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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 “Reflixive 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.)

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Filtered colimits are particularily well-behaved colimits (they have pretty much the same properties of increasing unions) The definition of filtered is 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. –  Mariano Suárez-Alvarez Jan 27 '11 at 23:12
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It is probably me, but I find the text of your question very confusing. Maybe you could make it more evident what exactly is that you are asking. –  Mariano Suárez-Alvarez Jan 27 '11 at 23:27
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It is confused, because I am confused. Is there a separate website for confused question? ;) –  beroal Jan 28 '11 at 21:49
    
Filtered colimits are (just) the kind of colimits that commuting by finite limits. Algebraic theory essentially is the translation of usually "algebraic object" (of universal algebra in sets) to "internally algebraic object" inside a category, by diagrams that involving finite limits (finite product and kernels). Then are "algebraic allowed" structures or property defined by such colimits. The importance of filtering colimits is very deep in mathematics, what said above is translatable internally to a topos, see also: flat presheaves , or C.4.2 of Sketches of an Elephant II. –  Buschi Sergio Aug 8 '13 at 14:28

5 Answers 5

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 :-)

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Good points, Tom. But I don't think that Wikipedia should be a source for this kind of math anyway. Some years ago I edited some articles, but now I think it's worthless. –  Martin Brandenburg Jan 28 '11 at 9:17
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it's even a little better than Tom says: if $J$ is filtered, and $D:K\to J$ is a finite diagram in $J$, the category of cocones over $D$ is not just non-empty but connected. –  Steve Lack Jan 28 '11 at 10:13
    
…and why finite limits in Set are important? –  beroal Jan 28 '11 at 20:49
    
Finite products are generally assumed when talking about models for a Lawvere theory, or finite limits more generally for essentially algebraic theories. –  David Roberts Jan 28 '11 at 22:14
    
@Tom Leinster “every finite diagram in $J$ admits a cocone” I guess your definition is an analogue of logical equivalences “the category has colimits ↔ the category has equalizers and products ↔ the category has pullbacks and a terminal object” and “the subset is directed ↔ every 2 elements in the subset has an upper bound and the subset has the least element”. That is a useful technical detail worth mentioning in an encyclopedia, but it is not problem-breaking. –  beroal Jan 29 '11 at 8:01

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.)

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The functor, sending a set $X$ to the set of words in $X$, is the free monoid functor? –  beroal Jan 28 '11 at 20:01
    
@beroal - Yes it is –  David Roberts Jan 28 '11 at 20:22
    
@beroal and David: "words" is ambiguous (my fault). In the context I was using it, I meant "words" in the sense of groups. For instance, $x y^{-2} z^3$ is a word-in-the-sense-of-groups in the letters x, y, z. But you can use "word" for any algebraic theory. In particular, you can use it for the theory of monoids, and in that setting, a word in a set X is simply a finite sequence of elements of X. For any algebraic theory, you can say that the free algebra functor sends a set X to the set of words in X (as long as you interpret the meaning of "words" correctly). –  Tom Leinster Jan 31 '11 at 14:36

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"

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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.

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That's too much, I did not ask about categorical algebra in its entirety, that's too much. :) The definition of a monadic functor refers to the Eilenberg–Moore category, I do not know any connection with Lawvere theories. It seems to sit aside of my question. Except that bit about finite limits in Set, which I will comment on the answer by Tom Leinster. –  beroal Jan 28 '11 at 20:48
    
I wasn't trying to heap a huge amount of information on you, beroal, but just give you a taste of some results for which filtered colimits are fairly central. My last sentence is not enough about Lawvere theories? –  David Roberts Jan 28 '11 at 22:16
    
Your edit of your comment is sufficient. –  beroal Jan 29 '11 at 8:35

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.

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I should note that most of the inspiration for the modern theory appears in SGA4.1.i (which is why I noted Grothendieck's involvement). –  Harry Gindi Jan 29 '11 at 0:03
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Locally presentable categories were first introduced by Gabriel-Ulmer in '70 and accessible categories were introduced by Lair in '81 under the name "catégories modelables" and baptized accessible by Makkai-Paré in their '89 book. I think it would be fairer to mention Tohoku instead of SGA in connection with Grothendieck, and for that one main motivation was the need of extending Cartan-Eilenberg (derived functors) to sheaves. To achieve this, G. needed to prove that there are enough injectives. It turns out that local presentability of Grothendieck abelian categories is the crucial point. –  Theo Buehler Jan 29 '11 at 1:34
    
But this is just a bit of nitpicking, your point remains valid, of course. –  Theo Buehler Jan 29 '11 at 1:34
    
Interesting! Thanks! –  Harry Gindi Jan 29 '11 at 1:51

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