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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeOct 19th 2010
   • (edited Oct 19th 2010)
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   Author: Harry Gindi
   Format: MarkdownItexCan every universal property be formulated in the language of partial adjoints? For instance, let $C$ be a $V$-category, and let $D$ be a small $V$-category. We can define the limit functor $lim:(C^D)'\to C^*$ to be the partial right adjoint to $\Delta:C^*\to C^D$ (induced by the unique functor $D\to *$) where $(C^D)'$ is the full subcategory of $C^D$ consisting of those $V$-functors $F:D\to C$ for which $C^D(F,\Delta(-)):C^* \toV$ is representable. This seems to characterize the limit exactly when it exists and determines the universal property pretty easily. Can every ordinary universal property (in the sense that we have uniqueness up to unique isomorphism) be characterized this way? For example, can we give the definition of a weighted limit as a partial adjoint to some functor? At least for the weighted limit, which we may define to be an object representing $L(b)=V^D(W(-),C(b,F(-))$ (here $b$ is just a second cipher). Then everything depends on our ability to give some $V$-functor $K_W:D\otimes C\to C$ assigned to the weight $W$ such that $C^D(K_W(b),F)\cong V^D(W(-),C(b,F(-))$ naturally in $b$. Is such a feat possible? Can it be described as something like left adjoint to the co-yoneda embedding? Is there a requirement that $C$ is tensored?

   Can every universal property be formulated in the language of partial adjoints?

   For instance, let $C$ be a $V$-category, and let $D$ be a small $V$-category. We can define the limit functor $lim:(C^D)'\to C^*$ to be the partial right adjoint to $\Delta:C^*\to C^D$ (induced by the unique functor $D\to *$) where $(C^D)'$ is the full subcategory of $C^D$ consisting of those $V$-functors $F:D\to C$ for which $C^D(F,\Delta(-)):C^* \toV$ is representable.

   This seems to characterize the limit exactly when it exists and determines the universal property pretty easily. Can every ordinary universal property (in the sense that we have uniqueness up to unique isomorphism) be characterized this way?

   For example, can we give the definition of a weighted limit as a partial adjoint to some functor?

   At least for the weighted limit, which we may define to be an object representing $L(b)=V^D(W(-),C(b,F(-))$ (here $b$ is just a second cipher).

   Then everything depends on our ability to give some $V$-functor $K_W:D\otimes C\to C$ assigned to the weight $W$ such that $C^D(K_W(b),F)\cong V^D(W(-),C(b,F(-))$ naturally in $b$. Is such a feat possible? Can it be described as something like left adjoint to the co-yoneda embedding? Is there a requirement that $C$ is tensored?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeOct 19th 2010
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   Author: Mike Shulman
   Format: MarkdownItexYour second paragraph only works when V is cartesian monoidal; otherwise there may be no functor $D\to *$. But that's a detail. The answer is yes in the following sense at least: if you have any V-functor $F\colon C^{op}\to V$ and you want to characterize objects potentially representing $F$ using a partial adjoint, regard $F$ as a V-profunctor $I ⇸ C$ and take its [[collage]] $|F|$. If the inclusion $C\to {|F|}$ has a partial right adjoint defined at the unique object $*\in I$, then the value of that adjoint is a representing object for $F$ and conversely.

   Your second paragraph only works when V is cartesian monoidal; otherwise there may be no functor $D\to *$. But that’s a detail. The answer is yes in the following sense at least: if you have any V-functor $F\colon C^{op}\to V$ and you want to characterize objects potentially representing $F$ using a partial adjoint, regard $F$ as a V-profunctor $I ⇸ C$ and take its collage $|F|$. If the inclusion $C\to {|F|}$ has a partial right adjoint defined at the unique object $*\in I$, then the value of that adjoint is a representing object for $F$ and conversely.

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeOct 19th 2010
   • (edited Oct 20th 2010)
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   Author: Harry Gindi
   Format: MarkdownItexAh, I see! So instead of thinking of representability of functors, we should think of representability of profunctorsS?

   Ah, I see! So instead of thinking of representability of functors, we should think of representability of profunctorsS?

