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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
I have touched formatting and wording of this entry, in the hope to increase readability.
In particular I have added explicit statement of the Kleisli equivalence as an explicit proposition (now here – previously there was just a proof, following no proposition statement).
Things left to do:
There is still a switch of notation from objects being denoted $M, N, \cdots$ to $X, Y, \cdots$.
The Idea-section states the universal property in a way hardly suitable for an Idea-section, but an essentially duplicate paragraph on the matter then does appear in the Properties section. I suggest the text in the Idea section be merged into that in the Properties section.
added pointer to:
I am wondering about the following somewhat vague question:
Given an adjoint pair $\Box \dashv \lozenge$ of a monad and a comonad on some category $\mathcal{C}$, I am looking at an application where one wants to “glue” (for lack of a better word) their Kleisli categories to a new category which fully contains both Kleisli categories, but in addition has morphisms going from one to the other by compositions of the $\Box$-counit with the $\lozenge$-unit.
While I can just define this, I am wondering if this construction has some good general abstract meaning. Is it just my intended application that makes me want to look at this construction, or do univeral algebraists arrive at the same notion (or something similar), on general grounds?
Asking Google this question, the engine suggests
On p. 2 (of 26) in this article it says that a category with morphisms of the form $\Box X \longrightarrow \lozenge Y$ has been considered in
Unfortunately, I have not found a copy of this article yet. But presumably the construction in question is that also found in
section 6 “Double Kleisli categories” of:
These double Kleisli categories might be what I need, using that necessity and possibility satisfy a distributive law in the ambidextrous case.
I found a link to Computational comonads and intensional semantics here. However, I don’t see the definition I would expect there (namely, that of “double Kleisli categories” in Monads and Comonads in Intensional Semantics), though their “computation comonads” in §4 seem related (consider a pointed functor rather than a monad).
Thanks!
I had seen that .ps
file earlier, but my network hadn’t allow me to access it, for some reason. Now I have gotten hold of it, have transformed it into a pdf
and have recorded it (here) at monad (in computer science) .
Yes, strange that they don’t say what Power and Watanabe credit them for.
Looking at their article now, for a moment I thought that their computational comonads include those obtained from ambidextrous adjunctions, with their “$\gamma$” being the additional unit map. But this does not seem to fit their axioms.
So when I express quantum measurement/state preparation via writer/reader-monads as shown here, then the construction looks quite reminiscent of the constructions involved in the “double Kleisli category” of Brookes & Van Stone 1993 (§6).
It feels like there should be more to this similarity. Possibly the “BvS double Kleisli category” for $\Box \dashv \lozenge$ on linear types over finite sets is the correct fully abstract incarnation of the category of quantum gates, in some sense.
But I still don’t understand the BvS double Kleisli category well enough (I mean, I certainly understand its definition and existence, but I am not sure yet about what its morphisms really “mean”).
[edit: I had two mistakes here: On the one hand my earlier diagram did not actually commute (this is fixed now), on the other hand the BvS construction does not actually apply to the situation (not sure what to make of that)]
31,32 the entire volume with the article in pdf is at http://library.lol/main/8D1FA6858DFA95CB60323AC67851C8C8
Mixed distributive laws are, of course, earlier, from early 1970s at least.
Many authors discuss distributive laws, but I was after the “double Kleisli category” induced by a distributive law. For this, the single reference that I am aware of, so far, remains Brookes & Van Stone (1993) §6.
@Urs: Harmer–Hyland–Melliès’s Categorical Combinatorics for Innocent Strategies and Garner’s Polycategories via pseudo-distributive laws are also references (there the construction is called the “two-sided Kleisli construction”). However, they are much later references and do not cite any other source for the construction.
One of the points of the Street’s 1972 JPAA article Formal theory of monads is that the distributive laws between monads are simply monads in the bicategory of monads, and the mixed distributive laws are simply monads in op-cop dual of that bicategory. So, in principle, one is just taking a Kleisli construction in that bicategory. But when writing explicitly out one just have comonad on a category with extra data writing out which is straightforward and than writing out the Kleisli (you call it co-Kleisli) category in this case.
P.S. In the case of algebras and coalgebras instead of general comonads and monads I have once written in detail the bicategory and some issues related to the bicategory of such mixed distributive laws in an unpublished preprint Bicategory of entwinings, arxiv:0805.4611 (The referee complained (in 2008) that the paper should be done with more categorical theory and less explicit methods and suggested to resubmit elsewhere with inclusion of such methods, but I left it as it is and did not publish.)
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