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    • CommentRowNumber1.
    • CommentAuthorAndrew Stacey
    • CommentTimeNov 5th 2010

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

    I have bifunctors H i:A i×B iCH_i: A_i \times B_i \to C which are a bit like a homhom-functor. I have a functor F:A 1A 2F \colon A_1 \to A_2 and G:B 2B 1G \colon B_2 \to B_1 which are “adjoint” in the sense that H 1(a,G(b))H 2(F(a),b)H_1(a, G(b)) \cong H_2(F(a), b) for all aA 1a \in A_1 and bB 2b \in B_2, naturally in both. Where do I look to learn about these things? In particular, are there conditions on FF or GG, 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, CC is actually the category of ZZ-graded sets for some fixed set ZZ, A 1A_1 is ZZ-graded B 1B_1s, H 1H_1 is the corresponding ZZ-graded hom-functor and similarly for the 2{}_2 versions.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 5th 2010

    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/ZSet/Z-enriched category theory then all theorems and everything hold as usual.

    • CommentRowNumber3.
    • CommentAuthorAndrew Stacey
    • CommentTimeNov 5th 2010

    It doesn’t take place in a Set/ZSet/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 DD be a cocomplete (locally small) category. A functor G:DSetG \colon D \to Set has a left adjoint if and only if it is representable.

    Proof: adjoint \implies representable: G(d)Set(,G(d))D(F(),d)G(d) \cong Set(\star, G(d)) \cong D(F(\star),d)
    representable \implies adjoint: if G(d)=D(d 0,d)G(d) = D(d_0,d), define F(x) xd 0F(x) \coloneqq \coprod_x d_0, then Set(x,G(d))G(d) x=D(d 0,d) xD( xd 0,d)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 DD. We consider D ZD^Z, the category of ZZ-graded objects in DD (aka the category of functors ZDZ \to D where ZZ is viewed as a small discrete category). We can extend the hom-functor on DD to a bifunctor D Z×DSet ZD^Z \times D \to Set^Z in the (hopefully) obvious way. Then the result is:

    A functor G:DSet ZG \colon D \to Set^Z has a left adjoint if and only if it is representable by an object in D ZD^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).