A bigroupoid’s topology
(or, Topologising the homotopy bigroupoid of a space)
The fundamental bigroupoid of a topological space is one way of capturing its homotopy 2-type. When the space is semilocally 2-connected, one can lift the construction to a bigroupoid internal to the category of topological spaces, as Brown and Danesh-Naruie lifted the fundamental groupoid to a topological groupoid. For locally relatively contractible spaces the resulting topological bigroupoid is locally trivial in a way analogous to the case of the topologised fundamental groupoid.
Key words and phrases:fundamental bigroupoid, homotopy 2-type, topological bigroupoid
2010 Mathematics Subject Classification:18D05; 22A22; 55Q05
\cczero This article is released under a CC0 license, https://creativecommons.org/publicdomain/zero/1.0/
One of the standard examples of a groupoid is the fundamental groupoid of a topological space , generalising the fundamental group at a basepoint to consider ‘all basepoints at once’. In [Brown-Danesh-Naruie_75] Brown and Danesh-Naruie showed that, under a mild assumption, the fundamental groupoid can be given the structure of a topological groupoid. That is, the sets of objects and arrows—points in the space and homotopy classes of paths, respectively—can be given topologies such that all the maps that make up the groupoid (source, target, composition etc) are continuous.
The mild assumption mentioned in the previous paragraph is exactly that which guarantees the existence of a universal covering space; a seemingly little-known fact is that said covering space can be constructed directly from the topologised fundamental groupoid as given in [Brown-Danesh-Naruie_75]. Moreover, this construction is formally analogous to the construction of the first stage of the Whitehead tower of a topological space.
Drawing inspiration from the celebrated Homotopy Hypothesis linking higher groupoids and homotopy types, we see that to extend these constructions to dimension 2 we need to consider some form of 2-dimensional groupoid. While there are several different algebraic models that completely capture the homotopy 2-type of a space, such as crossed modules (Whitehead, 1940s) and double groupoids (Brown–Higgins 1970s), here we choose to consider bigroupoids; Stevenson [Danny_phd] and Hardie–Kamps–Kieboom [HKK_01] constructed a fundamental bigroupoid of a space . The idea of such an object, albeit in the fully general case of weak -groupoids representing arbitrary homotopy -types, seems to go back to Grothendieck’s 1975 letters to Breen [Grothendieck_75].
The idea of a bigroupoid is illustrated nicely by considering this special case. Firstly, bigroupoids have object and arrows, as groupoids do, but also 2-arrows, which are arrows between arrows. Objects of are points in and arrows are paths . Paths can be composed, but since at this point there is no quotient by the relation of homotopy, such composition is not associative. Similar issues arise when composing by constant paths, or reverse paths, representing identity arrows and inverses repectively. This is where the 2-arrows come in: 2-arrows in are homotopy classes of homotopies of paths. Or, equivalently, homotopy classes of bigons, which are certain maps . As maps support pasting in two directions, we get the horizontal composition of bigons end-to-end (inducing composition on their boundary paths) and the vertical composition of bigons pasted along one of their boundary paths.
This article will give, under a mild local condition, topologies for all the sets involved in —points in , paths in and homotopy classes of bigons—such that every operation in the bigroupoid structure is continuous. One of the reasons that a strict model is not chosen is that they do not seem well-adapted to the application that motivated the present author, namely constructing geometrically and in a smooth fashion the second stage in the Whitehead tower of a manifold. Double groupoids and crossed modules over groupoids, both championed by Ronnie Brown, seem to work best in the context of computing with topologically discrete algebraic structures (see however the concluding remarks in section LABEL:sec:local_triv). Likewise the homotopy 2-groupoid of a Hausdorff space given in [HKK_00] uses thin homotopy classes which does not lead to a well-behaved space of arrows111Even worse, in the smooth setting, one does not even have a half-decent manifold structure on the set of thin homotopy classes of loops, see [Stacey_10]..
The approach of the paper is that one can in fact take the given topologies on the sets of objects and arrows of , namely the topology on and the compact-open topology on ; the main novelty is to define a very particular basis for the topology on the set of homotopy classes of bigons so that one can prove the required continuity of structure maps involving 2-arrows. This uses in an essential way the local assumptions on . The paper finishes by showing that , with the topology we define, satisfies analogues of the local triviality222Local triviality of topological groupoids is a condition, introduced by Ehresmann [Ehresmann_59], that relates them with locally trivial principal bundles. and discreteness properties that the topological groupoid has.
