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Quick 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.
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 $Set/Z$-enriched category theory then all theorems and everything hold as usual.
It 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).
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