Abstract Nonsense

Crushing one theorem at a time

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.

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January 10, 2012 - Posted by | Algebra, Category Theory | , , , ,

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