Weak units and homotopy 3 - types

We show that every braided monoidal category arises as End(I) for a weak unit I in an otherwise completely strict monoidal 2-category. This implies a version of Simpson’s weak-unit conjecture in dimension 3, namely that one-object 3-groupoids that are strict in all respects, except that the object has only weak identity arrows, can model all connected, simply connected homotopy 3-types. The proof has a clear intuitive content and relies on a geometrical argument with string diagrams and configuration spaces. 0 Introduction The subtleties and challenges of higher category theory start with the observation (in fact, not a trivial result) that not every weak 3-category is equivalent to a strict 3category. The topological counterpart of this is that not every homotopy 3-type can be realised by a strict 3-groupoid. The discrepancy between the strict and weak worlds can be pinpointed down to the case of connected, simply-connected 3-types, where it can be observed rather explicitly: such 3-types correspond to braided monoidal categories (in fact braided categorical groups), while connected, simply-connected strict 3-categories are essentially commutative monoidal categories — the braiding is forced to collapse, as a consequence of the Eckmann-Hilton argument. In precise terms, strict n-groupoids can realise only homotopy n-types with trivial Whitehead brackets. This collapse can be circumvented by weakening the structures. The notion of tricategory of Gordon, Power, and Street [1] is meant to be the weakest possible definition of 3-category. They show that a tricategory with only one object is equivalent to a Gray monoid, and in particular, a tricategory with one object and one arrow is equivalent to a braided monoidal category. Furthermore, every braided monoidal category arises in this way. The most general result relating higher categories to homotopy types is Tamsamani’s theorem [8], that weak n-groupoids (in the sense of Tamsamani) can realise all homotopy n-types. This result was conjectured by Grothendieck [2], or rather: it was stated as a desideratum for a to-be theory of weak higher categories. In Tamsamani’s theory, and in most other theories of higher categories, the essential weakening bears on the composition laws and their interchange laws. However, a careful analysis of the situation in strict 3-groupoids led Simpson [7] to observe that the collapse of the braiding, via the Eckmann-Hilton argument, can be traced back to the strictness of the identity arrows. He conjectured that (a suitable notion of) strict Joyal & Kock: Weak units and homotopy 3-types [2/20] n-groupoids with weak identity arrows should realise all homotopy n-types, and furthermore that the homotopy category of such n-groupoids should be equivalent to the homotopy category of Tamsamani n-groupoids. (In fact he went further and conjectured that the same homotopy equivalence should hold in the non-invertible case, i.e. for n-categories, not just for n-groupoids.) An ad hoc notion of weak identity arrows was sketched, but the details were not worked out, and it was acknowledged that it might not be the correct notion to fulfil the conjectures. Simpson’s conjectures are highly surprising: they go against all trends in higher category theory, where the emphasis is mostly on the composition laws, and questions about identity arrows are often swept under the carpet. A consequence of the conjectures is that every weak n-category should be equivalent to one with strict composition laws and strict interchange laws! In this workwe prove a version of Simpson’s conjecture in the crucial case of dimension 3. We restrict ourselves to the connected, simply-connected case, working with strict monoidal 2-categories with weak units. The basics of weak units in monoidal 2-categories is worked out in a companion paper [3], but in fact very little is needed in our proof. Our key result is this: Main Theorem. Let I be a weak unit of an otherwise completely strict monoidal 2-category. Then End(I) is a braided monoidal category, and every braided monoidal category arises in this way. Connected, simply-connected homotopy 3-types correspond to braided categorical groups. Under the correspondence of the Main Theorem, these correspond to strict 2-groupoids with invertible tensor product and weak units, which in turn can be regarded as one-object 3-groupoids. Hence we get the following version of Simpson’s conjecture in dimension 3: Main Corollary. One-object 3-groupoids that are strict in all respects, except that the object has only weak identity arrows, can model all connected, simply connected homotopy 3-types. The paper is organised as follows. In Section 1 we show that End(I) is braided, and explain the geometry of this braiding. In Section 2 we introduce the geometric language of train track diagrams and show that the space of all train track diagrams is acyclic. Finally in Section 3, given a braided monoidal category B, we use a geometrical construction to get a monoidal 2-category with weak unit I such that End(I) is equivalent to B. 1 From weak unit to braiding 1.1 Semi-monoidal 2-categories. A strict semimonoidal 2-category (or a 2-category with strict multiplication) is a (strict) 2-category C equipped with a strictly associative multiplication functor ⊗ : C × C → C . We write the tensor product by plain juxtaposition: (X,Y) 7→ XY. We use the symbol # to denote composition of arrows, written from the left to the right, writing for example f # g for the composite fgJoyal & Kock: Weak units and homotopy 3-types [3/20] and we use the same symbol for ‘horizontal’ composition of 2-cells. For ‘vertical’ composition of 2-cells we decorate the symbol with the locus along which the 2-cells are pasted together, writing for example U # g V for the composite 2-cell