{\overset{\mathbb{L}{|-|}}{\leftarrow}}{\stackrel{\overset{\mathbb{R}Sing}{\to}}{\underset{\mathbb{L}{|-|}}{\leftarrow}}}} \infty Grpd \,. \,.

In particular a presentation of the intrinsic fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos is given by the familiar singular simplicial complex construction

$$\Pi (X)\simeq ℝ\mathrm{Sing}X.$$\Pi(X) \simeq \mathbb{R} Sing X \,.

Remark

While degenerate, it is sometimes useful to make this example of a cohesive (∞,1)-topos explicit. For instance it allows to think of simplicial models for topological fibrations in terms of topological higher parallel transport. Some remarks on this are in Flat higher parallel transport in Top.

Remark

Notice that the topology that enters the explicit construction of the objects in Top here does not show up as cohesive structure. A topological space here is a model for a discrete $\mathrm{\infty }$-groupoid, the topology only serves to allow the construction of $\mathrm{Sing}X$. For discussion of $\mathrm{\infty }$-groupoids equipped with genuine topological cohesion see Euclidean-topological ∞-groupoid.

Cohomology and principal $\mathrm{\infty }$-bundles

We discuss the general abstract notion of cohomology and principal $\mathrm{\infty }$-bundles a in cohesive $\mathrm{\infty }$-toposes (see here) in the context of discrete cohesion.

Definition

Write $\mathrm{sGrp}=\mathrm{Grp}(\mathrm{sSet})$ for the category of simplicial groups.

A classical reference is section 17 of May.

Proposition

The category $\mathrm{sGrpd}$ inherits a model category structure transferred along the forgetful functor $F:\mathrm{sGrp}\to \mathrm{sSet}$.

The category ${\mathrm{sSet}}_0\hookrightarrow \mathrm{sSet}$ of reduced simplicial sets (simplicial sets with a single vertex) carries a model category structure whose weak equivalences and cofibrations are those of ${\mathrm{sSet}}_{\mathrm{Quillen}}$.

There is a Quillen equivalence

$$(G⊣\overline{W}):\mathrm{sGrp}\stackrel{\stackrel{G}{\leftarrow }}{\underset{\overline{W}}{\to }}{\mathrm{sSet}}_0$$(G \dashv \bar W) : sGrp \stackrel{\overset{G}{\leftarrow}}{\underset{\bar W}{\to}} sSet_{0}

which presents the abstract looping and delooping equivalence of $\mathrm{\infty }$-categories

$$(\Omega ⊣B):\mathrm{\infty }\mathrm{Grpd}\stackrel{\stackrel{\Omega }{\leftarrow }}{\underset{B}{\to }}\mathrm{\infty }{\mathrm{Grpd}}_{\mathrm{connected}},$$(\Omega \dashv \mathbf{B}) : \infty Grpd \stackrel{\overset{\Omega}{\leftarrow}}{\underset{B}{\to}} \infty Grpd_{connected} \,,

The model structures and the Quillen equivalence are classical, discussed in (GoerssJardine, section V)

This means on abstract grounds that for $G$ a simplicial group, $\overline{W}G\in \mathrm{sSet}$ is a model of the classifying delooping object $BG$ for discrte $G$-principal ∞-bundles. The following statements assert that these principal $\mathrm{\infty }$-bundles themselves can be modeled as ordinary simplicial principal bundles

Definition

For $G$ a simplicial group and $\overline{W}G$ the model for $BG$ given by the above proposition, write

$$WG\to \overline{W}G$$W G \to \bar W G

for the simplicial decalage on $\overline{W}G$.

This characterization of the object going by the classical name $WG$ is made fairly explicit in (Duskin, p. 85).

Proposition

The morphism $WG\to \overline{W}G$ is a Kan fibration resolution of the point inclusion $*\to \overline{W}G$.

This follows directly from the characterization of $WG\to \overline{W}G$ by decalage. Pieces of this statement appear in (May): lemma 18.2 there gives the fibration property, prop. 21.5 the contractibility of $WG$.

Corollary

For $G$ a simplicial group, the sequence of simplicial sets

$$G\to WG\to \overline{W}G$$G \to W G \to \bar W G

is a presentation of the fiber sequence

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

Hence $WG\to \overline{W}G$ is a model for the universal $G$-principal discrete $\mathrm{\infty }$-bundle (see universal principal ∞-bundle):

every $G$-principal discrete $\mathrm{\infty }$-bundle $P\to X$ in $\mathrm{\infty }\mathrm{Grpd}$, which by definition is a homotopy fiber

$$\begin{array}{ccc}P& \to & *\\ \downarrow & {⇙}_{\simeq }& \downarrow \\ X& \to & BG\end{array}$$\array{ P &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ X &\to& \mathbf{B}G }

in ∞Gpd, is presented in the standard model structure on simplicial sets by the ordinary pullback

$$\begin{array}{ccc}P& \to & WG\\ \downarrow & & \downarrow \\ X& \to & \overline{W}G\end{array}.$$\array{ P &\to& W G \\ \downarrow && \downarrow \\ X &\to& \bar W G } \,.

The explicit statement that the sequence $G\to WG\to \overline{W}G$ is a model for the looping fiber sequence appears on p. 239 of Crossed Menagerie . The universality of $WG\to \overline{W}G$ for $G$-principal simplicial bundles is the topic of section 21 in (May), where however it is not made explicit that the “twisted cartesian products” considered there are precisely the models for the pullbacks as above. This is made explicit on page 148 of Crossed Menagerie.

In Euclidean-topological ∞-groupoid we discuss how this model of discrete principal $\mathrm{\infty }$-bundles by simplicial principal bundles lifts to a model of topological principal $\mathrm{\infty }$-bundles by simplicial topological bundles principal over simplicial topological groups.

Related concepts

• discrete space

• discrete group

• discrete groupoid

• discrete ∞-groupoid

References

Simplicial groups and simplicial principal bundles are discussed in

• Peter May, Simplicial Objects in Algebraic Topology (djvu).

and section V of

• Paul Goerss and Rick Jardine, 1999, Simplicial Homotopy Theory, number 174 in Progress in Mathematics, Birkhauser. (ps)

The relation of $WG\to \overline{W}G$ to decalage is mentioned on p. 85 of

• John Duskin, Simplicial methods and the interpretation of “triple” cohomology, number 163 in Mem. Amer. Math. Soc., 3, Amer. Math. Soc. (1975)

Discrete cohesion is the topic of section 3.1 of

• Urs Schreiber, differential cohomology in a cohesive topos ,

where much of the above material is taken from.