nLab geometrically discrete infinity-groupoid




The term discrete \infty-groupoid or bare \infty-groupoid or geometrically discrete geometric homotopy type is essentially synonymous to just ∞-groupoid or just homotopy type. It is used for emphasis in contexts where one considers \infty-groupoids with extra geometric structure (e.g. cohesive structure) to indicate that this extra structure is being disregarded, or rather that the special case of discrete such structure is considered.



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


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

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

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

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


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

Structures in DiscGrpdDisc\infty Grpd

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

Geometric homotopy and Galois theory

We discuss the general absatract notion of geometric homotopy in cohesive (,1)(\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 (,1)(\infty,1)-toposes.

For that purposes consider the following


Define the quasi-categories

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


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

where on the right we have the standard model structure on topological spaces Top QuillenTop_{Quillen} and the standard model structure on simplicial sets sSet QuillensSet_{Quillen} and N(() )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.


(||Sing):Top QuillenSing||sSet 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||Sing):TopSing𝕃||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 ||{|-|} and SingSing 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


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

(ΠDiscΓcoDisc):Top𝕃||Sing𝕃||SingGrpd.. (\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

Π(X)SingX. \Pi(X) \simeq \mathbb{R} Sing X \,.

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.


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 \infty-groupoid, the topology only serves to allow the construction of SingXSing X. For discussion of \infty-groupoids equipped with genuine topological cohesion see Euclidean-topological ∞-groupoid.

Cohomology and principal \infty-bundles

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


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

A classical reference is section 17 of May.


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

The category sSet 0sSet\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 sSet Quillen\mathrm{sSet}_{\mathrm{Quillen}}.

There is a Quillen equivalence

(GW¯):sGrpW¯GsSet 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 \infty-categories

(ΩB):GrpdBΩGrpd 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 GG a simplicial group, W¯GsSet\bar W G \in \mathrm{sSet} is a model of the classifying delooping object BG\mathbf{B}G for discrte GG-principal ∞-bundles. The following statements assert that these principal \infty-bundles themselves can be modeled as ordinary simplicial principal bundles


For GG a simplicial group and W¯G\bar W G the model for BG\mathbf{B}G given by the above proposition, write

WGW¯G W G \to \bar W G

for the simplicial decalage on W¯G\bar W G.

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


The morphism WGW¯GW G \to \bar W G is a Kan fibration resolution of the point inclusion *W¯G{*} \to \bar W G.

This follows directly from the characterization of WGW¯GW G \to \bar 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 WGW G.


For GG a simplicial group, the sequence of simplicial sets

GWGW¯G G \to W G \to \bar W G

is a presentation of the fiber sequence

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

Hence WGW¯GW G \to \bar W G is a model for the universal GG-principal discrete \infty-bundle (see universal principal ∞-bundle):

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

P * X BG \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

P WG X W¯G. \array{ P &\to& W G \\ \downarrow && \downarrow \\ X &\to& \bar W G } \,.

The explicit statement that the sequence GWGW¯GG \to W G \to \bar W G is a model for the looping fiber sequence appears on p. 239 of Crossed Menagerie . The universality of WGW¯GW G \to \bar W G for GG-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 \infty-bundles by simplicial principal bundles lifts to a model of topological principal \infty-bundles by simplicial topological bundles principal over simplicial topological groups.

geometries of physics

A\phantom{A}(higher) geometryA\phantom{A}A\phantom{A}siteA\phantom{A}A\phantom{A}sheaf toposA\phantom{A}A\phantom{A}∞-sheaf ∞-toposA\phantom{A}
A\phantom{A}discrete geometryA\phantom{A}A\phantom{A}PointA\phantom{A}A\phantom{A}SetA\phantom{A}A\phantom{A}Discrete∞GrpdA\phantom{A}
A\phantom{A}differential geometryA\phantom{A}A\phantom{A}CartSpA\phantom{A}A\phantom{A}SmoothSetA\phantom{A}A\phantom{A}Smooth∞GrpdA\phantom{A}
A\phantom{A}formal geometryA\phantom{A}A\phantom{A}FormalCartSpA\phantom{A}A\phantom{A}FormalSmoothSetA\phantom{A}A\phantom{A}FormalSmooth∞GrpdA\phantom{A}


Simplicial groups and simplicial principal bundles are discussed in

and section V of

The relation of WGW¯GW G \to \bar 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

where much of the above material is taken from.

Last revised on June 25, 2018 at 13:00:32. See the history of this page for a list of all contributions to it.