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There is a slightly contentious statement at varieties of algebras namely: The variety of monoids is a subvariety of the variety of groups.
Because of the additional use of the term variety of groups perhaps some additional comment could be made here. (What do folks think?)
Frankly, what is stated at varieties of algebras is not what I understand by a “variety of algebras”. In the article, a variety is defined to be a certain theory (according to a traditional syntax involving signature and axioms); I quote:
Then a variety of algebras consists of a signature and a set of axioms in that signature.
I understand “variety” in universal algebra rather to be a class of algebraic structures (or, in categorical universal algebra, a category of such structures). For example, one has Birkhoff’s HSP theorem that characterizes varieties of algebras; see Wikipedia.
The distinction is crucial in sorting out what is meant by “subvariety”. Do we mean a theory, or a class of structures? Syntax or semantics?
Great , Todd. Your first sentence was exactly my reaction. I had been looking for some nPOV thoughts on varieties of groups, and or their higher dimensional analogues and the entry gave nothing in that direction at all. A variety in the class of algebraic objects leads to a set of words in the language that are in some sense satisfied by the structures. Monoids do not then form a subvariety of groups, but nilpotent groups of class n do.
Groups rather form a variety in monoids.
This also seems right by analogy with the meaning of “variety” in algebraic geometry, which is a subset of determined by satisfaction of some equations, and a “subvariety” is determined by a subset of the points (hence a superset of the equations), not a subset of the equations.
Never thought about the nomenclature in that way. But it certainly makes sense if we think of models as points in the “spectrum” of some theory.
I very much like thinking of the category of models of an algebraic theory as being its spectrum. There is even a duality theorem: the category of Lawvere theories is anti-equivalent to the category of finitary-monadic categories over (with morphisms the strictly commuting triangles).
I think what I’d like to do is copy the article to my personal web (the public one) and work it over a little, since we in this thread seem to be in some agreement about the concepts. (I don’t want to work it over right on the main site, since Toby and Urs are on vacation and might want to add their inputs before things get reworked.) I’ll leave a note here again when I’m done doing what I want.
Well, not done, but I whipped up something quick here. Still very rough, but gives an idea of at least some of the basic ingredients I’d like to have in there.
Or here… :-)
Ah! That is beginning to look good. Once it takes shape and goes into place I may be able to do a ‘varieties of groups’ page. (I have some difficutly typing due to some sort of ’mousitis’ at the moment but it is getting better.)
I’ve always assumed that the analogy to algebraic varieties was the origin of the term “variety of algebras” — I certainly can’t think of any other sensible origin of the phrase. (Laying aside the nonsensicality of the phrase “algebraic variety” itself, which I guess must be due to a mistranslation or a shift in the meanings of words over time.)
But, I think that 11 is right.
@Mike
I thought it came from translating Mannigfaltigkeit: after all, manifoldness is not so far from variety…
For ’variety of algebras’ remember that Cox’s Orange Pippin is a variety of apples, so in that use variety is almost ’type’. I have never known why the term was used in Alg. Geom. but think Zhen Lin is correct.
Well, whatever the actual case may be, I have to say that #12 also sounds quite plausible. A “variety of algebras” being akin to “species of structure”.
Regarding “manifold”: my impression has always been that this referred to the manifold or n-fold degrees of freedom possible at each point (having n degrees of freedom at each point = locally Euclidean of dimension n). From Wikipedia:
Using induction, Riemann constructs an n-fach ausgedehnte Mannigfaltigkeit (n times extended manifoldness or n-dimensional manifoldness) as a continuous stack of (n−1) dimensional manifoldnesses.
I don’t really believe “it’s the collective noun for algebras” as an origin of the word “variety”, because even if that’s true, universal algebra was invented in the last couple of centuries, so someone would have had to decide at some point that “variety” would be the collective noun for algebras, and why would that have been? And it doesn’t really mean the same thing either; I think #15 sounds more plausible.
I think #15 sounds more plausible.
Well, right, that’s what I actually meant when I was thinking of its being akin to “species of structure”: a type of algebraic structure. Except that I wanted to avoid the word ‘type’, for fear of misunderstanding.
There is a point that should come out later, e.g. in my (distantly planned) stub on varieties in Groups, and that is the ’verbal subgroup’ idea, where the ’words’ in the language that are true of the objects in the variety and no others thus going back towards presentations of theories, rewriting etc.
