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• CommentRowNumber1.
• CommentTimeJan 16th 2020

In the entry for lax natural transformations (https://ncatlab.org/nlab/show/lax+natural+transformation) the following axiom is given:

for each $A,B$, the $\alpha_f$ are the components of a (strict) natural transformation $\alpha_{A,B}: (\alpha_A)^* \circ G_{A,B} \dot{\to} (\alpha_B)_* \circ F_{A,B}$

however, $G_{A,B}$, $F_{A,B}$, $(-)^*$, and $(-)_*$ are left undefined.

Does any one know what their definitions are? I’m happy to extend the entry to make it more clear as well.

Thanks, Harley

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJan 17th 2020
• (edited Jan 17th 2020)

The $G_{A,B}$ and $F_{A,B}$ denote the hom-functors of the given 2-functors between the pair of objects $(A,B)$.

The $(-)^*$ and $(-)_*$ denote functors of left and right whiskering of hom-categories in a 2-category by 1-morphisms.

The $\alpha_{A,B}$ then is the collection of “tin-can” diagrams to be formed by the 2-cells of the lax natural transformation.

You are completely right that, as an explanation, the text in that page is not worth the electrons that it is written on. It can only be understood by people who already understand it. If you would add explanation (and diagrams!) you’d do a great service.

• CommentRowNumber3.