added (here) statement and proof that the compatibility of Kleisli composition under monad transformations passes to two-sided Kleisli categories if the transformation is compatible with the two distributive laws in the evident way

]]>added (here) statement of the two-sided (“double”) Kleisli category in the case of a comonad distributing over the given monad

]]>Have also expanded the proof (here).

]]>at the end of the proposition “Kleisli equivalence” (here) I have added a `tikzcd`

-diagram showing the component maps at a glance, including the reverse map on hom-sets (by precomposition with the unit)

and pointer to:

- Francis Borceux, pp. 191 in:
*Handbook of Categorical Algebra*, Vol 2*Categories and Structures*, Encyclopedia of Mathematics and its Applications**50**, Cambridge University Press (1994) [doi:10.1017/CBO9780511525865]

added pointer to:

- Thomas Streicher, pp. 54 in:
*Introduction to Category Theory and Categorical Logic*(2003) [pdf, Streicher-CategoryTheory.pdf:file]

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once written in detail the bicategory and some issues related to the bicategory of such mixed distributive laws in an unpublished preprint *Bicategory of entwinings*, arxiv:0805.4611 (The referee complained (in 2008) that the paper should be done with more categorical theory and less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

]]>