Conceptual Mathematics A First Introduction to Categories

Objects satisfying universal mapping properties are in a sense trivial if you look at them from one side, but not trivial if you look at them from the other side. For example, maps from an object to the terminal object 1 are trivial; but if, after establishing that 1 is a terminal object, one counts the maps whose domain is 1, 1 → X, the answer gives us valuable information about X. A similar remark is valid about products. Mapping into a product B1 × B2 is trivial in the sense that the maps X → B1 × B2 are precisely determined by the pairs of maps X → B1, X → B2 which we could study without having the product. However, specifying a map B1 × B2 → X usually cannot be reduced to anything happening on B1 and B2 separately, since each of its values results from a specific ‘interaction’ of the two factors.

Binary operations and actions

In this session we will study two important cases of mapping a product to an object. The first case is that in which the three objects are the same, i.e. maps B × B → B. Such a map is called a binary operation on the object B. The word ‘binary’ in this definition refers to the fact that an input of the map consists of two elements of B.