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I have tried to brush-up Kleisli category; also made Kleisli composition redirect to it and cross-linked with monad (in computer science)
You mean Kleisli composition.
I did some editing at Kleisli composition. Probably I should have checked in with Urs before doing this, but I believe that “algebra of a monad” is much more common and familiar than “module of a monad”, and so I interchanged the order of those two words throughout the article. We should probably discuss this anyway. I also fixed a few sentences (one was missing some words).
but I believe that “algebra of a monad” is much more common and familiar than “module of a monad”
This depends on a community. In pure category/algebra community yes, but in geometry community the other way around. But then module over a monad not “of a monad”.
At the definition of Kleisli composition, what does the phrase
as in the Grothendieck construction
mean?
The definition of composition in the Grothendieck construction bears some similarity to Kleisli composition, but I haven’t been able to see exactly why.
Probably because they are both lax colimits.
@Zoran #4: that’s interesting; I wasn’t aware of that. Do you know anything about the history of this? Because “algebra of a monad” (or over a monad) has been around for more than 45 years; since the geometry community presumably knew this, it sounds as if they deliberately decided to break with that usage. (This is not to say that I think “module” is an illogical choice, although there is some potential for confusion, as when one speaks of a module over an algebra of an operad.)
I may be wrong but I thought that the use of ’module over a monad’ crept in from the close link between operads and monads.
@Mike #7: that’s what I was thinking too.
I thought it would have been the link between monads and monoids myself. People never seem to say ’algebra over a monoid’ (although they do say ’algebra over an operad’).
I should have checked in with Urs before doing this,
I am fine with this. Did I even write the piece that you changed (maybe I did, I haven’t checked, I don’t rememeber).
The terminology issue with algebras/modules over monads is old I thought there is a discussion at algebra over a monad, but maybe there is not.
Anyway, both terms have their perfect justification given the two different perspectives on monads: externally its a monoid that has modules, internally it’s a something that has algebras. Seems to me to also match the two different points of views exposed at the very entry Kleisli category.
Yes, I think it is deliberate. Namely, Grothendieck has thought that the geometry should be concentrated not on the properties of spaces, but properties of morphisms of spaces (relative point of view). Thus one considers affine morphisms generalizing affine schemes (the latter means over Spec Z). Now the affine $k$-scheme is a spectrum of a $k$-algebra. Its category of quasicoherent sheaves of $\mathcal{O}$-modules is the category of modules over the monad induced by the algebra in the base category of quasicoherent sheaves over $Spec k$, what is nothing other than the category of $k$-vector spaces. Here clearly modules are the appropriate ones. Now if one relatives over any base scheme $S$ then the relative affine $S$-schemes will have quasicoherent sheaves given by a monad in the base category of quasicoherent modules. This point of view and terminology is most notably pronounced in Deligne’s 1988 Categories Tannakiennes in Grothendieck Festschrift. This or that way the rings and algebras in the geometry over a field, in relative setup become monads, and the modules over the former and modules over the latter are both the quasicoherent (sheaves of $\mathcal{O}$-) modules. The role of algebras as affine objects and the role of modules as quasicoherent modules are clearly distinguished in geometry and calling the latter ones algebras would make a mess in geometric terminology.
@Tim #9, I have always heard “algebra over an operad” too.
Zoran, that sounds very similar in spirit to the less elaborate example given earlier: that a module over an algebra $A$ is the same as an algebra of the monad $A \otimes_k -$, but to say ’algebra’ over an algebra is inviting confusion. If one’s focus is on such restricted types of monad, I can see why one would feel strongly about saying ’module’ instead.
Like Tim, I think I have heard ’module over an operad $C$’, particularly in the context of considering actions from the other side $- \circ C$ (where $\circ$ denotes the substitution product on species), but in my experience “algebra over an operad” is much more usual for actions from the ’usual’ side, $C \circ -$.
