Abstract Nonsense

Crushing one theorem at a time

Limits, Colimits, and Representable Functors (Pt. III)


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

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Well, call (in general) a functor R:\mathcal{D}\to\mathbf{Set} representable by X if X is an object of \mathcal{D} such that R\cong\text{Hom}_\mathcal{D}(X,\bullet)–in other words, R is secretly just the covariant Hom functor associated to X. In this case we say that R  represents X. Call R representable if it is represented by some object X of \mathcal{D}. Similarly, if R:\mathcal{D}\to\mathbf{Set} is a contravariant functor we say it’s representable if there exists Y such that R\cong\text{Hom}_\mathcal{D}(\bullet,Y)–the rest of the terminology also translates.

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So, back to thinking about our functor F:\mathcal{I}\to\mathcal{C}. Fix an object X in \mathcal{C} and \text{Hom}(X,F) denote the functor \mathcal{I}\to\mathbf{Set} given by i\mapsto \text{Hom}(X,F(i)) and i\xrightarrow{j}s is mapped to \text{Hom}(X,F(i))\xrightarrow{s_\ast}\text{Hom}(X,F(j)) with f\mapsto s\circ f. Since this is a functor into \mathbf{Set} from \mathcal{I} we can take its limit to get the set \lim\text{Hom}(X,F). To be explicit it is the following set

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\displaystyle \lim \text{Hom}(X,F)=\left\{(f_k)\in\prod_k\text{Hom}(Y,F(k)):s\circ f_i=f_j\text{ for all }i\xrightarrow{s}j\right\}

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Let now H be the contravariant functor \mathcal{I}\to\mathbf{Set} which on objects takesX\mapsto \lim \text{Hom}(X,F) and which is harder to define on arrows. Ok, so let X\xrightarrow{g}Y be a map. We then define a map H_i(g):\lim\text{Hom}(Y,F)\to\text{Hom}(X,F(i)) by H_i((f_k))=f_i\circ g which extends uniquely to a map H(g):\lim\text{Hom}(Y,F)\to\lim\text{Hom}(X,F).

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Ok, so we now have this somewhat crazy contravariant functor H:\mathcal{C}\to\mathbf{Set}. What we claim though is that if \lim F exists then it represents H. Indeed, we define the natural transformation \eta:\text{Hom}(\bullet,\lim F)\implies H with \eta_X:\text{Hom}(X,\lim F)\to H(X) given by the unique extension of the map \text{Hom}(X,\lim F)\to \text{Hom}(X,F(i)) given by f\mapsto \alpha_i\circ f where \{\alpha_i\} are the cone maps. We first claim that each \eta_X is a bijection. To see that \eta_X is an injection we merely note that if \eta_X(f)=\eta_X(g) then \alpha_i\circ f=\alpha_i\circ g for all i and so by definition of a limit we must have that f=g. To see that \eta_X is surjective we take an arbitrary (f_k)\in\lim \text{Hom}(X,F). We know then that f_i is a morphism X\to F(i) and so by the definition of \lim F there exists a morphism X\to\lim F such that \alpha\circ f=f_i and thus we see that \eta_X(f)=(f_k) as desired.

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Ok, now that we know the \eta_X‘s are bijections the only thing left to check is that the \eta_X‘s satisfy the naturality condition, namely that for each arrow X\xrightarrow{g}Y in \mathcal{C} the following diagram commutes:

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\begin{array}{ccc}\text{Hom}(Y,\lim F) & \overset{\text{Hom}(g,\lim F)}{\longrightarrow} & \text{Hom}(X,\lim F)\\ ^{\eta_X}\big\downarrow & & \big\downarrow^{\eta_Y}\\ H(Y) & \underset{H(g)}{\longrightarrow} & H(X)\end{array}

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To do this let Y\xrightarrow{f}\lim F be arbitrary. We see then that

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\eta_Y\left(\text{Hom}(Y,g)(f)\right)=\eta_Y(f\circ g)

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whose i^{\text{th}} coordinate is \alpha_i\circ (f\circ g). Now,

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H(g)(\eta_Y(f))=H(g)((\alpha_k\circ f))=(\alpha_k\circ f\circ g)

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which has i^{\text{th}}-coordinate \alpha_i\circ (f\circ g). Thus, we may conclude that the diagram we want commutes and thus \eta really is a natural isomorphism.

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While I won’t go through all of it one can verify that we had instead started with looking at H' defined on objects by \lim\text{Hom}(F,X) (where the functor \text{Hom}(F,X) is defined in the obvious way) and on arrows in the obvious way that \text{colim }F (if it exists) becomes a representing element of the covariant functor H'.

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Of course, we can dualize the property above to instead recover the definition of limit and colimit from these functors. Namely, it’s not hard to see that a representing element of H and H' is necessarily going to be a limit and colimit respectively. How? Well, clearly the representing object L of H, for example, will be the object \lim F and so we are just left trying to figure out what we should take the cone arrows to be. But, if there is any justice in the world (and there is–well, at least in math) we should be able to use the above construction to actually figure them out. In particular, we know that \eta_X:\text{Hom}(X,\lim F)\to\lim\text{Hom}(X,F) on the i^{\text{th}} coordinate is a map \text{Hom}(X,\lim F)\to\text{Hom}(X,F(i)) given by f\mapsto \alpha_i\circ f. Now, if we wanted to recover \alpha_i it seems pretty sensible to take f to be the identity map (so that \alpha_i\circ f=\alpha_i) but this only makes sense if we take X=\lim F. Thus, by tinkering, we discover that we should be able to recover the cone arrows L\to F(i) by \eta_L(1_L). And, while I won’t prove this, it’s true (check it yourself).

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The moral of the story is that we can think about limits/colimits as either their cone definition or as representing elements of certain functors–both ways can be useful at different points.

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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] Mitchell, Barry. Theory of Categories. New York: Academic, 1965. Print.

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

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April 13, 2012 - Posted by | Algebra, Category Theory | , , , ,

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