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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 30th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded at [[adjoint functor]] * more details in the section <a href="http://ncatlab.org/nlab/show/adjoint+functor#UniversalArrows">In terms of universal arrows</a>; * a bit in the section <a href="http://ncatlab.org/nlab/show/adjoint+functor#Examples">Examples</a>

   added at adjoint functor

   • more details in the section In terms of universal arrows;

   • a bit in the section Examples

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeApr 24th 2011
   • (edited Apr 24th 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexWhile $n$Lab is down I must write here some essential notes, which are actually of interest for discussion as well. * Samuel Eilenberg, John C. Moore, _Adjoint functors and triples_, Illinois J. Math. __9__, Issue 3 (1965), 381-398. Here *triple* is unfortunately in the sense of [[monad]]. Section 3 is called *Adjoint triples* but is not what we call [[adjoint triple]]s, but instead the case where the underlying endofunctor of a monad $\mathbf{T} = (T,\mu^T,\eta^T)$ has a *right* adjoint $G$. Then automatically $G$ is a part of a comonad $\mathbf{G} = (G,\delta^G,\epsilon^G)$ where $\delta^G$ and $\epsilon^G$ are in some sense dual to $\mu^T$ and $\eta^T$. Thus there is a correspondence between monads having right adjoint and comonads having left adjoint, what [[Alexander Rosenberg|Rosenberg]] calls **duality**. I am not sure that the terminology is optimal. In any case, it is a little more than a consequence of two general facts. 1. If $T\dashv G$ then $T^k \dashv G^k$ for every natural number $k$. 2. Given two adjunctions $S\dashv T$ and $S'\dashv T'$ where $S,S': B\to A$, then there is a bijection between the natural transformations $\phi:S'\Rightarrows S$ and natural transformations $\psi:T\Rightarrows T'$ such that $$\array{ A (S,-) &\to& B(-,T)\\ A(\phi,-)\downarrow &&\downarrow B(-,\psi)\\ A(S',-)&\to & B(-,T') }$$ where the horizontal arrows are the natural bijections given by the adjunctions. If $\eta,\eta'$ and $\epsilon,\epsilon'$ are their unit and counit of course the upper arrow is $(SM\stackrel{f}\to N)\mapsto Tf\circ \eta_M$ and the lower arrow $(S'M\stackrel{g}\to N)\mapsto T'g\circ\eta'_M$. Thus the condition renders as $$T'(f\circ\phi_M)\circ\eta'_M = \psi_N\circ Tf\circ\eta_M$$ or $T'f\circ T'\phi_M\circ\eta'_M = T'f\circ \psi_{SM}\circ\eta_M$. Given $\phi$, the uniqueness of $B(-,\psi)$ is clear from the above diagram, as the horizontal arrows are invertible. $B(-,\psi)$ determines $\psi$, namely $\psi_N = B(-,\psi)(id_N)$. For the existence of $\psi$ (given $\phi$) satisfying the above equation, one proposes that $\psi$ is the composition $\psi = T'\epsilon \circ T'\phi T \circ \eta'T$, i.e. $$ T\stackrel{\eta' T}\longrightarrow T'S' T\stackrel{T'\psi T}\longrightarrow T'ST \longrightarrow{T'\epsilon}\longrightarrow T' $$ and checks that it works. The inverse is similarly given by the composition $$ S'\stackrel{S'\eta}\longrightarrow S' TS\stackrel{S'\psi S}\longrightarrow S'T'S\longrightarrow(\epsilon' S} S $$ The mechanism strongly reminds of [[mate]]s, but it is not (classical) mates (in their case one starts with one adjunction). Maybe somebody can elucidate the connection, maybe in some framework it is the same. This now enables in a special case to dualize $\mu^T$ to $\delta^G$, and similarly unit to the counit. I guess one could do that kind of dualization for more general algebras over operads in the category of endofunctors. By the way, is this extension known ?

   While $S'\stackrel{S'\eta}\longrightarrow S' TS\stackrel{S'\psi S}\longrightarrow S'T'S\longrightarrow(\epsilon' S}n$Lab is down I must write here some essential notes, which are actually of interest for discussion as well.

   • Samuel Eilenberg, John C. Moore, Adjoint functors and triples, Illinois J. Math. 9, Issue 3 (1965), 381-398.

   Here triple is unfortunately in the sense of monad. Section 3 is called Adjoint triples but is not what we call adjoint triples, but instead the case where the underlying endofunctor of a monad $\mathbf{T} = (T,\mu^T,\eta^T)$ has a right adjoint $G$. Then automatically $G$ is a part of a comonad $\mathbf{G} = (G,\delta^G,\epsilon^G)$ where $\delta^G$ and $\epsilon^G$ are in some sense dual to $\mu^T$ and $\eta^T$. Thus there is a correspondence between monads having right adjoint and comonads having left adjoint, what Rosenberg calls duality. I am not sure that the terminology is optimal. In any case, it is a little more than a consequence of two general facts.

   1. If $T\dashv G$ then $T^k \dashv G^k$ for every natural number $k$.

   2. Given two adjunctions $S\dashv T$ and $S'\dashv T'$ where $S,S': B\to A$, then there is a bijection between the natural transformations $\phi:S'\Rightarrows S$ and natural transformations $\psi:T\Rightarrows T'$ such that

   $$\array{ A (S,-) &\to& B(-,T)\\ A(\phi,-)\downarrow &&\downarrow B(-,\psi)\\ A(S',-)&\to & B(-,T') }$$

   where the horizontal arrows are the natural bijections given by the adjunctions. If $\eta,\eta'$ and $\epsilon,\epsilon'$ are their unit and counit of course the upper arrow is $(SM\stackrel{f}\to N)\mapsto Tf\circ \eta_M$ and the lower arrow $(S'M\stackrel{g}\to N)\mapsto T'g\circ\eta'_M$. Thus the condition renders as

   $$T'(f\circ\phi_M)\circ\eta'_M = \psi_N\circ Tf\circ\eta_M$$

   or $T'f\circ T'\phi_M\circ\eta'_M = T'f\circ \psi_{SM}\circ\eta_M$. Given $\phi$, the uniqueness of $B(-,\psi)$ is clear from the above diagram, as the horizontal arrows are invertible. $B(-,\psi)$ determines $\psi$, namely $\psi_N = B(-,\psi)(id_N)$. For the existence of $\psi$ (given $\phi$) satisfying the above equation, one proposes that $\psi$ is the composition $\psi = T'\epsilon \circ T'\phi T \circ \eta'T$, i.e.

   $$T\stackrel{\eta' T}\longrightarrow T'S' T\stackrel{T'\psi T}\longrightarrow T'ST \longrightarrow{T'\epsilon}\longrightarrow T'$$

   and checks that it works. The inverse is similarly given by the composition

   The mechanism strongly reminds of mates, but it is not (classical) mates (in their case one starts with one adjunction). Maybe somebody can elucidate the connection, maybe in some framework it is the same.

