## Comma Categories (Pt. II)

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

As a fairly simple exercise try to convince one’s self that if is a category and is an initial object then and if is a terminal object then (intuitively, this is like a pointed but we aren’t given a choice which element is chosen).

As another example let’s find out what looks like, when is come commutative unital ring. We see that, almost by definition, we can think of the objects of as being commutative unital rings with a distinguished ring homomorphism . Ok, there’s no more “simplification” that can be done there. So, what do the arrows in this comma category look like? Well, if we have two objects and , we see that an arrow between them is an arrow (of course, technically it’s a pair of arrows, but as we’ve gone through for the last two paragraphs in the case when the left element of is discrete this is always the identity arrow, and so unimportant) such that . But, this precisely the formulation for commutative algebras over . In other words, we have commutative unital rings with specified -arrows and the morphisms between two such objects and is just a -arrow which respects and . Thus, we see that .

As a last example, what does where are objects in some fixed category ? Well, the objects are nothing more than arrows in and since the only arrows in the discrete categories and are the identity arrows we see the only arrows in are the identity arrows we can thus conclude that is nothing more than the discrete category whose class of objects is .

*Projections and Comma Categories Relation to Natural Transformations*

It’s not surprising, considering the “tuple like” quality of comma categories, that the categories from diagrams naturally come intact with “projections” and . Indeed, we define these *projections* by and and similarly for .

So, as stated before, we can use comma categories to restate the definition of natural transformation. Indeed, let and be categories with (indexed different just to tell them apart). Then, if we have functors we can consider the comma category . Suppose now that we had a natural transformation . We can then define a functor by sending to (or, in our arrow notation ) and to . The fact that, if well-defined, this really is a functor is easy since we compose arrows component-wise and and are functors. Really, the bulk of the observation is that are actually arrows in . Indeed, we need to check that , but this is precisely the definition of a natural transformation! Note that this functor satisfies .

Conversely, suppose that we had a functor such that . We claim then that we can create a natural transformation based on without much work. Indeed, for each object we know that is of the form where is some -arrow . We claim that is a natural transformation . Indeed, it is certainly an indexed set of arrows and so it suffices to check that the naturality condition holds. But, we know that if is an arrow in and then . But! Note that since we have that and so we have that which is precisely the statement of naturality.

From this we conclude that for functors a natural transformation is really, conceptually, nothing more than a functor such that the both of the natural projections precomposed with give .

**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.

[6] Herrlich, Horst, and George E. Strecker. *Category Theory: An Introduction*. Lemgo: Heldermann, 2007. Print.

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