<p>The definition of morphism of adjunctions may be found in MacLane's book. Let $F:\mathcal C\rightarrow\mathcal D$, $G:\mathcal D\rightarrow\mathcal C$, $F':\mathcal C'\rightarrow\mathcal D'$, $G':\mathcal D'\rightarrow\mathcal C'$, and let $(F,G,\eta,\varepsilon)$, $(F',G',\eta',\varepsilon')$ be adjunctions. A <em>map of adjunctions</em> from the first to the second is a pair of functors $L:\mathcal C\rightarrow\mathcal C'$, $K:\mathcal D\rightarrow\mathcal D'$ such that $KF=F'L$, $LG=G'K$ and $L\eta=\eta'L$ (or, equivalently, $\varepsilon K=K\varepsilon'$).</p>

<p>In the case of the category of adjunctions of a monad, the category $\mathcal C$ on which the endofunctor of the monad acts is fixed, so $\mathcal C'=\mathcal C$ and one considers the above definition only in case $L=Id_{\mathcal C}$.</p>

CC BY-SA 3.0
2015-04-03T17:20:57.223Z
I'm a bit disturbed by the $=$ sign between functors: surely this means naturally isomorphic?
cody
CC BY-SA 3.0
2015-04-03T18:03:33.737Z
@cody: MacLane's definition (Section IV.7, p.97) uses equality.  That seems to be the standard definition (see for instance [here](http://unapologetic.wordpress.com/2007/07/30/transformations-of-adjoints/)). I don't know what the consequences of a relaxed definition would be.
Damiano Mazza
CC BY-SA 3.0
2015-04-03T13:19:57.443Z
2015-04-03T13:19:57.443Z
Damiano Mazza
<p>The <a href="https://en.wikipedia.org/w/index.php?title=Monad_(category_theory)&amp;oldid=644350323#Monads_and_adjunctions" rel="nofollow">Wikipedia page for Monad</a> says just that for a monad $(T,\eta,\mu)$ we can define the category of all adjunctions that define the monad:</p>

<blockquote>
  <p>Let ￼$\textbf{Adj}(C,T)$ be the category whose objects are the adjunctions ￼$(F,G,e,\varepsilon)$ such that $(GF,e,G\varepsilon F)=(T,\eta,\mu)$￼ and whose arrows are the morphisms of adjunctions which are the identity on ￼$C$. Then this category has</p>
  
  <ul>
  <li>an initial object $(F_T,G_T,\eta,\mu_T) : C\to C_T$￼, where $C_T$￼ is the <a href="http://en.wikipedia.org/wiki/Kleisli_category" rel="nofollow">Kleisli category</a>,</li>
  <li>a terminal object $(F^T,G^T,\eta,\mu^T) : C\to C^T$￼, where ￼$C^T$ is the Eilenberg-Moore category.</li>
  </ul>
</blockquote>

<p>What are the morphisms of adjunctions in $\textbf{Adj}(C,T)$ and what is meant by <em>... which are the identity on $C$</em>?</p>

<ct.category-theory><monad>
What are the morphisms of Adj(C,T) - the category whose objects are the adjunctions of a given monad?
Petr
# What are the morphisms of Adj(C,T) - the category whose objects are the adjunctions of a given monad?

(Asked by: username_0 on 2015-04-03)

The [Wikipedia page for Monad](https://en.wikipedia.org/w/index.php?title=Monad_(category_theory)&oldid=644350323#Monads_and_adjunctions) says just that for a monad $(T,\eta,\mu)$ we can define the category of all adjunctions that define the monad:

> Let ￼$\textbf{Adj}(C,T)$ be the category whose objects are the adjunctions ￼$(F,G,e,\varepsilon)$ such that $(GF,e,G\varepsilon F)=(T,\eta,\mu)$￼ and whose arrows are the morphisms of adjunctions which are the identity on ￼$C$. Then this category has
>
> * an initial object $(F\_T,G\_T,\eta,\mu\_T) : C\to C\_T$￼, where $C\_T$￼ is the [Kleisli category](http://en.wikipedia.org/wiki/Kleisli_category),
> * a terminal object $(F^T,G^T,\eta,\mu^T) : C\to C^T$￼, where ￼$C^T$ is the Eilenberg-Moore category.

What are the morphisms of adjunctions in $\textbf{Adj}(C,T)$ and what is meant by *... which are the identity on $C$*?

**Tags:** ct.category-theory, monad


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## 1 answers

### Answer by username_1 on 2015-04-03. Score: 8 | Accepted answer

The definition of morphism of adjunctions may be found in MacLane's book. Let $F:\mathcal C\rightarrow\mathcal D$, $G:\mathcal D\rightarrow\mathcal C$, $F':\mathcal C'\rightarrow\mathcal D'$, $G':\mathcal D'\rightarrow\mathcal C'$, and let $(F,G,\eta,\varepsilon)$, $(F',G',\eta',\varepsilon')$ be adjunctions. A *map of adjunctions* from the first to the second is a pair of functors $L:\mathcal C\rightarrow\mathcal C'$, $K:\mathcal D\rightarrow\mathcal D'$ such that $KF=F'L$, $LG=G'K$ and $L\eta=\eta'L$ (or, equivalently, $\varepsilon K=K\varepsilon'$).

In the case of the category of adjunctions of a monad, the category $\mathcal C$ on which the endofunctor of the monad acts is fixed, so $\mathcal C'=\mathcal C$ and one considers the above definition only in case $L=Id\_{\mathcal C}$.

> **Comments:**
> **username_2**: I'm a bit disturbed by the $=$ sign between functors: surely this means naturally isomorphic?  
> **username_1**: @username_2: MacLane's definition (Section IV.7, p.97) uses equality.  That seems to be the standard definition (see for instance [here](http://unapologetic.wordpress.com/2007/07/30/transformations-of-adjoints/)). I don't know what the consequences of a relaxed definition would be.  

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