Timeline for Natural transformations as categorical homotopies

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May 11, 2011 at 0:23	comment	added	user13113		I assert that when one is thinking in terms of the 2-categories of categories, definitions are no longer important. (Amongst) what is important is the knowledge of various equivalent ways to capture a notion. I think describing natural transformations as functors $\mathcal{C} \times 2 \to \mathcal{D}$ doesn't become useful until thinking in terms of the 2-category of categories mainly because the only other way I see to use description is to unfold it into the ordinary description and working in terms of that.
May 10, 2011 at 20:02	comment	added	Giorgio Mossa		I agree with you when you say that it's more natural thinking homotopy as a path of function, and in this sense one could think about natural transformation as functor of kind $2 \to \text{Fun}(\mathcal C,\mathcal D)$, by the way this require the category of functors and so natural transformation as well. A similar problem seems to arise in topology, where one must define a topology on the space of function in order to define homotopies like path of function. Why do you think this definition of natural transformation is useful when one start to think in terms of the 2-categories of categories?
May 9, 2011 at 10:34	history	answered	user13113	CC BY-SA 3.0