• Images
• Videos
• Documents
• Books
• Articles

Definition

Let F : A × B o p × B → D {\displaystyle F:A\times B^{\mathrm {op} }\times B\rightarrow D} and G : A × C o p × C → D {\displaystyle G:A\times C^{\mathrm {op} }\times C\rightarrow D} be two functors of categories. A family η ( a , b , c ) : F ( a , b , b ) → G ( a , c , c ) {\displaystyle \eta (a,b,c):F(a,b,b)\rightarrow G(a,c,c)} is said to be natural in a and extranatural in b and c if the following holds:

• η ( − , b , c ) {\displaystyle \eta (-,b,c)} is a natural transformation (in the usual sense).
• (extranaturality in b) ∀ ( g : b → b ′ ) ∈ M o r B {\displaystyle \forall (g:b\rightarrow b^{\prime })\in \mathrm {Mor} \,B} , ∀ a ∈ A {\displaystyle \forall a\in A} , ∀ c ∈ C {\displaystyle \forall c\in C} the following diagram commutes

F ( a , b ′ , b ) → F ( 1 , 1 , g ) F ( a , b ′ , b ′ ) F ( 1 , g , 1 ) ↓ η ( a , b ′ , c ) ↓ F ( a , b , b ) → η ( a , b , c ) G ( a , c , c ) {\displaystyle {\begin{matrix}F(a,b',b)&\xrightarrow {F(1,1,g)} &F(a,b',b')\\_{F(1,g,1)}\downarrow \qquad &&_{\eta (a,b',c)}\downarrow \qquad \\F(a,b,b)&\xrightarrow {\eta (a,b,c)} &G(a,c,c)\end{matrix}}}

• (extranaturality in c) ∀ ( h : c → c ′ ) ∈ M o r C {\displaystyle \forall (h:c\rightarrow c^{\prime })\in \mathrm {Mor} \,C} , ∀ a ∈ A {\displaystyle \forall a\in A} , ∀ b ∈ B {\displaystyle \forall b\in B} the following diagram commutes

F ( a , b , b ) → η ( a , b , c ′ ) G ( a , c ′ , c ′ ) η ( a , b , c ) ↓ G ( 1 , h , 1 ) ↓ G ( a , c , c ) → G ( 1 , 1 , h ) G ( a , c , c ′ ) {\displaystyle {\begin{matrix}F(a,b,b)&\xrightarrow {\eta (a,b,c')} &G(a,c',c')\\_{\eta (a,b,c)}\downarrow \qquad &&_{G(1,h,1)}\downarrow \qquad \\G(a,c,c)&\xrightarrow {G(1,1,h)} &G(a,c,c')\end{matrix}}}

Properties

Extranatural transformations can be used to define wedges and thereby ends² (dually co-wedges and co-ends), by setting F {\displaystyle F} (dually G {\displaystyle G} ) constant.

Extranatural transformations can be defined in terms of dinatural transformations, of which they are a special case.³

See also

• Dinatural transformation

External links

• extranatural+transformation at the nLab

References

1. Eilenberg and Kelly, A generalization of the functorial calculus, J. Algebra 3 366–375 (1966) /wiki/Samuel_Eilenberg ↩

2. Fosco Loregian, This is the (co)end, my only (co)friend, arXiv preprint [1] https://arxiv.org/abs/1501.02503 ↩

3. Fosco Loregian, This is the (co)end, my only (co)friend, arXiv preprint [1] https://arxiv.org/abs/1501.02503 ↩