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How is this different from a variety of algebras?
Sorry, that was a mix-up. I had meant to clear the other page as per the request there. Re-instantiated now.
How is this different from a variety of algebras?
They refer to the same concept from different perspectives. Perhaps they could be merged, but I think it is helpful to have a page about locally strongly finitely presentable categories specifically in analogy to locally finitely presentable categories. Someone could well be interested in these without caring about the algebraic perspective at all. If they are merged, I would rather have variety of algebras redirect to locally strongly finitely presentable category rather than vice versa.
In fact, in its current form the entry variety of algebras gives no hint that it might be related to the entry here!
We have a tad more of a hint towards the relation at (∞,1)-algebraic theory, in this Prop., but this too is lacking commentary.
So I would say: If “varieties of algebras” managed live as an entry on until now without a hint of the relation, then apparently merging the entries is not compelling.
But merging entries or not is always a purely mechanical question that we don’t need to get hung up on much. More pressing seems to be to bring in some more of the missing content.
Where is the term “locally strongly finitely presentable category” used in research literature?
The book of Adamek, Rosicky, Vitale does talk about this perspective (free sifted cocompletions), but they do not use this terminology. Rather, they talk about algebraic categories and varieties of algebras.
By the way, the entry algebraic category is another duplicate of this article…
Where is the term “locally strongly finitely presentable category” used in research literature?
The terminology was introduced in Lack–Rosický’s 2008 paper Notions of Lawvere theory. I’ve seen it used in other papers, such as [1] and [2]. It aligns with the terminology “strongly finitary functor”, which has been around much longer.
By the way, the entry algebraic category is another duplicate of this article…
This is certainly strongly related, but that page proposes several definitions of algebraic category, only one of which (i.e. finitarily monadic over Set) is equivalent to local strong finitely presentability.
Dmitri, what are you after? You seem to be pushing varkor to merge (a) an entry that currently says exactly nothing about categories of algebras into (b) an entry which, albeit on algebras, also says exactly nothing about the properties of their categories – on the basis that you know that secretly both entries are related. :-)
What we need is somebody who actually adds the relevant information to either entry! Whether or not they are then merged seems rather secondary.
You seem to be well-versed in the matter you have in mind. If you add – to any entry – a section explaining the relation, that would be a great service to the $n$Lab.
Re #9: I would like to add this material to the entry algebraic category. However, that entry currently discusses a different concept, corresponding to a quasivariety of algebras.
The recent textbook of Adámek–Rosický–Vitale uses the term algebraic category to refer to a variety of algebras.
A much older book by Adámek–Herrlich–Strecker use the term algebraic category to refer to a quasivariety of algebras.
So if we are to add something to this entry, we have to decide how to handle the separation of these two incompatible concepts.
Mac Lane points out in his MathSciNet review that “Perhaps because of this isolation, the book uses considerable nonstandard terminology, as in the following partial list, with the terms from the book in quotes: generator (“separator”), cone (“natural source”), Stone-Čech (“Čech-Stone”), left adjoint (“coadjoint”), exponential (“power object”), B→G(A) (a “G-structured arrow”). The famous triangular conditions for an adjunction are not so titled, while Yoneda’s well-known lemma (Corollary 6.19) does not carry his name. Such esoteric terminology also tends to isolate readers.”
So perhaps we should stick to the terminology of the much more modern Adámek–Rosický–Vitale book, which also seems to be less esoteric.
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