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I have added some accompanying text to the list of links at monad (disambiguation).
One question: in the entry Gottfried Leibniz it is claimed that the term “monad” for a functor on a category with monoid structure also follows Leibniz’s notion of monads. Is this really so? What’s a reference for this claim?
I am asking because I don’t see how the notion of monoid in the endomorphisms of a category would be related to what Leibniz was talking about. What’s the idea, if there is one?
I can see how the nonstandard analysis sense of monad relates to Leibniz’s, but the other seems very unlikely. Wasn’t that monad chosen because of monoid, itself not the greatest of names? Isn’t it pure chance that the infinitesimal shape modality is a monad?
So that’s what I am thinking, too, but I don’t have a source. Why was “monoid” called “monoid”? Was it after realizing it as a one-object category (“mono-object category”)?
The etymological explanation that I’ve heard (probably from Albert Burroni, but I’m not sure) is that Bénabou coined the term not based on the point of view of monad as a monoid in endofunctors, but as a lax functor of type 1 -> Cat. The idea of monad as ‘indivisible unit’ is meant refer to the domain of the functor.
4 this is for a monad; for monoid, I think it is Bourbaki.
Jonas, thanks! That makes sense. I have added a remark to this extent to monad (disambiguation). But might you have a reference for this?
Benabou defines monads as lax functors in section 5.4 of “Introduction to Bicategories” (1967), this could be the earliest reference.
Urs, you call a lax functor of type 1 –> Cat a “point” of Cat viewed as 2-topos. Do you have any intuition behind that? It does not seem to be a categorification of “point of a Grothendieck topos” as a geometric morphism from Set.
We shouldn’t give the impression that Leibniz took a monad to be some smallest part of space. Rather for him it’s a mind-like simple substance. Space is something of an illusion. The sense of it arises in the monad which is our soul by this monad’s ability to reflect the functioning of the other monads. These monads are not in any kind of spatial contact with each other, and have no influence on each other.
It’s a very odd system, supremely confidently delivered in The Monadology, as though there’s nothing surprising about it. Yet,
Bertrand Russell, for example, famously remarked in the Preface to his book on Leibniz that he felt that “the Monadology was a kind of fantastic fairy tale, coherent perhaps, but wholly arbitrary.” And, in perhaps the wittiest and most biting rhetorical question asked of Leibniz, Voltaire gibes, “Can you really believe that a drop of urine is an infinity of monads, and that each of these has ideas, however obscure, of the universe as a whole?” (Oeuvres complètes, Vol. 22, p. 434)
I’ve slightly modified accordingly.
I suspect Urs was thinking of global elements as “points”, but I don’t think there is any 2-topos involving lax morphisms.
Mike, I just said “canonical 2-topos” to indicate that this point (in the sense of global element, yes) is not a point in an arbitrary random context, but in what might be called the canonical context for lax points to be in.
I should have written “the 2-category underlying the canonical 2-topos”. But I see you don’t like mentioning 2-toposes here, and I won’t insist.
David, thanks for making this more accurate. Of course we might add with the discussion at The monad of Leibniz that if following Hegel in the Lawverian perspective as discussed there, then monads are $\flat$-modal units (if we follow the tradition of nonstandard analysis) or maybe $\flat_{inf}$-modal units (if we follow Hegel’s complaint about monads just being “für sich” and not “für eins” (no halo!)) and these could in particular be atoms of space, but could also be something very different, depending on which differential cohesion we have realized.
So there could be non-spatial forms of differential cohesion? Is this related to the way modalities need not be taken as spatial, ’It is necessarily the case that…’ rather than ’It is locally the case that…’?
Something I mean to think about is Neel Krishnaswami’s comment on modalities as reflections of judgements in related categories.
I just mean that the axioms are rather general. Already in simple cases like super-cohesion people wouldn’t generally agree that the $\flat$-modal types are atoms of “space”. They might complain that this superspace is an imaginary, hypothetical “space”, only. And this is still very close to the standard model. More general notions of cohesion should encode more exotic notions of “space” even.
I don’t see how “the 2-category underlying the canonical 2-topos” is any better, since as I said, I don’t know of any 2-topos which contains lax functors.
This issue is maybe not worth heavy discussion, but just to clarify what I actually said: there is a 2-topos $Cat$ and it has an underlying 2-category $Cat$ and we can consider lax functors $\ast \to Cat$ and it’s not unreasonable to call them points of $Cat$.
Oh, I misunderstood. I can see “lax point of the 2-category Cat”.
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