added pointer to:

- David Jaz Myers, §2.3 of:
*Categorical systems theory*, book project [github, pdf]

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once written in detail the bicategory and some issues related to the bicategory of such mixed distributive laws in an unpublished preprint *Bicategory of entwinings*, arxiv:0805.4611 (The referee complained (in 2008) that the paper should be done with more categorical theory and less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.)

Thanks for the further references! I have added pointer to Garner’s article here.

]]>@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 However, they are much later references and do not cite any other source for the construction.

]]>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.

]]>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.

]]>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)]

]]>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.

]]>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).

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

- John Power, Hiroshi Watanabe,
*Combining a monad and a comonad*, Theoretical Computer Science**280**1–2 (2002) 137-162 [doi:10.1016/S0304-3975(01)00024-X]

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

- S. Brookes, S. Geva,
*Computational comonads and intensional semantics*, Proc. Durham Conf. Categories in Computer Science (1991)

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:

- Stephen Brookes, Kathryn Van Stone,
*Monads and Comonads in Intensional Semantics*(1993) [dtic:ADA266522, pdf]

These double Kleisli categories might be what I need, using that necessity and possibility satisfy a distributive law in the ambidextrous case.

]]>added pointer to:

- Mark Kleiner,
*Adjoint monads and an isomorphism of the Kleisli categories*, Journal of Algebra Volume**133**1 (1990) 79-82 [doi:10.1016/0021-8693(90)90069-Z]

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.

have restated with reference to proof as suggested in #27

]]>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.

]]>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)

]]>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.

]]>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.

]]>@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.

]]>@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?)

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.

]]>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.

]]>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?

]]>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.

]]>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”).

]]>