# Abstract Nonsense

## Comma Categories (Pt. II)

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

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As a fairly simple exercise try to convince one’s self that if $\mathcal{C}$ is a category and $x$ is an initial object then $(x\downarrow\mathcal{C})\cong\mathcal{C}$ and if $y$ is a terminal object then $(\mathcal{C}\downarrow y)\cong\mathcal{C}$ (intuitively, this is like a pointed $\mathcal{C}$ but we aren’t given a choice which element is chosen).

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As another example let’s find out what $(R\downarrow\mathbf{CRing})$ looks like, when $R$ is come commutative unital ring. We see that, almost by definition, we can think of the objects of $(R\downarrow\mathbf{CRing})$ as being commutative unital rings $A$ with a distinguished ring homomorphism $f:R\to A$. 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 $f:R\to A$ and $g:R\to B$, we see that an arrow between them is an arrow $h:A\to B$ (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 $(-\downarrow-)$ is discrete this is always the identity arrow, and so unimportant) such that $g=h\circ f$. But, this precisely the formulation for commutative algebras over $R$. In other words, we have commutative unital rings $A$ with specified $\mathbf{CRing}$-arrows $R\to A$ and the morphisms between two such objects $(R,A,f)$ and $(R,B,g)$ is just a $\mathbf{CRing}$-arrow $A\to B$ which respects $f$ and $g$. Thus, we see that $\left(R\downarrow\mathbf{CRing}\right)\cong R\text{-}\mathbf{CAlg}$.

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As a last example, what does $(x\downarrow y)$ where $x,y$ are objects in some fixed category $\mathcal{C}$? Well, the objects are nothing more than arrows $x\xrightarrow{f}y$ in $\mathcal{C}$ and since the only arrows in the discrete categories $\{x\}$ and $\{y\}$ are the identity arrows we see the only arrows in $(x\downarrow y)$ are the identity arrows $(1_x,1_y)$ we can thus conclude that $(x\downarrow y)$ is nothing more than the discrete category whose class of objects is $\text{Hom}_\mathcal{C}(x,y)$.

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Projections and  Comma Categories Relation to Natural Transformations

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It’s not surprising, considering the “tuple like” quality of comma categories, that the categories $(F\downarrow G)$ from diagrams $\mathcal{D}\overset{F}{\longrightarrow}\mathcal{C}\overset{G}{\longleftarrow}\mathcal{E}$ naturally come intact with “projections” $P_\mathcal{D}:(F\downarrow G)\to\mathcal{D}$ and $P_\mathcal{E}:(F\downarrow G)\to\mathcal{E}$. Indeed, we define these projections by $P_\mathcal{D}(x,y,f)=x$ and $P_\mathcal{D}(j,h)=j$ and similarly for $P_\mathcal{E}$.

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So, as stated before, we can use comma categories to restate the definition of natural transformation. Indeed,  let $\mathcal{C}$ and $\mathcal{D}_1,\mathcal{D}_2$ be categories with $\mathcal{D}_1=\mathcal{D}_2=\mathcal{D}$ (indexed different just to tell them apart). Then, if we have functors $\mathcal{D}_1\overset{F}{\longrightarrow}\mathcal{C}\overset{G}{\longleftarrow}\mathcal{D}_2$ we can consider the comma category $(F\downarrow G)$. Suppose now that we had a natural transformation $\eta:F\implies G$. We can then define a functor $E:\mathcal{D}_1\to (F\downarrow G)$ by sending $d\in\text{Obj}(\mathcal{D}_1)$ to $(d,d,\eta_d)$ (or, in our arrow notation $F(d)\xrightarrow{\eta_d}G(d)$) and $d\xrightarrow{f}d'$ to $(f,f)$. The fact that, if well-defined, this really is a functor is easy since we compose arrows component-wise and $F$ and $G$ are functors. Really, the bulk of the observation is that $(F(f),G(f))$ are actually arrows in $(F\downarrow G)$. Indeed, we need to check that $G(f)\circ\eta_d=\eta_{d'}\circ F(f)$, but this is precisely the definition of a natural transformation! Note that this functor $E$ satisfies $P_{\mathcal{D}_1}\circ E=P_{\mathcal{D}_2}\circ E=1_\mathcal{D}$.

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Conversely, suppose that we had a functor $E:\mathcal{D}_1\to (F\downarrow G)$ such that $P_{\mathcal{D}_1}\circ E=P_{\mathcal{D}_2}\circ E=1_\mathcal{D}$. We claim then that we can create a natural transformation based on $E$ without much work. Indeed, for each object $d\in\mathcal{D}_1$ we know that $E(d)$ is of the form $(d,d,\nu_d)$ where $\nu_d$ is some $\mathcal{C}$-arrow $F(d)\to G(d)$. We claim that $\nu=\{\nu_d\}$ is a natural transformation $F\implies G$. Indeed, it is certainly an indexed set of arrows $F(d)\to G(d)$ and so it suffices to check that the naturality condition holds. But, we know that if $d\xrightarrow{f}d'$ is an arrow in $\mathcal{C}$ and $E(f)=(j,h)$ then $\eta_{d'}\circ F(j)=G(h)\circ\eta_d$. But! Note that since $P_{\mathcal{D}_1}\circ E=P_{\mathcal{D}_2}\circ E=1_\mathcal{D}$ we have that $j=h=f$ and so we have that $\eta_{d'}\circ F(f)=G(f)\circ\eta_d$ which is precisely the statement of naturality.

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From this we conclude that for functors $F,D:\mathcal{D}\to\mathcal{C}$ a natural transformation $\eta:F\implies G$ is really, conceptually, nothing more than a functor $E:\mathcal{D}\to (F\downarrow G)$ such that the both of the natural projections precomposed with $E$ give $1_\mathcal{D}$.

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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.

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