# A 2-Categories Companion

@article{Lack2007A2C, title={A 2-Categories Companion}, author={Stephen Lack}, journal={arXiv: Category Theory}, year={2007}, pages={105-191} }

This paper is a rather informal guide to some of the basic theory of 2-categories and bicategories, including notions of limit and colimit, 2-dimensional universal algebra, formal category theory, and nerves of bicategories.

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## References

SHOWING 1-10 OF 68 REFERENCES

### Formal category theory: adjointness for 2-categories

- Mathematics, Philosophy
- 1974

Categories.- 2-categories.- Bicategories.- Properties of Fun(A,B) and Pseud(A,B).- Properties of 2-comma categories.- Adjoint morphisms in 2-categories.- Quasi-adjointness.

### Limits for Lax Morphisms

- MathematicsAppl. Categorical Struct.
- 2005

Limits in the 2- category of strict algebras and lax morphisms for a 2-monad are investigated, which includes both the2-category of monoidal categories and monoidal functors.

### Homotopy-theoretic aspects of 2-monads

- Mathematics
- 2006

We study 2-monads and their algebras using a Cat-enriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2-categorical points of view. Every 2-category…

### 2-nerves for bicategories

- Mathematics
- 2008

We describe a Cat-valued nerve of bicategories, which associates to every bicategory a simplicial object in Cat, called the 2-nerve. We define a 2-category NHom whose objects are bicategories and…

### A Quillen model structure for Gray-categories

- Mathematics
- 2002

A Quillen model structure on the category Gray-Cat of Gray-categories is described, for which the weak equivalences are the triequivalences. It is shown to restrict to the full subcategory Gray-Gpd…

### Elementary observations on 2-categorical limits

- MathematicsBulletin of the Australian Mathematical Society
- 1989

With a view to further applications, we give a self-contained account of indexed limits for 2-categories, including necessary and sufficient conditions for 2-categorical completeness. Many important…

### Complicial sets characterising the simplicial nerves of strict ω-categories

- Mathematics
- 2008

Simplicial operators and simplicial sets A little categorical background Double categories, 2-categories and $n$-categories An introduction to the decalage construction Stratifications and filterings…