homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
An $\infty$-anafunctor is a model for a morphism in an (∞,1)-category $\mathbf{H}$ of structured (n,r)-categories notably of ∞-stacks with values in (n,r)-categories in terms of spans of ordinary morphisms (of presheaves, notably). It generalizes the notion of 1-anafunctor.
The category with weak equivalences that models the $(n,r)$-categories in questions should be refined to the structure of a category of fibrant objects for $\infty$-anafunctors to be a good model
In such a model $[C^{op},(n,r)Cat]$ for $\mathbf{H}$, a morphism in $\mathbf{H}$ between $X$ to $Y$ is presented by a span
of presheaves. This is an $\infty$-anafunctor, generalizing the notion of anafunctor.
The point is that this is a special simple form of the more general and more complicated constructions in simplicial localization, which require zig-zags of morphisms of greater length than a single span.
An $\infty$-anafunctor $g : C \to D$ is a span
of whose left leg is an acyclic fibration with
Given two $\infty$-anafunctors $C \stackrel{g}{\to} D$ and $D \stackrel{h}{\to} E$ their composite $\infty$-anafunctor is given by the span
where the top left square is a pullback square. Notice that acyclic fibrations are preserved under pullback and closed under composition, so that the total vertical $\omega$-functor on the left is indeed again an acyclic fibration and hence the total span here indeed again an $\infty$-anafunctor.
While this definition of composition is in itself well defined, it is not quite associative: the two ways of composing three spans representing $\infty$-anafunctors this way in general produces two spans out of different – albeit isomorphic – hypercovers.
Of course the precise choice of hypercover $\hat C$ of $C$ is not an essential datum: two $\infty$-anafuntors should be regarded as equivalent if they become equal after pulled back to a joint hypercover. This is formalized in the following definition:
(…)
Any of the algebraic models for higher categories provide categories of fibrant objects, such as the folk model structures.
But also thopse simplicial sheaves that are stalkwise Kan complexes works, and notably the full subcategories on fibrant objects in one of the model structure on simplicial presheaves and more generally of model structures on homotopical presheaves model arbitrary $\infty$-stacks with values in (n,r)-categories.
For $G \in [C^{op}, sSet]$ a model in simplicial presheaves for an ∞-group in $\mathbf{H}$ and $\mathbf{B}G \in [C^{op}, sSet]$ its fibrant delooping object, an $\infty$-anafunctor
is a cocycle for a $G$-principal ∞-bundle. Here $\hat X \to X$ is essentially the hypercover on which the cocycle is defined.
More precisely, in order for this to be a useful cocycle presentation we either use the structure of a category of fibrant objects given by stalk-wise fibrant simplicial presheaves as in (Brown), or if we use the full subcategory on fibrant objects of a local model structure on simplicial presheaves then we should take the site $C$ to contain only “elementary test objects” so that $\mathbf{B}G$ has a chance of being fibrant. This issue of big versus small sites and how this shifts the amount of fibrant and cofibrant objects around is discussed in some detail at ∞-Lie groupoid in the context of sites of smooth test spaces.
An introductory exposition to how such spans are Cech cohomology cocycles see for instance ∞-Chern-Weil-theory introduction.
The principal ∞-bundle classified by such an $\infty$-anafuntor is the (ordinary!) pullback
where $\mathbf{E}G$ is the universal principal ∞-bundle, itself defined by the pullback
Ordinary anafunctors of Lie groupoids can equivalently be modeled by groupoid principal bundles that are bibundles.
The generalization of this statement to $\infty$-anafunctors is given in (BlohmannZhu).
For $(\infty,0)$-anafunctors modeling morphisms in a hypercomplete (∞,1)-topos this is in
A result relating $(\infty,0)$-anafunctors to ∞-groupoid-bibundles has been announced by