I am a syntactically oriented sort of mathematician, so I like thinking of a variety of algebras as given by a set of operations and a set of equations between them (while recognising that the origin of the term, which I have always thought was #11, refers literally to the collection of models). But this should not in any way affect what one means by ‘subvariety’! The meaning of ‘sub’ depends on the morphisms; and just like a sublocale is not the same thing as a subframe (although that is not the best example, since a sublocale can also be viewed as a subposet in the frame direction), so a subvariety is not the same thing as a subset of operations and equations, even if one chooses to define ‘variety’ in a syntactic manner.
So for starters: we are all agreed that the category of monoids is not a subvariety of the category of groups?
I think so! The arrow goes the other way.
That example was what made me uneasy!
I am only looking at this thread now. I’d rather sooner than later see Varieties of algebras (toddtrimble) infect the Lab entry variety of algebra.
I agree. I glanced at a few of the linked entries at the bottom of the current version and it seemed to me that there would be no problem if Todd copied his version across, but I have not checked that in detail.
When we were having this discussion before, I knew that Toby was away, and thought it would be better to wait for him to come back before writing all over (or rewriting) the article, since he authored most of the current version.
I do feel that the way ’variety’ is defined in the article does not well reflect the way others use the term. I am thinking particularly of how those who have worked in categorical universal algebra, such as Adamek and Rosicky, use the term, where it is essentially a category or subcategory of algebraic models. It is not a signature and a set of axioms in that signature.
It sounds like we all agree about that now, including Toby.
Well, if there are no objections then, I can start revisions perhaps a little later today, with the warning that it might wind up being a major rewrite (with material currently there exported to some other place).
I wasn’t clear on where Toby stood, and there are presently passages like “Sometimes a variety of algebras is identified with its category of models in , but this is probably not wise…” which pull in the opposite direction from what you say we all agree on. Maybe I’m not understanding something.
I can’t speak for Toby, but I think it’s consistent to think that a variety should not be regarded as being its category of models in Set, and yet that the notion of “subvariety” is better defined in the “model-theoretic” direction than the “syntactic” one. Just as Toby said in #21: a locale is defined to be a frame and is not identified with its topological space of points (i.e. its poset of “models” in ), and yet the notion of “sublocale” is different from that of “subframe”, the former going in the “model-theoretic” direction.
I've rewritten variety of algebras about how I would have if I'd only seen Todd's #1–3. I have not tried to integrate this with Todd's rewrite. I wouldn't like to completely lose the elementary stuff in my version, but it's possible that this belongs on a more syntactically oriented page such as algebraic theory might be.
Thanks very much, Toby! I’ll think about how to blend in what I wrote on my web.
There is one other small point which I think is kind of pretty. The question arises about how one should define a model of variety in other category . In the present draft we have
A variety of algebras is traditionally identified with its category of models in Set (or even with simply the class of objects of this category), but it then becomes unclear what an algebra in the variety would be in some other category .
In some sense this depends on an ambient doctrine to which belongs (e.g., we could consider the doctrine of cartesian monoidal categories, or the subdoctrine of complete categories, etc.). In the general case of cartesian monoidal categories , it’s true that one has to work a bit, and wend one’s way back to some syntax, e.g., get our hands on the Lawvere theory giving rise to (correct me if I’m wrong, but I think we could use , that is, consider functors that preserve all limits and sifted colimits), followed by product-preserving functors .
But it’s much prettier if we take to belong to the doctrine of complete categories (for example, could be another variety). Here a model of variety in is just a functor that preserves all limits (or what is the same, a right adjoint , since is totally cocomplete). For example, if and are both varieties, we can define a symmetric monoidal product on varieties by the formula
where the equivalence comes about by playing with mates. That is, an -object of is a -object in , and this is an object of the variety . This would be an analogue at the level of varieties of the symmetric monoidal product on Lawvere theories, sometimes called the Kronecker product I believe, which is symmetric monoidal and which is the coproduct when restricted to commutative theories.
Yes, that is a very nice idea. And if we don't want to restrict ourselves to finitary varieties [there are actually several places where one traditionally takes things to be finite that need not be], then complete categories (or at least, categories with all products) are where we want to be.
There is a nice idea that I tried to explore some years ago. In this, using simplicial group(oid)s as models for homotopy types, I tried to find varieties of such things that corresponded to various interesting classes of homotopy type. For instance, one can relatively easily give the variety that corresponds to products of Eilenberg-Mac Lane spaces (i.e. those representable by a crossed complex). It would be neat to look at analogues of varieties in less algebraic models to homotopy types, and to see which are homotopically significant. (If this was explored, of course, the varieties we would need would not be Set-based.)
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