A Quillen model structure for 2-categories

Stephen Lack

This appeared in K-Theory 26:171-205, 2002.

We describe a cofibrantly generated Quillen model structure on the locally finitely presentable category 2-Cat of (small) 2-categories and 2-functors; the weak equivalences are the biequivalences, and the homotopy relation on 2-functors is just pseudonatural equivalence. The model structure is proper, and is compatible with the monoidal structure on 2-Cat given by the Gray tensor product. It is not compatible with the cartesian closed structure, in which the tensor product is the product.

The model structure restricts to a model structure on the full subcategory PsGpd of 2-Cat, consisting of those 2-categories in which every arrow is an equivalence and every 2-cell is invertible. The model structure ono PsGpd is once again proper, and compatible with the monoidal structure given by the Gray tensor product.


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There is an error in the definition of the fibrations and the generating trivial cofibrations. This has been fixed in A Quillen model structure for bicategories. There is no change to the weak equivalences and trivial fibrations, and all the other results (and their proofs) remain valid.


Steve Lack
Last modified: Wed Jun 25 11:36:13 EST 2003