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeOct 20th 2010
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   Author: Mike Shulman
   Format: MarkdownItexWell, the two are really equivalent. It's just that thinking of it as a profunctor suggests building the collage.

   Well, the two are really equivalent. It’s just that thinking of it as a profunctor suggests building the collage.

5. • CommentRowNumber5.
   • CommentAuthorHarry Gindi
   • CommentTimeOct 20th 2010
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   Author: Harry Gindi
   Format: MarkdownItexIs there any treatment of weighted (co)limits, (co)ends, kan extensions, etc. from the point of view of profunctors and cographs?

   Is there any treatment of weighted (co)limits, (co)ends, kan extensions, etc. from the point of view of profunctors and cographs?

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeOct 21st 2010
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   Author: Mike Shulman
   Format: MarkdownItexSadly, no, not really at an introductory or comprehensive level.

   Sadly, no, not really at an introductory or comprehensive level.

7. • CommentRowNumber7.
   • CommentAuthorHarry Gindi
   • CommentTimeOct 21st 2010
   • (edited Oct 21st 2010)
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   Author: Harry Gindi
   Format: MarkdownItexThat's unfortunate. It seems like the most natural way to approach them. Is there a way to define composition of profunctors without relying on a coend? Here's how it seems like it should work: Given two V-profunctors $F:C ⇸ C'$ and $G:C' ⇸ C''$, consider their cographs. The cographs are naturally equipped with fixed $V$-functors into the free $V$-category on the quiver $\cdot \to \cdot$, which we will call $\Delta^1$. This gives us a natural map $f:D=C\star^F C'\coprod_{C'} C'\star^G C''\to \Delta^2$. Then the composite of their cographs should be given by the pullback by the coface $d^1:\Delta^1\to \Delta^2$, that is, $$D\times_{f,\Delta^2,d^1}\Delta^1\to \Delta^1.$$ Does this somehow fail? If that works, let's characterize the Kan extension and weighted limit like this. Edit: If it's not already clear, I am really uneasy about the end/coend having a universal property in a class of morphisms that doesn't even define a category and the weighted limit relying on the Yoneda embedding.

   That’s unfortunate. It seems like the most natural way to approach them.

   Is there a way to define composition of profunctors without relying on a coend?

   Here’s how it seems like it should work:

   Given two V-profunctors $F:C ⇸ C'$ and $G:C' ⇸ C''$, consider their cographs. The cographs are naturally equipped with fixed $V$-functors into the free $V$-category on the quiver $\cdot \to \cdot$, which we will call $\Delta^1$. This gives us a natural map $f:D=C\star^F C'\coprod_{C'} C'\star^G C''\to \Delta^2$. Then the composite of their cographs should be given by the pullback by the coface $d^1:\Delta^1\to \Delta^2$, that is,

   $$D\times_{f,\Delta^2,d^1}\Delta^1\to \Delta^1.$$

   Does this somehow fail?

   If that works, let’s characterize the Kan extension and weighted limit like this.

   Edit: If it’s not already clear, I am really uneasy about the end/coend having a universal property in a class of morphisms that doesn’t even define a category and the weighted limit relying on the Yoneda embedding.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeOct 21st 2010
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   Author: Mike Shulman
   Format: MarkdownItexWell, once again, a cograph doesn't come with a V-functor into $\Delta^1$ unless V is cartesian. However, if you get less addicted to simplices and just say "take the full subcategory of $|F| +_{C'} |G|$ on the objects of $C$ and $C''$" then it works fine. The Australians have studied this in much greater abstraction and generality, but they haven't written it out in an introductory fashion. It seems to me that Kan extensions and weighted limits are already essentially chararcterized in terms of profunctors, although that can and should be more explicit. But I see no reason to insist whenever we use composition of profunctors that it be described in this way, since there are also plenty of other perfectly adequate definitions. I don't know what you mean by "the end/coend having a universal property in a class of morphisms that doesn't even define a category," and I don't see where the weighted limit requires the Yoneda embedding.

