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I noticed some inconsistencies in the section outline at algebraic theory, that must have come from different people editing different pieces and mixing up some global entry structure.
I have briefly tried to reinstantiate consistent order. But the entry could probably do with somebody looking over in its entirety with an editor-hat on.
at Relation to monads I found that a link to the keyword “finitary monad” had been missing. So I added it briefly, at the end of the first paragraph.
I added this recent work
looking to unify
several variants of universal algebra, such as theories of symmetric operads, non-symmetric operads, generalised operads, PROPs, PROs, and monads.
The current definition says
A Lawvere theory or algebraic theory is a locally small category CC with small products that is equipped with an object xx such that the (unique-up-to-isomorphism) product-preserving functor i:Set op→C:[1]↦xi: Set^{op} \to C: [1] \mapsto x is essentially surjective.
The modern definition in the Adamek–Rosický–Vitale book is that an algebraic theory is a small category with finite products, whereas a Lawvere theory is an algebraic theory T together with a functor S*→T that preserves finite products and induces a bijection on objects, where S is a set and S* denotes the category whose objects are finite words in S and morphisms s_0…s_{a−1}→t_0…t_{b−1} are maps f:{0,…,b−1}→{0,…,a−1} such that s_f(j) = t_j for all j.
Should the article be adjusted accordingly?
(Sorry, I’ve been away for a while.) Yeah, maybe so, but while writing large chunks of the article, I had wanted to consider the doctrine of categories with small products, and the obvious analogue of Lawvere theory for this situation. I think earlier in the article I called that an infinitary Lawvere theory, and then I probably got tired of writing “infinitary”.
Clearly the notion of infinitary Lawvere theory, and its relation to the concept of monad on $Set$, is an important one. So, if there are no other terminology clashes, I propose that I could insert “infinitary” everywhere, or is there a more elegant way of handling this? Besides the axis of arities, there’s also the axis of sortedness, but “infinitary $S$-sorted” everywhere gets to be a mouthful.
The current article makes “Lawvere theory” synonymous with “algebraic theory”. But the original definition by Lawvere is that of a single-sorted algebraic theory.
The Adamek–Rosický–Vitale book makes a clear distinction between the two: an algebraic theory is just a small category with finite products, and they develop much of the abstract theory in this generality.
When they need sorts, they talk specifically about S-sorted algebraic theories.
(As far as I can see, they do not directly use “Lawvere theory”, possibly because of the ambiguity identified above.)
It appears that this book is now a more-or-less standard reference on this topic (by virtue of being the only book-length exposition), so it would make sense to align the terminology with it.
I’m aware of this. (And aware of it at the time I was writing this, although I was “playing Humpty Dumpty”.) The question is: how would you like to change the terms?
The article was written to take into account not just finite products, but infinite products as well. So “infinitary $S$-sorted algebraic theory” throughout? Something less verbose?
In my experience, “algebraic theory” is very overloaded (as Lawvere theories, cartesian categories, or equational presentations), and my suggestion would be to avoid using the term and instead disambiguate (e.g. “one-sorted algebraic theory” or “Lawvere theory”). “Infinitary S-sorted algebraic theory” is quite long, but it’s very clear.
Algebraic theories defined using finite products have rather different properties from algebraic theories defined using infinite products.
For instance, do any theorems about algebraic theories (with finite products) that involved sifted colimits have analogs for infinite products?
If not, I’d suggest to refer to the infinite-product version as “infinitary algebraic theory” and reserve “algebraic theory” for the finite-product case.
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