I am trying to move in categorical algebra beyond the basics. A Lawvere theory L is a small category with finite products. (I know that there also is a functor $(skeleton(FinSet))^{op}\to L$, which restricts a number of sorts in the algebraic theory to 1. Lets drop that requirement for now.) How to convert some variety to a Lawvere theory is pretty clear for me. The link (varieties ↦ Lawvere theories) is clear in some elementary operations, like

- mapping an algebra by some functor F ↦ postcomposing F;
- underlying functor ↦ precomposition of a functor between Lawvere theories.

Then filtered colimits come. Lets take for reference “Adámek. a categorical introduction to general algebra.” Chapter 2 “Sifted and filtered colimits” and chapter 3 “Reflexive coequalizers” are devoid of **mentioning varieties**. Why the definition of a filtered colimit is such? I suppose there should be more concrete explanations involving algebraic operations, this is called “algebra” after all. Google suggests few texts on this subject, but they are abstract too. Any references?

The claim “an arbitrary algebra is a filtered colimit of finitely generated algebras” is needed to construct the left adjoint to an underlying functor. Can anyone refer me to its proof? (Update 2011-01-29. Also I want a precise proof constructing that left adjoint.) (Update 2011-01-29. Thank you all for insightful answers and comments. I suspect that there is no direct link between filtered colimits and traditional algebra, i.e. it is an abstract thing that is needed for another abstract thing… I need to think it through to formulate further questions.)

filteredis like it is because experience has shown that it captures the usefulness and good properties of increasing unions. As for finitely generated subalgebras of an algebra: show that the set of finitely generated algebras is directed by inclusion, and then show that the colimit of the tautological functor defined on that set is the algebra you started with. $\endgroup$