In this chapter, we study the problem of the presentation of an algebraic category by different algebraic theories. This is inspired by the classical work of Kiiti Morita, who, in the 1950s, studied this problem for the categories R-Mod of left modules over a ring R. He completely characterized pairs of rings R and S such that R-Mod and S-Mod are equivalent categories; such rings are nowadays said to be Morita equivalent. We will recall the results of Morita subsequently, and we will show in which way they generalize from R-Mod to Alg T, where T is an algebraic theory. We begin with a particularly simple example.

Example In 1.4, we described a one-sorted algebraic theory N of Set: N is the full subcategory of Setop whose objects are the natural numbers. Here is another one-sorted theory of Set: T2 is the full subcategory of Setop whose objects are the even natural numbers 0, 2, 4, 6, … T2 obviously has finite products. Observe that T2 is not idempotent complete (consider the constant functions 2 → 2) and that N is an idempotent completion of T2: for every natural number n, we can find an idempotent function f: 2n → 2n with precisely n fixed points. Then n is obtained by splitting f. Following 6.14 and 8.12, Alg T2 ≃ Alg N ≃ Set.

In fact, we can repeat the previous argument for every natural number k > 0.