   Well, once again, a cograph doesn’t come with a V-functor into $\Delta^1$ unless V is cartesian. However, if you get less addicted to simplices and just say “take the full subcategory of $|F| +_{C'} |G|$ on the objects of $C$ and $C''$” then it works fine. The Australians have studied this in much greater abstraction and generality, but they haven’t written it out in an introductory fashion.

   It seems to me that Kan extensions and weighted limits are already essentially chararcterized in terms of profunctors, although that can and should be more explicit. But I see no reason to insist whenever we use composition of profunctors that it be described in this way, since there are also plenty of other perfectly adequate definitions. I don’t know what you mean by “the end/coend having a universal property in a class of morphisms that doesn’t even define a category,” and I don’t see where the weighted limit requires the Yoneda embedding.

9. • CommentRowNumber9.
   • CommentAuthorHarry Gindi
   • CommentTimeOct 21st 2010
   • (edited Oct 21st 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexThe only other definition I found of composition of profunctors involves ends and coends (sometimes these are hidden as equalizers and coequalizers), which in a perfect world would be definable in terms of completely primitive notions. Before my world came crashing down around me when I learned about enriched categories, this primitive concept was exactly the Kan extension (oh, how grand it would be if Mac Lane's pronouncement that all concepts are Kan extensions was actually true). Any interesting concept (except for the end and coend) have an explicit description in terms of partial "u-limits/u-colimits" in the terminology of DHKS. That is, partial adjoints of the functor $u^*:C^E\to C^D$ for some $u:D\to E$. It turns out (this was news to me, probably not to you) that enriched category theory breaks everything. I guess that my question is: When we're working with enriched categories, can we describe all of our universal constructions in terms of something as elegant (or almost as elegant) as the partial adjoint? This doesn't mean just redefining things in a way that looks similar. I mean to redefine things in a way so that they are only logically dependant on that one notion.

   The only other definition I found of composition of profunctors involves ends and coends (sometimes these are hidden as equalizers and coequalizers), which in a perfect world would be definable in terms of completely primitive notions.

   Before my world came crashing down around me when I learned about enriched categories, this primitive concept was exactly the Kan extension (oh, how grand it would be if Mac Lane’s pronouncement that all concepts are Kan extensions was actually true). Any interesting concept (except for the end and coend) have an explicit description in terms of partial “u-limits/u-colimits” in the terminology of DHKS. That is, partial adjoints of the functor $u^*:C^E\to C^D$ for some $u:D\to E$.

   It turns out (this was news to me, probably not to you) that enriched category theory breaks everything. I guess that my question is: When we’re working with enriched categories, can we describe all of our universal constructions in terms of something as elegant (or almost as elegant) as the partial adjoint? This doesn’t mean just redefining things in a way that looks similar. I mean to redefine things in a way so that they are only logically dependant on that one notion.

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeOct 21st 2010
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    Author: Mike Shulman
    Format: MarkdownItexIn ordinary unenriched category theory, ends and coends are also definable in terms of partial adjoints, since they can be expressed as special limits and colimits respectively. Mac Lane does this somewhere. It's fun that most concepts in category theory can be expressed in terms of each other, but I wouldn't attach too much importance to it. If you really want one basic elegant concept in terms of which all other concepts can be expressed, I would be inclined to single out representability of a functor (or, more generally, a profunctor). Pretty much any categorical construction you care to name is expressible in terms of that, whether enriched or not, and I think it's quite elegant: it also points out the basic role of the Yoneda Lemma in everything.

    In ordinary unenriched category theory, ends and coends are also definable in terms of partial adjoints, since they can be expressed as special limits and colimits respectively. Mac Lane does this somewhere.

    It’s fun that most concepts in category theory can be expressed in terms of each other, but I wouldn’t attach too much importance to it. If you really want one basic elegant concept in terms of which all other concepts can be expressed, I would be inclined to single out representability of a functor (or, more generally, a profunctor). Pretty much any categorical construction you care to name is expressible in terms of that, whether enriched or not, and I think it’s quite elegant: it also points out the basic role of the Yoneda Lemma in everything.