Ideas from my thesis will be seen here.
There were two motivating ideas for the thesis, one abstract and one concrete:
These run quickly into Grothendieck’s ideas from Pursuing stacks, albeit in a low-dimensional case.
The intersection of the two ideas is this: construct a suitably functorial 2-connected covering ‘space’, where now ‘spaces’ include topological groupoids. G. W. Whitehead gave us a construction of a 2-connected cover of a space, but it requires one to choose representatives of homotopy classes of generators of homotopy groups. We have already seen echoes of the desired construction in various descriptions of the universal central extension of the loop group (Mickelsson, Murray, Murray-Stevenson) and a 2-group model of the string group (Baez-Crans-Stevenson-Schreiber) (which is really a 3-connected cover…).
A quick wrap-up of the end product (missing all the exciting bits about bicategories) is in my talk linked on my HomePage.
My own viewpoint on anafunctors (see also the nLab page) between internal categories is this: that they are a way to restore to the status of equivalences those internal functors which are fully faithful and ‘essentially surjective’. This last condition needs a little bit more machinery than a general ambient category supplies, and so one has to let the ambient category be a site, with the coverage satisfying some conditions.
There is a close relation to the Everaert-Kieboom-van der Linden model structure on the category of internal categories in some site, using enriched model categories (since, of course, the category of internal categories is enriched over Cat), but this is outside my own expertise.
It is common to see results such as ‘such and such (bi)category of stacks is equivalent to internal groupoids with some morphisms inverted’, usually using the phrases Morita equivalence, Hilsum-Skandalis morphism, generalised morphism, essential equivalence and so on. These are all special cases of the following result
Let be a category equipped with a class of admissible maps . Then the 2-catgeory admits a bicategory of fractions inverting all -equivalences.
The class of admissible maps acts as the analogue of surjective maps in Set, and -equivalences are fully faithful functors such that the ‘usual arrow’ expressing essential surjectivity is in . An example of a class of admissible maps is those arrows split locally with respect to some coverage on , called -epimorphisms. Indeed, it follows from the definition of admissible maps that they themselves form a pretopology, but not necessarily a subcanonical one.
If there exists a subcanonical singleton pretopology on such that is precisely the class of -epimorphisms, then one can calculate this bicategory of fractions using anafunctors. It is important to note that different pretopologies can give rise to the same class of admissible maps, but they give rise to equivalent bicategories of anafunctors. Thus one can ‘tune’ anafunctors to the problem at hand.
In the thesis we will really only be concerned with Top, and possibly Manifolds. In the last case, Frechet manifolds will no doubt appear briefly.
The fundamental bigroupoid of a space was defined by Hardie-Kamps-Kieboom and is basically what one expects: objects are points of the space, 1-arrows are paths and 2-arrows are homotopy classes of relative homotopies between paths. This bigroupoid encodes the 1-type of that space (in the case that we have a Hausdorff space, HKK give a strictification based on quotienting out thin homotopies).
The fundamental bigroupoid of a topological groupoid is, formally, exactly the same: objects are points (i.e. functors ), 1-arrows are paths and 2-arrows are homotopies of paths (up to homotopy). Since the “real” 2-category of topological groupoids is localised (i.e. is some category of anafunctors), paths are actually spans (anafunctors) with domain and codomain . The general idea is to work with the pretopology of open covers, and since open covers of the interval is a bit of an overkill, these are replaced by partitions, which are cofinal in open covers (can be viewed as finite pro-open covers). Thus paths are paths-with-jumps, that is, finite unions of paths in the object space with arrows between successive endpoints.
So far so good. This will reproduce the paths considered by others (Haefliger, Moerdijk-Mrcun, Colman, Schreiber-Waldorf and so on) in defining a fundamental groupoid of a topological groupoid (usually in some special case, like Lie groupoids or foliation groupoids). Now instead of just postulating existence of some equivalence of paths, we need to consider them explicitly.
Just as paths were “anafunctors” with domain , homotopies will be “anafunctors” with domain . Now, we need to decide what will replace partitions as we go up dimensions. One could triangulate, cubify, cover by disks and so on. In an earlier version of this work, I had closed covers by polygons such that vertices of this cover were at most trivalent. This turned out to be difficult for constructing homotopies inductively, so now we let the covers of be cubical.