Coherence for weak units

We define weak units in a semi-monoidal 2-category C as cancellable pseudo-idempotents: they are pairs (I, α) where I is an object such that tensoring with I from either side constitutes a biequivalence of C , and α : I ⊗ I → I is an equivalence in C . We show that this notion of weak unit has coherence built in: Theorem A: α has a canonical associator 2-cell, which automatically satisfies the pentagon equation. Theorem B: every morphism of weak units is automatically compatible with those associators. Theorem C: the 2-category of weak units is contractible if non-empty. Finally we show (Theorem E) that the notion of weak unit is equivalent to the notion obtained from the definition of tricategory: α alone induces the whole family of left and right maps (indexed by the objects), as well as the whole family of Kelly 2-cells (one for each pair of objects), satisfying the relevant coherence axioms. Introduction The notion of tricategory, introduced by Gordon, Power, and Street [2] in 1995, seems still to represent the highest-dimensional explicit weak categorical structure that can be manipulated by hand (i.e. without methods of homotopy theory), and is therefore an important test bed for higher-categorical ideas. In this work we investigate the nature of weak units at this level. While coherence for weak associativity is rather well understood, thanks to the geometrical insight provided by the Stasheff associahedra [12], coherence for unit structures is more mysterious, and so far there seems to be no clear geometric pattern for the coherence laws for units in higher dimensions. Specific interest in weak units stems from Simpson’s conjecture [11], according to which strict n-groupoids with weak units should model all homotopy n-types. In the present paper, working in the setting of a strict 2-category C with a strict tensor product, we define a notion of weak unit by simple axioms that involve only the notion of equivalence, and hence in principle make sense in all dimensions. Briefly, a weak unit is a cancellable pseudo-idempotent. We work out the basic theory of such units, and compare with the notion extracted from the definition of tricategory. In the companion paper Weak units and homotopy 3-types [4] we employ this notion of unit to prove a version of Simpson’s conjecture for 1-connected homotopy 3-types, which is the first nontrivial case. The strictness assumptions of the present paper should be justified by that result. By cancellable pseudo-idempotent we mean a pair (I, α) where I is an object in C such that tensoring with I from either side is an equivalence of 2-categories, and α : I⊗ I ∼ → I is an equi-arrow (i.e. an arrow admitting a pseudo-inverse). The remarkable fact units.tex 2009-07-18 09:24 [2/37] about this definition is that α, viewed as a multiplication map, comes with canonical higher order data built in: it possesses a canonical associator A which automatically satisfies the pentagon equation. This is our Theorem A. The point is that the arrow α alone, thanks to the cancellability of I, induces all the usual structure of left and right constraints with all the 2-cell data that goes into them and the axioms they must satisfy. As a warm-up to the various constructions and ideas, we start out in Section 1 by briefly running through the corresponding theory for cancellable-idempotent units in monoidal 1-categories. This theory has been treated in detail in [8]. The rest of the paper is dedicated to the case of monoidal 2-categories. In Section 2 we give the definitions and state the main results: Theorem A says that there is a canonical associator 2-cell for α, and that this 2-cell automatically satisfies the pentagon equation. Theorem B states that unit morphisms automatically are compatible with the associators of Theorem A. Theorem C states that the 2-category of units is contractible if non-empty. Hence, ‘being unital’ is, up to homotopy, a property rather than a structure. Next follow three sections dedicated to proofs of each of these three theorems. In Section 3 we show how the map α : I I ∼ → I alone induces left and right constraints, which in turn are used to construct the associator and establish the pentagon equation. The left and right constraints are not canonical, but surprisingly the associator does not depend on the choice of them. In Section 4 we prove Theorem B by interpreting it as a statement about units in the 2-category of arrows, where it is possible to derive it from Theorem A. In Section 5 we prove Theorem C. The key ingredient is to use the left and right constraints to link up all the units, and to show that the unit morphisms are precisely those compatible with the left and right constraints; this makes them ‘essentially unique’ in the required sense. In Section 6 we go through the basic theory of classical units (i.e. as extracted from the definition of tricategory [2]). Finally, in Section 7 we show that the two notions of unit are equivalent. This is our Theorem E. A curiosity implied by the arguments in this section is that the left and right axioms for the 2-cell data in the Gordon-PowerStreet definition (denoted TA2 and TA3 in [2]) imply each other. (We have no Theorem D.) This notion of weak units as cancellable idempotents is precisely what can be extracted from themore abstract, Tamsamani-style, theory of fair n-categories [7] bymaking