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Added an entry on cocompleteness of varieties of algebras. It surely needs some improvements, but I hope that there are no fatal errors.
Yaron, I think this is fine, but there are alternative proofs which you might enjoy. First, the category of algebras of a monad on has coequalizers; see the proof of proposition 3.4 (page 278 of 303) here. Then, you can get coproducts easily: let be the underlying functor, with left adjoint . If you have a family of -algebras , then there are canonical coequalizers
and since is the coproduct in the category of algebras, i.e., since coproducts of free algebras exist, the coproduct of the is constructed as a coequalizer of an evident pair
obtained by summing over all the canonical pairs.
see the proof of proposition 3.4 (page 278 of 303) here
and add it to the nLab!
and add it to the nLab!
Done. I added two sections: (a), that the category of algebras of a monad on is cocomplete, and (b), that the category of algebras of a finitary monad on a cocomplete cartesian closed category is cocomplete. These could be useful in practice.
Done.
Thanks! . Very nice.
Let’s try to make sure that this entry is being linked to from where it needs to be linked to, so that one can find it when looking for information on this question.
Todd, thanks for the explanation and the references! It will take me some time to understand this (that is, why the coequalizer of the last pair is the required coproduct).
Also, just to be sure, is it true that the assumption that the monad is on was used only for (1) getting the coequalizers for the monad by the Theorem of Barr-Wells, and (2) For getting the coproducts of the free algebras?
I’m asking this because I’m a little confused by a similar proposition in Borceux (Vol. 2, Prop. 4.3.4, p. 200). This proposition asserts that, for a monad on a cocomplete category , the category of algebras is cocomplete iff it has coequlizers.
So, it seems that he is exactly in the situation of your proposition (he assumes coequlizers, and by cocompleteness of , he has coproducts of free objects), but his proof for the existence of the coproduct is more than 2 pages long. What’s going on here? :)
An original reference is “Coequalizers in categories of algebras” by Fred Linton (1969), in which he proves (essentially) that
I would think the general facts about algebras for a monad would fit better at Eilenberg-Moore category, maybe, or some other page not specifically titled as being about varieties of algebras.
Mike, thanks for the reference to Linton and for the review of his results. In view of what you say, I think that as a first step I’ll just read the proof in Borceux.
Thanks, Yaron. Let me run through the bit about the coequalizer of the last pair being the coproduct.
So, suppose is an algebra map which coequalizes that pair of arrows; this map corresponds to an -indexed tuple of maps . Now consider the composite
The adjunct of this is the (-indexed) family of maps
By similar reasoning, the adjunct of the other composite is the family
and these two adjuncts must be equal. So factors through a unique algebra map for all , and we have thus shown that maps out of the coequalizer corresponds to families of algebra maps , making the coequalizer the coproduct.
here is the broken record again: please somebody make sure to give all these new bits of information their proper place on the lab. Thanks!
Todd, thanks for the detailed explanation! I’ve just finished reading the proof in Borceux, and I’ll now read carefully your explanation.
Added some more material to colimits in categories of algebras, leading up to the conclusion that categories monadic over (or over , covering the multisorted case) are Barr-exact.
More edits at colimits in categories of algebras, trying to keep in mind results likely to be useful to customers of category theory. I think I’m done for the time being.
Thanks, that’s an beautiful page!
I went there and turned many of the
\to
in the diagrams into
\longrightarrow
Because, at least on my system (but I think that’s MathJax behaviour in general), the former doesn’t render as intended in
\stackrel
stacking (it comes out too short).
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