## Natural Transformations (Pt. II)

**Point of Post: **This is a continuation of this post.

**Homotopic Definition of Natural Transformation**

We now give the definition of natural transformation that is of the homotopical flavor. Namely, let us define the category to have two distinct objects, represented as , and have one non-identity morphism . Suppose then that we have two categories and functors . We then define a natural transformation to be a functor (where the product of categories is described as before) such that if we consider the functor (defined in the obvious way) then and similarly, .

To see how these two definitions jive, suppose first that we have a functor such that and on objects and and on morphisms. Define then, for each the morphism by (note that really is a morphism , since is a morphism and so is a morphism , but since we assumed that and the rest follows). Suppose then that we have two objects and some morphism . We need then to show that , but note that by assumption this is equivalent to verifying that

But, since is a functor the left hand side is just and the right is and so the desired equality holds. Thus, we see that induces the natural transformation .

Suppose now that we have the traditional example of a natural transformation . We can then define the functor by the rule and on objects, and on morphisms, and . One quickly checks that this, in fact, a functor.

In fact, one quickly realizes that the above was really redundant, at least for us, since this equivalence falls right out of the fundamental theorem of bifunctors.

Thus, we see that natural transformations in the traditional sense, and natural transformations in this homotopical sense are practically the same thing. The second, as the motivation indicated, is definitely more intuitively appealing, but the first is definitively more practical.

**References:**

[1] Mac, Lane Saunders. *Categories for the Working Mathematician*. New York: Springer-Verlag, 1994. Print.

[2] Adámek, Jirí, Horst Herrlich, and George E. Strecker. *Abstract and Concrete Categories: the Joy of Cats*. New York: John Wiley & Sons, 1990. Print.

[3] Berrick, A. J., and M. E. Keating. *Categories and Modules with K-theory in View*. Cambridge, UK: Cambridge UP, 2000. Print.

[4] Freyd, Peter J. *Abelian Categories.* New York: Harper & Row, 1964. Print.

[5] Rotman, Joseph J. *Introduction to Homological Algebra*. Springer-Verlag. Print.

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