structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
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 $(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : \infty Grpd \to \infty Grpd$ 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
the inverse image $Disc$ of the global section functor is a full and faithful (∞,1)-functor and hence embeds ∞Grpd as a full sub-(∞,1)-category of $\mathbf{H}$. A general object in $\mathbf{H}$ is a cohesive $\infty$-groupoid . We say $X \in \mathbf{H}$ is a discrete $\infty$-groupoid if it is in the image of $Disc$.
This generalizes the traditional use of the terms discrete space and discrete group:
a discrete space is equivalently a 0-truncated discrete $\infty$-groupoid;
a discrete group is equivalently a 0-truncated group object in discrete $\infty$-groupoids.
We discuss now some of the general abstract structures in a cohesive (∞,1)-topos realized in discrete $\infty$-groupoids.
We discuss the general absatract notion of geometric homotopy in cohesive $(\infty,1)$-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 $(\infty,1)$-toposes.
For that purposes consider the following
Define the quasi-categories
and
where on the right we have the standard model structure on topological spaces $Top_{Quillen}$ and the standard model structure on simplicial sets $sSet_{Quillen}$ 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
the standard Quillen equivalence of the homotopy hypothesis-theorem given by the singular simplicial complex-functor and geometric realization, write
for the corresponding derived functors (the image under the homotopy coherent nerve of the restriction of ${|-|}$ and $Sing$ 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
Top is exhibited a cohesive (∞,1)-topos over ∞Grpd by setting
In particular a presentation of the intrinsic fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos is given by the familiar singular simplicial complex construction
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 $Sing X$. For discussion of $\infty$-groupoids equipped with genuine topological cohesion see Euclidean-topological ∞-groupoid.
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 $\mathrm{sGrp} = \mathrm{Grp}(\mathrm{sSet})$ for the category of simplicial groups.
A classical reference is section 17 of May.
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
which presents the abstract looping and delooping equivalence of $\infty$-categories
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, $\bar W G \in \mathrm{sSet}$ is a model of the classifying delooping object $\mathbf{B}G$ for discrte $G$-principal ∞-bundles. The following statements assert that these principal $\infty$-bundles themselves can be modeled as ordinary simplicial principal bundles
For $G$ a simplicial group and $\bar W G$ the model for $\mathbf{B}G$ given by the above proposition, write
for the simplicial decalage on $\bar W G$.
This characterization of the object going by the classical name $W G$ is made fairly explicit in (Duskin, p. 85).
The morphism $W G \to \bar W G$ is a Kan fibration resolution of the point inclusion ${*} \to \bar W G$.
This follows directly from the characterization of $W G \to \bar W G$ by decalage. Pieces of this statement appear in (May): lemma 18.2 there gives the fibration property, prop. 21.5 the contractibility of $W G$.
For $G$ a simplicial group, the sequence of simplicial sets
is a presentation of the fiber sequence
Hence $W G \to \bar W G$ is a model for the universal $G$-principal discrete $\infty$-bundle (see universal principal ∞-bundle):
every $G$-principal discrete $\infty$-bundle $P \to X$ in $\infty \mathrm{Grpd}$, which by definition is a homotopy fiber
in ∞Gpd, is presented in the standard model structure on simplicial sets by the ordinary pullback
The explicit statement that the sequence $G \to W G \to \bar W G$ is a model for the looping fiber sequence appears on p. 239 of Crossed Menagerie . The universality of $W G \to \bar W G$ 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 $\infty$-bundles by simplicial principal bundles lifts to a model of topological principal $\infty$-bundles by simplicial topological bundles principal over simplicial topological groups.
discrete ∞-groupoid
Simplicial groups and simplicial principal bundles are discussed in
and section V of
The relation of $W G \to \bar W G$ to decalage is mentioned on p. 85 of
Discrete cohesion is the topic of section 3.1 of
where much of the above material is taken from.