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Adamek and Rosicky seem to agree with Mike: theorem 3.36 says that a category is locally -presentable iff it is (equivalent to) the category of models of a -ary essentially algebraic theory (so l.f.p. is the case).
I haven’t checked, but I think that when it says at essentially algebraic theory that a finitely complete category is an e.a. theory it means a finitary one. Certainly the syntactic definition there gives the finitary case.
Yes, the page essentially algebraic theory seems to be only about the finitary version. I replied at lfp category.
The problem is that sometimes people mean to include ‘finitary’ when they say ‘algebraic’, but it is best not to. So this can lead to confusion.
I think that I’ve edited essentially algebraic theory to fix everything that implied that such a theory must be finitary.
Oh, wow - I hadn’t guessed that a “finitary” essentially algebraic theory was a category with finite limits! Like many other people, I use “essentially algebraic theory” to mean a category with finite limits! So, I thought a “finitary” essentially algebraic theory obeyed some further finiteness condition, like finitely many generating objects or morphisms or something.
Thanks for clarifying that, Mike. I’m greatly relieved that everything I believed is still true. But I will add more clarifications, because I still find the term “finitary” pretty confusing when it’s used on some of these pages without explanation.
On the other hand, I think “finite limits theory” is pretty hard to be confused by.
I did a bit of looking around and it really seems true that everyone except Toby and perhaps Mike uses ’essentially algebraic theory’ to mean a category with finite limits. This makes the term ’finitary essentially algebraic theory’ confusing to anyone who has already studied this stuff… though I see why Toby likes this term. I would like to completely get rid of it, but I didn’t.
To reduce the confusion, I did a bit of editing on locally finitely presentable category and essentially algebraic theory. In the process I made some minor additions to exact functor and maybe some other pages too, like finitely cocomplete category.
I still think essentially algebraic theory is a bit confusing. For one thing, it’s not clear whether the ’traditional syntactic definition’ allows for infinitary operations or not: there’s an index set , but the size of this set is never specified, I think. There are some other small glitches like this…
Yeah, I’m not too fond of “essentially algebraic” myself; it doesn’t convey to me any intuition for how such a theory differs from an “algebraic” one. If the consensus in the literature is that it implies finitary, then I don’t see a really good reason to try to change that, since as you say we have a perfectly good unambiguous term “finite limit theory” to use instead, and we can again unambiguously say “limit theory” (or “small limit theory”) for the non-finitary version.
We are also using the not-necessarily-finite concept of algebraicity at algebraic theory. It would be confusing (to me) if an algebraic category might not be essentially algebraic. We should also want the terminology between ‘algebraic theory’ and ‘limits theory’ to be analogous, so that the latter is the category-theoretic characterisation of the former.
When I began algebraic category, I decided to use the definitions of Abstract and Concrete Categories. I considered using Johnstone’s definitions instead; he says that (at least over , which is the only case that he or we consider) an algebraic category is precisely one which is monadic over . This definition is more restrictive, but even it allows for non-finitary algebraic categories! This is necessary, for example, if you want to say that compact Hausdorff spaces form an algebraic category.
The argument that ‘algebraic’ should not include ‘finitary’ (and not even ‘monadic’) may be found in Section 24.11 of ACC, but it is the existence of the examples that follow it that really prove the point. (Note that we consider only concrete categories over , while ACC uses an arbitrary base, so some things are simpler for us. ACC uses the term ‘construct’ for a concrete category over .)
I’m more familiar with algebraic = monadic, although I suppose one might arge that since we have the word “monadic,” we should use “algebraic” to mean something different. I don’t have ACC with me right now, but the examples mentioned at algebraic category don’t really convince me – why should I think of cancellative monoids or Stone spaces as “algebraic”? Operations and equations seems to me closer to my intuitive meaning of “algebraic”.
Toby wrote:
We are also using the not-necessarily-finite concept of algebraicity at algebraic theory.
Yeah - and I don’t like that either, simply because most category theorists don’t use this term that way, and I don’t have any strong desire to change them. I believe that most category theorists use “algebraic theory” to mean “small category with finite products”. See for example this book, Algebraic Theories.
So, if you want to change the meaning of “algebraic theory”, you should at least tell people that this is new, and say what the old usage was.
@ Mike:
As Stephen Britton has been reminding us lately, ACC is free online.
Yeah, I’m not terribly convinced by their examples either. Mostly I was convinced by already having the word ‘monadic’, as you suggest. Another argument is that, according to ACC, the theory of algebraic functors is better behaved, especially over bases other than , and therefore deserves a term, but I don’t understand that very well.
@ John:
Well, this usage (the non-finitary restriction) is certainly not new, although it may be new to category theorists. It’s definitely not new to universal algebraists.
But it’s also not new to the category theorists whose work on algebraic theories that I know, such as Johnstone and the writers of ACC, with the exception of Lawvere. But now you’ve shown me another book (with, ironically, much overlap in authorship with ACC!).
I think we definitely need to add more warnings about the different usages of these terms. But I also think that limiting algebra to finitary operations is definitely the wrong way to go (for reasons that I won’t repeat).
I think if anything, this discussion is convincing me not to use the word “algebraic” with any precise meaning at all. We can say “monadic” when we mean that, “finite product theory” when we mean that, and “finite limit theory” when we mean that, and “Horn theory” when we mean that. Why introduce unnecessary ambiguity by using a word that has many different meanings?
Actually, there are also infinitary versions of ‘Horn theory’, although one could also say ‘infinitary Horn theory’.
Anyway, do you propose a word for ACC’s notion of algebraic theory? I suppose that ‘quasimonadic theory’ would make sense. (Although I worry that somebody has already defined ‘quasimonad’ in an incompatible way.)
I was thinking of “Horn” in that context, since I thought I read that their “algebraic” categories over Set are the “quasivarieties of algebras”, which look to me the same as Horn theories (am I wrong about that?).
Sorry, I was a bit mixed up. Infinitary Horn theories are infinitary in a different way, in that they allow infinitely many statements on the left-hand side of the clause.
So you are correct, Horn theories written using (ordinary finitary) Horn clauses in the equational language of a (possibly infinitary) algebraic signature correspond to quasivarieties of algebras in that signature (or else I’m missing something too).
So I think that you’ve answered all objections.
I haven’t been following this thread closely, but FWIW, Michael Barr has some pertinent information on the distinction between Horn theories and finite limit theories here.
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