The Unapologetic Mathematician

Mathematics for the interested outsider

An awful lot of natural maps

There are a bunch of natural (in both senses) maps we can consider now. Some look all but tautological over \mathbf{Set}, and we may have used them in the past without comment. In the enriched context, though, we should go over them.

  1. The family of arrows \eta_C:\mathbf{1}\rightarrow\hom_\mathcal{D}(S(C),T(C)) is extraordinarily \mathcal{V}-natural exactly when \eta_C:S(C)\rightarrow T(C) is \mathcal{V}-natural.
  2. For a \mathcal{V}-functor T:\mathcal{C}\rightarrow\mathcal{D}, the map T_{A,B}:\hom_\mathcal{C}(A,B)\rightarrow\hom_\mathcal{D}(T(A),T(B)) is natural in both A and B.
  3. Similarly, if T:\mathcal{A}\otimes\mathcal{B}\rightarrow\mathcal{C} is a \mathcal{V}-functor, then T(\underline{\hphantom{X}},1_B):\hom_\mathcal{A}(A,A')\rightarrow\hom_\mathcal{C}(T(A,B),T(A',B)) is \mathcal{V}-natural. And it’s also natural in B).
  4. In particular, \hom_\mathcal{C}(C,\underline{\hphantom{X}})_{A,B}:\hom_\mathcal{C}(A,B)\rightarrow\hom_\mathcal{C}(C,B)^{\hom_\mathcal{C}(C,A)} is natural in all three variables.
  5. Putting together naturalities 1 and 4 tells us that for a morphism f:\mathbf{1}\rightarrow\hom_\mathcal{C}(A,B), the transformation \hom_\mathcal{C}(1_C,f):\hom_\mathcal{C}(C,A)\rightarrow\hom_\mathcal{C}(C,B) is natural.
  6. The “evaluation” e:Z^Y\otimes Y\rightarrow Z is natural in both variables.
  7. Since we built compositions from evaluations and the closure adjunction, the arrow \hom_\mathcal{C}(B,C)\otimes\hom_\mathcal{C}(A,B)\rightarrow\hom_\mathcal{C}(A,C) is natural in all variables.
  8. The identity functor on \mathcal{C} has an identity natural transformation, so by naturality 1 we see that i_C:\mathbf{1}\rightarrow\hom_\mathcal{C}(C,C) is natural.
  9. All the monoidal structural isomorphisms — \alpha_{A,B,C}:(A\otimes B)\otimes C)\rightarrow A\otimes(B\otimes C), \lambda_A:\mathbf{1}\otimes A\rightarrow A, \rho:A\otimes\mathbf{1}\rightarrow A, and \gamma_{A,B}:A\otimes B\rightarrow B\otimes A — are natural in all variables.
  10. We can start with (Z^Y)^X and hit it with \underline{\hphantom{X}}\otimes Y to get (Z^Y\otimes Y)^{X\otimes Y}. Then we can evaluate to get Z^{X\otimes Y}. This is an isomorphism we call p_{X,Y,Z}^{-1}, corresponding to the adjunction \hom_\mathcal{C}(A\hom_\mathcal{C}(B,C))\cong\hom_\mathcal{C}(A\otimes B,C), and it’s natural in all variables.
  11. We can compose the following arrows:
    • \lambda_X^{-1}:X\rightarrow\mathbf{1}\otimes X
    • j_{X\otimes Y}\otimes1_X:\mathbf{1}\otimes X\rightarrow(X\otimes Y)^{X\otimes Y}\otimes X
    • p_{X,Y,X\otimes Y}\otimes1_X:(X\otimes Y)^{X\otimes Y}\otimes X\rightarrow((X\otimes Y)^Y)^X\otimes X
    • e_{X,(X\otimes Y)^Y}:((X\otimes Y)^Y)^X\otimes X\rightarrow(X\otimes Y)^Y

    and we get the “coevaluation” d_{X,Y} — the counit of the closure adjunction. And thus the coevaluation is \mathcal{V}-natural in all variables.

  12. We can compose the coevaluation d_{X,\mathbf{1}}:X\rightarrow(X\otimes\mathbf{1})^\mathbf{1}, and then use the right unit isomorphism to get a \mathcal{V}-natural isomorphism X\rightarrow X^\mathbf{1}.
  13. In general, a family f:T(D,D,A,B)\otimes S(E,E,A,C)\rightarrow R(F,F,B,C) is \mathcal{V}-natural in any of its variables if the corresponding variables are natural in \overline{f}:T(D,D,A,B)\rightarrow R(F,F,B,C)^{S(E,E,A,C)}

Whew. That’s a mouthful. It can be instructive to sit down and try to interpret some of these in the context of categories enriched over \mathbf{Set}, so when you recover I’d advise taking a look at that.

August 31, 2007 - Posted by | Category theory

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