an arbitrary choice of a fixed weak unit. In the theory of fair categories, the key object is a contractible space of all weak units, rather than any particular point in that space, and handling this space as a whole bypasses coherence issues. However, for the sake of understanding what the theory entails, and for the sake of concrete computations, it is interesting to make a choice and study the ensuing coherence issues, as we do in this paper. The resulting approach is very much in the spirit of the classical theory of monoidal categories, bicategories, and tricategories, and provides some new insight to these theories. To stress this fact we have chosen to formulate everything from scratch in such classical terms, without reference to the theory of fair categories. units.tex 2009-07-18 09:24 [3/37] In the case of monoidal 1-categories, the cancellable-idempotent viewpoint on units goes back to Saavedra [10]. The importance of this viewpoint in higher categories was first suggested by Simpson [11], in connection with his weak-unit conjecture. He gave an ad hoc definition in this style, as a mere indication of what needed to be done, and raised the question of whether higher homotopical data would have to be specified. The surprising answer is, at least here in dimension 3, that specifying α is enough, then the higher homotopical data is automatically built in. This paper was essentially written in 2004, in parallel with [4]. We are ourselves to blame for the delay of getting it out of the door. The present form of the paper represents only half of what was originally planned to go into the paper. The second half should contain an analysis of strong monoidal functors (along the lines of what wasmeanwhile treated just in the 1-dimensional case [8]), and also a construction of the ‘universal unit’, hinted at in [7]. We regret that these ambitions should hold back the present material for so long, and have finally decided to make this first part available as is, in the belief that it is already of some interest and can well stand alone. Acknowledgements. We thank Georges Maltsiniotis for pointing out to us that the cancellable-idempotent notion of unit in dimension 1 goes back to Saavedra [10], and we thank Josep Elgueta for catching an error in an earlier version of our comparison with tricategories. The first-named author was supported by the NSERC. The secondnamed author was very happy to be a CIRGET postdoc at the UQAM in 2004, and currently holds support from grants MTM2006-11391 and MTM2007-63277 of Spain. 1 Units in monoidal categories It is helpful first briefly to recall the relevant results for monoidal categories, referring the reader to [8] for further details of this case. 1.1. Semi-monoidal categories. A semi-monoidal category is a category C equipped with a tensor product (which we denote by plain juxtaposition), i.e. an associative functor C × C −→ C (X,Y) 7−→ XY. For simplicity we assume strict associativity, X(YZ) = (XY)Z. 1.2. Monoidal categories. (Mac Lane [9].) A semi-monoidal category C is a monoidal category when it is furthermore equipped with a distinguished object I and natural isomorphisms IX λX ≻ X ≺ ρX XI units.tex 2009-07-18 09:24 [4/37] obeying the following rules (cf. [9]): λI = ρI (1) λXY = λXY (2) ρXY = XρY (3) XλY = ρXY (4) Naturality of λ and ρ implies λIX = IλX, ρXI = ρX I, (5) independently of Axioms (1)–(4). 1.3 Remark. Tensoring with I from either side is an equivalence of categories. 1.4 Lemma. (Kelly [5].) Axiom (4) implies axioms (1), (2), and (3). Proof. (4) implies (2): Since tensoring with I on the left is an equivalence, it is enough to prove IλXY = IλXY. But this follows from Axiom (4) applied twice (swap λ out for a ρ and swap back again only on the nearest factor): IλXY = ρIXY = IλXY. Similarly for ρ, establishing (3). (4) and (2) implies (1): Since tensoring with I on the right is an equivalence, it is enough to prove λI I = ρI I. But this follows from (2), (5), and (4): λI I = λI I = IλI = ρI I. 2 The following alternative notion of unit object goes back to Saavedra [10]. A thorough treatment of the notion was given in [8]. 1.5. Units as cancellable pseudo-idempotents. An object I in a semi-monoidal category C is called cancellable if the two functors C → C X 7−→ IX X 7−→ XI are fully faithful. By definition, a pseudo-idempotent is an object I equipped with an isomorphism α : I I ∼ → I. Finally we define a unit object in C to be a cancellable pseudoidempotent. 1.6 Lemma. [8] Given a unit object (I, α) in a semi-monoidal category C , for each object X there are unique arrows IX λX ≻ X ≺ ρX XI such that (L) IλX = αX (R) ρX I = Xα. The λX and ρX are isomorphisms and natural in X. units.tex 2009-07-18 09:24 [5/37] Proof. Let L : C → C denote the functor defined by tensoring with I on the left. Since L is fully faithful, we have a bijection Hom(IX,X) → Hom(I IX, IX). Now take λX to be the inverse image of αX; it is an isomorphism since αX is. Naturality follows by considering more generally the bijection Nat(L, idC ) → Nat(L ◦ L,L); let λ be the inverse image of the natural transformation whose components are αX. Similarly on the right. 2 1.7 Lemma. [8] For λ and ρ as above, the Kelly axiom (4) holds: XλY = ρXY. Therefore, by Lemma 1.6 a semi-monoidal category with a unit object is a monoidal category in the classical sense. Proof. In the commutative square XIIY XIλY ≻ XIY