Motivation of filtered colimits 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.)
 A: About the proof that every algebra is a filtered colim of finitely presented ones: Every algebra has a presentation by generators and relations. You can just build your colim diagram by gathering finitely many generators at each stage and dividing out by the relations between those. Then every generator will occur in the diagram, hence in the colim, and since each relation is only between finitely many generators they all are introduced at some place in the diagram, too. A thorough proof is in Adamek/Rosicky's "Locally presentable and Accessible Categories"

A: Might I suggest http://ncatlab.org/nlab/show/monadicity+theorem and links there, where various theorems for a functor to be monadic (roughly, a forgetful functor from a category of algebras) are stated. This is essential for an understanding of categorical algebra. The crude monadicity theorem has a requirement on reflexive coequalisers, which could be why the text you are using mentions them so prominently.
Also, filtered colimits commute, in $Set$, with finite limits, so this is a very natural class of colimits to consider when dealing with Lawvere theories. Also, a monadic functor $C \to Set$ is the forgetful functor for (the algebras for) a Lawvere theory if and only if it preserves filtered colimits.
Edit: the link between finitary monads and Lawvere theories is explained here. The categories of finitary monads and Lawvere theories are equivalent.
A: The reason that Grothendieck originally considered filtered colimits and what is now known as the theory of accessible and locally presentable categories (I think named by Makkai and Paré) is as follows:
Let $x$ be an object of a category $C$ such that $hom_C(x,-)$ preserves $\alpha$-filtered colimits, then given any morphism $x\to colim F$ where $F:D\to C$ is an $\alpha$-filtered diagram, the morphism $x\to colim F$ factors through at least one $F(d)$ for some $d$ in $d$, and given any two factorizations through $F(d)$ and $F(d')$, there exists a majorant factorization through $F(d'')$ where $d''\geq d'$ and $d''\geq d$ extending the other two factorizations.  
I leave this as an exercise (it is, if you will, proof by introspection) (Hint: Use the corresponding statement for sets (which are valid because the statements hold for hom-sets) to perform the necessary manipulations).
This very powerful technique is used, for instance, in the modern generalizations of the small object argument and in situations regarding Bousfield localizations (the notion of accessibility is absolutely essential for results like Jeff Smith's theorem, for instance).  
See Clark Barwick's paper for a fairly detailed treatment with regard to its use in homotopy theory.  I would also suggest taking a look at Appendices 1 and 2 of Lurie's Higher topos theory as well as the book of Makkai-Paré, and also the standard modern reference on the subject by Adamek and Rosicky.
A: To expand on one of the points in David's answer, the absolutely crucial property of filtered colimits is that

Finite limits commute with filtered colimits in Set.

It's probably more important to know this than to know the definition of filtered colimit.  In fact, you can use it as a definition, in the following sense:
Theorem Let $J$ be a small category.  Then the following are equivalent:


*

*$J$ is filtered

*colimits over $J$ commute with finite limits in Set.


One weak point of the wikipedia article is that it gives the very concrete definition of filtered category, but it doesn't mention the following more natural-seeming formulation: a category $J$ is filtered if and only if every finite diagram in $J$ admits a cocone.  
(A finite diagram in $J$ is a functor $D: K \to J$ where $K$ is a finite category.  A cocone on $D$ is an object $j$ of $J$ together with a natural transformation from $D$ to the constant functor on $j$.  The three conditions stated in the Wikipedia article correspond to three particular values of $K$.)
If the last couple of paragraphs have helped you, you can balance your karma by incorporating them into the Wikipedia page :-)
A: Here are two further ways that one might motivate filtered colimits.  I'll put them in a different answer from my previous one, since they're separate thoughts, although they're still along the lines of "think about what filtered colimits do rather than what the definition is".  
First motivation
The functors from Set to Set that appear in universal algebra often have the property that they are "determined" by their values on finite sets.  To be more precise: given any functor FinSet $\to$ Set, there is a canonical way of extending it to a functor Set $\to$ Set (namely, left Kan extension).  A functor Set $\to$ Set is called finitary if when you restrict down to FinSet and then extend back up to Set again, you get back the functor that you started with.  
For example, the free group functor $T:$ Set $\to$ Set, sending a set $X$ to the set $T(X)$ of words in $X$, is finitary.  Informally, this is because the theory of groups involves only finitary operations: each operation takes only finitely many arguments.  Thus, each element of the free group on $X$ touches only finitely many elements of $X$.  The same is true for any other finitary algebraic theory: rings, lattices, Lie algebras, etc.
So finitary functors are useful.  Now the key fact is that a functor from Set to Set is finitary if and only if it preserves filtered colimits.  This immediately suggests that filtered colimits are interesting.  This fact is also rather useful: for example, it tells us that the class of finitary functors is closed under composition, which wasn't obvious from the definition.
Second motivation 
It's not a bad approximation to think of the class of filtered colimits as the complement of the class of finite colimits.  
For example, every category with both finite and filtered colimits has all (small) colimits. Similarly, every functor preserving both finite and filtered colimits preserves all colimits.  Moreover, the two classes are in some sense disjoint: there are very few colimits that are both filtered and finite.  
(One way to make this precise is the following: given a small category A, if you freely adjoin finite colimits to A and then freely adjoin filtered colimits to that, the end result is the same as if you'd freely adjoined all small colimits to A.)
