# 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