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1. • CommentRowNumber1.
   • CommentAuthorAndrew Stacey
   • CommentTimeNov 5th 2010
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   Author: Andrew Stacey
   Format: MarkdownItexQuick question as to which nLab pages I should be reading for the following situation: I have bifunctors $H_i: A_i \times B_i \to C$ which are a bit like a $hom$-functor. I have a functor $F \colon A_1 \to A_2$ and $G \colon B_2 \to B_1$ which are "adjoint" in the sense that $H_1(a, G(b)) \cong H_2(F(a), b)$ for all $a \in A_1$ and $b \in B_2$, naturally in both. Where do I look to learn about these things? In particular, are there conditions on $F$ or $G$, similar to the adjoint functor theorems, that guarantee that they have an adjoint (ie so if I only know one, when can I deduce that the other exists?)? In case it helps to be more specific, $C$ is actually the category of $Z$-graded sets for some fixed set $Z$, $A_1$ is $Z$-graded $B_1$s, $H_1$ is the corresponding $Z$-graded hom-functor and similarly for the ${}_2$ versions.

   Quick question as to which nLab pages I should be reading for the following situation:

   I have bifunctors which are a bit like a -functor. I have a functor and which are “adjoint” in the sense that for all and , naturally in both. Where do I look to learn about these things? In particular, are there conditions on or , similar to the adjoint functor theorems, that guarantee that they have an adjoint (ie so if I only know one, when can I deduce that the other exists?)?

   In case it helps to be more specific, is actually the category of -graded sets for some fixed set , is -graded s, is the corresponding -graded hom-functor and similarly for the versions.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 5th 2010
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   Author: Urs
   Format: MarkdownItexMaybe in your last line some words are missing or something? Not sure, but if your setup can be thought of as taking place entirely in $Set/Z$-enriched category theory then all theorems and everything hold as usual.

   Maybe in your last line some words are missing or something? Not sure, but if your setup can be thought of as taking place entirely in -enriched category theory then all theorems and everything hold as usual.

3. • CommentRowNumber3.
   • CommentAuthorAndrew Stacey
   • CommentTimeNov 5th 2010
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexIt doesn't take place in a $Set/Z$-enriched category. The thing to think about is graded Lawvere theories. I think that my generalisation isn't quite right, so here's some more details of the specific situation. First, the simple ungraded lemma: > Let $D$ be a cocomplete (locally small) category. A functor $G \colon D \to Set$ has a left adjoint if and only if it is representable. > > Proof: adjoint $\implies$ representable: $G(d) \cong Set(\star, G(d)) \cong D(F(\star),d)$ > representable $\implies$ adjoint: if $G(d) = D(d_0,d)$, define $F(x) \coloneqq \coprod_x d_0$, then $Set(x,G(d)) \cong G(d)^x = D(d_0,d)^x \cong D(\coprod_x d_0, d)$. Now the graded version. We have a cocomplete (locally small) category $D$. We consider $D^Z$, the category of $Z$-graded objects in $D$ (aka the category of functors $Z \to D$ where $Z$ is viewed as a small discrete category). We can extend the hom-functor on $D$ to a bifunctor $D^Z \times D \to Set^Z$ in the (hopefully) obvious way. Then the result is: > A functor $G \colon D \to Set^Z$ has a left adjoint if and only if it is representable by an object in $D^Z$. The proof isn't that much more complicated than the ungraded version, but I'd like to know the general context into which it fits (and thus if there's a Main Result that can be quoted to avoid having to give the proof at all).

   It doesn’t take place in a -enriched category. The thing to think about is graded Lawvere theories. I think that my generalisation isn’t quite right, so here’s some more details of the specific situation. First, the simple ungraded lemma:

   Let be a cocomplete (locally small) category. A functor has a left adjoint if and only if it is representable.

   Proof: adjoint representable:
   representable adjoint: if , define , then .

   Now the graded version. We have a cocomplete (locally small) category . We consider , the category of -graded objects in (aka the category of functors where is viewed as a small discrete category). We can extend the hom-functor on to a bifunctor in the (hopefully) obvious way. Then the result is:

   A functor has a left adjoint if and only if it is representable by an object in .

   The proof isn’t that much more complicated than the ungraded version, but I’d like to know the general context into which it fits (and thus if there’s a Main Result that can be quoted to avoid having to give the proof at all).