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started a Properties-section at Lawvere theory with some basic propositions.
Would be thankful if some experts looked over this.
Also added the example of the theory of sets. (A longer list of examples would be good!) And added the canonical reference.
I think it looks good. A few comments:
(1) Why not say explicitly that the theory of sets (or theory of equality) is $\mathcal{S} \simeq FinSet^{op}$? This is the free category with products on one generator.
(2) Why do you need to mention $A(\pi_i)$ in describing congruences? These projections are automatically among the morphisms $f: n \to 1$ in the syntactic category.
(3) Besides the formulation of free algebras in terms of formal expressions, you could also describe them as filtered colimits of finitely generated free algebras (induced by the filtered poset of finite subset inclusions), which are represented by the objects $n$ of the syntactic category $\mathcal{T}$:
$F(X) = colim_{[n] \subseteq X} \hom_{\mathcal{T}}(n, -)$and of course this filtered colimit is computed objectwise. (Each function $f: [m] \to [n]$ in $FinSet$ induces a map $m \to n$ in $\mathcal{T}^{op}$ by applying the functor $i^{op}: FinSet \to \mathcal{T}^{op}$; cf. point (1) above.)
(4) The relatively free construction $T_1$-Alg $\to T_2$-Alg induced by a theory map $T_1 \to T_2$ has an elegant decription in terms of the associated monads: for $X$ an $M_1$-algebra = model of $T_1$, just construct the tensor product $M_2 \circ_{M_1} X$ as the evident reflexive coequalizer
$M_2 M_1 X \stackrel{\to}{\to} M_2 X \to M_2 \circ_{M_1} X$in the category of $M_2$-algebras. This point came up before in a discussion you and I had about the adjunction between algebras and $C^\infty$-algebras.
Thanks, Todd.
(1) Why not say explicitly that the theory of sets (or theory of equality) is $\mathcal{S} \simeq FinSet^{op}$? This is the free category with products on one generator.
Did this now.
(2) Why do you need to mention $A(\pi_i)$ in describing congruences? These projections are automatically among the morphisms $f: n \to 1$ in the syntactic category.
Maybe I said this awkwardly (or maybe I am misunderstanding you): the way I put it was to say if equivalence holds for all projections, then it holds for all functions.
(3) Besides the formulation of free algebras in terms of formal expressions, you could also describe them as filtered colimits of finitely generated free algebras (induced by the filtered poset of finite subset inclusions), which are represented by the objects $n$ of the syntactic category $\mathcal{T}$:
$F(X) = colim_{[n] \subseteq X} \hom_{\mathcal{T}}(n, -)$and of course this filtered colimit is computed objectwise. (Each function $f: [m] \to [n]$ in $FinSet$ induces a map $m \to n$ in $\mathcal{T}^{op}$ by applying the functor $i^{op}: FinSet \to \mathcal{T}^{op}$; cf. point (1) above.)
Sure (this is actually the point of view that I emphasized recently elsewhere on the nLab at function algebras on infinity-stacks, since this is what shows that homming into the “$T$-line object” produces again a T-algebra). Okay, so I expanded on this aspect in the Lawvere-theory entry now.
4) […] This point came up before in a discussion you and I had about the adjunction between algebras and $C^\infty$-algebras.
Ah, right, thanks for reminding me. I had almost forgotten about that again. Will put that into the Lawvere-entry later.
Thanks, Todd.
Sorry if I came off as overly critical in my comments.
maybe I am misunderstanding you): the way I put it was to say if equivalence holds for all projections, then it holds for all functions.
Okay. I only meant that if the equivalence holds for all $f: n \to 1$, then of course it holds for all the projections, since these are among the $f$.
Sure (this is actually the point of view that I emphasized recently
In that case, sorry if I’m belaboring the obvious. It just seemed to me to be an elegant way of putting things.
No-no I wasn’t complaining! Quite the opposite. I am very grateful indeed for your comments! Hope you continue making comments!
Okay. I only meant that if the equivalence holds for all $f : n \to 1$, then of course it holds for all the projections, since these are among the $f$.
Let me see if I have this definition mixed up, I am not looking at the standard sources at the moment:
I want to say when an equivalence relation on the set $A(1)$ is called a congruence relative to the $T$-algebra structure on $A(1)$. So I want to say that the condition is that if all pairs of arguments of an operation are equivalent, then the condition is that also the result of the operation on the two sets of arguments are equivalent.
I still don’t see what you suggest should be or could be said differently. Sorry, I am probably being dense.
Could you just write out here exactly the way you would write the definition of congruence? Thanks.
My apologies, Urs – I completely misread what you wrote. Your definition is absolutely correct; I was the one who was dense here.
Each $a \in A(n) \cong A(1)^n$ may be regarded as an $n$-tuple $(a_1, \ldots, a_n)$. You are merely saying that if $a_i \sim b_i$ for $1 \leq i \leq n$, then we should have $f(a_1, \ldots, a_n) \sim f(b_1, \ldots, b_n)$ for every $n$-ary operation, and that of course is exactly the condition needed for congruence.
