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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJan 14th 2019
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   Author: Mike Shulman
   Format: MarkdownItexAdd a note about "Daniel's answer" to the semantics-structure question. The discussion on this page should really be merged into the main text and archived at the forum. <a href="https://ncatlab.org/nlab/revision/diff/monadic+adjunction/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/monadic+adjunction/18">v18</a>, <a href="https://ncatlab.org/nlab/show/monadic+adjunction">current</a>

   Add a note about “Daniel’s answer” to the semantics-structure question. The discussion on this page should really be merged into the main text and archived at the forum.

   diff, v18, current

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 14th 2019
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   Author: David_Corfield
   Format: MarkdownItexI was just wondering yesterday when the Kleisli and Eilenberg-MacLane catgories coincide. Is the answer [here](https://math.stackexchange.com/questions/2316104/when-is-the-kleisli-category-equivalent-to-the-eilenberg-moore) the best one can say, where what you call "cosemantics" earlier coincides with semantics?

   I was just wondering yesterday when the Kleisli and Eilenberg-MacLane catgories coincide. Is the answer here the best one can say, where what you call “cosemantics” earlier coincides with semantics?

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJan 14th 2019
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   Author: Mike Shulman
   Format: MarkdownItexI don't know any better condition.

   I don’t know any better condition.

4. • CommentRowNumber4.
   • CommentAuthorAlec Rhea
   • CommentTimeJan 15th 2019
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   Author: Alec Rhea
   Format: MarkdownItexAdded discussion to main body of page as suggested by Mike, archived discussion at nForum page. <a href="https://ncatlab.org/nlab/revision/diff/monadic+adjunction/19">diff</a>, <a href="https://ncatlab.org/nlab/revision/monadic+adjunction/19">v19</a>, <a href="https://ncatlab.org/nlab/show/monadic+adjunction">current</a>

   Added discussion to main body of page as suggested by Mike, archived discussion at nForum page.

