Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorAndrew Stacey
   • CommentTimeNov 5th 2010
   • PermaLink
   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 $H_i: A_i \times B_i \to C$ 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.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 5th 2010
   • PermaLink
   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 $Set/Z$-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 $Set/Z$-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).