This chapter will be a basic introduction to the theory of Gray-categories. There are a variety of natural ways to motivate the Gray-tensor product of 2-categories, and I would like to mention a few of them briefly without worrying about proofs of the various technical results that make this theory work. To be clear, I do not believe any of the material in this chapter is new; I have only collected together material on the Gray tensor product and Gray-categories that we will need later in studying either coherence for tricategories or the general coherence problem for algebras over Gray-monads. The main references are Gray's (1974, 1976) work, although the handwritten notes of Street (1988) provide another perspective. I have also drawn heavily from the material in Gordon–Power–and Street (1995). I do not know of a reference for the explanation of the Gray-tensor product in terms of a factorization, although it is mentioned in passing by Lack (2010b), and it was certainly from the lectures upon which that article is based that I learned that the Gray-tensor product could be expressed in this way.

This chapter proceeds as follows. First, I will give the generators-andrelations definition of the Gray-tensor product. While this definition will not be particularly useful in the discussion of coherence for tricategories, it will be used with regularity when we turn to discussing algebras for Graymonads.