   diff, v19, current

5. • CommentRowNumber5.
   • CommentAuthorAlec Rhea
   • CommentTimeJan 15th 2019
   • PermaLink
   Author: Alec Rhea
   Format: TextI merged the contents of the discussion into the main body of the page -- I am not sure if this is where you meant for the conversation to be archived, but for posterity I'm copy-pasting it here. ## Discussion [[John Baez]]: We read above: "One can turn [[monads]] into [[adjunctions]] and adjunctions into monads, but one doesn\'t always return where one started." This suggests that there is something like an _adjunction_ between monads and adjunctions! What's the precise story? I imagine something like this: there's a functor (or 2-functor) <latex> [adjunctions] \to [monads] </latex> and this has left and right adjoints (or 2-adjoints) <latex> Kleisli : [monads] \to [adjunctions] </latex> <latex> Eilenberg Moore: [monads] \to [adjunctions] </latex> As evidence, note that the [[Kleisli category]] gives the initial object among adjunctions that give rise to a specified monad, while the [[Eilenberg-Moore category]] gives the terminal one. See the [Wikipedia article](http://en.wikipedia.org/wiki/Monad_%28category_theory%29#Monads_and_adjunctions). Zoran &#352;koda: the best-well-known article on monads, R. Street, _Formal theory of monads_, JPAA 2, 149--168, 1972, has the basic fact that Eilenberg-Moore construction as a correspondence from monads to categories, extends to a 2-functor which is right 2-adjoint to the trivial monad 2-functor and the left adjoint is the underlying 2-functor (forgetting actions). Now you are trying to view EM construction as an adjoint to adjunctions to monads correspondence. First of all, I could imagine several different 2-categories of adjunctions, hence several different questions in the game (and monads make 2-category in more than one way, but with lessvariety than adjunctions). But there are better people to ask about this... [[Mike Shulman]]: This adjunction does exist (although you sometimes have to be careful about size issues) and is called the **semantics-structure adjunction**. The EM-category functor $Mnd(C) \to RAdj/C$ is called the "semantics" functor (here $RAdj/C$ is the subcategory of $Cat/C$ consisting of the right adjoints) and its left adjoint (the monad underlying an adjunction) is called the "structure" functor. In fact, the structure functor is defined on a larger subcategory of $Cat/C$, namely those functors $g:A\to C$ such that $Ran_g g$ exists (if $g$ has a left adjoint $f$ then $Ran_g g$ always exists and is equal to $g f$). In this case $Ran_g g$ is the image of $g$ under "structure", also called its *codensity monad*. Presumably by duality, "structure" also has a left adjoint "cosemantics" given by the Kleisli construction. The semantics-structure adjunction can be found in Chapter II of Dubuc's "Kan Extensions in Enriched Category Theory" and also in section 2 of "The Formal Theory of Monads". [[Emily Riehl]]: I'm interrupting Mike to comment on duality. The Kleisli construction is really a functor $Mnd(C)^{op} \to C/LAdj$. A monad map $(C,S) \to (C,T)$ consists of a 2-cell $T \Rightarrow S$ satisfying conditions. This defines a map of Kleisli categories $C_T \to C_S$ that commutes with the left adjoints but not with the right adjoints (that apply the monads to objects to get back to $C$). I didn't check all the details but it does seem to be the case that "structure" and "cosemantics" are mutual left adjoints, with adjunct maps defined in exactly the way you'd expect. [[Mike Shulman]]: If anyone can give a nice conceptual explanation of the terms "semantics" and "structure" in this context, Daniel Schaeppi is currently writing a paper which could benefit from such an explanation. I've never found anyone who really explains the words (nor is it entirely clear who pioneered their use in this context---Dubuc perhaps?) It makes sense to me that the E-M category can be called the "semantics" of a monad, but my intuition for "structure" is fuzzier, except that of course any monad can be regarded as a notion of [[stuff, structure, property|structure]] with which one can equip objects. But why is that "the" structure associated to an adjunction? [[Mike Shulman]]: In case anyone is following this, Daniel has come up with what seems to be the right answer to this question; I'm hoping that eventually he will write about it here. [[Aron Fischer]]: I'm interested in this too. Who is Daniel? or rather: Hi Daniel! can you write up your answer please? ;) [[Mike Shulman]]: The important parts of this discussion should be merged into the page, and the discussion archived at the nForum. But I don't have time to do that right now, so I'll just record the fact that Daniel's answer appeared in [Tannaka duality for comonoids in cosmoi](https://arxiv.org/abs/0911.0977), at the beginning of section 5 (quoted in full [here](https://mathoverflow.net/a/320835/49)).

   I merged the contents of the discussion into the main body of the page -- I am not sure if this is where you meant for the conversation to be archived, but for posterity I'm copy-pasting it here.
   
   ## Discussion
   
   John Baez: We read above: "One can turn monads into adjunctions and adjunctions into monads, but one doesn\'t always return where one started." This suggests that there is something like an _adjunction_ between monads and adjunctions! What's the precise story?
   
   I imagine something like this: there's a functor (or 2-functor)
   
   $$ [adjunctions] \to [monads] $$
   
   and this has left and right adjoints (or 2-adjoints)
   
   $$ Kleisli : [monads] \to [adjunctions] $$
   
   $$ Eilenberg Moore: [monads] \to [adjunctions] $$
   
   As evidence, note that the Kleisli category gives the initial object among adjunctions that give rise to a specified monad, while the Eilenberg-Moore category gives the terminal one. See the [Wikipedia article](http://en.wikipedia.org/wiki/Monad_%28category_theory%29#Monads_and_adjunctions).
   
