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New stub for the notion of clone in universal algebra, including a reference relating it to operads.
Zoran: the categorical form of the full clone on a functor was one dual form of categorical shape theory if I remember it rightly.
I rediscovered this page from an MO citation.
Where on earth did the word “clone” come from?
And if a clone is really the same as a Lawvere theory, why not merge it with Lawvere theory and add a redirect?
I don’t know where ’clone’ came from, but it’s been around a rather long time, before Lawvere’s thesis I believe.
Clones and Lawvere theories and cartesian operads are virtually the same; you could say that a clone is to a Lawvere theory as a multicategory is to a monoidal category (so a clone is something like a cartesian multicategory with one object). Thus I believe clones and cartesian operads are synonyms, with maybe a slight difference in how they are usually packaged. I have notes on this here.
The ‘full clone’ on a functor was a term that occurred in describing categorical shape theory, but I would have to check up how! I seem to remember the use involved Fred Linton’s work on theories. (Edit: It seems I have repeated myself … At least I said the same thing both times!)
Todd, maybe you could add some of these remarks as comments on MO. What I wrote in my answer is literally everything that I know about clones (I’ve learned about them solely because I wanted to understand the example of affine spaces), so I’m not familiar with all these subtleties.
Well, we also have cartesian multicategory, so if that would be more appropriate we could redirect “clone” there. But it doesn’t make sense to me to have two pages with different names about essentially the same object.
Before we do that, let’s pause and consider again the analogy, written as the proportion clone: cartesian multicategory :: Lawvere theory: cartesian monoidal category, where redirecting clone to cartesian multicategory would be analogous to redirecting Lawvere theory to cartesian monoidal category. I wouldn’t think we’d want to do the latter.
An even better analogy is clone : cartesian multicategory :: operad : symmetric multicategory, and the latter two are separate pages. So, I guess. But I hate to have so much duplication; I notice that we also have Lawvere theory and algebraic theory as separate pages. At least the pages should link to each other; I added some.
Mike, thanks for adding those links.
As for Lawvere theory and algebraic theory: the latter was written after the former as a separate article because the full generality of algebraic theory (of unbounded rank) could not of course be written under Lawvere theory which is traditionally refers just to the finitary single-sorted case. The connection between the two is implicitly acknowledged by adopting the phrase “large Lawvere theory” (red herring-like). Personally, I think it’s okay on occasion to admit some redundancy; the articles are slightly different in style, with the Lawvere theory article giving some extra intuition and motivation – appropriate for readers encountering the categorical concept first in its baby form – whereas the algebraic theory article is somewhat more hard-core in style.
There’s yet another article, infinitary Lawvere theory. I believe Andrew Stacey started that (and I see you were the last editor). I had wanted to pursue a different approach but didn’t want to write all over what I regarded as Andrew’s (couldn’t think of how to do it gracefully), so I started algebraic theory also for that reason.
Edit: Apologies; it wasn’t started by Andrew; it was started by Toby.
You make good points. I would like it, though, if we could have some more obvious way of pointing out at the beginning of an article that some other page(s) are about “almost the same subject”, as opposed to merely a related notion, so that someone finding the page from elsewhere won’t miss out on something that they would probably be interested in just because it’s on a page with a different name.
The Idea and Definition sections of the page clone are a bit contradictory. Is there a difference between an “abstract” clone (= one-object cartesian multicategory) and a “concrete” clone (consisting of operations on a particular set)?
The definition section reads weirdly to me. It might be possible to embed any abstract clone into an endomorphism clone (even that’s not so clear to me), but if that’s so, I’d think there should then be many ways of doing it. Do we then make a distinction between the resulting subclone supported on an set $S$ and the resulting one supported on a different set $S'$? (There could even be abstractly isomorphic but different such subclones supported on the same set.)
I’m thinking the definition section should be rewritten. Miles Gould’s thesis seems as good a source as any.
Hm, I’m not sure I agree. A group is a group, regardless of whether we define it in terms of multiplication and inversion or in terms of some other operation like $(g,h)\mapsto g h^{-1}$.
Or, closer to home, an operad is an operad, whether we present it with May-style composition $g \circ (f_1,\dots, f_n)$ or Markl-style composition $g \circ_i f$.
That’s true. Perhaps it would fairer to say that an abstract clone is an alternate presentation of a cartesian operad (cf. the one on [cartesian multicategory]). I do think “abstract clone” should be taken to mean the specific presentation: otherwise there is no distinction between abstract clones, cartesian operads, and Lawvere theories — whereas, in practice, the different perspectives are helpful. In this case, perhaps the page for abstract clone and for cartesian multicategory should be merged, with abstract clone given its own subsection on the cartesian multicategory page?
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