   This now enables in a special case to dualize $\mu^T$ to $\delta^G$, and similarly unit to the counit. I guess one could do that kind of dualization for more general algebras over operads in the category of endofunctors. By the way, is this extension known ?

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeApr 24th 2011
   • (edited Apr 24th 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexNo, above is the old form, the newest update has been lost in refresh.

   No, above is the old form, the newest update has been lost in refresh.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeApr 24th 2011
   • (edited Apr 25th 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexWhile $n$Lab is down I must write here some essential notes, which are actually of interest for discussion as well. * Samuel Eilenberg, John C. Moore, _Adjoint functors and triples_, Illinois J. Math. __9__, Issue 3 (1965), 381-398. Here *triple* is unfortunately in the sense of [[monad]]. Section 3 is called *Adjoint triples* but is not what we call [[adjoint triple]]s, but instead the case where the underlying endofunctor of a monad $\mathbf{T} = (T,\mu^T,\eta^T)$ has a *right* adjoint $G$. Then automatically $G$ is a part of a comonad $\mathbf{G} = (G,\delta^G,\epsilon^G)$ where $\delta^G$ and $\epsilon^G$ are in some sense dual to $\mu^T$ and $\eta^T$. Thus there is a correspondence between monads having right adjoint and comonads having left adjoint, what [[Alexander Rosenberg|Rosenberg]] calls **duality**. I am not sure that the terminology is optimal. In any case, it is a little more than a consequence of two general facts. 1. If $T\dashv G$ then $T^k \dashv G^k$ for every natural number $k$. 2. Given two adjunctions $S\dashv T$ and $S'\dashv T'$ where $S,S': B\to A$, then there is a bijection between the natural transformations $\phi:S'\Rightarrow S$ and natural transformations $\psi:T\Rightarrow T'$ such that $$ \array{ A (S,-) &\to& B(-,T)\\ A(\phi,-)\downarrow &&\downarrow B(-,\psi)\\ A(S',-)&\to & B(-,T') }$$ where the horizontal arrows are the natural bijections given by the adjunctions. If $\eta,\eta'$ and $\epsilon,\epsilon'$ are their unit and counit of course the upper arrow is $(SM\stackrel{f}\to N)\mapsto Tf\circ \eta_M$ and the lower arrow $(S'M\stackrel{g}\to N)\mapsto T'g\circ\eta'_M$. Thus the condition renders as $$ T'(f\circ\phi_M)\circ\eta'_M = \psi_N\circ Tf\circ\eta_M $$ or $T'f\circ T'\phi_M\circ\eta'_M = T'f\circ \psi_{SM}\circ\eta_M$. Given $\phi$, the uniqueness of $B(-,\psi)$ is clear from the above diagram, as the horizontal arrows are invertible. $B(-,\psi)$ determines $\psi$, namely $\psi_N = B(-,\psi)(id_N)$. For the existence of $\psi$ (given $\phi$) satisfying the above equation, one proposes that $\psi$ is the composition $\psi = T'\epsilon \circ T'\phi T \circ \eta'T$, i.e. $$ T\stackrel{\eta' T}\longrightarrow T'S' T \stackrel{T'\psi T}\longrightarrow T'ST \stackrel{T'\epsilon}\longrightarrow T' $$ and checks that it works. The inverse is similarly given by the composition $$ S'\stackrel{S' \eta}\longrightarrow S' TS \stackrel{S'\psi S}\longrightarrow S'T'S \stackrel(\epsilon' S}\longrightarrow S $$ The mechanism strongly reminds of [[mate]]s, but it is not (classical) mates (in their case one starts with one adjunction). Maybe somebody can elucidate the connection, maybe in some framework it is the same. This now enables in a special case to dualize $\mu^T$ to $\delta^G$, and similarly unit to the counit. I guess one could do that kind of dualization for more general algebras over operads in the category of endofunctors. By the way, is this extension known ?

   While $S'\stackrel{S' \eta}\longrightarrow S' TS \stackrel{S'\psi S}\longrightarrow S'T'S \stackrel(\epsilon' S}n$Lab is down I must write here some essential notes, which are actually of interest for discussion as well.

   • Samuel Eilenberg, John C. Moore, Adjoint functors and triples, Illinois J. Math. 9, Issue 3 (1965), 381-398.

   Here triple is unfortunately in the sense of monad. Section 3 is called Adjoint triples but is not what we call adjoint triples, but instead the case where the underlying endofunctor of a monad $\mathbf{T} = (T,\mu^T,\eta^T)$ has a right adjoint $G$. Then automatically $G$ is a part of a comonad $\mathbf{G} = (G,\delta^G,\epsilon^G)$ where $\delta^G$ and $\epsilon^G$ are in some sense dual to $\mu^T$ and $\eta^T$. Thus there is a correspondence between monads having right adjoint and comonads having left adjoint, what Rosenberg calls duality. I am not sure that the terminology is optimal. In any case, it is a little more than a consequence of two general facts.

   1. If $T\dashv G$ then $T^k \dashv G^k$ for every natural number $k$.

   2. Given two adjunctions $S\dashv T$ and $S'\dashv T'$ where $S,S': B\to A$, then there is a bijection between the natural transformations $\phi:S'\Rightarrow S$ and natural transformations $\psi:T\Rightarrow T'$ such that

   $$\array{ A (S,-) &\to& B(-,T)\\ A(\phi,-)\downarrow &&\downarrow B(-,\psi)\\ A(S',-)&\to & B(-,T') }$$

   where the horizontal arrows are the natural bijections given by the adjunctions. If $\eta,\eta'$ and $\epsilon,\epsilon'$ are their unit and counit of course the upper arrow is $(SM\stackrel{f}\to N)\mapsto Tf\circ \eta_M$ and the lower arrow $(S'M\stackrel{g}\to N)\mapsto T'g\circ\eta'_M$. Thus the condition renders as

   $$T'(f\circ\phi_M)\circ\eta'_M = \psi_N\circ Tf\circ\eta_M$$

   or $T'f\circ T'\phi_M\circ\eta'_M = T'f\circ \psi_{SM}\circ\eta_M$. Given $\phi$, the uniqueness of $B(-,\psi)$ is clear from the above diagram, as the horizontal arrows are invertible. $B(-,\psi)$ determines $\psi$, namely $\psi_N = B(-,\psi)(id_N)$. For the existence of $\psi$ (given $\phi$) satisfying the above equation, one proposes that $\psi$ is the composition $\psi = T'\epsilon \circ T'\phi T \circ \eta'T$, i.e.

   $$T\stackrel{\eta' T}\longrightarrow T'S' T \stackrel{T'\psi T}\longrightarrow T'ST \stackrel{T'\epsilon}\longrightarrow T'$$

   and checks that it works. The inverse is similarly given by the composition

   The mechanism strongly reminds of mates, but it is not (classical) mates (in their case one starts with one adjunction). Maybe somebody can elucidate the connection, maybe in some framework it is the same.