   Zoran Škoda: the best-well-known article on monads, R. Street, _Formal theory of monads_, JPAA 2, 149--168, 1972, has the basic fact that Eilenberg-Moore construction as a correspondence from monads to categories, extends to a 2-functor which is right 2-adjoint to the trivial monad 2-functor and the left adjoint is the underlying 2-functor (forgetting actions). Now you are trying to view EM construction as an adjoint to adjunctions to monads correspondence. First of all, I could imagine several different 2-categories of adjunctions, hence several different questions in the game (and monads make 2-category in more than one way, but with lessvariety than adjunctions). But there are better people to ask about this...
   
   Mike Shulman: This adjunction does exist (although you sometimes have to be careful about size issues) and is called the **semantics-structure adjunction**. The EM-category functor $Mnd(C) \to RAdj/C$ is called the "semantics" functor (here $RAdj/C$ is the subcategory of $Cat/C$ consisting of the right adjoints) and its left adjoint (the monad underlying an adjunction) is called the "structure" functor. In fact, the structure functor is defined on a larger subcategory of $Cat/C$, namely those functors $g:A\to C$ such that $Ran_g g$ exists (if $g$ has a left adjoint $f$ then $Ran_g g$ always exists and is equal to $g f$). In this case $Ran_g g$ is the image of $g$ under "structure", also called its *codensity monad*. Presumably by duality, "structure" also has a left adjoint "cosemantics" given by the Kleisli construction. The semantics-structure adjunction can be found in Chapter II of Dubuc's "Kan Extensions in Enriched Category Theory" and also in section 2 of "The Formal Theory of Monads".
   
   Emily Riehl: I'm interrupting Mike to comment on duality. The Kleisli construction is really a functor $Mnd(C)^{op} \to C/LAdj$. A monad map $(C,S) \to (C,T)$ consists of a 2-cell $T \Rightarrow S$ satisfying conditions. This defines a map of Kleisli categories $C_T \to C_S$ that commutes with the left adjoints but not with the right adjoints (that apply the monads to objects to get back to $C$).
   
   I didn't check all the details but it does seem to be the case that "structure" and "cosemantics" are mutual left adjoints, with adjunct maps defined in exactly the way you'd expect.
   
   Mike Shulman: If anyone can give a nice conceptual explanation of the terms "semantics" and "structure" in this context, Daniel Schaeppi is currently writing a paper which could benefit from such an explanation. I've never found anyone who really explains the words (nor is it entirely clear who pioneered their use in this context---Dubuc perhaps?) It makes sense to me that the E-M category can be called the "semantics" of a monad, but my intuition for "structure" is fuzzier, except that of course any monad can be regarded as a notion of structure with which one can equip objects. But why is that "the" structure associated to an adjunction?
   
   Mike Shulman: In case anyone is following this, Daniel has come up with what seems to be the right answer to this question; I'm hoping that eventually he will write about it here.
   
   Aron Fischer: I'm interested in this too. Who is Daniel? or rather: Hi Daniel! can you write up your answer please? ;)
   
   Mike Shulman: The important parts of this discussion should be merged into the page, and the discussion archived at the nForum. But I don't have time to do that right now, so I'll just record the fact that Daniel's answer appeared in [Tannaka duality for comonoids in cosmoi](https://arxiv.org/abs/0911.0977), at the beginning of section 5 (quoted in full [here](https://mathoverflow.net/a/320835/49)).
6. • CommentRowNumber6.
   • CommentAuthorDavidRoberts
   • CommentTimeJan 16th 2019
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   Author: DavidRoberts
   Format: MarkdownItexMade better link for the recent MO question. <a href="https://ncatlab.org/nlab/revision/diff/monadic+adjunction/20">diff</a>, <a href="https://ncatlab.org/nlab/revision/monadic+adjunction/20">v20</a>, <a href="https://ncatlab.org/nlab/show/monadic+adjunction">current</a>

   Made better link for the recent MO question.

   diff, v20, current

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeJan 16th 2019
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexAlec: thanks! Yes, that's what I meant.

   Alec: thanks! Yes, that’s what I meant.