Extending these results further up the ladder of higher groupoids needs to take a different approach, because even weak 3-groupoids—strict 3-groupoids are known to be insufficient—are quite complicated. After that, the explicit algebraic definitions are no longer practical if one wants to capture the full homotopy type. One could consider however other models for higher groupoids, such as operadic definitions of weak -groupoids; the approach of Trimble [Trimble_99] seems like it may be appropriate, given the approach of the sequel [Roberts_15b] to this paper. The analogue of the results in the current paper would be that, under suitable local connectivity assumptions, the algebras for the operads involved in the definitions would be topological, i.e. algebras in the category of spaces rather than in the category of sets. Alternatively one might use Kan complexes with certain unique filler conditions and then consider internal Kan complexes in , or even simplicial sheaves on , as models for higher topological groupoids.
In [Bakovic_phd] Bakovic gives a recipe, partly building on [Roberts-Schreiber_08], for taking an internal bigroupoid (for instance in topological spaces) and giving a principal 2-bundle. Topological bigroupoids with non-discrete object space do not seem to be very common, so this paper gives at least one family of examples for Bakovic’s general machinery. In fact the resulting principal 2-bundle is the desired second stage in the Whitehead tower, as was constructed in the author’s thesis [Roberts_phd].
Thanks are due to several anonymous referees who helped beat this article into shape over several iterations, and to Tim Porter for both inviting its submission to this volume and his subsequent patience. Thanks also to Ronnie Brown, whose lovely book [RBrown_groupoids] on groupoids and topology was influential in my thesis work (of which this paper formed a small part) in ways that are not apparent to the casual observer: Happy Birthday Ronnie!
2. Topological groupoids and bigroupoids
Recall that a topological groupoid is a groupoid with a space of objects and a space of arrows such that all the structure maps are continuous. Functors between topological groupoids and consist of continuous maps and commuting with all the groupoid structure. The reason that we do not use the term ‘continuous functor’ here is that this has a separate meaning for functors unrelated to topology. The category of topological groupoids will be denoted by .
Recall that there is a full inclusion , sending a topological space to the topological groupoid , with arrows and objects both given by with all structure maps the identity. All ‘spaces’ will be topological spaces in what follows, unless otherwise specified.
To describe the topological fundamental bigroupoid of a space , we first need to define topological bigroupoids. Such a thing may be defined using the full diagrammatic definition of an internal bicategory in as in Bénabou’s [Benabou], which gives all the structure maps and spaces explicitly together with many commuting diagrams. We will adopt instead a more compact approach. For those familiar with such things, the definition below is the internal analogue of weak enrichment in groupoids. For the uninitiated, one can think of the definition of a (topological) bigroupoid as being a generalisation of the following reworking of the definition of groupoid.
An ordinary groupoid is given by a set together with a family of sets , the set of arrows from the object to the object . One should think of this as a set parameterised by , or in other words, a set over , written . Composition is given by a function
respecting the maps down to . Here the pullback is , considered as a set over via . Associativity can be enforced by asking that a certain diagram in sets over commutes. Likewise, inversion in the groupoid is an endomorphism of covering the swap map on , and if is considered as a set over by the diagonal map, then the function assinging identity arrows is the map .
Moving to bigroupoids, the hom-sets are replaced by hom-groupoids, as can be seen by considering the case of . For two fixed objects and —points in —we have a set of paths from to , and a set of (homotopy classes of) bigons with vertices and , and such bigons can be pasted vertically along a common edge. This, together with degenerate bigons and reversal of orientation gives a groupoid. Now, allowing and to vary we see that what we have is a family of groupoids parameterised by , or, in other words, a groupoid equipped with a functor to . Horizontal composition can then be encoded by a functor, and this composition is now not associative. The commuting diagram of functions between sets that encodes associativity is now a diagram of functors between groupoids and only commutes up to a natural isomorphism, which of course needs to satisfy coherence conditions. A generalisation of this approach was used by Trimble [Trimble_99], for instance, to define a general notion of weak higher groupoid.
The definition of topological bigroupoid takes this idea of a family of hom-groupoids and replaces it by a continuous family of topological groupoids over the space , or in other words, a functor between topological groupoids. This definition can be unpacked to recover the standard definition of a bigroupoid, but would take up a fair amount of space.333For the sake of consiseness, any pullbacks or iterated pullbacks over the space will follow the following convention: letting or , pullbacks of the form use the functor or its object component, and pullbacks of the form use the map or its object component. In the following definition is a stand-in for when necessary to save space.
A topological bigroupoid is a topological space (the space of objects) and a topological groupoid (the hom-groupoid, with source and target maps denoted , respectively) equipped with a functor , together with:
(composition and identity, respectively) over and a functor
(inverse) covering the swap map for ;
letting and , continuous maps
that are the component maps of natural isomorphisms