   This now enables in a special case to dualize $\mu^T$ to $\delta^G$, and similarly unit to the counit. I guess one could do that kind of dualization for more general algebras over operads in the category of endofunctors. By the way, is this extension known ?

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeApr 24th 2011
   • (edited Apr 24th 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexWell now I got it correct in source code. Number 3. Except no formulas in the output!!

   Well now I got it correct in source code. Number 3. Except no formulas in the output!!

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeApr 24th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItex$$T'(f\circ\phi_M)\circ\eta'_M = \psi_N\circ Tf\circ\eta_M$$

   $$T'(f\circ\phi_M)\circ\eta'_M = \psi_N\circ Tf\circ\eta_M$$
7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeApr 24th 2011
   • PermaLink
   Author: zskoda
   Format: Text<latex>T'(f\circ\phi_M)\circ\eta'_M = \psi_N\circ Tf\circ\eta_M</latex>

   
8. • CommentRowNumber8.
   • CommentAuthorzskoda
   • CommentTimeApr 24th 2011
   • PermaLink
   Author: zskoda
   Format: TextStrange 6 has a formula in double dollar sign in Markdown+Itex and it does not appear. 7 has the same formula as text output. Over there, the machine has changed the dollar signs into latex in brackets front and back and appeared as a formula. Is somebody experimenting ?

   Strange 6 has a formula in double dollar sign in Markdown+Itex and it does not appear. 7 has the same formula as text output. Over there, the machine has changed the dollar signs into latex in brackets front and back and appeared as a formula. Is somebody experimenting ?
9. • CommentRowNumber9.
   • CommentAuthorzskoda
   • CommentTimeApr 24th 2011
   • (edited Apr 24th 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexMore experimenting $$ T'(f\circ\phi_M)\circ\eta'_M = \psi_N\circ Tf\circ\eta_M $$

   More experimenting

   $$T'(f\circ\phi_M)\circ\eta'_M = \psi_N\circ Tf\circ\eta_M$$
10. • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeApr 24th 2011
    • PermaLink
    Author: zskoda
    Format: MarkdownItexUrs's formula $$ \Delta^n \times A^n \simeq \coprod_{\sigma \in A^n} \Delta^n $$

    Urs’s formula

    $$\Delta^n \times A^n \simeq \coprod_{\sigma \in A^n} \Delta^n$$
11. • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeApr 24th 2011
    • PermaLink
    Author: zskoda
    Format: MarkdownItex$$ T'(f\circ\phi_M)\circ\eta'_M = \psi_N\circ Tf\circ\eta_M $$

    $$T'(f\circ\phi_M)\circ\eta'_M = \psi_N\circ Tf\circ\eta_M$$
12. • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeApr 24th 2011
    • PermaLink
    Author: zskoda
    Format: MarkdownItex$$ T'(f\circ\phi_M)\circ\eta'_M = \psi_N\circ Tf\circ\eta_M $$

    $$T'(f\circ\phi_M)\circ\eta'_M = \psi_N\circ Tf\circ\eta_M$$
13. • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeApr 24th 2011
    • PermaLink
    Author: zskoda
    Format: MarkdownItex$$ \Delta^n \times A^n \simeq \coprod_{\sigma \in A^n} \Delta^n $$ * Eilenberg-Moore, [euclid](http://projecteuclid.org/euclid.ijm/1256068141)

    $$\Delta^n \times A^n \simeq \coprod_{\sigma \in A^n} \Delta^n$$

    • Eilenberg-Moore, euclid
14. • CommentRowNumber14.
    • CommentAuthorzskoda
    • CommentTimeApr 24th 2011
    • PermaLink
    Author: zskoda
    Format: MarkdownItex$$ T'(f \circ \phi_M) \circ \eta'_M = \psi_N \circ Tf \circ \eta_M $$

    $$T'(f \circ \phi_M) \circ \eta'_M = \psi_N \circ Tf \circ \eta_M$$
15. • CommentRowNumber15.
    • CommentAuthorzskoda
    • CommentTimeApr 24th 2011
    • (edited Apr 24th 2011)
    • PermaLink
    Author: zskoda
    Format: MarkdownItex$$ T(f \circ \phi_M) \circ \eta_M $$ I am not getting it...

    $$T(f \circ \phi_M) \circ \eta_M$$

    I am not getting it…

16. • CommentRowNumber16.
    • CommentAuthorzskoda
    • CommentTimeApr 24th 2011
    • PermaLink
    Author: zskoda
    Format: MarkdownItex$$ T'(f \phi_M) \eta $$

    $$T'(f \phi_M) \eta$$
17. • CommentRowNumber17.
    • CommentAuthorzskoda
    • CommentTimeApr 24th 2011
    • PermaLink
    Author: zskoda
    Format: MarkdownItex$$ A $$

    $$A$$
18. • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeApr 25th 2011
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexI still cannot see the math in any of the above comments.

    I still cannot see the math in any of the above comments.

19. • CommentRowNumber19.
    • CommentAuthorzskoda
    • CommentTimeApr 25th 2011
    • (edited Apr 25th 2011)
    • PermaLink
    Author: zskoda
    Format: MarkdownItexLet me try to reinput, that should rerender...I hope. $$ T\stackrel{\eta' T}\longrightarrow T'S' T\stackrel{T'\psi T}\longrightarrow T'ST \stackrel{T'\epsilon}\longrightarrow T' $$

    Let me try to reinput, that should rerender…I hope.

    $$T\stackrel{\eta' T}\longrightarrow T'S' T\stackrel{T'\psi T}\longrightarrow T'ST \stackrel{T'\epsilon}\longrightarrow T'$$
20. • CommentRowNumber20.
    • CommentAuthorzskoda
    • CommentTimeApr 25th 2011
    • (edited Apr 25th 2011)
    • PermaLink
    Author: zskoda
    Format: MarkdownItexYes, finally! Edit: well no. I made some changes to 4 and some formulas now appeared but some did not.

    Yes, finally!

