An -anafunctor is a model for a morphism in an (∞,1)-category 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.
of presheaves. This is an -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 -anafunctor is a span
of whose left leg is an acyclic fibration with
Given two -anafunctors and their composite -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 -functor on the left is indeed again an acyclic fibration and hence the total span here indeed again an -anafunctor.
While this definition of composition is in itself well defined, it is not quite associative: the two ways of composing three spans representing -anafunctors this way in general produces two spans out of different – albeit isomorphic – hypercovers.
Of course the precise choice of hypercover of is not an essential datum: two -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 -stacks with values in (n,r)-categories.
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 to contain only “elementary test objects” so that 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.
where is the universal principal ∞-bundle, itself defined by the pullback
The generalization of this statement to -anafunctors is given in (BlohmannZhu).