Theory and Applications of Categories, Vol. 18, No. 1, 2007, pp. 1–3.

BICAT IS NOT TRIEQUIVALENT TO GRAY

STEPHEN LACK

Abstract. Bicat is the tricategory of bicategories, homomorphisms, pseudonatural transformations, and modifications. Gray is the subtricategory of 2-categories, 2functors, pseudonatural transformations, and modifications. We show that these two tricategories are not triequivalent.

1. Background. Weakening the notion of 2-category by replacing all equations between 1-cells by suitably coherent isomorphisms gives the notion of bicategory [1]. The analogous weakening of a 2-functor is called a homomorphism of bicategories, and the weakening of a 2-natural transformation is a pseudonatural transformation. There are also modifications between 2-natural or pseudonatural transformations, but this notion does not need to be weakened. The bicategories, homomorphisms, pseudonatural transformations, and modifications form a tricategory (a weak 3-category) called Bicat.

The subtricategory of Bicat containing only the 2-categories as objects, and only the 2-functors as 1-cells, but with all 2-cells and 3-cells between them, is called Gray. As well as being a particular tricategory, there is another important point of view on Gray. The category 2-Cat of 2-categories and 2-functors is cartesian closed, but it also has a different symmetric monoidal closed structure [3], for which the internal hom [𝒜 , ℬ] is the 2-category of 2-functors, pseudonatural transformations, and modifications between 𝒜 and ℬ. A category enriched over 2-Cat with respect to this closed structure is called a Gray-category. A Gray-category has 2-categories as hom-objects, so is a 3-dimensional categorical structure, and it can be seen as a particular sort of tricategory. The closed structure of 2-Cat gives it a canonical enrichment over itself and the resulting Graycategory is just Gray. Gray is also sometimes used as a name for 2-Cat with this monoidal structure.

A homomorphism of bicategories T : 𝒜 → 𝒞 is called a biequivalence if it induces equivalences T_{A,B} : 𝒜 (A, B) → ℬ(T A, T B) of hom-categories for all objects A, B ∈ 𝒞 (T is locally an equivalence), and every object C ∈ 𝒞 is equivalent in 𝒞 to one of the form T A (T is biessentially surjective on objects). We then write 𝒜 ∼ ℬ. Every bicategory is equivalent to a 2-category [5].

A trihomomorphism of tricategories T : 𝒜 → 𝒞 is called a triequivalence if it induces biequivalences T_{A,B} : 𝒜 (A, B) → ℬ(T A, T B) of hom-bicategories for all objects A, B ∈ 𝒜 (T is locally a biequivalence), and every object C ∈ 𝒞 is biequivalent in 𝒞 to one of the form T A (T is triessentially surjective on objects). It is not the case that every tricategory is triequivalent to a 3-category, but every tricategory is triequivalent to a Gray-category [2].

The support of the Australian Research Council and DETYA is gratefully acknowledged.

Received by the editors 2006-12-11 and, in revised form, 2007-01-05.

Transmitted by Ross Street. Published on 2007-01-08.

2000 Mathematics Subject Classification: 18D05.

Key words and phrases: bicategory, tricategory, Gray-category, coherence.

\begin{matrix}c \\ ◯ \end{matrix} Stephen Lack, 2007. Permission to copy for private use granted.

Perhaps since a Gray-category is a category enriched in the monoidal category Gray, and a tricategory can be seen as some sort of “weak Bicat-category”, it has been suggested that Bicat might be triequivalent to Gray, and indeed Section 5.6 of [2] states that this is the case. We prove that it is not. First we prove:

2. Lemma. The inclusion Gray → Bicat is not a triequivalence.

Proof. If it were then each inclusion Gray(𝒜 , ℬ) → Bicat(𝒜 , ℬ) would be a biequivalence, and so each homomorphism (pseudofunctor) between 2-categories would be pseudonaturally equivalent to a 2-functor. This is not the case. For example (see [4, Example 3.1]), let 𝒜 be the 2-category with a single object ∗, a single non-identity morphism f : ∗ → ∗ satisfying f^{2} = 1, and no non-identity 2-cells (the group of order 2 seen as a one-object 2-category); and let ℬ be the 2-category with a single object ∗, a morphism n : ∗ → ∗ for each integer n, composed via addition, and an isomorphism n ≅ m if and only if n − m is even (the “pseudo-quotient of ℤ by 2ℤ”). There is a homomorphism 𝒜 → ℬ sending f to 1; but the only 2-functor 𝒜 → ℬ sends f to 0, so this homomorphism is not pseudonaturally equivalent to a 2-functor.