    Edit: well no. I made some changes to 4 and some formulas now appeared but some did not.

21. • CommentRowNumber21.
    • CommentAuthorzskoda
    • CommentTimeApr 25th 2011
    • PermaLink
    Author: zskoda
    Format: MarkdownItexIt is strange that the math code from my long comment 4 works when copied and pasted into [[adjoint monad]] (new entry announced [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=2649)).

    It is strange that the math code from my long comment 4 works when copied and pasted into adjoint monad (new entry announced here).

22. • CommentRowNumber22.
    • CommentAuthorMike Shulman
    • CommentTimeApr 26th 2011
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexAs far as I can tell from what I can see, this is exactly the notion of mate, in the special case when the functors without adjoints are identities. What do you mean by "in their case one starts with one adjunction"?

    As far as I can tell from what I can see, this is exactly the notion of mate, in the special case when the functors without adjoints are identities. What do you mean by “in their case one starts with one adjunction”?

23. • CommentRowNumber23.
    • CommentAuthorzskoda
    • CommentTimeApr 26th 2011
    • (edited Apr 26th 2011)
    • PermaLink
    Author: zskoda
    Format: MarkdownItexOf course, I expected Mike to find the connection. Hm, now I look into [[mate]] and there you indeed start with two adjunctions. I took the notion of mates from Leinster's book [math.CT/0305049](http://arxiv.org/abs/math.CT/0305049) where he starts with one adjunction only, page 150 (180 of the file). He starts with ONE adjunction $P\dashv Q$, $P: D'\to D$, $Q:D\to D'$ and two functors $T:D\to D$, $T':D'\to D'$. Then there is a correspondence between the transformations $\phi:PT'\Rightarrow TP$ and the transformations $\psi: T'Q\Rightarrow QT$. The $n$Lab entry [[mate]], as I now see, has two adjunctions and two 1-cells to start with. I did not see that general version. Thanks.

    Of course, I expected Mike to find the connection.

    Hm, now I look into mate and there you indeed start with two adjunctions. I took the notion of mates from Leinster’s book math.CT/0305049 where he starts with one adjunction only, page 150 (180 of the file).

    He starts with ONE adjunction $P\dashv Q$, $P: D'\to D$, $Q:D\to D'$ and two functors $T:D\to D$, $T':D'\to D'$. Then there is a correspondence between the transformations $\phi:PT'\Rightarrow TP$ and the transformations $\psi: T'Q\Rightarrow QT$.

    The $n$Lab entry mate, as I now see, has two adjunctions and two 1-cells to start with. I did not see that general version. Thanks.

24. • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexspelled out at [[adjoint functor]] in detail the basic proof that a right adjoint is faithful precisely if the counit has epi components.

    spelled out at adjoint functor in detail the basic proof that a right adjoint is faithful precisely if the counit has epi components.

25. • CommentRowNumber25.
    • CommentAuthoreparejatobes
    • CommentTimeMar 12th 2012
    • PermaLink
    Author: eparejatobes
    Format: MarkdownItexadded at [[adjoint functor]] the definition in terms of extensions/liftings, plus when a right/left adjoint is full in terms of the counit/unit

    added at adjoint functor the definition in terms of extensions/liftings, plus when a right/left adjoint is full in terms of the counit/unit

26. • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeMar 12th 2012
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks. For the sake of those readers who come across this and don't (yet) know what "$Lan$" stands for and who don't (yet) know what "absolute" refers to, I have slightly edited to read as follows: > Given $L \colon C \to D$, we have that it has a [[right adjoint]] $R\colon D \to C$ precisely if the [[left Kan extension]] $Lan_L 1_C$ of the [[identity]] along $L$ exists and is [[absolute colimit|absolute]], in which case $R \simeq \mathop{Lan}_L 1_C$. What do you think?

    Thanks.

    For the sake of those readers who come across this and don’t (yet) know what “$Lan$” stands for and who don’t (yet) know what “absolute” refers to, I have slightly edited to read as follows:

    Given $L \colon C \to D$, we have that it has a right adjoint $R\colon D \to C$ precisely if the left Kan extension $Lan_L 1_C$ of the identity along $L$ exists and is absolute, in which case $R \simeq \mathop{Lan}_L 1_C$.

    What do you think?

27. • CommentRowNumber27.
    • CommentAuthoreparejatobes
    • CommentTimeMar 12th 2012
    • PermaLink
    Author: eparejatobes
    Format: MarkdownItexgreat, thanks. I was in a hurry and didn't gave a lot of thought to the phrasing. It's clearer now. I've also changed $=$s for $\simeq$s. Btw, is there a way to link to specific sections within a page? here for example, I'd rather have a link to a (to be added) section of [[Kan extension]] detailing when one such is absolute, than to [[absolute colimit]]

    great, thanks. I was in a hurry and didn’t gave a lot of thought to the phrasing. It’s clearer now. I’ve also changed $=$s for $\simeq$s.

    Btw, is there a way to link to specific sections within a page? here for example, I’d rather have a link to a (to be added) section of Kan extension detailing when one such is absolute, than to absolute colimit

28. • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeMar 12th 2012
    • PermaLink
    Author: Urs
    Format: MarkdownItex> is there a way to link to specific sections within a page? Yes, but unfortunately not in a nice wiki-way. You can link to any URl by writing [follow this link](http://url.com/see/here.html) and this may of course include anchors, such as [follow this link](http://url.com/see/here.html#anchor) So [absolute Kan extensiuon](http://ncatlab.org/nlab/show/Kan+extension#AbsoluteKanExtension) produces the link that you want: _[absolute Kan extensiuon](http://ncatlab.org/nlab/show/Kan+extension#AbsoluteKanExtension)_ (and see the source code of _[[Kan extension]]_ for how I produced that anchor to that subsection).

    is there a way to link to specific sections within a page?

    Yes, but unfortunately not in a nice wiki-way.

    You can link to any URl by writing

     [follow this link](http://url.com/see/here.html)
    

    and this may of course include anchors, such as

     [follow this link](http://url.com/see/here.html#anchor)
    

    So

    [absolute Kan extensiuon](http://ncatlab.org/nlab/show/Kan+extension#AbsoluteKanExtension)
    

    produces the link that you want: absolute Kan extensiuon

    (and see the source code of Kan extension for how I produced that anchor to that subsection).

29. • CommentRowNumber29.
    • CommentAuthoreparejatobes
    • CommentTimeMar 12th 2012
    • PermaLink
    Author: eparejatobes
    Format: MarkdownItexOk, many thanks Urs

    Ok, many thanks Urs

30. • CommentRowNumber30.
    • CommentAuthorMike Shulman
    • CommentTimeMar 12th 2012
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexBTW, you can leave off the "http://ncatlab.org" from the URL and it will still work (and even continue to work if the nLab's domain name had to be changed).

    BTW, you can leave off the “http://ncatlab.org” from the URL and it will still work (and even continue to work if the nLab’s domain name had to be changed).

31. • CommentRowNumber31.
    • CommentAuthoreparejatobes
    • CommentTimeMar 13th 2012
    • PermaLink
    Author: eparejatobes
    Format: MarkdownItexdone! added a rel link to the absolute kan extension section

    done! added a rel link to the absolute kan extension section

32. • CommentRowNumber32.
    • CommentAuthorUrs
    • CommentTimeNov 28th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded more historical [references](http://ncatlab.org/nlab/show/adjoint+functor#References)

    added more historical references

33. • CommentRowNumber33.
    • CommentAuthorUrs
    • CommentTimeJun 2nd 2018
    • PermaLink
    Author: Urs
    Format: MarkdownItexI am trying to bring this page into better expositional shape. First, I have now rewritten and expanded, with full proofs, the section _[In terms of hom-isomorphisms](https://ncatlab.org/nlab/show/adjoint+functor#InTermsOfHomIsomorphism)_. <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/70">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/70">v70</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    I am trying to bring this page into better expositional shape. First, I have now rewritten and expanded, with full proofs, the section In terms of hom-isomorphisms.

    diff, v70, current

34. • CommentRowNumber34.
    • CommentAuthorUrs
    • CommentTimeJun 2nd 2018
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have also changed the Idea-section. Removed the would-be attempt at a definition, since that is now discussed in the main text. Then I added an absolute minimum of words to put the concept in perspective. Should be expanded. Could be expanded to a long conceptual section, optimally.

    I have also changed the Idea-section. Removed the would-be attempt at a definition, since that is now discussed in the main text. Then I added an absolute minimum of words to put the concept in perspective. Should be expanded. Could be expanded to a long conceptual section, optimally.

35. • CommentRowNumber35.
    • CommentAuthorUrs
    • CommentTimeJun 3rd 2018
    • (edited Jun 3rd 2018)
    • PermaLink
    Author: Urs
    Format: MarkdownItexNow I have polished and expanded the next section: _[In terms of representable functors](https://ncatlab.org/nlab/show/adjoint+functor#InTermsOfRepresentableFunctors)_ <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/71">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/71">v71</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    Now I have polished and expanded the next section: In terms of representable functors

    diff, v71, current

36. • CommentRowNumber36.
    • CommentAuthorUrs
    • CommentTimeJun 3rd 2018
    • PermaLink
    Author: Urs
    Format: MarkdownItexnow I have similarly touched the next section: _[In terms of universal factorization through a (co)unit](https://ncatlab.org/nlab/show/adjoint+functor#UniversalArrows)_ <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/71">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/71">v71</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    now I have similarly touched the next section: In terms of universal factorization through a (co)unit

    diff, v71, current

37. • CommentRowNumber37.
    • CommentAuthorUrs
    • CommentTimeJun 3rd 2018
    • PermaLink
    Author: Urs
    Format: MarkdownItexafter the lead-in sentence claiming a key role of the concept of adjoint functors, I added this as a footnote: > "In all those areas where category theory is actively used the categorical concept of adjoint functor has come to play a key role." (first line from _[An interview with William Lawvere](https://ncatlab.org/nlab/show/William+Lawvere#Interview07)_, paraphrasing the first paragraph of _[Taking categories seriously](William+Lawvere#TakingCategoriesSeriously)_) <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/73">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/73">v73</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    after the lead-in sentence claiming a key role of the concept of adjoint functors, I added this as a footnote:

    “In all those areas where category theory is actively used the categorical concept of adjoint functor has come to play a key role.” (first line from An interview with William Lawvere, paraphrasing the first paragraph of Taking categories seriously)

    diff, v73, current

38. • CommentRowNumber38.
    • CommentAuthorUrs
    • CommentTimeJun 20th 2018
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have polished and expanded the discussion of left adjoints via pointwise limits over comma categories, now [this Prop.](https://ncatlab.org/nlab/show/adjoint+functor#PointwiseExpression). <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/80">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/80">v80</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    I have polished and expanded the discussion of left adjoints via pointwise limits over comma categories, now this Prop..

    diff, v80, current

39. • CommentRowNumber39.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 2nd 2018
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexCorrected the order of adjoint functors in one place. <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/84">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/84">v84</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    Corrected the order of adjoint functors in one place.

    diff, v84, current

40. • CommentRowNumber40.
    • CommentAuthorUrs
    • CommentTimeJan 28th 2019
    • (edited Jan 28th 2019)
    • PermaLink
    Author: Urs
    Format: MarkdownItexreplaced the first occurence of an Instikiti `array`-hack for adjoint pair notation by a `tikzcd`-version \begin{center} \begin{tikzcd} \mathcal{D} \arrow[r, shift right=6pt, "R"', "\bot"] & \mathcal{C} \arrow[l, shift right=6pt, "L"'] \end{tikzcd} \end{center} Suggestions for improvements welcome. What's the `tikzcd`-analog of what in `xymatrix` is \ar@{<-}[r] ? <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/87">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/87">v87</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    replaced the first occurence of an Instikiti array-hack for adjoint pair notation by a tikzcd-version

      \begin{center}
        \begin{tikzcd}
          \mathcal{D}
           \arrow[r, shift right=6pt, "R"', "\bot"]
          & 
          \mathcal{C}
           \arrow[l, shift right=6pt, "L"']
        \end{tikzcd}
      \end{center}
    

    Suggestions for improvements welcome.

    What’s the tikzcd-analog of what in xymatrix is

      \ar@{<-}[r]
    

    ?

    diff, v87, current

41. • CommentRowNumber41.
    • CommentAuthorMike Shulman
    • CommentTimeJan 28th 2019
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItex`\ar[r,<-]` or `\ar[from=r]` (I'm not actually sure if these is a difference between these)

    \ar[r,<-] or \ar[from=r] (I’m not actually sure if these is a difference between these)

42. • CommentRowNumber42.
    • CommentAuthorUrs
    • CommentTimeJan 29th 2019
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks! I just wanted to add this to the HowTo [here](https://ncatlab.org/nlab/show/HowTo#tikz), as another escaped-code example, but it seesm that generally the parser gets confused when there is escaped tikz-code, so I am removing it again.

    Thanks!

    I just wanted to add this to the HowTo here, as another escaped-code example, but it seesm that generally the parser gets confused when there is escaped tikz-code, so I am removing it again.