Okay, thanks. But clearly the way I wote it makes a very simple idea look overly opaque. I see if I can fix it now. Might have to go offline any second, though…
I had added one more reference to Lawvere theory.
But there should be more references given here. Any suggestions?
Sorry for the dumb question: Lie algebras are the algebras over a Lawvere theory, right?
Our entry Lawvere theory could do with some more discussion of classes of (non-)examples.
Yes, any structure specified by finitary operations on a single sort, and subject to universally quantified equations, is describable as an algebra over a Lawvere theory.
I’ll have a look later at examples.
I have added some more examples to Lawvere theory – Examples – Other examples.
(Of course there are many many examples, but it would be good to list those in common use, and it would be better to also list some non-examples in common use).
I added some non-examples, with commentary, including the HSP theorem.
Thanks Todd!
Would you mind if I move
the recognition theorem that you added here
its lead-in paragraph a bit further up
the HSP paragraph following it
alltogether to the section of Examples? In the section on the non-examples we could then still point to these statements.
Also, I made Birkhoff’s HSP theorem a link and created a stub for it.
Sure, that’s no problem. I’m at an airport and don’t have great access, otherwise I could do it myself. Anyway, please feel free!
Thanks!
So I went ahead and created one more subsection Characterization of examples built from part of the material that you had provided.
(Please double check one thing: the three clauses in HSP are not just necessary but also sufficient?)
Please also check what the subsection on the Non-examples now looks like.
Okay, I did a little bit of fine-tuning. Yes, those three clauses are necessary and sufficient (and sufficiency is the hard part). I titled the HSP theorem a theorem and not a proposition. Some minor rewordings. But thanks – the reorganization looks good!
I’d like to note that, interestingly, the first-order analogue of HSP is the characterization (see e.g. Chang and Keisler’s original text on continuous model theory) of elementary classes of structures: they’re precisely those closed under elementary substructures, elementary embeddings, ultraproducts, and ultraroots (if an ultrapower of something is in your class, that something was in your class.)
Thanks, Todd. I have added hyperlinks to “language” and to “structure. Then I have copied the statement of the HSP theorem also to Birkhoff’s HSP theorem.
Just to check if I am following: the clause”closed under homomorphic images” means that if $A$ and $B$ are elements of the class, and $\phi \colon A \to B$ is any homomorphism, then $im(\phi)$ should also be in the class, right?
Just to check if I am following: the clause “closed under homomorphic images” means that if $A$ and $B$ are elements of the class, and $\phi \colon A \to B$ is any homomorphism, then $im(\phi)$ should also be in the class, right?
Right. Of course normally we don’t say “homomorphic image” around here, we just say “image” (homomorphism or map being implied), but it was added just to help explain why it’s called the HSP theorem.
Could we say something about algebras with co-algebraic aspects to them being models of Lawvere theories. Never? Or sometimes after all?
Urs, I don’t think I understand the question. Could you give an example of what you mean?
Todd, sorry, i was just lazily shooting questions.
What I am really wondering is whether the model category of complete rational reduced simplicial commutative Hopf algebras is simplicial.
In that context I was wondering for a bit whether commutative Hopf algebras are models of a Lawvere theory. But I suppose from the criterion you gave it’s clear that they are not: the forgetful functor to underlying sets is not monadic, due to th co-operations in the algebra. Right?
Quite right. The terminal object in that category ought to be the ground field as Hopf algebra, and the underlying set of that is not terminal in $Set$, so the forgetful functor doesn’t preserve limits.
But the question you’re really wondering about is roughly speaking in an area I’d like to be studying soon myself (just as part of my general education).
Re #24: ah, that simplicial rational commutative Hopf algebras form a simplicial model category follows with Theorem 4 in Quillen’s “Homotopical Algebra” II.4 together with Proposition 2.24 in Appendix B of his “Rational Homotopy Theory”. I am adding this to the entry on model structure on simplicial algebras here.
And on p. 265 of “Rational homotopy theory”, Quillen comments on the issue with the Lawvere theories:
By a theorem of Lawvere a category closed under limits and having a small projective generator is a category of universal algebras and conversely. Therefore although $[$ complete Hopf algebras$]$ is not a category of universal algebras, it is not far from being one.
Where I gather the point is that complete rational Hopf algebras are closed under limits and have a projective generator, just not a small one.
A model of a Lawvere theory is a quantity modelled on said theory.
The article Lawvere theory so far does not mention this basic connection.
Should it mention it?
If so, I would add a few words on this, and a link to space and quantity.
this basic connection.
What do you have in mind?
Oh, maybe I see what you had in mind.