15: well, the term “algebra of/over a monad” is also from a restricted class of examples of monads: finitary monads in $Set$ aka algebraic theories which lead to algebras in the sense of universal algebra. That is the reason for the term, as stressed by Janelidze. The monads $A\otimes_k$ exhaust all monads if $k$ is a field, but the quasicoherent sheaves picture is true (modules in the monad sense correspond to qcoh sheaves over the relative affine scheme) much more generally. In fact there is a slight catch: the affine morphism correspond to monads which have a right adjoint functor (hence come from an adjoint triple). For cohomological purposes the case of monads without a right adjoint is equally good (Rosenberg calls that case “almost affine”).
This looks as if we should check that both terminology is used and explained somewhere in the entry. (I have not check to see if it has been.) There is the fact that operads were more often linear in their uses in algebraic topology and that May (pun intended) be why the linearised ‘module’ was introduced. Clearly both are used.
well, the term “algebra of/over a monad” is also from a restricted class of examples of monads: finitary monads in $Set$
But that’s just not true! From the very beginning (Eilenberg-Moore, 1965, at the very least), it’s meant something much broader: the operations can be infinitary (maybe even a proper class of arities), and over many other categories besides $Set$. The way you write, it sounds like you might be thinking of Lawvere theory.
Or maybe you’re just talking about where the motivation to use the word ’algebra’ came from. Partly from universal algebra, surely – but I cannot believe Janelidze completely here since equational varieties with infinitary operations were considered long before the categorical concepts came along. And the scope of the general idea, extending beyond the case over $Set$, was surely appreciated well before Eilenberg-Moore. Where exactly does Janelidze say this?
Or maybe you’re just talking about where the motivation to use the word ’algebra’ came from
Yes, that what we are talking about, the historical explanation why choosing one or another terminology. I discussed with him using term module and he is very much against what I consider the geometric terminology, because “these are algebras”, because they are algebras in universal algebra what it the principal historical class of examples in his view.
since equational varieties with infinitary operations were considered long before the categorical concepts
Were these also called varieties of algebras ? If so, an argument in his favor.
Zoran, it sounds like our wires are crossed. In #17 you said, “well, the term “algebra of/over a monad” is also from a restricted class of examples of monads: finitary monads in Set” (my emphases), and I was arguing against that restricted class as the sole source of the term ’algebra’. If Janelidze thought that the etymology referred to that restricted class, then I would say he is wrong, since for one thing infinitary algebras were well-known to everyone in 1965.
My guess is now that he had no such restriction in mind.
@Mike #7: Ah, so in fact they are both the same construction? I didn’t know lax colimits could be so easy to compute! (Is there a reason why Grothendieck construction only talks about pseudofunctors instead of lax functors in general?)
@Zhen: If you take a monad T as a lax functor $\mathbf{1} \to Cat$, then its Grothendieck construction is indeed the Kleisli category (as long as the morphisms are of the form $a \to T f (b)$, not $T f (a) \to b$, of course), although I can’t say off the top of my head what exactly its universal property is. The Grothendieck construction for lax functors, and more generally normal lax functors into Prof, as described at Conduche functor, isn’t really talked about much, possible because it’s not as useful or important as the sort that gives rise to fibrations. But maybe that stuff at Conduche functor could be moved to or linked to by Grothendieck construction.
Yes, they are instances of the same construction. Finn is probably right that being less useful is why the version for lax functors isn’t discussed as much. There is even a version for functors valued in Prof rather than Cat.
Thomason in his famous paper uses Grothendieck construction for lax functors.
Universal property of Kleisli and Eilenberg-Moore constructions in 2-categorical world can be found in Street’s 1972 paper Formal theory of monads in JPAA, see ref. under monad. Lack has written a paper few years ago in which he studies these constructions in terms of more elementary lax limits.
It might be that Janelidze included varieties of infinitary algebras if they are also called algebras, I do not know, maybe our discussion was incomplete in this respect and I had a bit more restricted impression. Still it is a different class of examples.
added to Ideas section about how the Kleisli category answers the converse question to the result that every adjunction gives rise to a monad (this is the context in which Kleisli introduced this notion)
Thanks!
I have taken the liberty or re-ordering the Idea-section, keeping the simple description at the beginning and your universal characterization afterwards.
In fact the universal characterization deserves to be (re-)stated in the Properties-section of the entry with some indication as to its proof, or at least with a reference.
have restated with reference to proof as suggested in #27
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