43. • CommentRowNumber43.
    • CommentAuthorRichard Williamson
    • CommentTimeJan 29th 2019
    • (edited Jan 29th 2019)
    • PermaLink
    Author: Richard Williamson
    Format: MarkdownItexYes, currently the Tikz code cannot be escaped, either by a ```<nowiki>``` block or in a code block. It is tricky to get this to work due to the interaction between the new and old renderer. I will try to fix it when I get the chance. In the meantime, one can always link to the source.

    Yes, currently the Tikz code cannot be escaped, either by a <nowiki> block or in a code block. It is tricky to get this to work due to the interaction between the new and old renderer. I will try to fix it when I get the chance. In the meantime, one can always link to the source.

44. • CommentRowNumber44.
    • CommentAuthorRichard Williamson
    • CommentTimeFeb 2nd 2019
    • PermaLink
    Author: Richard Williamson
    Format: MarkdownItexThe issues with escaping should now be fixed.

    The issues with escaping should now be fixed.

45. • CommentRowNumber45.
    • CommentAuthornLab edit announcer
    • CommentTimeJun 6th 2019
    • PermaLink
    Author: nLab edit announcer
    Format: MarkdownItexFixes typo in diagram: the discussion that follows implies C should be the domain of the functor L and the codomain of the functor R, and not vice versa. There may be other similar typos in the remainder of the article, I haven't done an exhaustive search (and am not good enough with LaTeX to be confident making extensive edits). Asad <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/89">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/89">v89</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    Fixes typo in diagram: the discussion that follows implies C should be the domain of the functor L and the codomain of the functor R, and not vice versa. There may be other similar typos in the remainder of the article, I haven’t done an exhaustive search (and am not good enough with LaTeX to be confident making extensive edits).

    Asad

    diff, v89, current

46. • CommentRowNumber46.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 7th 2019
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexYes, that's right. I corrected another one. <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/90">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/90">v90</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    Yes, that’s right. I corrected another one.

    diff, v90, current

47. • CommentRowNumber47.
    • CommentAuthorDean
    • CommentTimeNov 24th 2019
    • PermaLink
    Author: Dean
    Format: MarkdownItexThere is a certain lemma about adjoint functors that makes some of these things more manifest. See "Adjunctions" here https://edeany.com. Are people ok with me adding this lemma here?

    There is a certain lemma about adjoint functors that makes some of these things more manifest. See “Adjunctions” here https://edeany.com. Are people ok with me adding this lemma here?

48. • CommentRowNumber48.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 24th 2019
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexNot clear what lemma you're talking about. You should probably just state your lemma here rather than getting people to copy and paste your address into a separate window and then read a bunch of stuff.

    Not clear what lemma you’re talking about. You should probably just state your lemma here rather than getting people to copy and paste your address into a separate window and then read a bunch of stuff.

49. • CommentRowNumber49.
    • CommentAuthorMike Shulman
    • CommentTimeNov 25th 2019
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexIf you write `<https://edeany.com>` it will automatically be a link (<https://edeany.com>).

    If you write <https://edeany.com> it will automatically be a link (https://edeany.com).

50. • CommentRowNumber50.
    • CommentAuthornLab edit announcer
    • CommentTimeDec 23rd 2019
    • PermaLink
    Author: nLab edit announcer
    Format: MarkdownItexfixed typos Spencer Dembner <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/93">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/93">v93</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    fixed typos

    Spencer Dembner

    diff, v93, current

51. • CommentRowNumber51.
    • CommentAuthornLab edit announcer
    • CommentTimeDec 23rd 2019
    • PermaLink
    Author: nLab edit announcer
    Format: MarkdownItexfixed typos Spencer Dembner <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/94">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/94">v94</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    fixed typos

    Spencer Dembner

    diff, v94, current

52. • CommentRowNumber52.
    • CommentAuthoraleks
    • CommentTimeJun 2nd 2020
    • PermaLink
    Author: aleks
    Format: MarkdownItexFixed typos in diagrams. <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/96">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/96">v96</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    Fixed typos in diagrams.

    diff, v96, current

53. • CommentRowNumber53.
    • CommentAuthoraleks
    • CommentTimeJun 2nd 2020
    • PermaLink
    Author: aleks
    Format: MarkdownItexFix of the typo fix... <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/96">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/96">v96</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    Fix of the typo fix…

    diff, v96, current

54. • CommentRowNumber54.
    • CommentAuthoraleks
    • CommentTimeJun 2nd 2020
    • PermaLink
    Author: aleks
    Format: MarkdownItexFix of the fix of the fix <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/96">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/96">v96</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    Fix of the fix of the fix

    diff, v96, current

55. • CommentRowNumber55.
    • CommentAuthorJohn Baez
    • CommentTimeJun 23rd 2020
    • PermaLink
    Author: John Baez
    Format: MarkdownItexSwitched D and C in Prop. 1.10 to make it match the rest. <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/97">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/97">v97</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    Switched D and C in Prop. 1.10 to make it match the rest.

    diff, v97, current

56. • CommentRowNumber56.
    • CommentAuthorJohn Baez
    • CommentTimeJun 23rd 2020
    • PermaLink
    Author: John Baez
    Format: MarkdownItexWhoops! It was Prop. 1.9 that needed to have C and D switched. Done. <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/97">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/97">v97</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    Whoops! It was Prop. 1.9 that needed to have C and D switched. Done.

    diff, v97, current

57. • CommentRowNumber57.
    • CommentAuthorJas
    • CommentTimeSep 12th 2020
    • PermaLink
    Author: Jas
    Format: MarkdownItexIs there a typo in diagram (2) (the commutative square)? I don't see how the vertical arrows are induced by $g:c_2\rightarrow c_1$ and $h:d_1\rightarrow d_2$ - I have read the definition of hom-functor. It works if $Hom(L(c_1), d_1$ is considered instead of $Hom(L(c_2), d_1)$, and same goes for other hom sets.

    Is there a typo in diagram (2) (the commutative square)? I don’t see how the vertical arrows are induced by $g:c_2\rightarrow c_1$ and $h:d_1\rightarrow d_2$ - I have read the definition of hom-functor. It works if $Hom(L(c_1), d_1$ is considered instead of $Hom(L(c_2), d_1)$, and same goes for other hom sets.