3. Theorem. Gray is not triequivalent to Bicat.

Proof. Suppose there were a triequivalence Φ : Gray → Bicat. We show that Φ would be biequivalent to the inclusion, so that the inclusion itself would be a triequivalence; but by the lemma this is impossible.

The terminal 2-category 1 is a terminal object in Gray, so must be sent to a “triterminal object” Φ1 in Bicat; in other words, Bicat(ℬ, Φ1) must be biequivalent to 1 for any bicategory ℬ. For any 2-category 𝒜, we have biequivalences

𝒜 ∼ Gray(1, 𝒜) ∼ Bicat(Φ1, Φ𝒜) ∼ Bicat(1, Φ𝒜) ∼ Φ𝒜

where the first is the isomorphism coming from the monoidal structure on Gray, the second is the biequivalence on hom-bicategories given by Φ, the third is given by composition with the biequivalence Φ1 ∼ 1, and the last is a special case of the biequivalence Bicat(1, ℬ) ∼ ℬ for any bicategory, given by evaluation at the unique object ∗ of 1. All of these biequivalences are “natural” in a suitably weak tricategorical sense, and so Φ is indeed biequivalent to the inclusion.

4. Remark. The most suitable weak tricategorical transformation is called a tritransformation. The axioms are rather daunting, but really the coherence conditions are not needed here. We only need the obvious fact that for any 2-functor T : 𝒜 → ℬ, the square

\xymatrix{𝒜 \ar[d]_{T} \ar[r]^{∼} & Φ𝒜 \ar[d]^{ΦT} \\ ℬ \ar[r]_{∼} & Φℬ}

commutes up to equivalence.

The fact that every bicategory is biequivalent to a 2-category is precisely the statement that the inclusion Gray → Bicat is triessentially surjective on objects, but as we saw in the lemma, it is not locally a biequivalence. On the other hand Gordon, Power, and Street construct in [2] a trihomomorphism st : Bicat → Gray which is locally a biequivalence (it induces a biequivalence on the hom-bicategories). They do this by appeal to their Section 3.6, but this does not imply that st is a triequivalence, as they claim, and by our theorem it cannot be one. In fact Section 5.6 is not used in the proof of the main theorem of [2], it is only used to construct the tricategory Bicat itself, and this does not need st to be a triequivalence.

By the coherence result of [2], Bicat is triequivalent to some Gray-category; and by the fact that st is locally a biequivalence, Bicat is triequivalent to a full sub-Gray-category of Gray, but it is not triequivalent to Gray itself.

References

[1] Jean Bénabou. Introduction to bicategories. In Reports of the Midwest Category Seminar, pages 1–77. Springer, Berlin, 1967.

[2] R. Gordon, A. J. Power, and Ross Street. Coherence for tricategories. Mem. Amer. Math. Soc., 117(558):vi+81, 1995.

[3] John W. Gray. Formal category theory: adjointness for 2-categories. Springer-Verlag, Berlin, 1974.

[4] Stephen Lack. A Quillen model structure for 2-categories. K-Theory, 26(2):171–205, 2002.

[5] Saunders Mac Lane and Robert Paré. Coherence for bicategories and indexed categories. J. Pure Appl. Algebra, 37(1):59–80, 1985.

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Richard Blute, Université d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: r.brown@bangor.ac.uk Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it Valeria de Paiva, Xerox Palo Alto Research Center: paiva@parc.xerox.com

Ezra Getzler, Northwestern University: getzler(at)math(dot)northwestern(dot)edu

Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk G. Max Kelly, University of Sydney: maxk@maths.usyd.edu.au Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, University of Western Sydney: s.lack@uws.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Brooke Shipley, University of Illinois at Chicago: bshipley@math.uic.edu James Stasheff, University of North Carolina: jds@math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca