nLab canonical model structure on 2-categories

1. Idea

Just as the category Cat of categories can be equipped with a canonical model structure, namely the canonical model structure on categories, so too can 2Cat, the category of strict 2-categories. This extends to weak 2-categories too, although in order to actually have a category on which to put the model structure, one has to work with strict 2-functors.

The characterising feature of a canonical model structure on 2Cat is that the weak equivalences should be equivalences of 2-categories, categorifying the fact that the equivalences in the canonical model structure on categories are equivalences of categories. What the fibrations and cofibrations should be, or more specifically how to categorify isofibrations and iso-cofibrations, is one of the main points to resolve to obtain the model structure.

What is usually referred to as the canonical model structure on 2-categories is a model structure first described by Lack in Lack2002 and Lack2004. In this model structure, the fibrations are Lack fibrations, and every object is fibrant, but not every object is cofibrant, in contrast to the canonical model structure on categories.

However, it is possible to put a different canonical model structure on 2Cat in which every object is both fibrant and cofibrant, using the thesis Williamson2011. The fibrations in this case are, roughly speaking, those 2-functors in which semi-strict equivalences? can be lifted.

2. Lack model structure

The following is Theorem 4 in Lack2004.

Theorem 2.1. There is a model structure on 2Cat in which the equivalences are equivalences of 2-categories, and the fibrations are equiv-fibrations.

Note that by “equivalence of 2-categories” we mean the fully non-strict notion. In particular, although the morphisms in $2Cat$ are strict 2-functors, those that are equivalences may have inverses that are only pseudofunctors and hence not morphisms in $2Cat$.

Every object in this model category is fibrant, but not every object is cofibrant. The cofibrant 2-categories are those whose underlying 1-category is the free category on a quiver. This property ensures that any pseudofunctor with cofibrant domain is equivalent to a strict 2-functor, so that the “homotopy theory” of the model category is bicategorically correct.

This is also a monoidal model category with respect to the Gray tensor product.

3. Canonical model structure in which every object is both fibrant and cofibrant

Throughout this section, let $\mathcal{E}_{semi}$ denote the free-standing semi-strict equivalence.

Definition 3.1. A semi-strict equiv-fibration is a 2-functor $p: \mathcal{A} \rightarrow \mathcal{B}$ such that, for any (strictly) commutative diagram

in 2Cat, there is a functor $l: \mathcal{X} \times \mathcal{E}_{semi} \rightarrow \mathcal{A}$ such that the following diagram in 2Cat (strictly) commutes.

Definition 3.2. A semi-strict equiv-cofibration is a 2-functor $j: \mathcal{A} \rightarrow \mathcal{B}$ such that, for any (strictly) commutative diagram

in 2Cat, there is a functor $k: \mathcal{B} \times \mathcal{E}_{semi} \rightarrow \mathcal{X}$ such that the following diagram in 2Cat (strictly) commutes.

Definition 3.3. A 2-functor $f: \mathcal{A} \rightarrow \mathcal{B}$ is a semi-strict equivalence of 2-categories if there is a 2-functor $g : \mathcal{B} \to \mathcal{A}$ and 2-functors $h: \mathcal{A} \times \mathcal{E}_{semi} \rightarrow \mathcal{A}$ and $k : \mathcal{B}\times \mathcal{E}_{semi} \to \mathcal{B}$ exhibiting “homotopies” from $f\circ g$ and $g\circ f$ to identities.

Theorem 3.4. The category 2Cat can be equipped with a model structure in which the weak equivalences are semi-strict equivalences of 2-categories, the fibrations are semi-strict equiv-fibrations, and the cofibrations are semi-strict equiv-cofibrations.

Proof. By the section ‘Structured interval’ of the page walking equivalence, $\mathcal{E}_{semi}$ can be equipped with the structure of an interval object, and embellished with all the structures of Williamson2011 which are required for Corollary XV.6 and/or Corollary XV.7 of this work, satisfying all the hypotheses of these corollaries. Applying these corollaries, we immediately obtain the theorem.  ▮

4. Related concepts

• model structure on DblCat

5. References

• Stephen Lack, A Quillen model structure for 2-categories, K-Theory 26, No. 2, 171-205 (2002). Zentralblatt review author’s webpage

• Stephen Lack, A Quillen model structure for bicategories, K-Theory 33, No. 3, 185-197 (2004). Zentralblatt review author’s webpage

• Richard Williamson, Cylindrical model structures, DPhil thesis, University of Oxford, 2011. author’s webpage arXiv:1304.0867