58. • CommentRowNumber58.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2020
    • PermaLink
    Author: Urs
    Format: MarkdownItexYes, thanks for catching. Fixed now. <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/99">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/99">v99</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    Yes, thanks for catching. Fixed now.

    diff, v99, current

59. • CommentRowNumber59.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 14th 2022
    • (edited Aug 14th 2022)
    • PermaLink
    Author: J-B Vienney
    Format: MarkdownItexAdded a proof that a pair of adjoint functors induces a monad. <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/112">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/112">v112</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    Added a proof that a pair of adjoint functors induces a monad.

    diff, v112, current

60. • CommentRowNumber60.
    • CommentAuthorUrs
    • CommentTimeAug 14th 2022
    • (edited Aug 14th 2022)
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks, this is good material ([here](https://ncatlab.org/nlab/show/adjoint+functor#RelationToMonads)). * In the first lines I have fixed an "$L$" to "$F$" and an "$R$" to "$U$", since that's what you are using in the following. (On the other hand, myself, I prefer "$L \dashv R$" for the generic pair of adjoints, for what I find are obvious reasons: e.g. it is not generally the case that an *U*nderlying functor is a right adjoint.) * Then I took the liberty of removing the bullet item markup and replacing it by numeration by hand. This is a hack to workaround a shortcoming of the nLab software: Once you include `tikzcd` diagrams in an `Instiki` bullet item, Instiki gives up and ends the itemize typesetting. As a result, the left alignment in your text did not come out as intended. (At some point in the future we are going to hire a programmer to fix this and related software issues.) <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/113">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/113">v113</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    Thanks, this is good material (here).

    • In the first lines I have fixed an “$L$” to “$F$” and an “$R$” to “$U$”, since that’s what you are using in the following. (On the other hand, myself, I prefer “$L \dashv R$” for the generic pair of adjoints, for what I find are obvious reasons: e.g. it is not generally the case that an Underlying functor is a right adjoint.)

    • Then I took the liberty of removing the bullet item markup and replacing it by numeration by hand. This is a hack to workaround a shortcoming of the nLab software: Once you include tikzcd diagrams in an Instiki bullet item, Instiki gives up and ends the itemize typesetting. As a result, the left alignment in your text did not come out as intended.

      (At some point in the future we are going to hire a programmer to fix this and related software issues.)

    diff, v113, current

61. • CommentRowNumber61.
    • CommentAuthorHurkyl
    • CommentTimeAug 14th 2022
    • (edited Aug 14th 2022)
    • PermaLink
    Author: Hurkyl
    Format: MarkdownItexHuh. I'm surprised that indenting sections in the source doesn't fix the "instiki gives up" issue. I guess this system works differently than others I've used. (or maybe I accidentally made a different error when trying this fix)

    Huh. I’m surprised that indenting sections in the source doesn’t fix the “instiki gives up” issue. I guess this system works differently than others I’ve used. (or maybe I accidentally made a different error when trying this fix)

62. • CommentRowNumber62.
    • CommentAuthorUrs
    • CommentTimeAug 14th 2022
    • (edited Aug 14th 2022)
    • PermaLink
    Author: Urs
    Format: MarkdownItexYes, Instiki does not have markup for plain indentation. In fact, markup given by indenting by 6 or more whitespaces is interpreted differently altogether: it produces (indented, yes, but) verbatim output of the source.

    Yes, Instiki does not have markup for plain indentation. In fact, markup given by indenting by 6 or more whitespaces is interpreted differently altogether: it produces (indented, yes, but) verbatim output of the source.

63. • CommentRowNumber63.
    • CommentAuthorvarkor
    • CommentTimeDec 17th 2022
    • PermaLink
    Author: varkor
    Format: MarkdownItexMention that the comma category definition generalises to relative adjunctions. <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/116">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/116">v116</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    Mention that the comma category definition generalises to relative adjunctions.

    diff, v116, current

64. • CommentRowNumber64.
    • CommentAuthorncfavier
    • CommentTimeFeb 19th 2023
    • (edited Feb 19th 2023)
    • PermaLink
    Author: ncfavier
    Format: MarkdownItexAdded composition of adjunctions (L' ∘ L ⊣ R ∘ R'), precomposite adjunctions (— ∘ R ⊣ — ∘ L) and postcomposite adjunctions (L ∘ — ⊣ R ∘ —). Connected this to the fact that adjoints are absolute Kan extensions/lifts of the identity. I couldn't find this in the literature (but I didn't look very hard), so if this is known under a different name, or if you have a reference, please add. Formalisation of the pre/postcomposite adjunctions in Agda for the 1lab: <https://github.com/plt-amy/1lab/pull/193/files#diff-b2e8196987b851f006e0efde215ecbf35f972e7eeee3ae3a1612cb4f8a480b61> <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/119">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/119">v119</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    Added composition of adjunctions (L’ ∘ L ⊣ R ∘ R’), precomposite adjunctions (— ∘ R ⊣ — ∘ L) and postcomposite adjunctions (L ∘ — ⊣ R ∘ —). Connected this to the fact that adjoints are absolute Kan extensions/lifts of the identity.

    I couldn’t find this in the literature (but I didn’t look very hard), so if this is known under a different name, or if you have a reference, please add.

    Formalisation of the pre/postcomposite adjunctions in Agda for the 1lab: https://github.com/plt-amy/1lab/pull/193/files#diff-b2e8196987b851f006e0efde215ecbf35f972e7eeee3ae3a1612cb4f8a480b61

    diff, v119, current

65. • CommentRowNumber65.
    • CommentAuthorUrs
    • CommentTimeFeb 19th 2023
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks. If there is a good/stable link to the Agda formalization, then let's add it to the entry.

    Thanks. If there is a good/stable link to the Agda formalization, then let’s add it to the entry.

66. • CommentRowNumber66.
    • CommentAuthorncfavier
    • CommentTimeFeb 19th 2023
    • PermaLink
    Author: ncfavier
    Format: MarkdownItexSure, I can add that once the pull request gets merged. In general it would be nice to systematically have links from nlab pages to their formalised equivalents in projects like the [cubical](https://github.com/agda/cubical) library, the [1lab](https://1lab.dev), and/or [agda-unimath](https://unimath.github.io/agda-unimath).

    Sure, I can add that once the pull request gets merged.

    In general it would be nice to systematically have links from nlab pages to their formalised equivalents in projects like the cubical library, the 1lab, and/or agda-unimath.

67. • CommentRowNumber67.
    • CommentAuthorUrs
    • CommentTimeFeb 19th 2023
    • PermaLink
    Author: Urs
    Format: MarkdownItexYes, absolutely. We have some such referencing already, such as * [here](https://ncatlab.org/nlab/show/group#TTFormalizations) for group structure * [here](https://ncatlab.org/nlab/show/ring#TTFormalizations) for ring structure but it could be developed much further.

    Yes, absolutely. We have some such referencing already, such as

    • here for group structure

    • here for ring structure

    but it could be developed much further.