Yeah, I dunno. So the idea I suppose is to treat the objects of the theory as “test spaces”. The usual examples of space vs. quantity seem to cluster around theories of commutative algebra type, such as in algebraic geometry where we are considering the theory of commutative algebras over a ground field $k$. Of course the theory itself is the category opposite to the category of finitely generated free objects, so here the opposite of the category of polynomial rings $k[x_1, \ldots, x_n]$, which can be considered the category of affine spectra $k^n$ as the category of test spaces.
Now it may be (I actually don’t know) that Lawvere is in the habit of pushing the envelope and considering not just the usual examples, but arbitrary theories. Are there useful theorems in that generality? If there are, then I would definitely consider adding such to the article. But if not, then I guess one should weigh how useful such remarks would be.
Re #29:
this basic connection.
What do you have in mind?
For the time being I have in mind only rather superficial remarks, pointing out
A model of a Lawvere theory is a quantity modelled on said theory.
What I wrote there seems literally true (relative to the two nLab articles space and quantity and Lawvere theory, if “is” is taken literally, not to mean “is equivalent to”; rather, “is” is only to say: “every model of a Lawvere theory $T$ is a quantity modelled on $T$, but with the additional requirement that the quantity be product-preserving.
For the time being, I will not have time for doing much work on this, but my suggestion was meant in the sense that
For the time being, the suggestion was only to sprinkle a few connecting words here and there in both articles.
Oh, maybe I see what you had in mind.
It should be the following simple and general idea:
Given any small category $\mathcal{C}$ thought of as a category of “local models for some type of space” then
presheaves on $\mathcal{C}$ (possibly satisfying conditions, such as being sheaves) are generalized spaces modeled on $\mathcal{C}$
co-presheaves on $\mathcal{C}$ (possibly satisfying conditions, such as preserving products) are generalized quantities modeled on $\mathcal{C}$.
The theorem which is meant to generalize the familiar relation between commutative rings and varieties to this generality of quantity and space is Isbell duality.
It’s long ago that I looked into editing the relevant entries. But if this state of affairs does not come out in (all) of them, then I agree it would be good to add missing comments to this extent!
I think it’s a beautiful and important picture that helps put into context a lot of what is going on in mathematics and in mathematical physics.
Urs: yes. But this is in the context of any small category $C$. What sort of extra mileage is there when $C$ is a Lawvere theory? To what extent does it throw extra light on Lawvere theories?
It’s just that models of Lawvere theories are examples of quantities in this sense. My understanding was that this is what Peter wanted to highlight in the entry.
Well, it obviously wouldn’t do any harm to add a few remarks. But I’m still interested in learning of answers to the extra mileage question. And it could be there’s something there.
For example, what are the fixed points for the Isbell adjunction in the case where $C$ is a Lawvere theory? Do they include models of the theory, in general?
That would be good, yes. I don’t know.
Back then Andrew was thinking about something related when he wrote Isbell envelope and Frölicher spaces and Isbell envelopes.
The question I asked aloud in #35 seems to turn up nothing too great so far. I was vaguely thinking that the contravariant Isbell adjunction for a Lawvere theory $T$ whose functors might suggestively be denoted
$Spec: Set^T \to Set^{T^{op}}, \qquad \mathcal{O}: Set^{T^{op}} \to Set^T$that are defined by $Spec(F)(t) = Set^T(F, T(t, -))$ and $\mathcal{O}(X)(t) = Set^{T^{op}}(X, T(-, t))$, might have $T$-algebras $A$ (product-preserving functors $A: T \to Set$) among its fixed points. This seems hopelessly wrong, even in the classical case of $T$ = theory of commutative rings, and even if we change the doctrine from finite product theories to say finite limit theories.
So at issue is whether the unit
$i_A: A \to \mathcal{O} Spec A$is an isomorphism for $T$-algebras $A$. This is the same as asking whether
$Spec: Alg_T^{op} \to Set^{T^{op}}$is fully faithful. But it’s just not: if for example we consider $T$ the theory of commutative rings, and a simple commutative ring (a field) $A$ of large cardinality, then there are no $T$-algebra maps $A \to T(t, -)$. This argument is robust whether we consider that the theory of commutative rings $T$ is qua the doctrine of finite product theories (i.e., as a Lawvere theory) or the doctrine of finite limit theories. It also applies to other theories like the theory of groups, either as finite product or finite limit theories.
The situation probably improves if we jump up to small limit theories, although we might have to watch out for set-theoretic issues.
I corrected the comment on the terminal Lawvere theory to this:
“The terminal Lawvere theory has exactly one morphism $f \colon m \to n$ for each $m,n$. This is not the theory of idempotent commutative monoids, as some have conjectured. Since there is a unique morphism between each pair of objects, each object is in fact isomorphic. Thus, this theory is a terminal category. Algebras of the terminal Lawvere theory are terminal sets, singletons.”
Thanks, Joe. “… as some have conjectured” seems a little too grand for what I think is a case of “… as someone hastily assumed” – I wouldn’t even mention the mistake.
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