Abstract Nonsense

Crushing one theorem at a time

Natural Transformations (Pt. II)


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

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Homotopic Definition of Natural Transformation

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We now give the definition of natural transformation that is of the homotopical flavor. Namely, let us define the category \mathbf{2} to have two distinct objects, represented as \bullet_0,\bullet_1, and have one non-identity morphism \bullet_0\xrightarrow{a_0}\bullet_1. Suppose then that we have two categories \mathcal{C},\mathcal{D} and functors \mathcal{C}\overset{\displaystyle \overset{F}{\longrightarrow}}{\underset{G}{\longrightarrow}}\mathcal{D}. We then define a natural transformation F\implies G to be a functor N:\mathcal{C}\times\mathbf{2}\to\mathcal{D} (where the product of categories is described as before) such that if we consider the functor N(-,\bullet_0):\mathcal{C}\to\mathcal{D} (defined in the obvious way) then N(-,\bullet_0)=F and similarly, N(-,\bullet_1)=G.

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To see how these two definitions jive, suppose first that we have a functor N:\mathcal{C}\times\mathbf{2}\to\mathcal{D} such that N(-,\bullet_0)=F and N(-,\bullet_1)=G on objects and N(-,1_{\bullet_0})=F and N(-,1_{\bullet_1})=G on morphisms. Define then, for each A\in\text{Obj}(\mathcal{C}) the morphism F(A)\xrightarrow{\eta_A}G(A) by \eta_A=N(1_A,a_0) (note that N(1_A,a_0) really is a morphism F(A)\to G(A), since (1_A,a_0) is a morphism (A,\bullet_0)\to(A,\bullet_1) and so N(1_A,a_0) is a morphism N(A,\bullet_0)\to N(A,\bullet_1), but since we assumed that N(-\bullet_0)=F and N(A,\bullet_1)=G the rest follows). Suppose then that we have two objects A,B\in\text{obj}(\mathcal{C}) and some morphism A\xrightarrow{f}B. We need then to show that G(f)\circ\eta_A=\eta_B\circ F(f), but note that by assumption this is equivalent to verifying that

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N(f,1_{\bullet_1})\circ N(1_A,a_0)=N(1_B,a_0)\circ N(f,1_{\bullet_0})

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But, since N is a functor the left hand side is just N(f\circ1_A,1_{\bullet_1}\circ a_0)=N(f,a_0) and the right is N(1_B\circ f,a_0\circ1_{\bullet_0})=N(f,a_0) and so the desired equality holds. Thus, we see that N induces the natural transformation \eta=\{N(1_A,a_0)\}.

Suppose now that we have the traditional example of a natural transformation \eta:F\implies G. We can then define the functor N:\mathcal{C}\times\mathbf{2}\to\mathcal{D} by the rule N(-,\bullet_0)=F and N(-,\bullet_1)=G on objects, N(-,1_{\bullet_0})=F and N(-,1_{\bullet_1})=G on morphisms, and N(A\xrightarrow{f}B,a_0)=F(f)\circ\eta_B. One quickly checks that this, in fact, a functor.

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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.

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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.

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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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December 28, 2011 - Posted by | Category Theory | , , ,

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