Talk:Monad (category theory)

Module (category theory)

Hi there! Does the concept of Module (category theory) make any sense to you? If so, would anybody write an article about this? This article seems to be requested on Wikipedia:Requested articles/mathematics. I suspect the person requesting this article confused something, but I could be wrong. Thanks. Oleg Alexandrov 21:45, 28 Feb 2005 (UTC)

Traditional modules can be thought of as functors by taking a functor from a ring (considered as a category with one element) to abelian groups. One can generalize to modules in other categories and this is mentioned on the module page under generalizations.

And more -- see module at the nLab linas (talk) 15:22, 2 September 2012 (UTC)

Examples

Can people provide more examples? 137.111.90.68 23:30, 1 November 2007 (UTC)

Confusing sentence?

"While monads are quite common, making them explicit is less so (the language belongs to the school of Mac Lane, and has rarely been used in the school of Grothendieck, which prefers to write out monads and comonads longhand)." What does it mean, to "write out monads and comonads longhand"? Can anyone clarify? Thanks Sam Staton (talk) 15:19, 23 May 2008 (UTC)

It just means writing down formulae like f_*f^*, for operations on sheaves (usually), rather than saying explicitly that these are adjoint functors and we are taking the (co)monad of the adjunction. Charles Matthews (talk) 07:23, 25 May 2008 (UTC)

Ah, thanks. I've had a go at clarifying that sentence... Sam Staton (talk) 10:23, 25 May 2008 (UTC)

"See the article on natural transformations for the explanation of the notations Tμ and μT"

can't find the definition there. — Preceding unsigned comment added by StarlingX (talk • contribs) 08:03, 28 November 2011 (UTC)

Tμ and μT are the natural transformations which come from the horizontal compositions of μ with the identity natural transformation on T. Alternatively, they can be thought of in terms of components, so TμX is T(μX) and μTX is μ(TX). Hope this helps.

Monad and Kleisli triple are different structures

In the introduction to this article it is said that a Kleisli triple is also an endofunctor together with two natural transformations just like a monad. But this is factually wrong although we can construct a Kleisli triple from a monad and vice versa. A Kleisli triple is a map of objects (not a functor which would also comprise a map of morphisms) together with a map between hom sets and a natural transformation. From my point of view it would make sense to either have a section devoted to Kleisli triples and their relationship with monads or remove the mention of a Kleisli triple completely. — Preceding unsigned comment added by 134.106.29.176 (talk) 14:57, 31 October 2012 (UTC)

Looks like its fixed to me, and Kleisli triple now covers the topic. 99.153.64.179 (talk) 19:10, 25 June 2013 (UTC)

Etymology?

Are we sure that MacLane intended the term in its philosophical sense? I'd always heard that it was simply a portmanteau of monoid and triad (as triad and triple were common terms for this concept early on, and the connection to monoids is obvious). Cgibbard (talk) 04:41, 23 March 2014 (UTC)

I agree with Cgibbard. The quote referenced in the article explains nothing about Mac Lane's etymological intention. Suppose the concept of monad in category theory has any implications for monad in philosophy, it has nothing to do with its etymology by itself. In terms of etymology, a comment in this page is a good summary, and at least, there seems to be no reason to credit monad in category theory to monad in philosophy. There should be some concrete source to refer to monad in philosophy in the article. --Nolze (talk) 13:50, 13 December 2014 (UTC)

Does Eilenberg-Moore category deserve its own article?

Kleisli category is a separate article, whereas Eilenberg-Moore category redirects to Monad. I'd like to add an article for E-M too, but I'd like some feedback first on whether it's warranted. askyle (talk) 15:50, 20 May 2014 (UTC)

Unclear subsection

"The importance of the definition comes in a class of theorems from the categorical (and algebraic geometry) theory of descent. What was realised in the period 1960 to 1970 is that recognising the categories of coalgebras for a comonad was an important tool of category theory (particularly topos theory). The results involved are based on Beck's theorem. Roughly what goes on is this: while it is simple set theory that a surjective mapping of sets is as good as the equivalence relation "x is in the same fiber as y" on the domain of the mapping, for geometric morphisms what you should do is pass to such a coalgebra subcategory."

This might sound nice to an expert, but it actually makes no sense to someone who is not already on a high level in category AND topos theory. I would recommend to rewrite or cancel it. Saves this gibberisch for the nLab. — Preceding unsigned comment added by 37.120.66.60 (talk) 16:48, 3 November 2015 (UTC)

Co-algebras and comonads

Hi everyone,

Does anybody know of a reference which discusses coalgebras over (co)monads? I'm wondering how hopf-algebras integrate into this framework. Wundzer (talk) 16:43, 21 January 2021 (UTC)