We denote by the category defined uniquely (up to isomorphism) by the following.
1) There are exactly two objects, which we shall denote by and .
2) There are exactly two arrows .
3) There is exactly one arrow .
4) There are no non-identity arrows .
5) There are exactly two non-identity arrows , which are and .
In particular, because of 4) in Notation 1, the diagram
commutes in , and the diagram
commutes in .
The category can also be constructed by beginning with the free category on the directed graph defined uniquely by the fact that 1), 2), and 3) in Notation 1 hold, and by the fact that there are no other non-identity arrows. One then takes a quotient of this free category which forces the diagrams in Remark 2 to commute.
This quotient can be expressed as a colimit in the category of small categories, or, which ultimately amounts to the same, by means of the equivalence relation on the arrows of the free category generated by requiring that and , and by requiring that if and .
The category of cubes is the free strict monoidal category? on whose unit object is .
We denote the category of cubes by .
We refer to as the category of cubes.
It is not the case that is the free strict monoidal category on . Rather, is the free strict monoidal category with specified unit on , where the unit is specified to be .
Let be an integer. We often denote the object of by .
There are several useful variations of , to be described on other pages in the future.
For expository and other material, see category of cubes - exposition.