4 - Algebraic categories as free completions

Summary

In this chapter, we prove that every algebraic category has colimits. Moreover, the category Alg T is a free completion of T^op under sifted colimits. This shows that algebraic categories can be characterized by their universal property: they are precisely the free sifted-colimit completions of small categories with finite coproducts. This is analogous to the classical result of Gabriel and Ulmer characterizing locally finitely presentable categories as precisely the categories Ind C, where C is a small category with finite colimits and Ind is the free completion under filtered colimits (see Example 4.12).

Remark For the existence of colimits, since we already know that Alg T has sifted colimits and, in particular, reflexive coequalizers (see 2.5 and 3.3), all we need to establish is the existence of finite coproducts. Indeed, coproducts then exist because they are filtered colimits of finite coproducts. And coproducts and reflexive coequalizers construct all colimits (see 0.7). The first step toward the existence of finite coproducts has already been done in Lemma 1.13: finite coproducts of representable algebras, including an initial object, exist in Alg T. In the next lemma, we use the category of elements El A of a functor A: T → Set introduced in 0.14.

LemmaGiven an algebraic theory T, for every functor A in Set^T, the following conditions are equivalent:

1. A is an algebra.

2. El A is a sifted category.

3. A is a sifted colimit of representable algebras.

4. […]