68. • CommentRowNumber68.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2023
    • PermaLink
    Author: Urs
    Format: MarkdownItexProdded by discussion [here](https://nforum.ncatlab.org/discussion/8578/adjunction/?Focus=108199#Comment_108199) I have added to the very top of this entry highlighting of the parallel entry *[[adjunction]]*. <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/122">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/122">v122</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    Prodded by discussion here I have added to the very top of this entry highlighting of the parallel entry adjunction.

    diff, v122, current

69. • CommentRowNumber69.
    • CommentAuthorUrs
    • CommentTimeMay 5th 2023
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded a section "Transformation of adjoints" ([here](https://ncatlab.org/nlab/show/adjoint+functor#TransformationOfAdjoints)) with currently (just) the least general (but maybe most important) version of the definition (MacLane's "conjugate" transformations) with a brief comment on the relation to bifibrations. Since this probably deserves an entry of its own, I will also copy these paragraphs to a new entry *[[transformation of adjoints]]*. <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/123">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/123">v123</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    added a section “Transformation of adjoints” (here) with currently (just) the least general (but maybe most important) version of the definition (MacLane’s “conjugate” transformations) with a brief comment on the relation to bifibrations.

    Since this probably deserves an entry of its own, I will also copy these paragraphs to a new entry transformation of adjoints.

    diff, v123, current

70. • CommentRowNumber70.
    • CommentAuthorUrs
    • CommentTimeMay 6th 2023
    • (edited May 6th 2023)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have moved the lead-in sentences out of the Definition-section into a new Idea-section (now [here](https://ncatlab.org/nlab/show/adjoint+functor#Idea)) then expanded there a good bit, such as in making explicit the similarity to adjoint linear operators and moved the Lawvere quote out of a footnote into a quote-environment and added to it a paragraph quoted from p. 103 in MacLane71 <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/126">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/126">v126</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    I have moved the lead-in sentences out of the Definition-section into a new Idea-section (now here)

    then expanded there a good bit, such as in making explicit the similarity to adjoint linear operators

    and moved the Lawvere quote out of a footnote into a quote-environment

    and added to it a paragraph quoted from p. 103 in MacLane71

    diff, v126, current

71. • CommentRowNumber71.
    • CommentAuthorvarkor
    • CommentTimeMay 31st 2023
    • PermaLink
    Author: varkor
    Format: MarkdownItexAdded a link to a MathOverflow question with a list of properties of left/right adjoints. In the future, it would be useful to add all the examples directly to the nLab page, though I don't have time to do so right now. <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/129">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/129">v129</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    Added a link to a MathOverflow question with a list of properties of left/right adjoints. In the future, it would be useful to add all the examples directly to the nLab page, though I don’t have time to do so right now.

    diff, v129, current

72. • CommentRowNumber72.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2023
    • PermaLink
    Author: Urs
    Format: MarkdownItex[here](https://ncatlab.org/nlab/show/adjoint+functor#FurtherProperties) <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/130">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/130">v130</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    here

    diff, v130, current

73. • CommentRowNumber73.
    • CommentAuthornonemenon
    • CommentTimeJun 27th 2023
    • PermaLink
    Author: nonemenon
    Format: MarkdownItexthe opposite adjoint functors had their (co)domains reversed <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/131">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/131">v131</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    the opposite adjoint functors had their (co)domains reversed

    diff, v131, current

74. • CommentRowNumber74.
    • CommentAuthornonemenon
    • CommentTimeJun 27th 2023
    • PermaLink
    Author: nonemenon
    Format: MarkdownItexthe variables for the unit and counit components reflected the error concerning reversed (co)domains of opposite functors (see prior edit) <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/131">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/131">v131</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    the variables for the unit and counit components reflected the error concerning reversed (co)domains of opposite functors (see prior edit)

    diff, v131, current

75. • CommentRowNumber75.
    • CommentAuthornLab edit announcer
    • CommentTimeJul 17th 2023
    • PermaLink
    Author: nLab edit announcer
    Format: MarkdownItexSwapped categories C, D in the introduction for consistency with the rest of the article. Previously, the introduction introduced functors L : D -> C, R : C -> D while the rest of the article used L : C -> D, R : D -> C functors, which was confusing. Julius Marozas <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/135">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/135">v135</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    Swapped categories C, D in the introduction for consistency with the rest of the article.

    Previously, the introduction introduced functors L : D -> C, R : C -> D while the rest of the article used L : C -> D, R : D -> C functors, which was confusing.

    Julius Marozas

    diff, v135, current

76. • CommentRowNumber76.
    • CommentAuthorUrs
    • CommentTimeAug 10th 2023
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded another early historical reference: * [[Peter J. Huber]], §4 in: *Homotopy theory in general categories*, Mathematische Annalen **144** (1961) 361–385 &lbrack;[doi:10.1007/BF01396534](https://doi.org/10.1007/BF01396534)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/136">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/136">v136</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    added another early historical reference:

    • Peter J. Huber, §4 in: Homotopy theory in general categories, Mathematische Annalen 144 (1961) 361–385 [doi:10.1007/BF01396534]

    diff, v136, current

77. • CommentRowNumber77.
    • CommentAuthornLab edit announcer
    • CommentTimeJun 2nd 2024
    • PermaLink
    Author: nLab edit announcer
    Format: MarkdownItexC *^k D is the category under discussion for the hom sets not C^op x D. Jon <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/141">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/141">v141</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    C *^k D is the category under discussion for the hom sets not C^op x D.

    Jon

    diff, v141, current

78. • CommentRowNumber78.
    • CommentAuthornLab edit announcer
    • CommentTimeJun 2nd 2024
    • PermaLink
    Author: nLab edit announcer
    Format: MarkdownItexAdded set brackets to be explicit that the homset for k(X,Y) consists of a single morphism(k(X,Y)). Jon <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/141">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/141">v141</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    Added set brackets to be explicit that the homset for k(X,Y) consists of a single morphism(k(X,Y)).

    Jon

    diff, v141, current

79. • CommentRowNumber79.
    • CommentAuthornLab edit announcer
    • CommentTimeJun 2nd 2024
    • PermaLink
    Author: nLab edit announcer
    Format: MarkdownItexIt is better to use superscripts for contravariant action and subscripts for covariant action since this is relatively common in other areas. Jon <a href="https://ncatlab.org/nlab/revision/diff/adjoint+functor/141">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjoint+functor/141">v141</a>, <a href="https://ncatlab.org/nlab/show/adjoint+functor">current</a>

    It is better to use superscripts for contravariant action and subscripts for covariant action since this is relatively common in other areas.

    Jon

    diff, v141, current